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\begin{document}

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\title[Factor-Lasso and K-Step Bootstrap Supplement]{Supplementary Appendix for The Factor-Lasso and K-Step Bootstrap Approach for
Inference in High-Dimensional Economic Applications}\thanks{Christian Hansen, Booth School of Business, University of Chicago, Chicago, IL 60637. Christian.Hansen@chicagobooth.edu.  Yuan Liao, Department of Economics, Rutgers University, New Brunswick, NJ 08901. yuan.liao@rutgers.edu}

 

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\author{Christian Hansen}
\address{Booth School of Business, University of Chicago, Chicago, IL 60637, Christian.Hansen@chicagobooth.edu}
\email{Christian.Hansen@chicagobooth.edu} 
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\author{Yuan Liao}
\address{Department of Economics, Rutgers University, New Brunswick, NJ 08901}
\email{yuan.liao@rutgers.edu}
%\urladdr{www.math.sc.edu/$\sim$howard} % Delete if not wanted.


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%Columbia, SC 29208}
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\begin{abstract}
\singlespacing
This supplement contains additional proofs, simulation results, and empirical results for the paper ``The Factor-Lasso and K-Step Bootstrap Approach for Inference in High-Dimensional Economic Applications.''
\end{abstract} 


\maketitle

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\small


\appendix

\setcounter{section}{3}

%\textcolor{red}{In the appendix, bits in red are changes that I think should be made but should be double-checked.}

%\textcolor{blue}{In the appendix, bits in blue are questions or comments that should be deleted after being addressed.} 
  
  
  
\section{Estimating Factors Using Principal Components Analysis}\label{Sec: PC}

%In this section, we discuss estimation of factors and factor residuals using principal components (PC).  We also provide low-level conditions under which the high-level conditions used in establishing Theorem \ref{t3.1} are satisfied for PC. 

%\subsection{Principal Components Estimator}
Let
$$
\tilde X=\begin{pmatrix}
\tilde X_{i1}&\cdots &\tilde X_{n1}\\
\vdots&&\vdots\\
\tilde X_{iT}&\cdots &\tilde X_{nT}
\end{pmatrix}_{pT\times n},
 \quad  \tilde \Lambda=\begin{pmatrix}
\tilde \Lambda_1\\
\vdots \\
\tilde\Lambda_T
\end{pmatrix}_{pT\times K},\quad \tilde F=\begin{pmatrix}
\tilde f'_1\\
\vdots \\
\tilde f'_n
\end{pmatrix}_{n\times K},
$$
and define $\tilde U$ similarly.  The matrix form of the factor model is then
$$
\tilde X=\tilde\Lambda \tilde F'+\tilde U,
$$
where the individual and time effects have already been removed.
  Let $\widehat F=(\widehat f_1,...,\widehat f_n)'$ denote the $n\times K$ matrix of the estimated factors.  The columns of $\widehat F/\sqrt{n}$ are the eigenvectors of the first $K$ eigenvalues of  $\tilde X'\tilde X/(npT)$. We call this the ``PC estimator''.\footnote{We choose to focus on the PC estimator as a concrete example because it is relatively simple and is free of tuning parameters. One could consider other options which would also satisfy our assumed high-level conditions.  For example, the weighted PC estimator (e.g. \cite{choi, BL13}), can be more efficient than the standard PC estimator but requires additional tuning parameters for practical application.}
We now present regularity conditions which are sufficient to verify that the PC estimator satisfies the high-level sufficient conditions for the quality of factor estimation given in Assumption \ref{ca.1} in Section \ref{sec: hl factors}.  These regularity conditions are standard for high-dimensional approximate factor models.

 \begin{assumption}[Pervasiveness]\label{pervasive}
 There are $c, C>0$ so that
 $$
c< \lambda_{\min}\left(\frac{1}{T}\sum_{t=1}^T\frac{1}{p}\tilde\Lambda_t'\tilde\Lambda_t\right)\leq\lambda_{\max}\left(\frac{1}{T}\sum_{t=1}^T\frac{1}{p}\tilde\Lambda_t'\tilde\Lambda_t\right) <C.
 $$
 where $\lambda_{\min}(A)$ and $\lambda_{\max}(A)$ respectively denote the minimum and maximum eigenvalues of $A$.
 \end{assumption}
 
%Assumption \ref{pervasive} effectively implies that the factors do not load on a small number of series but rather are related to a large number of the available $X$-variables.  The use of this assumption in high-dimensional factor models provides part of the motivation for the factor-lasso approach where at least some forms of association between factors that are not pervasive but instead load on only a few elements in $X$ and an outcome can be captured through the presence of the factor residuals in the equations of interest. 

  \begin{assumption}[Second Order Weak Dependence]\label{adep}
There is $C>0$,
\begin{eqnarray*} 
&&\max_{m,t,i}    \sum_{s=1}^T \sum_{v=1}^p\Cov(U_{it,m} ^2, U_{is,v} ^2 )<C,\cr
&& \max_{i,m,s,t,v}\sum_{h=1}^T\sum_{l=1}^p   |\Cov( U_{it,v}  U_{is,m}, U_{ih,l}  U_{is,m})|<C,\cr
&&\max_{i} \frac{1}{T^2p} \sum_{m,l\leq p} \sum_{t,s, h,v\leq T} \Cov (U_{it,m}U_{is, m}, U_{ih,l}U_{iv, l})<C,\cr
&&\max_{i,m} \frac{1}{T^2p} \sum_{k,l\leq p} \sum_{t,s, h,v\leq T} |\Cov (U_{it,k}U_{is, m}, U_{ih,l}U_{iv, m})|<C.
\end{eqnarray*}
 
 \end{assumption}
   
%The  left hand side of the third condition equals $\max_i \Var(\frac{1}{\sqrt{p}}  \sum_{m=1}^p(\sqrt{T}\bar U_{i\cdot, m})^2)$. In addition, if we ignore the absolute value, then the left hand side of the fourth condition equals $\max_{im}  \Var (\frac{1}{\sqrt{p}} (\sqrt{T}\bar U_{i\cdot ,k})(\sqrt{T}\bar U_{i\cdot, m}))$.  Hence the third condition means that the variance of the standardized squared average should be bounded, and  the fourth  condition is slightly stronger than requiring  $\max_{im}  \Var (\frac{1}{\sqrt{p}} (\sqrt{T}\bar U_{i\cdot ,k})(\sqrt{T}\bar U_{i\cdot, m}))<C$. In the special case when $\{U_t\}$  is serially independent across $t$, all of the four conditions in Assumption \ref{adep} can be directly verified under various notions of weak cross-sectional dependence. 
 
 



  \begin{assumption}[Uniform cross-sectional convergence]\label{ad.3ad}
\begin{eqnarray*} 
&\max_{i\leq n} |\frac{1}{pT}\sum_{t=1}^T\sum_{m=1}^p(U_{it,m}^2-EU_{it,m}^2)|=O_P(1).\cr
&\max_{i\leq n}\|\frac{1}{pT}\sum_{t=1}^T\sum_{m=1}^p\tilde\lambda_{t,m}U_{it,m}\|_2=O_P\left(\sqrt{\frac{\log n}{pT}}\right).
\end{eqnarray*}
 \end{assumption}
%Suppose $\{U_{it,m}\}_{m\leq p}$ are cross-sectionally independent, then the sufficient condition of Assumption \ref{ad.3ad} is that $\log n=O(pT)$.

\subsection{High-level assumptions on the estimated factors}\label{sec: hl factors}
 
 More generally, the estimated factors should satisfy the following conditions in the original and the bootstrap data.
 
\begin{assumption}[Quality of Factor Estimation in Original Data] \label{ca.1} Suppose  there is an invertible $\dim(f_i)\times \dim(f_i)$ matrix $H$  with   $\|H\|+\|H^{-1}\|=O_P(1)$,  and  non-negative sequences $\Delta_F$, $\Delta_{eg}$,  $\Delta_{ud}$,   $\Delta_{fum}$, $ \Delta_{fe},   \Delta_{\max}$,   so that for $\tilde z_{it}\in\{\tilde \epsilon_{it}, \tilde \eta_{it}\}$, $\tilde w_{tm}\in\{\tilde\Lambda_t'\gamma_d, \tilde\Lambda_t'\gamma_y, \tilde\delta_{dt},  \tilde \delta_{yt}, \tilde \lambda_{tm}\}, $ $\tilde h_{tk}\in\{\tilde \delta_{dt}, \tilde \delta_{yt},  \tilde\lambda_{tk}\}$, and $\gamma\in\{\gamma_d,\gamma_y\},$
 \begin{eqnarray*}
 &&\max_{i\leq n}\|\widehat f_i-H'\tilde f_i\|_2=O_P(\Delta_{\max}),\quad
\frac{1}{n}\sum_{i=1}^n\|\widehat f_i-H'\tilde f_i\|_2^2=O_P(\Delta_F^2)\cr
&& \frac{1}{T} \sum_{t=1}^T\|\frac{1}{n}\sum_{i=1}^n (\widehat  f_i-H'\tilde f_i)\tilde z_{it}\|^2_2
 =O_P(\Delta_{fe}^2),\cr
 %\quad \frac{1}{T} \sum_{t=1}^T\|\frac{1}{n}\sum_{i=1}^n (\widehat  f_i-H'\tilde f_i)\tilde U_{it}'\gamma\|^2_2 =O_P(\Delta_{fu}^2)\cr
   &&\max_{m\leq p} \| \frac{1}{{nT}}\sum_{i=1}^n\sum_{t=1}^T(\widehat  f_i -H'\tilde f_i) \tilde z_{it}  \tilde w_{tm} '\|_F =O_P(\Delta_{eg}),\cr
&&
  \max_{m, k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat  f_i -H'\tilde f_i) \tilde U_{it,m}  \tilde h_{tk}'  \|_F=O_P(\Delta_{ud}), \cr
    && \max_{m\leq p, t\leq T}\|\frac{1}{n}\sum_{j=1}^n(\widehat f_j -H'\tilde f_j)\tilde U_{jt, m}\|_2=O_P(\Delta_{fum}).
\end{eqnarray*}
These sequences satisfy the following restrictions:
\begin{align*} 
&\sqrt{nT}|J|^2_0\Delta_F^2=o(1), \ \Delta_{eg}=o\left(\frac{1}{\sqrt{nT}}\right), \ \Delta_{ud}=o\left(\sqrt{\frac{\log p}{nT}}\right), \ |J|_0^2 \sqrt{\log p } \Delta_{ud}=o(1), \\
&\Delta_{fum}^2=o\left(\frac{\log p}{T|J|_0^2\log(pT)}\right), \ \Delta_{fe}^2=o\left(\frac{\log p}{T\log (pT)}\right), \ \Delta_{\max}^2=O(\log(n)), \ \text{and} \\
&\Delta_{\max}^2|J|_0^2T\left(\lambda^2_n|J|_0+ \Delta_F^2|J|^2_0+ {\frac{|J|_0}{n}}\right)=o(1).
\end{align*}
  
\end{assumption}


\begin{assumption} [Quality of Factor Estimation in Bootstrap Data] \label{a6.2} Suppose  there is an invertible $\dim(f_i)\times \dim(f_i)$ matrix $H^*$  with   $\|H^*\|+\|H^{*-1}\|=O_{P^*}(1)$,    
and  non-negative sequences $\Delta_F^*$, $\Delta_{eg}^*$,  $\Delta_{ud}^*$,    $ \Delta_{fe}^*$, so that for  $\tilde z_{it}^*  \in\{\tilde \eta_{it}^*, \tilde \epsilon_{it}^* \},   $    
  $\widehat g_{tm}\in\{\widehat\Lambda_t'\widehat\gamma_d, \widehat\Lambda_t'\widehat\gamma_y, \widehat\delta_{dt},\widehat\delta_{yt} , \widehat\lambda_{tm}\}$, and  $\widehat h_{tm}\in\{ \widehat\delta_{dt},\widehat\delta_{yt} , \widehat\lambda_{tm}\}$,
   \begin{eqnarray*}
&&  \frac{1}{n}\sum_{i=1}^n\|\widehat f_i^*-H^{*'}\widehat f_i\|_2^2=O_P(\Delta_F^{*2})\cr
%&&\frac{1}{T}\sum_{t=1}^T\|\frac{1}{n}\sum_{i=1}^n\tilde z_{it}^*  	 ( \widehat  f_i^{*}- H^{*'}\widehat f_i)\|_2  ^2=O_{P^*}(\Delta_{fe}^{*2})\cr
&&\max_{m\leq p}\| \frac{1}{{nT}}\sum_{i=1}^n\sum_{t=1}^T \tilde z_{it}^*  \widehat g_{tm}(\widehat  f_i ^*-H{^*}\widehat f_i)'\|_F=O_{P^*}(\Delta_{eg}^*)\cr
 %  && \max_{m\leq p, t\leq T}\|\frac{1}{n}\sum_{j=1}^n(\widehat f_j^*-H^{*'}\widehat f_j)\tilde U^*_{jt, m}\|_2=O_P(\Delta_{fum}^*).\cr
   &&  \max_{m, k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat f_i^*-H^{*'}\widehat f_i) \tilde U_{it,m}^*  \widehat h_{tk}'  \|_F=O_P(\Delta^*_{ud}).
\end{eqnarray*}
 These sequences satisfy the following restrictions:
\begin{align*}
&\sqrt{nT}|J|^2_0\Delta_F^{*2}=o(1), \ \Delta_{eg}^*=o\left(\frac{1}{\sqrt{nT}}\right), \ \Delta_{ud}^*=o\left(\sqrt{\frac{\log p}{nT}}\right), \ |J|_0^2 \sqrt{\log p } \Delta_{ud}^*=o(1), \\ 
&\Delta_{F}^{*2}=o\left(\frac{\log p}{T \log (pT)}\right), \ \text{and } \Delta_{\max}^2|J|_0^2T\Delta_{F}^{*2}=o(1).
\end{align*} 
\end{assumption}


The following proposition shows that the high-level Assumptions \ref{ca.1}  and \ref{a6.2} are satisfied by the PC estimator.

\begin{proposition}\label{p4.1} Assume $|J|_0^4=o(nT^3)$,  $|J|_0^4n=o(p^2T)$ and $|J|_0^2\log n=o(p)$. 
Then  Assumptions \ref{pervasive} - \ref{ad.3ad} imply Assumptions  \ref{ca.1} and  \ref{a6.2} about $\widehat F $ and $\widehat F^*$.\end{proposition}
    
  

%\begin{proposition}\label{p6.2} Consider the PC estimator $\widehat F$.  Further suppose that $|J|_0^4=o(nT^3)$, $|J|_0^4n=o(p^2T)$ Then Assumption  is satisfied. 
%\end{proposition}
 
The conditions  $|J|_0^4n=o(p^2T)$  and $|J|_0^2\log n=o(p)$ require lower bounds on the growth of $p$.  These conditions differ from those used in the literature on inference in purely sparse high-dimensional models, e.g. \cite{belloni2014inference}, in that lower bounds on $p$ are not required in the purely sparse setting.  These lower bounds arise because accurately  estimating the unknown factors using PCA requires a large number of observed series.   In the special case $|J|_0=O(1)$, these conditions require 
$$
T \ll n \ll p^2T, \ \log ^{3}p=O(n), \ \text{and} \ \log n=o(p).
$$
%The results developed in this paper will thus be inappropriate in settings where $p$ is quite small relative to $n$.  Of course, in the setting with $p$ small relative to $n$, a simple approach is to just use all of the available variables without dimension reduction.



       
% One of the major technical tasks of this paper is to show that the effects of estimating $(\tilde f_i,\tilde U_{it})$ are stochastically dominated by the  plug-in  tuning parameter $\kappa_n$ in (\ref{tuning}). Since $\kappa_n\asymp\sqrt{\frac{\log p}{nT}}$, this is a strong requirement. It turns out that the existing results in the literature on asymptotic properties for the factor estimators are not directly applicable to yield sufficiently sharp bound to control the effect of estimating $(\tilde F, \tilde U)$.  Thus we establish many new convergence rates that are more relevant to the current context.
 

Technically,  existing results in the literature on estimating factor models are not directly applicable to prove Proposition \ref{p4.1}. 
In Appendix \ref{sf}, we show that  
\begin{align*}
\frac{1}{n}\sum_{i=1}^n\|\widehat f_i-H'\tilde f_i\|_2^2&=O_P\left(\frac{1}{pT}+\frac{1}{n^2}+\frac{1}{nT^2}\right)
\end{align*}
when $\widehat f_i$ is estimated via PCA.  This rate is fairly standard (see, e.g., \cite{bai03}). On one hand, the estimation of factors depends on the number of ``units'' related to the factors.  In the current model $\tilde X_{it}=\tilde \Lambda_t\tilde f_i+\tilde U_{it}$, this number of ``units'' is $pT$ as the factors are common to the $p$ observed covariates and the $T$ time periods. On the other hand, the rate should also depend on the number of observations available for estimating the coefficients on the factors.  This number of observations enters as $O_P(n^{-2})$. (Note that in the current notation, the rate in  \cite{bai03} would be $O_P(n^{-1})$ under serial correlation. We achieve a faster rate $O_P(n^{-2})$ due to the independence across $i$.)  The term $O_P(\frac{1}{nT^2})$ is a nonstandard component, which appears due to the use of demeaned $\tilde U_{it}$ in the model.  Specifically, we need to bound $\frac{1}{n^2}\sum_iE[\frac{1}{p}\sum_{m=1}^p \bar U_{i\cdot, m}^2]^2$, which leads to an extra term $O_P(\frac{1}{nT^2})$.
 



The above rate  allows conditions involving $\Delta_F$ to be directly verified.  However, it does not imply the uniform convergence condition $\max_{t\leq T}\|\widehat f_i-H'\tilde f_i\|_2$. Nor is this result sufficient to verify the other stated conditions because other terms,  e.g. $\Delta_{eg}$, $\Delta_{fum}$, and $\Delta_{fe}$, involve ``weighted averages" of $\{\widehat f_i- H'\tilde f_i\}$ whose rates of convergence can be derived and shown to be faster than that of $\Delta_F^2=\frac{1}{pT}+\frac{1}{n^2}+\frac{1}{nT^2}$. For instance, if we use a simple Cauchy-Schwarz inequality to bound $\Delta_{ud}$, we would have
$$
\max_{m, k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat  f_i -H'\tilde f_i) \tilde U_{it,m}  \tilde h_{tk}'  \|_F^2\leq \frac{1}{n}\sum_{i=1}^n\|\widehat f_i-H'\tilde f_i\|_2^2 \max_{m,k\leq p}\frac{1}{n}\sum_{i=1}^n\|\frac{1}{T}\sum_{t=1}^T \tilde U_{it,m}  \tilde h_{tk}\|_2^2.
$$
It can be shown that $\max_{m,k\leq p}\frac{1}{n}\sum_{i=1}^n\|\frac{1}{T}\sum_{t=1}^T \tilde U_{it,m}  \tilde h_{tk}\|_2^2=O_P(\frac{\log p}{T})$, so this crude bound gives us $\Delta_{ud}=\Delta_F\sqrt{\frac{\log p}{T}}$. Unfortunately, this bound is not sharp enough to verify that $ \Delta_{ud}=o(\sqrt{\frac{\log p}{nT}})$ unless $n=o(pT)$. In the special case that $T$ is fixed, requiring $n=o(p)$ is a restrictive condition. 
 Rather than relying on these crude bounds, we achieve sharper bounds by directly deriving the rate of convergence for each required term in Appendix \ref{sf}, which relies on some novel technical work.  Using these sharper bounds, we only require  $n=o(p^2T)$ which provides much more freedom on the ratio $n/p$.  


\section{Additional Numerical Results} 
This section contains additional simulation results and an additional empirical example based on a cross-sectional version of the model generalized to allow for an endogenous variable.  Implementing the factor-lasso estimation algorithm in this setting requires a small adaptation to the algorithm in Section 2.2 of the main paper.  First, we outline the modification of the basic procedure in the panel data case with a single instrument, called $Z$, for clarity.  We then note that, in examples with a single cross-section, we simply drop all of the individual specific effects in the obvious way as estimation with unrestricted individual-specific heterogeneity cannot proceed with a single cross-section.

Specifically, we add an additional equation to the model to obtain 
\begin{align}\label{IVPPFM:y}
y_{it} &= \alpha d_{it} + \xi_t'f_i + U_{it}'\theta + g_i + \nu_t + \epsilon_{it} \\
\label{IVPPFM:d}
d_{it} &= \pi z_{it} + \delta_{dt}'f_i + U_{it}'\gamma_d + \zeta_i + \mu_t + \eta_{it} \\
\label{IVPPFM:z}
z_{it} &= \delta_{zt}'f_i + U_{it}'\gamma_z + a_i + b_t + \eta_{it} \\
\label{IVPPFM:x}
X_{it} &= \Lambda_t f_i +w_i+\rho_t+U_{it},
\end{align}
and adapt the algorithm in the following manner.

\begin{framed}

\noindent Algorithm (\textbf{IV Factor-Lasso Estimation of $\alpha$}.) 

\begin{enumerate}

\item Obtain $\{\widehat f_i, \widehat U_{it}\}_{i\leq n, t\leq T}$ by extracting factors from the model $\tilde X_{it}=\tilde\Lambda_t \tilde f_i+\tilde U_{it}$. 

\item For $\widehat \delta_{yt}=(\widehat F'\widehat F)^{-1}\widehat F'\tilde Y_t$, $\widehat \delta_{dt}=(\widehat F'\widehat F)^{-1}\widehat F'\tilde D_t$, and $\widehat\delta_{zt} = (\widehat F'\widehat F)^{-1}\widehat F'\tilde Z_t$, run the cluster-lasso programs (\ref{lassoY}) and (\ref{lassoD}) to obtain $\tilde \gamma_y$ and $\tilde \gamma_d$ and the cluster-lasso program
\begin{align}\label{lassoZ}
\tilde \gamma_z &= \arg\min_{\gamma\in\mathbb{R}^{p}}\frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n( \tilde z_{it}-   \widehat\delta_{zt}'\widehat f_i-\widehat U_{it}'\gamma)^2+\kappa_n\|\widehat\Psi^z\gamma\|_1
\end{align}
with tuning parameters as in (\ref{lassoY}) and (\ref{lassoD}).

\item Obtain the estimator $\widehat\alpha$ and corresponding estimated standard error as the coefficient on $\tilde d_{it}- \widehat\delta_{dt}'\widehat f_i-\widehat U_{it,\widehat J}'\widehat\gamma_d$ and associated clustered standard error from the instrumental variables regression of $\tilde y_{it}- \widehat\delta_{yt}'\widehat f_i-\widehat U_{it,\widehat J}'\widehat\gamma_y$ on $ \tilde d_{it}- \widehat\delta_{dt}'\widehat f_i-\widehat U_{it,\widehat J}'\widehat\gamma_d$ using $ \tilde z_{it}- \widehat\delta_{zt}'\widehat f_i-\widehat U_{it,\widehat J}'\widehat\gamma_z$ as the instrumental variable where $\widehat U_{it, \widehat J}$ is the subvector of $\widehat U_{it}$ whose elements are $\{\widehat U_{it, j}: j\in\widehat J\}$ and $\widehat J = \{j \leq p: \tilde \gamma_{y,j} \neq 0 \} \cup \{j \leq p: \tilde \gamma_{d,j} \neq 0\} \cup \{j \leq p: \tilde \gamma_{z,j} \neq 0\}$.  

\end{enumerate}

\end{framed}


\subsection{Instrumental Variables Model Simulations}

We generate data from the model

\begin{align*}
y_{i} &= \alpha d_{i} + (c_{\xi} \xi)'f_i + U_{i}'(c_{\theta} \theta) + \nu + \epsilon_{i} \\
d_{i} &= \pi z_{i} + (c_{\delta_d}\delta_{d})'f_i + U_{i}'(c_{\gamma_d} \gamma_d) + \mu + \eta_{i} \\
z_{i} &= (c_{\delta_z}\delta_{z})'f_i + U_{i}'(c_{\gamma_z} \gamma_z) + \zeta + v_{i} \\
X_{i} &= (c_{\Lambda}\Lambda) f_i + \rho + U_{i}
\end{align*}
with $n = 100$, $K = 2$, and $p = 100$.  Within this model, $d_i$ is an endogenous variable with coefficient of interest $\alpha$ and $z_i$ is an instrumental variable.  We generate $\epsilon_{i} \sim N(0,1)$ and $\eta_{i} \sim N(0,1)$ with $\Ep[\epsilon_i\eta_i] = .8$ i.i.d. across $i$ and independent of all other random variables.  We generate i.i.d. draws for $U_{i}$ as in the simulations in the main text, and  $v_i \sim N(0,1)$ independently from  $U_i$.  We also generate $(\nu,\mu,\rho, \xi,\delta_d,\Lambda, c_{\Lambda}, c_{\gamma_z}, c_{\delta_d}, c_{\xi}, c_{\theta})$ exactly as in the simulations in the main text.  We set $\theta = \gamma_d = \gamma_z$ to be vectors with $j^{\text{th}}$ entry given by $\theta_j = \gamma_{d,j} = \gamma_{z,j} = \frac{1}{j^2}$.   To control the strength of the instrument,  we choose $(c_{\delta_z},c_{\gamma_z})$ so that the $R^2$ of the infeasible regression of $z_{i} - \zeta$ on $(c_{\delta_z}\delta_{z})'f_i + U_{i}'(c_{\gamma_z} \gamma_z)$ is 0.7 and the factors account for 50\% of the explanatory power in this regression.  
 We set $\pi$ so that the fraction of variation accounted for by $z_i$ in the regression of $d_i$ on $z_i$, $f_i$ and $U_i$ is 25\%.  Finally, we set $\alpha = 1$. 

We again estimate $\alpha$ using six different IV procedures similar to those implemented in the main simulation results with one exception.  As the number of features is equal to the sample size in these simulations, we consider an infeasible ``oracle'' estimator that estimates $\alpha$ from IV regression of $y_{i} - (c_{\xi} \xi)'f_i - U_{i}'(c_{\theta} \theta) - \nu$ on $d_{i} - (c_{\delta_d}\delta_{d})'f_i - U_{i}'(c_{\gamma_d} \gamma_d) - \mu$ using $z_{i} - (c_{\delta_z}\delta_{z})'f_i - U_{i}'(c_{\gamma_z} \gamma_z) - \zeta$ as instrument (Oracle).  This  estimator provides a type of best-case benchmark and allows us to ascertain that instruments are strong enough that the usual asymptotic approximation provides a reasonable approximation in the idealized scenario where one is able to perfectly remove the effect of confounding from all variables.

 %We consider estimating $\alpha$ using IV based on the assumption the confounding is entirely captured by latent factors. To implement this procedure, We then obtain the estimator of $\alpha$ as the coefficient on $d_i$ in the standard IV regression of $y_i$ on $d_i$ using $z_i$ as instrument and controlling for $\widehat f_i$ (Factor).  We directly apply the double-selection procedure for an IV model with many controls described in \cite{BCH:JEP} (Double Selection), and, as in the previous case, we then consider two \textit{ad hoc} variants of this approach.  In the first, we extract the first 20 principal components and apply the procedure of \cite{BCH:JEP} to the data $(Y,D,X^*)$ where $X^*$ denotes the original $X$ variables augmented to include the 20 extracted principal components (Double Selection F).  The second \textit{ad hoc} procedure extracta factors, $\widehat f_i$, from $X_{it}$ by PCA using the \cite{AH} procedure to select the number of factors exactly as in the ``Factor'' procedure.  We then obtain estimates $\widehat U_{i}$ as in (\ref{Eq: hatU}) and apply the procedure of \cite{BCH:JEP} to the data $(Y,D,\hat{U}^*)$ where $\hat{U}^*$ denotes the matrix formed by combining $\widehat U$ with the selected factors (Double Selection U).  Finally, we directly apply the factor-lasso approach outlined in this paper modified to the IV setting (Factor Lasso).  The modification simply makes use of the additional reduced form equation for the instrument $z_i$ and adapts (\ref{Eq: deltas}) by adding estimation of $\delta_z$ and adding an additional lasso step of $z_i - \widehat\delta_z'\widehat f_i$ on $\widehat U_i$.  The final estimator is then simply IV of $y_i$ and $d_i$, $\widehat f_i$, and the union of the variables selected in the three lasso steps using $z_i$ as instrument.  
 

Figure \ref{Fig: IV RMSE} gives simulation RMSEs for the estimator of $\alpha$ resulting from applying each procedure.  The RMSEs are truncated at 0.1 for readability of the figure.\footnote{Theoretically, the MSE of the IV estimator does not exist in this context.  We report root mean truncated squared error with a truncation point of 1.}  Again, we see that the factor lasso procedure delivers good performance regardless of the relative strength of the factors and factor residuals in this simulation design.  Each of the other procedures exhibits behavior that depends strongly on the exact strength of the factors in the different equations.  It might be noted that the dominance of the factor-lasso estimator, in terms of RMSE, over the ``Oracle'' procedure is due to the definition of the oracle that we use which fully removes the variation in each variable due to factors and factor residuals even in situations in which some of these variables produce no confounding.  For example, one need not remove the variation in the instruments due to the factors in cases where the factors have zero loadings in the outcome equation, but this variation is always removed due to the way we have defined the oracle model.



We report size of 5\% level tests based on standard asymptotic approximations for each of the six procedures considered in Figure \ref{Fig: IV SIZE} with size truncated at 0.3 for readability of the figure.  In each panel, we report the rejection frequency of the standard t-test of the null hypothesis that $\alpha = 1$ using heteroscedasticity robust standard errors.  Here, we see that the only procedure that uniformly controls size is the infeasible oracle.  Among the feasible procedures, the proposed factor lasso approach performs relatively well in keeping size distortions small across the majority of combinations of relative strengths of the factors.  In this case, we do see that the factor-lasso procedure suffers from reasonably large size distortions when the factors account for all of the confounding in the outcome equation and a moderate amount of counfounding in the treatment equation.  We also see that the pure factor model controls size well in this case, but performs very poorly once all variation in the outcome equation is not due to the factors. 



We again conclude by looking at the performance of the k-step bootstrap in Figure \ref{Fig: IV BOOT}. %we report size of 5\% level tests using the factor-lasso estimator and the asymptotic approximation provided in Theorem \ref{t3.1}, the k-step bootstrap, and a score bootstrap based on \cite{BCFH:Policy}.  For the bootstrap procedures, we construct a 95\% level confidence interval using the bootstrap quantiles as described in Section \ref{Sec: Bootstrap} and reject if the true value falls outside of the interval. 
We see that there is a modest, but clearly visible, improvement from using the k-step bootstrap relative to the asymptotic approximation.  The score based bootstrap, on the other hand, lines up reasonably well with the asymptotic approximation.  


\subsection{Estimating the Effects of Institutions on Output}


We revisit the example considered in \cite{acemoglu:colonial}.  \cite{acemoglu:colonial} are interested in the parameter $\alpha$ in a structural model of the form
\begin{align*}
\log(\textrm{GDP per capita}_i) = \alpha (\textrm{Protection from Expropriation}_i) + x_i'\beta + \varepsilon_i
\end{align*}
based on aggregate country level data where ``Protection from Expropriation'' is a measure of the strength of individual property rights that is used as a proxy for the strength of institutions and $x_i$ is a set of variables that are meant to control for geography.  \cite{acemoglu:colonial} adopt an IV strategy where they instrument for institution quality using early European settler mortality to estimate $\alpha$ as institutions are clearly potentially endogenous.  They point out that their instrument would be invalid if there were other factors that are highly persistent and related to the development of institutions within a country and to the country's GDP.  A leading candidate for such a factor that they discuss is geography.  To address this possibility, \cite{acemoglu:colonial} control for the distance from the equator in their baseline specifications and consider different sets of geographic controls such as continent dummies within their robustness checks (e.g. \cite{acemoglu:colonial} Table 4).

%\cite{acemoglu:colonial} argue that this instrument is associated with institutions because settlers were more likely to set up better institutions in places where they were going to establish long term settlements and the likelihood of establishing a long term settlement was related to mortality at the time of initial colonization.  They then argue that present institutions are related to these initial conditions because institutions are extremely sticky.  The exclusion restriction is then motivated by the argument that shocks to GDP are unlikely to be so persistent that any shocks around the time of European colonization that were correlated to early settler mortality are strongly related to current GDP except through their influence on institutions.


There are, of course, many other ways to measure geography besides distance to the equator or continent where a country is found.  Rather than \textit{ex ante} choosing a small number of variables to proxy for geography, we put a large number of variables that potentially capture geography in $x_i$ and then use the data to reduce dimension.  Specifically, we consider dummies for Africa, Asia, North America, and South America as well as longitude, renewable water, land boundary, land area, amount of coastline, territorial seas, amount of arable land, average temperature, average high temperature, average low temperature, average precipitation, elevation of highest point, elevation of lowest point, fraction of area that is low-lying, latitude, and spherical distance from London.\footnote{We apply the PCA to extract common factors, recognizing that it may not be the ideal method in the presence of dummy variables. Several approaches have been developed in the literature to extend   the PCA to factor models with categorical data.  We refer to  \cite{keller1983multivariate},  \cite{ng2015constructing}, and references therein.}


We adapt the analysis of \cite{acemoglu:colonial} to the present setting by considering estimation of a partial factor instrumental variables model 
\begin{align*}
\log(\textrm{GDP per capita}_i) &= \alpha (\textrm{Protection from Expropriation}_i) + f_i'\xi + U_i'\theta + \varepsilon_i \\
\textrm{Protection from Expropriation}_i &= \pi \textrm{Early Settler Mortality}_i + f_i'\delta_d + U_i'\gamma_d + \eta_i \\
\textrm{Early Settler Mortality}_i &= f_i'\delta_z + U_i'\gamma_z + v_i \\
X_i = \Lambda f_i + U_i
\end{align*}
using our 20 geography measures as $x_i$ and the 64 countries from the original \cite{acemoglu:colonial} data.  The factor-lasso approach seems quite sensible in this setting.  Each of the observed geography measures could reasonably be taken as a noisy proxy for a country's geography.  This relationship is likely to be complicated and uneven with the chief features leading to association between the geography proxies plausibly being only weakly related to the notions of geography that are important predictors of mortality and institutions.  The factor-lasso approach, by allowing a small number of elements of $U_i$ to enter the equation of interest in addition to any common geography factors, readily accommodates this latter possibility in a parsimonious, data-dependent way.


We report estimation results for the first stage coefficient on the instrument in Table \ref{AJRTableResults}.  We report results from the factor-lasso approach in the row ``Factor-Lasso''. For comparison, we also report results from a few natural alternative models.  The row labeled ``Latitude'' uses the single variable distance from the equator to control for geography as in the baseline results from \cite{acemoglu:colonial}.  We report results based on using all 20 available geographic controls without dimension reduction in the row labeled ``All Controls.'' We apply the double selection approach of \cite{belloni2014inference} which would be appropriate if the relationship between geographic controls and the variables of interest were well-approximated by a sparse linear model in ``Double Selection.''  Finally, ``Factor'' reduces dimension through positing a conventional factor model.  All factors are estimated using PCA with number of factors selected by applying the procedure from \cite{ahn2013eigenvalue}.

The first-stage results using only the latitude control suggest there is a fairly strong relationship between the instrument and endogenous variable if latitude is a sufficient control for geography.  The first stage F-statistic using just latitude is 10.9 which many would take to indicate that the instrument is sufficiently strong to identify the effect of interest.\footnote{A benchmark that is commonly used in the applied literature to assess whether there is sufficient variation in the instrument to identify the effect of interest is to compare the first stage F-statistic to 10, with smaller values indicating weak identification.}  The results change in a potentially substantive way after allowing for the possibility that geography is not adequately captured by latitude.  For each of the remaining approaches considered, the first-stage F-statistic drops substantially below 10, with all methods besides applying the pure factor model returning first-stage coefficients that are statistically insignificant at the 5\% level.  

One might dismiss the lack of significance after including all controls without dimension reduction as it seems likely that a model with 20 covariates in addition to the variables of interest and only 64 observations is overfit.  The next strongest result is from the pure factor model which makes use of a single extracted component and produces a first-stage F-statistic of 7.5.  As evidenced in the simulation example, inference results based on a pure factor model may be misleading when elements of $U_i$ also have explanatory power.  It is then interesting that the double-selection approach and the factor-lasso approach deliver almost identical results indicating a weak association between the endogenous variable and instrument after controlling parsimoniously for geography.  The double-selection procedure selects four variables\footnote{These variables are the Africa dummy, average temperature, average high temperature, and amount of arable land.}, and the factor-lasso approach uses one factor and two additional variables.\footnote{The two selected variables in addition to the factor are the Africa and Asia dummies.}  One might take this to mean that the four variables selected in the double-selection procedure approximately capture the same information as the single factor and two variables used in the factor-lasso results.  In either case, the results suggest that, at best, identification of the structural effect of institutions as measured by ``Protection from Expropriation'' using settler mortality as instrument is weak after geography is controlled for in a parsimonious, data-dependent way.  Given this apparent weak identification, we do not report second stage estimates of the structural effect.\footnote{We note that it would be straightforward to adapt the weak-identification robust procedure of \cite{ch:WeakId} to the present setting.  We do not pursue this extension for brevity.}



\section{The Lasso estimator in the presence of latent factors}\label{App: FactorLasso}

\subsection{The convergence of the lasso estimator on the original data}
\begin{proposition} \label{p2.1} Under Assumptions \ref{a3.2}-\ref{a3.4} and Assumption \ref{ca.1}, for $x \in \{d,y\}$, the lasso estimator $\tilde\gamma_x$ satisfies: 
(i)   \begin{eqnarray*}
%&& \frac{1}{n}\sum_{t=1}^T\|\widehat U_t \Delta_{\gamma}\|_2^2= O_P(\kappa_n^2|J|_0),\cr
&& \frac{1}{n}\sum_{t=1}^T\|\widehat U_t (\gamma_x-\tilde\gamma_x)\|_2^2= O_P(\kappa_n^2|J|_0),\cr
% &&\|\Delta_{\gamma}\|_1=O_P( \kappa_n|J|_0),\cr
 &&\|\gamma_x-\tilde\gamma_x\|_1=O_P( \kappa_n|J|_0+\|R_x\|_1).
\end{eqnarray*}
(ii) 
$
|\widehat J|_0=O_P(|J|_0).
$
\end{proposition}



  
 \textbf{Proof of Proposition \ref{p2.1} (i)}
 
In the following, we show the argument for $\tilde\gamma_{y}$.  The argument for $\tilde\gamma_{d}$ is similar.

 Let 
$$ l(v):=\frac{1}{nT}\sum_{t=1}^T\|\tilde Y_t-\widehat F\widehat\delta_{yt}-\widehat U_tv\|_2^2.$$
 Recall that the solution of the lasso program is
 $
\tilde \gamma_y=\arg\min_{\gamma\in\mathbb{R}^{p}}l(\gamma)+\kappa_n\|\widehat\Psi^y\gamma\|_1.
$
We shall  use $J$ to denote the active set.
Let $\Psi_{J}^y$ denote a $|J|_0\times |J|_0$ diagonal matrix whose entries are the elements of $\Psi^y$ in the set $J.$   Recall that  $\gamma_y$ can be decomposed as 
$$
\gamma_y=\underbrace{\gamma_y^0}_{\text{exactly sparse}}+\underbrace{R_y}_{\text{remainder}}.$$
Recall that $\tilde Y_t=\tilde F\tilde\delta_{yt}+\tilde U_t\gamma_y+\tilde e_t$.  Let 
$$
M_t=\tilde F\tilde\delta_{yt}-\widehat F\widehat\delta_{yt}+(\tilde U_t-\widehat U_t) \gamma_y,\quad \Delta_{\gamma}=\gamma_y^0-\tilde\gamma_y.$$ 
Then $l(\tilde\gamma_y)+\kappa_n\|\widehat\Psi^y\tilde\gamma_y\|_1\leq l(\gamma_y^0)+\kappa_n\|\widehat\Psi^y\gamma_y^0\|_1$ implies  
\begin{equation}\label{ea.1}
\frac{1}{nT}\sum_{t=1}^T\left(\|\widehat U_t \Delta_{\gamma}\|_2^2+2 (R_y'\widehat U_t'+\tilde e_t'+M_t')\widehat U_t \Delta_{\gamma} \right)+\kappa_n\|\widehat\Psi^y\tilde\gamma_y\|_1
\leq  \kappa_n\|\widehat\Psi^y\gamma_y^0\|_1.
\end{equation}
Recall that 
$$
\Psi^y=\diag\left\{ \left({\frac{1}{n}\sum_i W_{im}^2}\right)^{1/2}: m\leq p \right\}
$$ 
where 
$
W_{im}=\frac{1}{\sqrt{T}}\sum_{t=1}^T(U_{it,m}-\bar U_{i\cdot, m})(e_{it}-\bar e_{i\cdot})
$
are independent across $i$. Hence, by the Bernstein inequality for independent data,  
 for $y=M\sqrt{\frac{\log p}{n}}$ and large enough $M>0$,
\begin{align*}
&P\left(\max_{m\leq p}|\frac{1}{n}\sum_{i=1}^nW_{im}^2-EW_{im}^2  |>y\right)
 \leq  p\max_mP\left(\left|\frac{1}{n}\sum_{i=1}^nW_{im}^2-EW_{im}^2\right|>y\right)\cr
 &\leq  p\exp\left( -\frac{ny^2}{4\max_{im}\Var(W^2_{im})}\right)\leq \exp\left(\log p -\frac{ny^2}{4 C}\right)=o(1).
\end{align*}
Hence $
\max_m|\frac{1}{n}\sum_i(W_{im}^2-EW_{im}^2)|=o_P(1).
$ 
Also, recall that  $\widehat\Psi^y $ is defined in (\ref{LoadingsY}), which satisfies $\|\Psi^y-\widehat \Psi^y\|_{\infty}=o_P(1)$. 
By the assumption that $\frac{1}{n}\sum_iEW_{im}^2$ is bounded away from zero, we have, $\|(\Psi^{y})^{-1}\|_{\infty}=O_P(1)$.
In addition, by Lemma \ref{la.5new}, 
$$
\left\|(\Psi^{y})^{-1}\frac{1}{nT}\sum_{t=1}^T\tilde U_t '\tilde e_t \right\|_{\infty}
\leq   \frac{1}{\sqrt{nT}}\Phi^{-1}\left(1-\frac{q_n}{2p}\right)+o_P\left(\sqrt{\frac{\log p}{nT}}\right)
$$ 
with probability approaching one, and 
$$
\frac{1}{nT}\sum_{t=1}^T\left( 2 (R_y'\widehat U_t'+M_t')\widehat U_t \Delta_{\gamma} \right)=o_P\left(\sqrt{\frac{\log p}{nT}}\right)\|\Delta_{\gamma}\|_1.
$$ 




Hence, for any $c_0>1$,
\begin{align*}
|\frac{2}{nT}\sum_{t=1}^T   (R_y'\widehat U_t'&+\tilde e_t'+M_t')\widehat U_t \Delta_{\gamma}| \\
&\leq o_P\left(\sqrt{\frac{\log p}{nT}}\right)\|\Psi^{y-1}\|_{\infty}\|\Psi^y\Delta_{\gamma}\|_1+ \frac{2}{\sqrt{nT}}\Phi^{-1}\left(1-\frac{q_n}{2p}\right)\|\Psi^y\Delta_{\gamma}\|_1 \\
&\leq \frac{c_0+1}{\sqrt{nT}}\Phi^{-1}\left(1-\frac{q_n}{2p}\right)\|\Psi^y\Delta_{\gamma}\|_1 \\
&=\frac{c_0+1}{2c_0}  \kappa_n\|\Psi^y\Delta_{\gamma}\|_1
\end{align*}
 with probability approaching one where the second inequality follows from the fact that there is $C>0$ such that $\Phi^{-1}\left(1-\frac{q_n}{2p}\right) \asymp \sqrt{\log p+\log\frac{1}{q_n}} \geq \sqrt{\log p}$ and we have used the definition 
$$
\kappa_n=\frac{2c_0}{\sqrt{nT}}\Phi^{-1}\left(1-\frac{q_n}{2p}\right),\quad \log(q_n^{-1})=O(\log p)
$$
in the last equality.
%  \textcolor{blue}{We do not use the phrase iterative lasso anywhere else in the paper.  In addition, I didn't notice a definition for $\widehat W_{im}$.  This section needs to be clarified.}  
We thus also have, with probability approaching one, 
\begin{align*}
|\frac{2}{nT}\sum_{t=1}^T   (R_y'\widehat U_t'&+\tilde e_t'+M_t')\widehat U_t \Delta_{\gamma} | \\
& \leq \frac{c_0+1}{2c_0}  \kappa_n\|\widehat\Psi^y \Delta_{\gamma}\|_1+\frac{c_0+1}{2c_0}  \kappa_n \| \widehat\Psi^y\Delta_{\gamma}\|_1\|\widehat\Psi^y- \Psi^y\|_{\infty}\|\widehat\Psi^{y-1}\|_{\infty} \\
&\leq \frac{c_0+1}{2c_0} \kappa_n\|\widehat\Psi^y \Delta_{\gamma}\|_1\left(1+\frac{c_0-1}{4c_0}\right) \\
&=\frac{3c_0+1}{4c_0} \kappa_n\|\widehat\Psi^y \Delta_{\gamma}\|_1.
\end{align*}
(\ref{ea.1}) then implies 
$ \frac{1}{nT}\sum_{t=1}^T \|\widehat U_t \Delta_{\gamma}\|_2^2 -\frac{3c_0+1}{4c_0} \kappa_n\|\widehat\Psi^y \Delta_{\gamma}\|_1 +\kappa_n\|\widehat\Psi^y\tilde\gamma_y\|_1
\leq  \kappa_n\|\widehat\Psi^y\gamma_y^0\|_1.$ 
Thus  
\begin{equation}\label{ed.2ad}
\frac{1}{nT}\sum_{t=1}^T \|\widehat U_t \Delta_{\gamma}\|_2^2  +\frac{c_0-1}{4c_0}\kappa_n\|(\widehat\Psi^y\Delta_{\gamma})_{J^c}\|_1
\leq \frac{7c_0+1}{4c_0}\kappa_n\|(\widehat\Psi^y\Delta_{\gamma})_{J}\|_1.
\end{equation}
Now, note that the sparse eigenvalue assumption \ref{a3.4} implies that,  for some $ \underline{\phi}>0$, $$
\underline{\phi}(m):=\inf_{\delta\in\mathbb{R}^{p}: \|\delta_{J^c}\|_1\leq m\|\delta_{J}\|_1}\mathcal{R}(\delta)\geq \underline{\phi}.
$$
And for some generic $C>0$, $\|\Delta_{\gamma}\|_1^2\leq C\|\Delta_{\gamma, J}\|_1^2\leq C\|\Delta_{\gamma}\|_2^2|J|_0.$  Hence $ \frac{1}{nT}\sum_{t=1}^T \|\tilde U_t \Delta_{\gamma}\|_2^2\geq  \underline{\phi}\|\Delta_{\gamma}\|_2^2$. In addition, by Lemma \ref{la.5new},   
$$
|J|_0\max_{m,k\leq p}|\frac{1}{nT}\sum_{it}(\widehat U_{it,m}\widehat U_{it,k}-\tilde U_{it,m}\tilde U_{it,k})|=o_P(1).
$$ 
Hence, 
\begin{align*}
\frac{1}{nT}\sum_{t=1}^T \|\widehat U_t \Delta_{\gamma}\|_2^2 &\geq \frac{1}{nT}\sum_{t=1}^T \|\tilde U_t \Delta_{\gamma}\|_2^2-\|\Delta_{\gamma}\|_1^2\max_{m,k\leq p}|\frac{1}{nT}\sum_{it}(\widehat U_{it,m}\widehat U_{it,k}-\tilde U_{it,m}\tilde U_{it,k})| \\
&\geq \underline{\phi}\|\Delta_{\gamma}\|_2^2- C\|\Delta_{\gamma}\|_2^2|J|_0\max_{m,k\leq p}|\frac{1}{nT}\sum_{it}(\widehat U_{it,m}\widehat U_{it,k}-\tilde U_{it,m}\tilde U_{it,k})| \\
&\geq \underline{\phi}\|\Delta_{\gamma}\|_2^2/2,
\end{align*}
with probability approaching one.  Also, $\max_m|\widehat \Psi^y_m|=O_P(2)$.   So (\ref{ed.2ad}) implies for some $C>0,$
$$
 \|\Delta_{\gamma}\|_2^2\leq  C\kappa_n\| (\Delta_{\gamma})_{J}\|_1\leq \sqrt{|J|_0} \kappa_n\| (\Delta_{\gamma})_{J}\|_2\leq \sqrt{|J|_0} \kappa_n\|  \Delta_{\gamma} \|_2.
$$
Hence $\|  \Delta_{\gamma} \|_2\leq O_P(\sqrt{|J|_0} \kappa_n)$, and 
 $\|\Delta_{\gamma}\|_1\leq C\|\Delta_\gamma\|_2\sqrt{|J|_0}\leq O_P( \kappa_n|J|_0)$. Still by (\ref{ed.2ad}), 
$$\frac{1}{nT}\sum_{t=1}^T\|\widehat U_t \Delta_{\gamma}\|_2^2  = O_P(\kappa_n^2|J|_0).$$ Moreover, $\gamma_y-\tilde\gamma_y=\Delta_\gamma+R_y$, and by Lemma \ref{la.5new}, $\max_{m,k\leq p}|\frac{1}{nT}\sum_{it}\widehat U_{it,m}\widehat U_{it,k}|=O_P(1)$, so 
  $\frac{1}{nT}\sum_{t=1}^T\|\widehat U_t R_y\|_2^2\leq \|R_y\|_1^2\max_{m,k\leq p}|\frac{1}{nT}\sum_{it}\widehat U_{it,m}\widehat U_{it,k}|=O_P(1) \|R_y\|_1^2$. We have
\begin{align*}
\frac{1}{nT}\sum_{t=1}^T\|\widehat U_t (\gamma_y-\tilde\gamma_y)\|_2^2 &= O_P(\|R_y\|_1^2+\kappa_n^2|J|_0), \\
\|\gamma_y-\tilde\gamma_y\|_1 &= O_P( \kappa_n|J|_0+\|R_y\|_1).
\end{align*}
 where $\frac{1}{nT}\sum_{t=1}^T\|\widehat U_t (\gamma_y-\tilde\gamma_y)\|_2^2$ follows from (\ref{ed.2ad}).
 


 
 \textbf{Proof of Proposition \ref{p2.1} (ii)}
 
We prove the convergence for $|\tilde\gamma_y|_0$ only; the result for $|\tilde\gamma_d|_0$ follows similarly.  
Write $\widehat J = \text{supp}(\tilde\gamma_y)$. Let $\widehat U_{it, \widehat J}$    be the $|\widehat J|_0\times 1$ subvector of $\widehat U_{it}$ whose elements correspond to the indices in $\widehat J$. Also, let $\widehat U_{t, \widehat J}$   be the  $n\times |\widehat J|_0$ matrix whose $i^{\text{th}}$ row is $\widehat U_{it, \widehat J}'$. 
 %For any $\varepsilon>0$, there is $C>0$, with probability at least  $1-\varepsilon,$ Write  submatrices of $\widehat U_t$ and $\tilde U_t$, whose columns correspond to the indices in $\widehat J$. 
In addition, write
$$
\widehat U_{\widehat J}=(\widehat U_{1, \widehat J}',...,\widehat U_{T, \widehat J}')',\quad nT\times |\widehat J|_0.
$$
Define $\tilde U_{it, \widehat J}$, $\tilde U_{t, \widehat J}$ and $ \tilde  U_{\widehat J}$ similarly.
Then by (\ref{ea.7}),    
\begin{align}
\nonumber
\| \frac{1}{nT}\sum_{t=1}^T\widehat U_{t, \widehat J}'\widehat U_{t, \widehat J}\| &= \frac{1}{nT}\|\widehat U_{\widehat J}'\widehat U_{\widehat J}\| = \frac{1}{nT}\|\widehat U_{\widehat J}\|^2 \\
\nonumber
& \leq \frac{2}{nT}\|\tilde U_{\widehat J}\|^2+\frac{2}{nT}\|\widehat U_{\widehat J}-\tilde U_{\widehat J} \|^2 \\
\nonumber
&=\|\frac{2}{nT}\sum_{t=1}^T\tilde  U_{t, \widehat J}'\tilde U_{t, \widehat J}\|+\|\frac{2}{nT}\sum_{t=1}^T(\widehat U_{t, \widehat J}-\tilde  U_{t, \widehat J})' (\widehat U_{t, \widehat J}-\tilde  U_{t, \widehat J})\| \\
\nonumber
&\leq\|\frac{2}{nT}\sum_{t=1}^T\tilde  U_{t, \widehat J}'\tilde U_{t, \widehat J}\|+\frac{2}{nT}\sum_{t=1}^T\sum_{i=1}^n\sum_{m\in\widehat J}(\widehat U_{it, m}-\tilde  U_{it, m})^2 \\
\label{ea.0}
&\leq 2\phi_{\max}(|\widehat J|_0)+ O_P(|\widehat J|_0\Delta_F^2+|\widehat J|_0\frac{\log (pT)}{n}),
\end{align}
where the last inequality follows from (\ref{ea.7}).
 
 
 
  Let $\Psi^y(\widehat J)$ be a $|\widehat J|_0\times 1$ subvector of the diagonal entries of $\Psi^y$, with elements in $\widehat J$.   Then, using the Karush-Kuhn-Tucker (KKT) condition and recalling that $ \tilde Y_t=\tilde F\tilde \delta_{yt}+\tilde U_t\gamma_y+\tilde e_t$, and that $M_t:=\tilde F \tilde \delta_{yt}-\widehat F\widehat\delta_{yt}+(\tilde U_t-\widehat U_t) \gamma_y$, we have that, with probability approaching one,
\begin{align*}
\kappa_n\|\widehat\Psi^y(\widehat J)\|_2 &= \|\frac{2}{nT}\sum_t\widehat U_{t,\widehat J}'(\tilde Y_t-\widehat F\widehat\delta_{yt}-\widehat U_t\tilde\gamma_y)\|_2 \\
%=\|\frac{2}{nT}\sum_t\widehat U_{t,\widehat J}'(\tilde F\delta_{yt}-\widehat F\widehat\delta_{yt}+\tilde U_t\gamma_y-\widehat U_t\tilde\gamma_y+\tilde e_t)\|_2\cr
%&\leq&\|\frac{2}{nT}\sum_t\widehat U_{t,\widehat J}'(\tilde F\delta_{yt}-\widehat F\widehat\delta_{yt})\|_2+\|\frac{2}{nT}\sum_t\widehat U_{t,\widehat J}'( \tilde U_t\gamma_y-\widehat U_t\tilde\gamma_y)\|_2+\|\widehat\Psi_{\widehat J}^{y}\widehat\Psi_{\widehat J}^{y-1}\frac{2}{nT}\sum_t\widehat U_{t,\widehat J}'\tilde e_t\|_2\cr
&= \|\frac{2}{nT}\sum_t\widehat U_{t,\widehat J}'(M_t+\tilde e_t+\widehat U_t(\gamma_y- \tilde\gamma_y))\|_2 \\
&\leq \frac{2}{nT}\|\sum_t\widehat U_{t,\widehat J}'\widehat U_{t,\widehat J}\|^{1/2} (\sum_t\|  \widehat U_t\gamma_y-\widehat U_t\tilde \gamma_y \|_2^2)^{1/2} \\
& \qquad + \|\widehat\Psi^{y}(\widehat J)\|_2\| \widehat\Psi_{\widehat J}^{y-1}\frac{2}{nT}\sum_t\widehat U_{t,\widehat J}'(M_t+\tilde e_t)\|_{\infty} \\
&\leq (2\phi_{\max}(|\widehat J|_0) + O_P\left(|\widehat J|_0\Delta_F^2+|\widehat J|_0 \frac{\log (pT)}{n}\right)^{1/2} O_P (\|R_y\|_1^2+\kappa_n^2|J|_0)^{1/2} \\
& \qquad + \|\widehat\Psi^{y}(\widehat J)\|_2\frac{3c_0+1}{4c_0}\kappa_n,
\end{align*}
where the last inequality follows from (\ref{ea.0}), Lemma \ref{la.5new}, and Proposition \ref{p2.1}. We then have
\begin{equation}\label{ea.3add}
\kappa_n|\widehat J|_0^{1/2} \leq O_P\left(2\phi_{\max}(|\widehat J|_0)+|\widehat J|_0\Delta_F^2+|\widehat J|_0\frac{\log(pT)}{n}\right)^{1/2}(\|R_y\|_1^2+\kappa_n^2|J|_0)^{1/2}.
\end{equation}
 
 
Case 1: $\phi_{\max}(|\widehat J |_0)<|\widehat J |_0\Delta_F^{2}+|\widehat J |_0\frac{\log(pT)}{n}$. In this case, (\ref{ea.3add}) implies 
$$
\kappa_n^2  \leq O_P  \left(  \Delta_F^2+ \frac{\log(pT)}{n} \right) \|R_y\|_1^2+O_P \left(  \Delta_F^2+ \frac{\log(pT)}{n}\right)\kappa_n^2|J|_0.
$$
However, this result contradicts the assumptions that $(  \Delta_F^2+ \frac{\log(pT)}{n})|J|_0=o(1)$ and that $ (  \Delta_F^2+ \frac{\log(pT)}{n}) \|R_y\|_1^2=o(\kappa_n^2)$.  Thus, we have that this case cannot happen.  
 
 
Case 2: $\phi_{\max}(|\widehat J |_0)\geq |\widehat J |_0\Delta_F^{2}+|\widehat J |_0\frac{\log(pT)}{n}$. 
In this case, (\ref{ea.3add}) implies 
$$
 |\widehat J|_0 \leq O_P   (\|R_y\|_1^2/\kappa_n^2+ |J|_0)\phi_{\max}(|\widehat J|_0)=O_P( |J|_0\phi_{\max}(|\widehat J|_0))
$$
where the equality is due to $\|R_y\|_1^2=O(\kappa_n^2|J|_0)$.  By Assumption \ref{a3.4}, there is an $l_n\to\infty$ such that $\phi_{\max}(l_n|J|_0)<c_2$. Now, let $g_n=l_n|J|_0.$ We now show $g_n \geq |\widehat J|_0$. Suppose otherwise, and let $l=|\widehat J|_0/g_n$.  Then $l\geq 1$. Now by Lemma 3 of \cite{belloni2013least}, $\phi_{\max}([lg_n])\leq [l]\phi_{\max}(g_n)$ for $l\geq 1$.  Hence, we have, with probability arbitrarily close to one, that there is $C>0$ such that  
  $$|\widehat J|_0\leq C\phi_{\max}(|\widehat J|_0)|J|_0\leq C\phi_{\max}(g_n)|J|_0 \frac{|\widehat J|_0}{g_n}< C c_2|J|_0 \frac{|\widehat J|_0}{g_n},
  $$
implying $l_n|J|_0=g_n\leq Cc_2|J|_0$. However, this result contradicts $l_n\to\infty$. Hence, $g_n\geq |\widehat J|_0.$  Then, we have $\phi_{\max}(|\widehat J|_0)\leq \phi_{\max}(g_n)$ because $\phi_{\max}(.)$ is non-decreasing. Thus, $$|\widehat J|_0\leq C\phi_{\max}(|\widehat J|_0)|J|_0 \leq C\phi_{\max}(g_n)|J|_0<Cc_2|J|_0.$$
 
 
 
\begin{lemma}\label{la.5new} Let $M_t=\tilde F\tilde \delta_{xt}-\widehat F\widehat\delta_{xt}+(\tilde U_t-\widehat U_t) \gamma_x$, with $x=y$ or $x=d$. Then \\
(i)
  $ \|\Psi^{y-1}\frac{1}{nT}\sum_{t=1}^T\tilde U_t '\tilde e_t \|_{\infty}
\leq   \frac{1}{\sqrt{nT}}\Phi^{-1}(1-\frac{q_n}{2p})+o_P(\sqrt{\frac{\log p}{nT}})$ with probability approaching one, provided   $\log p+\log q_n^{-1}=o(n^{1/3})$.\\
(ii) $
 \|\frac{1}{nT}\sum_{t=1}^T(\tilde U_t -\widehat U_t)'\tilde e_t \|_{\infty}= o_P(\sqrt{\frac{\log p}{nT}}).\\
$
(iii) $\|\frac{1}{nT}\sum_{t=1}^T M_t'\widehat U_t     \|_{\infty}=o_P(\sqrt{\frac{\log p}{nT}})$, \\
(iv) $|J|_0\max_{m,k\leq p}|\frac{1}{nT}\sum_{it}(\widehat U_{it,m}\widehat U_{it,k}-\tilde U_{it,m}\tilde U_{it,k})|=o_P(1)$.\\
(v) $\max_{k,m}|\frac{1}{nT}\sum_t\sum_{i=1}^n\widehat U_{it,k}\widehat U_{it,m}|=O_P(1)$. \\
(vi) $\|\frac{1}{nT}\sum_tR_x'\widehat U_t'\widehat U_t\|_{\infty}=o_P(\sqrt{\frac{\log p}{nT}}).$

 

\end{lemma}
\begin{proof}
 (i) We have $\frac{1}{nT}\sum_{t=1}^T\tilde U_t'\tilde e_t=
 \frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n(U_{it}-\bar U_{i\cdot})(e_{it}-\bar e_{i\cdot})-\frac{1}{T}\sum_{t=1}^T\bar U_{\cdot t}\bar e_{\cdot t}+\bar{\bar U}\bar{\bar e}.
 $
Define $W_{im}=\frac{1}{\sqrt{T}}\sum_{t=1}^T(U_{it,m}-\bar U_{i\cdot, m})(e_{it}-\bar e_{i\cdot})$.  Then by Assumption \ref{a3.3},  
$$
\max_{i\leq n, m\leq p}\frac{(E|W_{im}|^3)^{1/3}  }{  (EW_{im}^2)^{1/2}}\leq C,\quad EW_{im}=0.
$$ Now note that $\Psi_m^y=\sqrt{\frac{1}{n}\sum_iW_{im}^2}$.
It follows from Lemma 5 of \cite{belloni2012sparse}, 
$$
P\left(
\max_{m\leq p}\frac{|\frac{1}{n}\sum_iW_{im}|}{\sqrt{\frac{1}{n}\sum_iW_{im}^2}}<\frac{1}{\sqrt{n}}\Phi^{-1}\left(1-\frac{q_n}{2p}\right)
\right)\leq q_n(1+C/l_n^3)
$$
for some $l_n\to\infty$,  provided that $\log p+\log q_n^{-1}=o(n^{1/3}/l_n^2)$. Hence as long as $\log p+\log q_n^{-1}=o(n^{1/3})$, such $l_n\to\infty$ always exist, and thus  $\max_{m\leq p}\frac{|\frac{1}{n}\sum_iW_{im}|}{\Psi_m^y}<\frac{1}{\sqrt{n}}\Phi^{-1}(1-\frac{q_n}{2p})$ with probability approaching one.   Hence, with probability approaching one,
\begin{align*}
\| (\Psi^{y})^{-1}\frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n(U_{it}-\bar U_{i\cdot})(e_{it}-\bar e_{i\cdot})\|_{\infty}
 &= \frac{1}{\sqrt{T}}\max_m(\Psi_m^{y})^{-1}|\frac{1}{n}\sum_iW_{im}| \\
&< \frac{1}{\sqrt{nT}}\Phi^{-1}\left(1-\frac{q_n}{2p}\right).
\end{align*}
In addition,  we show $\|\frac{1}{T}\sum_{t=1}^T\bar U_{\cdot t}\bar e_{\cdot t}\|_{\infty} =o_P\left(\sqrt{\frac{\log p}{nT}}\right).$  If $T $ does not grow, then $$\|\frac{1}{T}\sum_t\bar U_{\cdot t}\bar e_{\cdot t}\|_\infty\leq \max_t\|\bar U_{\cdot t}\|_\infty \max_t|\bar e_{\cdot t}|\leq O_P(\frac{\sqrt{\log p}}{n})=o_P\left(\sqrt{\frac{\log p}{nT}}\right)$$ 
   where the last equality follows since $T\geq 1$ is a fixed constant.  
   
   
   On the other hand, if $T\to\infty$, then
     for any $k\leq p$, let $\mathcal X_{tk}=\bar U_{\cdot t, k}\bar e_{\cdot t}\Var(\bar U_{\cdot t, k}\bar e_{\cdot t})^{-1/2}$, which satisfies the strong mixing condition across $t$ (Assumption \ref{a3.2}).  To apply the Bernstein inequality for strong mixing data,       let $\bar r= \min\{r, 1\}$, $r_1=(0.5 +\bar r^{-1})^{-1}$,  $c=0.5(\gamma+1)$, then  $\gamma \bar r\geq 2$, $r_1<1$, $cr_1> 1$ and $2c\geq 1.$ Because $\bar r\leq r$, the strong mixing condition in Assumption \ref{a3.2} also holds with $\bar r$ in place of $r$. 
The     Bernstein inequality for weakly dependent data of \cite{merlevede2011bernstein} requires (a) the exponential tails (fulfilled by the sub-Gaussian condition in Assumption \ref{a3.2}), and (b) the strong mixing condition (also assumed in Assumption \ref{a3.2}). 
    The introduction of $\bar r$ is to ensure that the so-defined $r_1<1$, another requirement of applying the     Bernstein inequality for strong mixing sequences. Then by Theorem 1 of  \cite{merlevede2011bernstein} (proved using a ``coupling argument" of \cite{dedecker2004coupling} - see also \cite{wang2016homogeneity} for a similar argument),  for $y=M\frac{(\log p)^c}{\sqrt{T}}$, and sufficiently large $M>0$, we have
     $$
     P\left(\max_{k\leq p}\left|\frac{1}{T}\sum_t\mathcal X_{tk} \right|>y\right) \leq   p \max_{k\leq p}P\left(\left|\frac{1}{T}\sum_t\mathcal X_{tk} \right|>y\right) \leq  A_1+A_2+A_3
     $$
     where
     \begin{eqnarray*}
     A_1&=& pT\exp ( -C (Ty)^{r_1})=\exp(\log(pT) - CM^{r_1} T^{r_1/2} \log^{cr_1} p) =o(1),\quad (cr_1\geq 1)\cr
     A_2&=& p\exp\left(  -C\frac{(Ty)^2}{T} \exp\left(\frac{(Ty)^{r_1(1-r_1)}}{C\log^{r_1}(Ty)}\right)  \right)=o(1),\quad (2c> 1, Ty\to\infty)\cr
     A_3&=& p \exp\left( -CTy^2 \right)= \exp(\log p-CM^2\log^{2c}p) =o(1).
     \end{eqnarray*}
     Hence  $\max_{k\leq p}|\frac{1}{T}\sum_t\mathcal X_{tk} |=O_P\left(\frac{(\log p)^c}{\sqrt{T}}\right)$. In addition, 
     $\max_k\Var(\bar U_{\cdot t, k}\bar e_{\cdot t})=O(n^{-2})$. Hence 
     $$
     \|\frac{1}{T}\sum_t\bar U_{\cdot t}\bar e_{\cdot t}\|_\infty\leq \max_{k\leq p}|\frac{1}{T}\sum_t\mathcal X_{tk} | \sqrt{\max_k\Var(\bar U_{\cdot t, k}\bar e_{\cdot t})}=O_P\left(\frac{(\log p)^c}{\sqrt{T}}\frac{1}{n}\right)=o_P\left(\sqrt{\frac{\log p}{nT}}\right)
     $$
     where the last equality is due to $\log^{2c-1}p=\log^\gamma p=o(n)$.
   Also, $\|\bar{\bar U}\bar{\bar e}\|_{\infty}=o_P\left(\sqrt{\frac{\log p}{T}}\frac{1}{n}\right)$. Thus, 
$$\|\frac{1}{T}\sum_{t=1}^T\bar U_{\cdot t}\bar e_{\cdot t}\|_{\infty}+\|\bar{\bar U}\bar{\bar e}\|_{\infty}
= O_P\left(\sqrt{\frac{\log p}{T}}\frac{1}{n}\right)=o_P\left(\sqrt{\frac{\log p}{nT}}\right).
$$


(ii) Throughout the paper, we denote $(\widehat\lambda_{t1},...,\widehat\lambda_{tp})$ as the columns of the $K\times p$ matrix $\widehat\Lambda_t'$. Also, denote $( \lambda_{t1},..., \lambda_{tp})$ as the columns of the $K\times p$ matrix $ \Lambda_t'$. Note that for each $m\leq p$, $\widehat\lambda_{tm}$ estimates the demeaned loading $\tilde\lambda_{tm}$ up to a matrix transformation.


 For $m\leq p$, $\tilde U_{it, m}-\widehat U_{it, m}=(\widehat \lambda_{tm}'-\tilde\lambda_{tm}'(H')^{-1})\widehat f_i + \tilde\lambda_{tm}'(H')^{-1}(\widehat  f_i -H'\tilde f_i).$  Also, note
$\widehat\lambda_{tm}-H^{-1}\tilde \lambda_{tm}=(\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm}+(\widehat F'\widehat F)^{-1}\sum_{i=1}^n\widehat f_i \tilde U_{it, m}$.
Hence, 
\begin{align}
\nonumber
\|\frac{1}{nT}\sum_{t=1}^T(\tilde U_t &-\widehat U_t)'\tilde e_t \|_{\infty} =  \|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T \tilde e_{it}(\tilde U_{it}-\widehat U_{it})\|_{\infty} \\
\nonumber
 &\leq \max_m|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T \tilde e_{it}\widehat  f_i' (\widehat\lambda_{tm}- H^{-1}\tilde\lambda_{tm}) | \\
\nonumber
 & \qquad + O_P(1)\max_m    \| \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}(\widehat  f_i -H'\tilde f_i)\tilde \lambda_{tm}'\|_F \\
\nonumber
 &\leq \max_m \|(\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde \lambda_{tm} \tilde e_{it}\widehat  f_i' \|_F \\
\nonumber
 & \qquad + \max_m\| (\widehat F'\widehat F)^{-1}\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n\widehat f_j \tilde U_{jt, m}  \tilde e_{it}\widehat  f_i'\|_F
 +o_P\left(\sqrt{\frac{\log p}{nT}}\right) \\
\begin{split}\label{A.ii}
&\leq o_P(1)\max_m\|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde \lambda_{tm} \tilde e_{it}\tilde  f_i' \|_F \\
& \qquad + \max_m\|  \frac{1}{n^2T}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n\widehat f_j \tilde U_{jt, m}  \tilde e_{it}\widehat  f_i'\|_F + o_P\left(\sqrt{\frac{\log p}{nT}}\right),
\end{split}
\end{align}
where we used Assumption \ref{ca.1} that 
$$
\max_m    \| \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}(\widehat  f_i -H'\tilde f_i)\tilde \lambda_{tm}'\|_F=o_P\left(\sqrt{\frac{\log p}{nT}}\right).
$$

We now respectively bound the first two terms in (\ref{A.ii}).  For the first term, we have
$$
\max_{m\leq p}\|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde \lambda_{tm} \tilde e_{it}\tilde  f_i'\|_F=\max_m\|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde \lambda_{tm} \tilde e_{it}  f_i' \|_F
\leq l_1+...+l_4, 
$$
for $l_1-l_4$ defined in the following.
      
Let $ W_{im}= \frac{1}{ T} \sum_{t=1}^T\tilde\lambda_{tm} e_{it}f_i'$. For simplicity, we assume $\dim(f_i)=\dim(\tilde\lambda_{im})=1$ so $W_{im}$ is a scalar. The multivariate case follows from an argument that investigating it element-by-element. By the strong mixing condition,    $\max_{im}\Var(W_{im})\leq C\frac{1}{T}$ for some $C>0$. Using the Bernstein inequality for independent data, we can obtain, for $y=M\sqrt{\frac{\log p}{nT}}$ and large enough $M>0$,
\begin{align*}
&P\left(\max_{m\leq p}|\frac{1}{n}\sum_{i=1}^nW_{im}|>y\right)
 \leq  p\max_mP(|\frac{1}{n}\sum_{i=1}^nW_{im}|>y)\cr
 &\leq  p\exp\left( -\frac{ny^2}{4\max_{im}\Var(W_{im})}\right)\leq \exp\left(\log p -\frac{nTy^2}{4 C}\right)=o(1).
\end{align*}
Therefore, 
\begin{align*}
l_1 &= \max_m\|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde \lambda_{tm} e_{it}   f_i'\|_F = \max_m\|\frac{1}{n}\sum_{i=1}^nW_{im}\|_F  =O_P\left(\sqrt{\frac{\log p}{nT}}\right) \\
l_2 &= \max_m\|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde \lambda_{tm} \bar e_{i\cdot } f_i' \|_F = O_P(1)
\|\frac{1}{n}\sum_{i=1}^n \bar e_{i\cdot } f_i' \|_F \\
    &= O_P\left(\max_{kl\leq K}(\frac{1}{n^2T^2} \sum_i\sum_{ts}Ef_{ik}f_{il}e_{it}e_{is})^{1/2}\right) \\
		&\leq O_P\left(\max_{it}\sup_{f_i}\sum_s|E(e_{it}e_{is}|f_i)|\right)^{1/2}O_P\left(\frac{1}{\sqrt{nT}}\right)\max_{ik}(Ef_{ik}^2)^{1/2}=O_P\left(\frac{1}{\sqrt{nT}}\right) ,
		\end{align*}
		in the last equality we used $\max_{it}\sup_{f_i}\sum_s|E(e_{it}e_{is}|f_i)|=O_P(1)$, because given $\{f_i\}$, the process $\{e_t=\alpha\eta_{t}+\epsilon_{t}\}_{t\leq T}$ is weakly dependent satisfying the strong mixing condition in Assumption \ref{a3.2}.
		\begin{align*}
l_3 &= \max_m\|\frac{1}{T}\sum_{t=1}^T\tilde \lambda_{tm} \bar e_{\cdot t} \bar f' \|_F \\
    &= O_P(1)\max_m\|\frac{1}{nT}\sum_t\sum_i\tilde\lambda_{tm}e_{it}\|_2 = O_P\left(\sqrt{\frac{\log p}{nT}}\right) \\
l_4 &= \max_m\|\frac{1}{T}\sum_{t=1}^T\tilde \lambda_{tm} \bar {\bar e} \bar f' \|_F=O_P(1)|\bar{\bar e}| = O_P\left(\frac{1}{\sqrt{nT}}\right).
\end{align*} Here $l_3$ follows from the same argument as $l_1$, using the Bernstein inequality for independent data.
Hence, $o_P(1)\max_m\|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde \lambda_{tm} \tilde e_{it}\tilde  f_i' \|_F=o_P\left(\sqrt{\frac{\log p}{nT}}\right)$. For the second term, we have 
$$
\max_m\|  \frac{1}{n^2T}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n\widehat f_j \tilde U_{jt, m}  \tilde e_{it}\widehat  f_i'\|_F\leq a_1+...+a_4,
$$
where, up to a $\|H\|=O_P(1)$ term,  
\begin{align*}
a_1 &= \max_m\|  \frac{1}{n}\sum_{j=1}^n(\widehat f_j-H'\tilde f_j) \frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n\tilde U_{jt, m}  \tilde e_{it}(\widehat  f_i-H'\tilde f_i)'\|_F \\
    &\leq \left( \frac{1}{T}\sum_t\|\frac{1}{n}\sum_{j=1}^n(\widehat f_j-H'\tilde f_j) \tilde U_{jt, m}\|_2^2\right)^{1/2}\left( \frac{1}{T}\sum_t\|\frac{1}{n}\sum_{i=1}^n  \tilde e_{it}(\widehat  f_i-H'\tilde f_i)'\|_2^2\right)^{1/2} \\
    &= O_P\left(\Delta_{fum}\Delta_{fe}\right) =^{(a)} o_P\left(\sqrt{\frac{\log (p)}{nT}}\right) \\
a_2 &= \max_m\|  \frac{1}{n^2T}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n\tilde f_j \tilde U_{jt, m}  \tilde e_{it} \tilde f_i'\|_F \leq^{(b)} o_P\left(\sqrt{\frac{\log (p)}{nT}}\right), \\
a_3 &= \max_m\|  \frac{1}{n^2T}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n \tilde f_j \tilde U_{jt, m}  \tilde e_{it}(\widehat  f_i-H'\tilde f_i)'\|_F \\
    & \leq \max_{mt} \|\frac{1}{n}\sum_{j=1}^n \tilde f_j \tilde U_{jt, m}\|_2 \left(\frac{1}{T} \sum_{t=1}^T\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}(\widehat  f_i-H'\tilde f_i)\|^2_2\right)^{1/2} \\
		&=^{(c)} O_P\left(\sqrt{\frac{\log (pT)}{n}}\right)o_P\left(\sqrt{\frac{\log p}{T\log (pT)}}\right) \\
a_4 &= \max_m\|  \frac{1}{n^2T}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n(\widehat f_j -H'\tilde f_j)\tilde U_{jt, m}  \tilde e_{it} \tilde f_i'\|_F \\
    &\leq \max_{mt}\|\frac{1}{n}\sum_{j=1}^n(\widehat f_j -H'\tilde f_j)\tilde U_{jt, m}\|_2 \left(\frac{1}{T}\sum_t\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it} \tilde f_i\|_2^2\right)^{1/2} \\
		&=^{(d)} O_P\left(\frac{1}{\sqrt{n}}\right)o_P\left(\sqrt{\frac{\log p}{T}}\right),
 \end{align*}
where 
(a) follows from $\Delta_{fe}\Delta_{fum}\leq\Delta_F^2=o\left(\sqrt{\frac{\log (p)}{nT}}\right)$ due to $ \frac{1}{T} \sum_{t=1}^T\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}(\widehat  f_i-H'\tilde f_i)\|^2_2:=O_P(\Delta_{fe}^2) $ and $\max_{mt}\|\frac{1}{n}\sum_{j=1}^n(\widehat f_j -H'\tilde f_j)\tilde U_{jt, m}\|_2:=O_P(\Delta_{fum})$; (b) follows from Lemma \ref{la.3june}; 
(c) follows using the assumption $\Delta_{fe}^2=o\left({\frac{\log p}{T\log (pT)}}\right);$ and
(d) uses the assumption $\Delta_{fum}=o\left(\sqrt{\frac{\log p}{T}}\right).$
 


(iii) For, $M_t=\tilde F\tilde \delta_{xt}-\widehat F\widehat\delta_{xt}+(\tilde U_t-\widehat U_t) \gamma_x$, we have
\begin{align}
\nonumber
\|\frac{1}{nT}\sum_{t=1}^T M_t'\widehat U_t \|_{\infty}
&= \| \frac{1}{n} \sum_{i=1}^n\widehat U_{it} \frac{1}{T} \sum_{t=1}^T M_{it} \|_{\infty} \\
\label{iii.eq1}
&\leq \| \frac{1}{n} \sum_{i=1}^n\widehat U_{it}  \frac{1}{T} \sum_{t=1}^T  (\widehat \delta_{xt}-H^{-1}\tilde\delta_{xt})'\widehat f_i\|_{\infty} \\ 
\label{iii.eq2}
&\qquad + \| \frac{1}{n} \sum_{i=1}^n\tilde U_{it}  \frac{1}{T} \sum_{t=1}^T  \tilde  \delta_{xt}'(H')^{-1}(\widehat f_i-H'\tilde f_i)\|_{\infty} \\
\label{iii.eq3}
&\qquad + \| \frac{1}{n} \sum_{i=1}^n(\widehat U_{it} -\tilde U_{it}) \frac{1}{T} \sum_{t=1}^T  \tilde  \delta_{xt}' (H')^{-1}(\widehat f_i-H'\tilde f_i)\|_{\infty} \\
\label{iii.eq4}
&\qquad + \|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it}  -\tilde U_{it} )  (\tilde U_{it} -\widehat U_{it} ) '\gamma_x \|_{\infty} \\ 
\label{iii.eq5}
&\qquad + \|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T\tilde U_{it}    (\tilde U_{it} -\widehat U_{it} ) '\gamma_x \|_{\infty}.
\end{align}
(\ref{iii.eq1}) is $ \|\frac{1}{n}\widehat U_t'\widehat F \frac{1}{T} \sum_{t=1}^T  (\widehat \delta_{xt}-H^{-1}\tilde\delta_{xt})\|_{\infty}=0$ due to Lemma \ref{la0}.  (\ref{iii.eq2}) is $o_P\left(\sqrt{\frac{\log p}{nT}}\right)$ due to the assumption $\Delta_{ud}=o\left(\sqrt{\frac{\log p}{nT}}\right)$.
Terms (\ref{iii.eq3})-(\ref{iii.eq5}) are all $o_P\left(\sqrt{\frac{\log p}{nT}}\right)$ due to  Lemma  \ref{la.4imp}. Hence $\|\frac{1}{nT}\sum_{t=1}^T M_t'\widehat U_t      \|_{\infty} = o_P\left(\sqrt{\frac{\log p}{nT}}\right)$.

  
 
(iv) We have 
\begin{align}
\nonumber
\max_{m,k\leq p}|\frac{1}{nT}\sum_{it}(\widehat U_{it,m}\widehat U_{it,k}&-\tilde U_{it,m}\tilde U_{it,k})| \\
\label{v.a}
&\leq \max_{m,k\leq p}|\frac{2}{nT} \sum_{it}\tilde U_{it,m}(\widehat U_{it,k} - \tilde U_{it,k})| \\
\label{v.b}
&\qquad + \max_{m,k\leq p}|\frac{1}{nT}\sum_{it}(\widehat U_{it,m} - \tilde U_{it,m})(\widehat U_{it,k}-\tilde U_{it,k})|.
\end{align}
We also have the equality 
$$
\tilde U_{it, m}-\widehat U_{it, m}= \widehat  f_i' (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm} +   \widehat  f_i' 		 (\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m}+\tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i).
$$
Hence, we may bound (\ref{v.a}) by
\begin{align*}
|J|_0&\max_{m,k\leq p}|\frac{1}{nT} \sum_{it}\tilde U_{it,m}(\widehat U_{it,k} - \tilde U_{it,k})| \\
&\leq |J|_0\max_{m,k\leq p}|\frac{1}{nT} \sum_{it}\tilde U_{it,k} \widehat f_i'\tilde \lambda_{tm} |O_P(\Delta_F) + |J|_0 \max_{m,k\leq p}|\frac{1}{nT} \sum_{it}\tilde U_{it,k}  \widehat  f_i' 		 (\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m}| \\
&\qquad + |J|_0\max_{m,k\leq p}|\frac{1}{nT} \sum_{it}\tilde U_{it,k} \tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i)| \\
&\leq O_P\left(\Delta_{ud}+\sqrt{\frac{\log p}{nT}}\right)|J|_0(\Delta_F)+|J|_0 \max_{m\leq p} \frac{1}{T} \sum_t  \|\frac{1}{n}\widehat F'\tilde U_{t, m} \|_2^2 + O_P(|J|_0 \Delta_{ud}) \\
&= o_P(1) + O_P\left({\frac{\log (pT)}{n}}+\Delta_{fum}^2\right)|J|_0 = o_P(1)
\end{align*}
where we used Lemma \ref{la.1} (i) that $\max_{m\leq p} \left(\frac{1}{n^2T}\sum_t \|\tilde U_{t,m}'\widehat F   \|_2^2\right) = O_P\left({\frac{\log (pT)}{n}}+\Delta_{fum}^2\right)$, Lemma \ref{la.1} (vi) that $\max_{m,k\leq p}|\frac{1}{nT} \sum_{it}\tilde U_{it,k}   \widehat  f_i'  \tilde \lambda_{tm} |=O_P\left(\Delta_{ud}+\sqrt{\frac{\log p}{nT}}\right)$, and the assumptions that $\left(\sqrt{\frac{\log p}{nT}}\right)|J|_0\Delta_F+\left({\frac{\log (pT)}{n}}+\Delta_{fum}^2\right)|J|_0=o(1)$ and that $|J|_0 \Delta_{ud}\leq |J|_0 \Delta_F=o(1).$
The last inequality is due to the following argument: 

\textbf{Proof of $\Delta_{ud}\leq  \Delta_F$:} By definition, $\Delta_{ud}, \Delta_F$ are sequences such that 
   $$
     \max_{m, k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat  f_i -H'\tilde f_i) \tilde U_{it,m}  \tilde h_{tk}'  \|_F=O_P(\Delta_{ud}),\quad \frac{1}{n}\sum_{i=1}^n\|\widehat f_i-H'\tilde f_i\|_2^2=O_P(\Delta_F^2).
   $$ where $\tilde h_{tk}\in\{\tilde \delta_{dt}, \tilde \delta_{yt},  \tilde\lambda_{tk}\}$ is bounded. 
   Using the Cauchy-Schwarz inequality, we have
\begin{equation}\label{e.13add}
    \max_{m, k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat  f_i -H'\tilde f_i) \tilde U_{it,m}  \tilde h_{tk}'  \|_F
    \leq O_P(\Delta_F)(  \max_{m\leq p}\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T  \tilde U_{it,m}   ^2)^{1/2}. 
\end{equation}
 In the proof of Lemma \ref{l4.1} below, we show that 
$$
\max_{mv}|\frac{1}{nT}   \sum_{i}\sum_t\tilde U_{it, m}\tilde U_{it,v}-\frac{1}{nT}\sum_{i}\sum_t( U_{it, m}-\bar U_{i\cdot, m})(U_{it,v}-\bar U_{i\cdot,v})| =o_P(1)
$$
and 
$$
\max_{mv}\left|\frac{1}{nT} \sum_{i}\sum_t( U_{it, m}-\bar U_{i\cdot, m})(U_{it,v}-\bar U_{i\cdot,v})- E\left[\frac{1}{nT}\sum_{i}\sum_t( U_{it, m}-\bar U_{i\cdot, m})(U_{it,v}-\bar U_{i\cdot,v})\right]\right|=o_P(1).
$$
In addition, $\max_{mv}|E\left[\frac{1}{nT}\sum_{i}\sum_t( U_{it, m}-\bar U_{i\cdot, m})(U_{it,v}-\bar U_{i\cdot,v})\right]|=O(1)$ due to Assumption \ref{a3.2}.  Hence
\begin{equation}\label{e.13ad}
\max_{mv}|\frac{1}{nT}   \sum_{i}\sum_t\tilde U_{it, m}\tilde U_{it,v}\leq o_P(1)+\max_{mv}\left|E\left[\frac{1}{nT}\sum_{i}\sum_t( U_{it, m}-\bar U_{i\cdot, m})(U_{it,v}-\bar U_{i\cdot,v})\right]\right|=O_P(1).
\end{equation}
It also implies $\max_{m\leq p}\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T  \tilde U_{it,m}   ^2=O_P(1)$.
Hence (\ref{e.13add}) implies $$ \max_{m, k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat  f_i -H'\tilde f_i) \tilde U_{it,m}  \tilde h_{tk}'  \|_F
    \leq O_P(\Delta_F).$$
Then by the definition of $\Delta_{ud}$, we can take $\Delta_{ud}$ to be at least $\Delta_F$ in order to upper bound $ \max_{m, k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat  f_i -H'\tilde f_i) \tilde U_{it,m}  \tilde h_{tk}'  \|_F$, this means $\Delta_F$ is the least favorable choice for $\Delta_{ud}$. Hence $\Delta_{ud}\leq\Delta_F.$

\vspace{-0.3em}
As for (\ref{v.b}), we have 
\begin{align}\label{ea.7}
\begin{split}
|J|_0\max_{m,k\leq p}|\frac{1}{nT} \sum_{it} (\widehat U_{it,m} - \tilde U_{it,m})(\widehat U_{it,k} - \tilde U_{it,k})|  &\leq |J|_0  O_P\left(\Delta_F^2+\frac{\log (pT)}{n}\right) \\
&= o_P(1).
\end{split}
\end{align}
The desired result follows. 

In fact, we have just shown
\begin{align}\label{ea.10}
\begin{split}
\max_{m,k\leq p}|\frac{1}{nT} \sum_{it} \tilde U_{it,m} (\widehat U_{it,k} - \tilde U_{it,k})| &\leq  O_P\left(  \sqrt{\frac{\log p}{nT}}\Delta_F+ \Delta_{ud}+{\frac{\log (pT)}{n}}+\Delta_{fum}^2\right) \\
\max_{m,k\leq p}|\frac{1}{nT} \sum_{it} \widehat U_{it,m} (\widehat U_{it,k} - \tilde U_{it,k})| &\leq  O_P\left(  \sqrt{\frac{\log p}{nT}}\Delta_F+ \Delta_{ud}+{\frac{\log (pT)}{n}}+\Delta_{fum}^2\right).
\end{split}
\end{align}

(v) By part (iv), it suffices to prove $\max_{k,m}|\frac{1}{nT}\sum_t\sum_{i=1}^n\tilde U_{it,k}\tilde U_{it,m}|=O_P(1)$.



Hence $\max_{k,m}|\frac{1}{nT}\sum_t\sum_{i=1}^n\widehat U_{it,k}\widehat U_{it,m}|\leq o_P(1)+\max_{mk}|E\left[\frac{1}{nT}\sum_{i}\sum_t( U_{it, m}-\bar U_{i\cdot, m})(U_{it,k}-\bar U_{i\cdot,k})\right]|$, which is $O_P(1)$.

(vi) Under the assumption that $\|R_x\|_1 = o\left(\sqrt{\frac{\log p}{nT}}\right)$, by part (v), $\max_{k,m}|\frac{1}{nT}\sum_t\sum_{i=1}^n\widehat U_{it,k}\widehat U_{it,m}|=O_P(1)$, so
\begin{align*}
\|\frac{1}{nT}\sum_tR_x'\widehat U_t'\widehat U_t\|_{\infty}
&= \max_m|\frac{1}{nT}\sum_t\sum_{k=1}^p\sum_{i=1}^nR_{x,k}\widehat U_{it,k}\widehat U_{it,m}| \\
&\leq \|R_{x}\|_1\max_{k,m}|\frac{1}{nT}\sum_t\sum_{i=1}^n\widehat U_{it,k}\widehat U_{it,m}| = o_P\left(\sqrt{\frac{\log p}{nT}}\right).
\end{align*}



\end{proof}


\subsection{Proof of  Lemma \ref{l4.1}} 
    
\proof We start by observing
\begin{align*}
\max_{mv}|\frac{1}{nT} & \sum_{i}\sum_t\tilde U_{it, m}\tilde U_{it,v}-\frac{1}{nT}\sum_{i}\sum_t( U_{it, m}-\bar U_{i\cdot, m})(U_{it,v}-\bar U_{i\cdot,v})|  \\
&\leq \max_{mv}|\frac{1}{nT}\sum_{i}\sum_t(\tilde U_{it, m}- U_{it, m}+\bar U_{i\cdot, m})\tilde U_{it,v}| \\
& \qquad + \max_{mv}|\frac{1}{nT}\sum_{i}\sum_t(U_{it,m}-\bar U_{i\cdot, m})(\bar U_{it,v}-U_{it,v}+\bar U_{i\cdot, v})| \\
&= \max_{mv}|\frac{1}{T}\sum_t(\bar U_{\cdot t,m}-\bar {\bar U}_m)(\bar U_{\cdot t,v} -\bar{\bar U}_v)| \\
&= \max_{mv}|\frac{1}{T}\sum_t\bar U_{\cdot t,m}(\bar U_{\cdot t,v} -\bar{\bar U}_v)| \\
&\leq \max_m\frac{1}{T}\sum_t\bar U_{\cdot t,m}^2+\max_m\bar{\bar U}_m^2 \\
&= O_P\left(\frac{\log(p)}{nT}\right) + \max_m\frac{1}{T}\sum_tE\bar U_{\cdot t,m}^2+\max_m|\frac{1}{T}\sum_t\bar U_{\cdot t,m}^2-E\bar U_{\cdot t,m}^2| \\
&= O_P\left(\frac{\log(p)}{nT}+\frac{1}{n}+\sqrt{\frac{\log p}{n^2T}}\right).
\end{align*}
Also, let $\mathcal{U}_{i,mv}=\frac{1}{T}\sum_t( U_{it, m}-\bar U_{i\cdot, m})(U_{it,v}-\bar U_{i\cdot,v})=  \frac{1}{T}\sum_t U_{it, m}U_{it,v}-\bar U_{i\cdot,m}\bar U_{i\cdot,v}$.    Note that 
\begin{align*}
\max_{mv} \frac{1}{n}\sum_i\Var( \frac{1}{T}\sum_t U_{it, m}U_{it,v}) &= \max_{mv} \frac{1}{n}\sum_i\frac{1}{T^2}\sum_t\sum_s\Cov(U_{it, m}U_{it,v}, U_{is, m}U_{is,v}) \\
&=O\left(\frac{1}{T}\right)
\end{align*}
and that
\begin{align*}
\max_{mv} \frac{1}{n}\sum_i\Var( \bar U_{i\cdot,m}\bar U_{i\cdot,v}) &=\max_{mv} \frac{1}{n}\sum_i\frac{1}{T^4}\sum_{tskl}\Cov( U_{it,m}U_{is,v}, U_{ik,m}U_{il,v}) \\
&= O\left(\frac{1}{T^2}\right).
\end{align*}
Thus,
$\max_{mv} \frac{1}{n}\sum_i\Var(\mathcal{U}_{i,mv})\leq O(\frac{1}{T})+O(\frac{1}{T^2})+\sqrt{O(\frac{1}{T^2})O(\frac{1}{T})}=O(\frac{1}{T}).$
Hence,
\begin{align*}
\max_{mv}&\left|\frac{1}{nT} \sum_{i}\sum_t( U_{it, m}-\bar U_{i\cdot, m})(U_{it,v}-\bar U_{i\cdot,v})- E\left[\frac{1}{nT}\sum_{i}\sum_t( U_{it, m}-\bar U_{i\cdot, m})(U_{it,v}-\bar U_{i\cdot,v})\right]\right| \\
&=\max_{mv}|\frac{1}{n}\sum_i\mathcal{U}_{i,mv}-E\mathcal{U}_{i,mv}|=O_P\left(\sqrt{\frac{\log p}{nT}}\right).
\end{align*}
    Write $A_{mv}=\frac{1}{nT}\sum_{i}\sum_t\tilde U_{it, m}\tilde U_{it,v}-E\frac{1}{nT}\sum_{i}\sum_t( U_{it, m}-\bar U_{i\cdot, m})(U_{it,v}-\bar U_{i\cdot,v})$. We have proven
$$
\max_{mv}|A_{mv}| = O_P\left(\sqrt{\frac{\log p}{nT}}+\frac{\log(p)}{nT}+\frac{1}{n}+\sqrt{\frac{\log p}{n^2T}}\right)=O_P\left(\sqrt{\frac{\log p}{nT}}\right).
$$
Therefore, for any $\delta\in\mathbb{R}^p/\{0\}$,
\begin{align}\label{eb.111}
\begin{split}
\delta'\frac{1}{nT}\sum_{it} [\tilde U_{it}\tilde U_{it}' &- E ( U_{it}-\bar U_{i\cdot})(U_{it}-\bar U_{i\cdot}) ' ]\delta \\
&\leq \|\delta\|_1^2\max_{mv}|A_{mv}|= \|\delta\|_1^2 O_P\left(\sqrt{\frac{\log p}{nT}}\right)
\end{split}
\end{align}
where the $O_P\left(\sqrt{\frac{\log p}{nT}}\right)$ term is independent of $\delta$.
    
We are now ready to verify Assumptions \ref{a3.4}  and \ref{a6.2new}.

 
\textbf{Verification of Assumption \ref{a3.4} } Uniformly on the set $\{\delta: \|\delta\|_0\leq l_T|J|_0\}$, we have
$\|\delta\|_1\leq  \|\delta\|_2\sqrt{l_T|J|_0}$. Hence as long as $l_T^2|J|_0^2\log p=o(n)$ (for instance, $l_T=\log p$), 
\begin{eqnarray*}
\mathcal{R}(\delta)&\geq& c-O_P\left(\sqrt{\frac{\log p}{nT}}\right)\frac{\|\delta\|_1^2}{\|\delta\|_2^2}\geq c-O_P\left(l_T|J|_0\sqrt{\frac{\log p}{nT}}\right)\geq c/2\cr
\mathcal{R}(\delta)&\leq& C+O_P\left(\sqrt{\frac{\log p}{nT}}\right)\frac{\|\delta\|_1^2}{\|\delta\|_2^2}\leq C+O_P\left(l_T|J|_0\sqrt{\frac{\log p}{nT}}\right)\leq 2C,
\end{eqnarray*}
where $C$ denotes the maximum eigenvalue of $ E\left[ ( U_{it}-\bar U_{i\cdot})(U_{it}-\bar U_{i\cdot}) '\right]$.
  

\textbf{Verification of Assumption \ref{a6.2new}  }  The result follows immediately from (\ref{eb.111}).
 

 $\blacksquare$
 
    
 
\section{Proof of Theorem  \ref{t6.1} }

Define $TK\times 1$ matrices $\widehat\Xi = (\widehat\xi_1',...,\widehat\xi_T')'$ and $\widehat\Delta_d = (\widehat\delta_{d1}',...,\widehat\delta_{dT}')'$.
%\textcolor{blue}{$\widehat\Xi$ was not defined.  Double-check that definition lines up correctly.}


Note that
$
\tilde Y^*=\tilde D^*\widehat\alpha+(I_T\otimes \widehat F)\widehat \Xi+\tilde U^*\widehat\theta+\tilde \epsilon^*
$,  and that
$
\widehat\eta^*=M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^*})\tilde D^*.$
 Hence
 \begin{eqnarray*}
\widehat\alpha^*&=&
(\widehat\eta^{'*}\widehat\eta^*)^{-1}\widehat\eta^{'*}M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^*})\tilde Y
\cr
&=&\widehat\alpha
+(\widehat\eta^{'*}\widehat\eta^*)^{-1}\widehat\eta^{'*}M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^*})[(I_T\otimes \widehat F)\widehat\Xi+\tilde U^*\widehat\theta+\tilde \epsilon^*].
\end{eqnarray*}
 Hence
 \begin{eqnarray}\label{ee.1}
&&\sqrt{nT}(\frac{1}{nT}\widehat\eta^{'*}\widehat\eta^*)( \widehat\alpha^*-\widehat\alpha)= \frac{1}{\sqrt{nT}} \tilde\eta^{*'} \tilde \epsilon^*+\sum_{i=1}^6A_i^*
 \end{eqnarray}
  where
 \begin{eqnarray*}
 A_1^*&=&  \frac{1}{\sqrt{nT}}
 (\widehat\eta^*-\tilde\eta^*)'M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^*})\tilde \epsilon^*, \quad A_2^*=\frac{1}{\sqrt{nT}} \widehat\eta^{'*}M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^*}\widehat F) \widehat\Xi\cr
 A_3^*&=&-\frac{1}{\sqrt{nT}}\tilde\eta^{*'}(I_T\otimes P_{\widehat F^*})\tilde \epsilon^*,\quad A_4^*=\frac{1}{\sqrt{nT}}\widehat\eta^{'*}M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^*})
\tilde U^*\widehat\theta\cr
A_5^*&=&-\frac{1}{\sqrt{nT}} \tilde\eta^{*'}P_{\widehat U^*_{\widehat J^*}} \tilde \epsilon^*,\quad A_6=\frac{1}{\sqrt{nT}}\tilde\eta^{*'}P_{\widehat U^*_{\widehat J^*}}(I_T\otimes P_{\widehat F^*})\tilde \epsilon^*=0.
  \end{eqnarray*}


We shall prove that  $A^*_i=o_{P^*}(1)$ for $i=1,...,6$ and $\frac{1}{nT}\widehat\eta^{*'}\widehat\eta^*-\frac{1}{nT}\tilde\eta^{'}\tilde\eta=o_{P^*}(1)$.  Similarly to (\ref{eb.2ag}), it can be shown that 
\begin{align}\label{ee.2ag}
\begin{split}
\frac{1}{\sqrt{nT}}(\widehat\eta^* -\tilde\eta^*)
&= \frac{1}{\sqrt{nT}}M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^*}(\widehat FH^*-\widehat F^*)H^{*-1}) \widehat\Delta_d \\
& \qquad + \frac{1}{\sqrt{nT}}M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^*})\tilde U^*\widehat\gamma_d \\
& \qquad - \frac{1}{\sqrt{nT}}P_{\widehat U^*_{\widehat J^*}} \tilde\eta^* \\
& \qquad - \frac{1}{\sqrt{nT}}  M_{\widehat U^*_{\widehat J^*}}(I_T\otimes P_{\widehat F^*})\tilde\eta^*.
\end{split}
\end{align}

 

Recall that $a_n$ is defined in Assumption \ref{a6.1}, which quantifies the minimization error of the k-step lasso. Write
\begin{eqnarray}\label{ee.3a}
\psi_n^*:=\kappa_n|J|^{1/2}_0 +\Delta_F^*|J|_0+  \sqrt{\frac{|J|_0}{n}}+\sqrt{a_n}.
\end{eqnarray}
\begin{proposition}\label{pf.1} (i)
$
\frac{1}{\sqrt{nT}}\|\widehat\eta^*-\tilde\eta^*\|_2=O_P( \psi_n^*).$\\
(ii)   $A_l^*=o_{P^*}(1), l=1,...,6$.
\end{proposition}

 
 
 
\proof  The proof for (i) is similar to that of Proposition \ref{pb.1}, based on (\ref{ee.2ag}) and Lemma \ref{le.1} below.  The proof for (ii) follows the same arguments employed in Sections \ref{sc2}-\ref{sc4} using on Lemmas \ref{le.1} and \ref{le.4} below.   The only difference is that $A_4^*=o_{P^*}(1)$ requires $a_n\sqrt{nT}=o(1)$.  We omit the details for brevity.  $\blacksquare$
 
 
 
 
 
 \subsection{Proof of Theorem \ref{t6.1}}
 
 It follows from equation (\ref{ee.1}) and Proposition \ref{pf.1} that 
$$
 \sqrt{nT}(\frac{1}{nT}\widehat\eta^{'*}\widehat\eta^*)( \widehat\alpha^*-\widehat\alpha)= \frac{1}{\sqrt{nT}} \tilde\eta^{*'} \tilde \epsilon^*+o_{P^*}(1).
$$
Note that $ \frac{1}{\sqrt{nT}} \tilde\eta^{*'} \tilde \epsilon^*= \frac{1}{\sqrt{n}}\sum_i \varpi_iw_i^Yw_i^D$, where $\varpi_i=\frac{1}{\sqrt{T}}\sum_t \widehat \eta_{it} \widehat  \epsilon_{it}$.   Let 
$$
b_n= \left[\Var^*(\frac{1}{\sqrt{n}}\sum_i\varpi_iw_i^Yw_i^D)\right]^{-1/2},
$$
so
$$
b_n \sqrt{nT}(\frac{1}{nT}\widehat\eta^{'*}\widehat\eta^*)( \widehat\alpha^*-\widehat\alpha)= \frac{b_n}{\sqrt{nT}} \tilde\eta^{*'} \tilde \epsilon^*+o_{P^*}(b_n).
$$
  
Now note that Step 1 below shows $b_n=O_P(1)$. Step 2 shows  $ \frac{b_n}{\sqrt{nT}} \tilde\eta^{*'} \tilde \epsilon^*\to^{d^*}\mathcal{N}(0,1)$ and hence $ b_n\sqrt{nT} ( \widehat\alpha^*-\widehat\alpha)=O_{P^*}(1)$.  Step 3 shows 
$|  \sigma^2_{\eta}- \frac{1}{nT}\widehat\eta^{'*}\widehat\eta^* |=o_{P^*}(1)$.
Therefore,   $ b_n\sqrt{nT}\frac{1}{nT}(\widehat\eta^{'*}\widehat\eta^*- \sigma^2_{\eta})( \widehat\alpha^*-\widehat\alpha)=o_{P^*}(1)$ which yields
$$
b_n \sqrt{nT}\sigma^2_\eta( \widehat\alpha^*-\widehat\alpha)= \frac{b_n}{\sqrt{nT}} \tilde\eta^{*'} \tilde \epsilon^*\to^{d^*}\mathcal{N}(0,1).
$$
In addition, we also show in Step 1 below that $ \sqrt{nT}(\sigma_{\eta}^2)( \widehat\alpha^*-\widehat\alpha)=O_{P^*}(b_n^{-1})=O_{P^*}(1)$. Hence,  
$$
(b_n-\sigma_{\eta\epsilon}^{-1/2}) \sqrt{nT}(\sigma_{\eta}^2)( \widehat\alpha^*-\widehat\alpha)=o_{P^*}(1).
$$
Thus, we have both
\begin{align}\label{ee.4}
\begin{split}
\sqrt{nT}\sigma_{\eta\epsilon}^{-1/2}\sigma_{\eta}^2( \widehat\alpha^*-\widehat\alpha) &\to^{d^*} \mathcal{N}(0,1) \\
\sqrt{nT}\sigma_{\eta\epsilon}^{-1/2}\sigma_{\eta}^2(\widehat\alpha-\alpha) &\to^d \mathcal{N}(0,1).
\end{split}
\end{align}
where the second line follows from Theorem \ref{t3.1}.
 
Now let $q_\tau^*$ be such that $ P^*(\sqrt{nT}|\widehat\alpha^*-\widehat\alpha|\leq q_{\tau}^*)=1-\tau$, and let
\begin{eqnarray*}
a_1&:=&P(\sqrt{nT}|\widehat\alpha-\alpha|\leq q_{\tau}^*)=P(\sigma_{\eta\epsilon}^{-1/2}\sigma_{\eta}^2\sqrt{nT}|\widehat\alpha-\alpha|\leq \sigma_{\eta\epsilon}^{-1/2}\sigma_{\eta}^2q_{\tau}^*)\cr
a_2&:=&P^*(\sqrt{nT}|\widehat\alpha^*-\widehat\alpha|\leq q_{\tau}^*)=P^*(\sigma_{\eta\epsilon}^{-1/2}\sigma_{\eta}^2\sqrt{nT}|\widehat\alpha^*-\widehat\alpha|\leq \sigma_{\eta\epsilon}^{-1/2}\sigma_{\eta}^2q_{\tau}^*)
\end{eqnarray*}
The results summarized in (\ref{ee.4}) then implies $a_1\to a_2=1-\tau$.    
 
 
 
\textbf{Step 1: } Show $b_n=O_P(1)$ and $b_n^{-1}=O_P(1)$.  

Let $Z_i= \sum_t (\widehat \eta_{it} \widehat  \epsilon_{it}-  \tilde \eta_{it}\tilde\epsilon_{it})$. Then by Lemma \ref{lb.4}, $\frac{1}{nT}\sum_iZ_i^2=o_P(1)$. Hence,
\begin{align*}
b_n^{-2} &= \Var^*(\frac{1}{\sqrt{n}}\sum_i\varpi_iw_i^Yw_i^D) \\
&= \frac{1}{n}\sum_i\varpi_i^2E^*(w_i^Yw_i^D)^2 = \frac{1}{n}\sum_i\varpi_i^2 \\
&= \frac{1}{n}\sum_i(\frac{1}{\sqrt{T}}\sum_t \widehat \eta_{it} \widehat  \epsilon_{it})^2 = \frac{1}{nT}\sum_i(Z_i+\sum_t\tilde \eta_{it}\tilde\epsilon_{it})^2 \\
&=\frac{1}{nT}\sum_iZ_i^2+\frac{1}{nT}\sum_i(\sum_t\tilde \eta_{it}\tilde\epsilon_{it})^2+\frac{2}{nT}\sum_iZ_i\sum_t\tilde \eta_{it}\tilde\epsilon_{it} \\
&\leq o_P(1)+\frac{1}{nT}\sum_i(\sum_t\tilde \eta_{it}\tilde\epsilon_{it})^2+o_P(1)\sqrt{\frac{1}{nT}\sum_i(\sum_t\tilde \eta_{it}\tilde\epsilon_{it})^2}.
\end{align*}
By (\ref{eb.7a}) and (\ref{eb.91}), $\frac{1}{nT}\sum_{i=1}^n(\sum_{t=1}^T\tilde\eta_{it}\tilde\epsilon_{it})^2\to^P\sigma_{\eta\epsilon}$. Hence, $b_n^{-2}\to^P\sigma_{\eta\epsilon}>c$.  It follows that $b_n=O_P(1)$, $b_n\to^P\sigma_{\eta\epsilon}^{-1/2},$ and $b_n^{-1}=O_P(1)$ since $\sigma_{\eta\epsilon}<C$.
 
 
  
\textbf{Step 2: } Apply the CLT to $ \frac{b_n}{\sqrt{nT}} \tilde\eta^{*'} \tilde \epsilon^*$.
 
Let  $g_{n,i}=b_n\varpi_iw_i^Yw_i^D$ and $s_n^2=\sum_i\Var^*(g_{n,i})=n$.  Note that $E^*g_{n,i}=0$.
 
 

We now verify the Lindeberg condition for the triangular array $\{g_{n,i}\}$. For any $\varepsilon>0$,
$$
E^*\left(\frac{1}{n}\sum_ig_{n,i}^21\{|g_{n,i}|>\varepsilon \sqrt{n}\}\right)\leq E^*\left[\frac{1}{n}\sum_ig_{n,i}^2\right]=1.
$$
Hence, by the dominated convergence theorem,
$$
s_n^{-2}\sum_iE^*g_{n,i}^21\{|g_{n,i}|>\varepsilon s_n\}=E^*\left(\frac{1}{n}\sum_ig_{n,i}^21\{|g_{n,i}|>\varepsilon \sqrt{n}\}\right)\to0.
$$
We then have, by the Lindeberg central limit theorem, 
 \begin{equation}
 \frac{1}{s_n}\sum_ig_{n,i}=\frac{b_n}{\sqrt{n}}\sum_i\varpi_iw_i^Yw_i^D=
 \frac{b_n}{\sqrt{nT}}\tilde\eta^{*'}\tilde\epsilon^*\to^{d^*}\mathcal{N}(0,1).
 \end{equation}
  
    
\textbf{Step 3: } Show  $|\sigma_{\eta}^2- \frac{1}{nT}\widehat\eta^{'*}\widehat\eta^* |=o_{P^*}(1)$. 

First,
\begin{align*}
|\frac{1}{nT}\widehat\eta^{'*}\widehat\eta^*- \frac{1}{nT}\sum_{i,t}\tilde\eta_{it}^2(w_i^D)^2|
&= | \frac{1}{nT}\sum_{i,t}(\widehat\eta_{it}^2-\tilde\eta_{it}^2)(w_i^D)^2| \\
&\leq \frac{1}{nT}\sum_{i,t}|\widehat\eta_{it}-\tilde\eta_{it}||\widehat\eta_{it}+\tilde\eta_{it}| \max_i(w_i^D)^2 \\
&\leq \left(\frac{1}{nT}\sum_{i,t}(\widehat\eta_{it}-\tilde\eta_{it})^2\right)O_{P^*}(\log n) \\
&= O_{P^*}(\psi_n\log n)=o_{P^*}(1).
\end{align*}
% Display above was previously
%\begin{align*}
%|\frac{1}{nT}\widehat\eta^{'*}\widehat\eta^*- \frac{1}{nT}\sum_{i,t}\tilde\eta_{it}^2(w_i^D)^2|
%&= | \frac{1}{nT}\sum_{i,t}(\widehat\eta_{it}^2-\tilde\eta_{it}^2)(w_i^D)^2| \\
%&\leq \frac{1}{nT}\sum_{i,t}|\widehat\eta_{it}-\tilde\eta_{it}||\widehat\eta_{it}+\tilde\eta_{it}| \max_i(w_i^D)^2 \\
%&\leq \left(\frac{1}{nT}\sum_{i,t}(\widehat\eta_{it}-\tilde\eta_{it})^2\right)^{1/2}\left(\frac{1}{nT}\sum_{i,t}(\widehat\eta_{it}+\tilde\eta_{it})^2\right)^{1/2}O_{P^*}(\log n) \\
%&= O_{P^*}(\psi_n\log n)=o_{P^*}(1).
%\end{align*}
Then, using $E^*(w_i^D)^2=1$ and 
\begin{align*}
E^* | \frac{1}{nT}\sum_{i,t}\tilde\eta_{it}^2((w_i^D)^2- E^*(w_i^D)^2)|^2 
&=\Var^*\left(\frac{1}{nT}\sum_{i,t}\tilde \eta^{2}_{it}(w_i^D)^2\right) \\
&=\frac{1}{n^2T^2}\sum_i\Var^*\left((w_i^D)^2\right)\left(\sum_{t}\tilde \eta^{2}_{it}\right)^2 \\
&\leq \frac{C}{n^2}\sum_i \left(\frac{1}{T}\sum_{t}\tilde \eta^{2}_{it}\right)^2 = O_P\left(\frac{1}{n}\right),
\end{align*}
we have  $|\frac{1}{nT}\sum_{i,t}\tilde\eta_{it}^2(w_i^D)^2-\frac{1}{nT}\sum_{i,t}\tilde\eta_{it}^2|=o_{P^*}(1)$, 
from which
$$
|\frac{1}{nT}\widehat\eta^{'*}\widehat\eta^*-\frac{1}{nT}\sum_{i,t}\tilde\eta_{it}^2|=o_{P^*}(1)
$$
follows.
Moreover, equation (\ref{eb.2new}) in the proof of Theorem \ref{t3.1} shows $|\frac{1}{nT}\tilde\eta'\tilde\eta-\frac{1}{nT}\sum_{i,t}E\iota_{it}|=o_P(1)$ for $\iota_{it} = (\eta_{it} - \bar{\eta}_{i\cdot})^2$. 
 Hence $|\frac{1}{nT}\sum_{i,t}E\iota_{it}- \frac{1}{nT}\widehat\eta^{'*}\widehat\eta^* |=o_{P^*}(1)$. Finally, note that $\sigma_{\eta}^2:=\frac{1}{nT}\sum_{i,t}E\iota_{it}$. $\blacksquare$

 

 %Proposition \ref{pf.1} yields $|\frac{1}{nT}\tilde \eta^{*'}\tilde\eta^*- \frac{1}{nT}\widehat\eta^{'*}\widehat\eta^* |=o_{P^*}(1)$. Also, $|\frac{1}{nT}\widehat\eta'\widehat\eta-\frac{1}{nT}\sum_{it}Ew_{it}|=o_P(1)$ by (\ref{eb.2new}).   It remains to show $|\frac{1}{nT}\tilde \eta^{*'}\tilde\eta^*- \frac{1}{nT}\widehat\eta^{'}\widehat\eta |=o_{P^*}(1)$. In fact, 
 
   
\section{Technical lemmas for the main theorems}
 
 
 
\subsection{Technical Lemmas for Theorem \ref{t3.1}}


 

\begin{lemma}\label{lb.1}   For $\gamma \in \{\gamma_y,\gamma_d,\theta\}$,\\
 (i)   $\|\frac{1}{\sqrt{nT}}M_{\widehat U_{\widehat J}} \tilde U\gamma\|_2^2=O_P\left(\|R_y\|_1^2+\kappa_n^2|J|_0+|J|_0^2\Delta_F^2+\frac{|J|_0}{n}\right)$.
\\
 (ii) $\|\frac{1}{\sqrt{nT}} (I_T\otimes P_{\widehat F})\tilde U\gamma\|_2^2=O_P\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^2\right).$
\\
(iii) $ \|  \frac{1}{\sqrt{nT}}  (I_T\otimes P_{\widehat F})\tilde\eta\|_2^2 = O_P\left(\Delta_{fe}^2+\frac{1}{n}\right)$ and $\|  \frac{1}{\sqrt{nT}}  (I_T\otimes P_{\widehat F})\tilde\epsilon\|_2^2 = O_P\left(\Delta_{fe}^2+\frac{1}{n}\right).$\\
(iv) $\|\frac{1}{\sqrt{nT}}P_{\widehat U_{\widehat J}} \tilde\eta\|_2^2 = O_P\left(|J|_0\frac{\log p}{nT}\right)$ and $\|\frac{1}{\sqrt{nT}}P_{\widehat U_{\widehat J}} \tilde\epsilon\|_2^2 = O_P\left(|J|_0\frac{\log p}{nT}\right)$.
%(v) $\frac{1}{T}\|P_{\widehat U_I}\eta\|_2^2=O_P(s^2\nu^2+s^2\frac{\Delta_F^4}{\nu^2})=\frac{1}{T}\|P_{\widehat U_I}\epsilon\|_2^2$.
\end{lemma}
 
\proof (i)  First, consider the following constrained problem:
\begin{equation}\label{eb.2}
\widehat m=\arg\min_m\|\widehat U(\gamma-m)\|_2^2, \quad m_j=0, j\notin\widehat J.
\end{equation}
The solution satisfies $\widehat U\widehat m=P_{\widehat U_{\widehat J}}\widehat U\gamma$. Hence, from  Proposition \ref{p2.1}, \\ $ \frac{1}{nT}\sum_{t=1}^T\|\widehat U_t (\gamma_y-\tilde\gamma_y)\|_2^2= O_P(\|R_y\|_1^2+\kappa_n^2|J|_0)$, and 
\begin{align*}
\|\frac{1}{\sqrt{nT}}M_{\widehat U_{\widehat J}} \widehat U\gamma\|_2^2 &=\|\frac{1}{\sqrt{nT}}( \widehat U\gamma-P_{\widehat U_{\widehat J}} \widehat U\gamma)\|_2^2 \\
&=\|\frac{1}{\sqrt{nT}}\widehat  U(\gamma- \widehat m)\|_2^2 \leq \|\frac{1}{\sqrt{nT}}\widehat  U(\gamma- \tilde\gamma)\|_2^2 \\
&=\frac{1}{nT}\sum_{t=1}^T\|\widehat  U_t(\gamma- \tilde\gamma)\|_2^2 = O_P(\|R_y\|_1^2+\kappa_n^2|J|_0).
\end{align*}
Next, $\frac{1}{T}\sum_t\|\frac{1}{n}\widehat F'\tilde U_t\gamma\|_2^2=O_P\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^2\right)$ by Lemma \ref{la.1}. Also, using the equality
\begin{align*}
\tilde U_{it, m}-\widehat U_{it, m} &= \widehat  f_i' (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm} + \widehat  f_i'(\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m} \\
& \qquad  +\tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i),
\end{align*}
we have
\begin{align}\label{eb.9add}
\begin{split}
\left\|\frac{1}{\sqrt{nT}}M_{\widehat U_{\widehat J}} (\widehat U-\tilde U)\gamma\right\|_2^2 &\leq \frac{1}{nT}\|(\widehat U-\tilde U)\gamma\|_2^2 \\
&= \frac{1}{nT}\sum_{t=1}^T\sum_i\left[\sum_m(\widehat U_{it,m}-\tilde U_{it,m})\gamma_m\right]^2 \\
&\leq |J|_0^2 O_P(\Delta_F^2) + \frac{1}{T}\sum_{t=1}^T \left\|\frac{1}{n}\widehat F'\tilde U_{t}\gamma\right\|_2^2+|J|_0^2O_P(\Delta_F^2) \\
&= O_P\left(|J|_0^2\Delta_F^2+\frac{|J|_0}{n}\right).
\end{split}
\end{align}  
    where we used, by Lemma \ref{la.1},  $
\frac{1}{T}\sum_t\|\frac{1}{n}\widehat F'\tilde U_t\gamma\|_2^2 = O_P\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^2\right).$ 
Combining these results yields $\left\|\frac{1}{\sqrt{nT}}M_{\widehat U_{\widehat J}} \tilde U\gamma\right\|_2^2=O_P\left(\|R_y\|_1^2+\kappa_n^2|J|_0+|J|_0^2\Delta_F^2+\frac{|J|_0}{n}\right)$.
 

(ii) By Lemma \ref{la.1},
\begin{align*}
\left\|\frac{1}{\sqrt{nT}} (I_T\otimes P_{\widehat F})\tilde U\gamma\right\|_2^2  &= \frac{1}{nT}\sum_t\left\|P_{\widehat F}\tilde U_t\gamma\right\|_2^2 \\
&\leq \frac{1}{n^2T}\sum_t\left\|\widehat F'\tilde U_t\gamma\right\|_2^2 \\
&=O_P\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^2\right).
\end{align*}

 
(iii)  By Lemma \ref{la.1}, $\frac{1}{T}\sum_t\left\|\frac{1}{n}\widehat F'\tilde \eta_t\right\|_2^2= O_P\left(\frac{1}{n}+\Delta_{fe}^2\right)$. Thus,
$$ \left\|  \frac{1}{\sqrt{nT}}  (I_T\otimes P_{\widehat F})\tilde\eta \right\|_2^2
=\frac{1}{nT}\sum_t\left\|P_{\widehat F}\tilde\eta_t\right\|_2^2 \leq   \frac{1}{n^2T}\sum_t\left\|{\widehat F}'\tilde\eta_t\right\|_2^2 = O_P\left(\Delta_{fe}^2+\frac{1}{n}\right).
$$
 

(iv) The same argument as employed in the proof of Lemma \ref{la.5new}  yields
\begin{align*}
\max_m\left|\frac{1}{{nT}}\sum_t\sum_i\tilde U_{it,m}\tilde\eta_{it}\right| = O_P\left(\sqrt{\frac{\log p}{nT}}\right)
\end{align*}
and
\begin{align*}
\max_m\left|\frac{1}{{nT}}\sum_t\sum_i(\tilde U_{it,m}-\widehat U_{it,m})\tilde\eta_{it}\right|=o_P\left(\sqrt{\frac{\log p}{nT}}\right).
\end{align*}
Hence, using $|\widehat J|_0=O_P(|J|_0)$ from Proposition \ref{p2.1}, we have
\begin{align*}
\left\|\frac{1}{{nT}}{\tilde U_{\widehat J}}' \tilde\eta\right\|_2 &= \left\|\frac{1}{{nT}}{\sum_t\tilde U'_{t,\widehat J}}\tilde\eta_t\right\|_2 \\
&\leq \max_m\left|\frac{1}{{nT}}\sum_t\sum_i\tilde U_{it,m}\tilde\eta_{it}\right|\sqrt{|\widehat J|_0} \\
&= O_P\left(\sqrt{|J|_0\frac{\log p}{nT}}\right)
\end{align*}
and 
$\left\|\frac{1}{{nT}}(\tilde U_{\widehat J}-\widehat U_{\widehat J})' \tilde\eta\right\|_2 
\leq \left\|\frac{1}{{nT}}\sum_t(\tilde U'_{t,\widehat J}-\widehat U'_{t,\widehat J})\tilde\eta_t\right\|_{\infty}\sqrt{|\widehat J|_0} = o_P\left(\sqrt{|J|_0\frac{\log p}{nT}}\right).
$
Hence, 
\begin{equation}\label{eb.7new}
\left\|\frac{1}{{nT}}{\widehat U_{\widehat J}}' \tilde\eta\right\|_2 = O_P\left(\sqrt{|J|_0\frac{\log p}{nT}}\right).
\end{equation}
Next, by Lemma \ref{la.5new}, 
$$
\left\|\frac{1}{nT}\widehat U_{\widehat J}'\widehat U_{\widehat J}-\frac{1}{nT}\tilde U_{\widehat J}'\tilde U_{\widehat J}\right\| \leq O_P( |J|_0)\max_{m,k\leq p}\left|\frac{1}{nT}\sum_{it}(\widehat U_{it,m}\widehat U_{it,k}-\tilde U_{it,m}\tilde U_{it,k})\right|=o_P(1).
$$
Therefore, $\|\frac{1}{\sqrt{nT}}\widehat U_{\widehat J}\|^2=\|\frac{1}{nT}\widehat U_{\widehat J}'\widehat U_{\widehat J}\|\leq \|\frac{1}{nT}\tilde U_{\widehat J}'\tilde U_{\widehat J}\|=O_P(\phi_{\max}(|J|_0))+o_P(1)=O_P(1)$ and
\begin{equation}\label{eb.4add}
\lambda_{\min}\left(\frac{1}{nT}\widehat U_{\widehat J}'\widehat U_{\widehat J}\right) \geq \lambda_{\min}\left(\frac{1}{nT}\tilde U_{\widehat J}'\tilde U_{\widehat J}\right)-o_P(1) \geq \phi_{\min}(|\widehat J|_0)-o_P(1),
\end{equation}
which is bounded away from zero.  Thus, $\left\|\left(\frac{1}{nT}\widehat U_{\widehat J}'\widehat U_{\widehat J}\right)^{-1}\right\|=O_P(1)$.

 
% in the proof of (\ref{ea.0}), we showed
%$$\frac{1}{nT}\|\widehat U_{\widehat J}\|^2\leq 2\phi_{\max}(|\widehat J|_0)+ O_P(|\widehat J|_0\Delta_F^2+|\widehat J|_0\frac{\log (pT)}{n})=O_P(1),$$
%where the last equality follows from $|\widehat J|_0=O_P(|J|_0)$, $\phi_{\max}(| J|_0)=O_P(1)$ and the assumptions that $|J|_0\Delta_F^2$ and $|J|_0\frac{\log pT}{n}=O(1)$. Also, $\frac{1}{nT}\|\tilde U_{\widehat J}\|^2=O_P(1)$. Hence (\ref{ea.7}) implies
%\begin{eqnarray*}
%&&\|\frac{1}{nT}\widehat U_{\widehat J}'\widehat U_{\widehat J}-\frac{1}{nT}\tilde U_{\widehat J}'\tilde U_{\widehat J}\|\leq \|\frac{1}{\sqrt{nT}}(\widehat U_{\widehat J}-\tilde U_{\widehat J} )\|\frac{1}{\sqrt{nT}}(\|\widehat U_{\widehat J} \|+\|\tilde U_{\widehat J} \|)\cr
%&=&O_P(1)(|J|_0\max_m\frac{1}{{nT}}\sum_{it}(\tilde U_{it,m}-\widehat U_{it,m})^2)^{1/2} =o_P(1).
%\end{eqnarray*}

Finally,
$\left\|\frac{1}{\sqrt{nT}}P_{\widehat U_{\widehat J}} \tilde\eta\right\|_2 \leq \left\|\frac{1}{\sqrt{nT}}  \widehat U_{\widehat J} (\frac{1}{nT}\widehat U_{\widehat J} '\widehat U_{\widehat J} )^{-1}\frac{1}{nT}\widehat U_{\widehat J} '\tilde\eta\right\|_2 = O_P\left(\sqrt{|J|_0\frac{\log p}{nT}}\right).$ $\blacksquare$



%The following lemmas are useful in bounding $A_1$.

\begin{lemma}\label{lb.2new}
(i) $\|\frac{\sqrt{T}}{n\sqrt{n}}\sum_{m=1}^p\widehat F'\frac{1}{T}\sum_t U_{t,m}\epsilon_t'\widehat F\gamma_{dm}\|_F=o_P(1)$\\
(ii) $ \|\frac{1}{n}\sum_{j=1}^n\widehat f_j  	
\bar U_{j\cdot}'\gamma_{d}\|_2=O_P\left(\Delta_F|J|_0+\sqrt{\frac{|J|_0}{nT}}\right)$, and $ \|\frac{1}{n}\sum_{i=1}^n\bar \epsilon_{i\cdot}
\widehat f_i'\|_2=O_P\left(\Delta_F+\frac{1}{\sqrt{nT}}\right)$.\\
(iii) $\|\frac{\sqrt{nT}}{n}\sum_{m=1}^p\sum_{j=1}^n \widehat f_j  	
\frac{1}{T} \sum_{t=1}^T U_{jt,m} \bar \epsilon_{\cdot t}
\gamma_{dm}\|_2=o_P(1)$.\\
(iv) $\|\frac{\sqrt{nT}}{n}  \sum_{i=1}^n 
 \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t}'\gamma_d\epsilon_{it}
\widehat f_i'\|_2=o_P(1)$ and $\sqrt{nT}| \sum_{m=1}^p 	
 \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t,m} \bar\epsilon_{\cdot t}
\gamma_{dm}|=o_P(1)$.\\
%v) $\frac{1}{T}\sum_t\sum_{m\in\widehat J}\|\frac{1}{n}\sum_i\widehat U_{it,m}\widehat f_i\|_2^2=O_P(\frac{\log(pT)}{n}+\Delta_{fum}^2+\Delta_F^4+\Delta_f^2)|J|_0. $\\
%(vi) $\|\frac{1}{\sqrt{nT}}\sum_t\tilde\epsilon_t'P_{\widehat F}\widehat U_{t, \widehat J}\|_2=
%O_P( a_n).$\\
(v) $ \|\frac{1}{nT}\sum_t\widehat U_{t,\widehat J}' (\tilde FH-\widehat F)H^{-1}\tilde\delta_{dt}\|_2 = O_P\left(\Delta_{ud}+   \Delta_F\sqrt{\frac{\log p}{nT}}\right)\sqrt{|J|_0}.$
%$\max_m\|\frac{1}{nT}\sum_t\sum_i\tilde\delta_{dt}\widehat U_{it,m}\widehat f_i'\|_F=O_P(\Delta_{ud}+ \Delta_F^2+  \sqrt{\frac{\log p}{nT}}).$
%$ \|\frac{1}{nT}\sum_t\widehat U_{t,\widehat J}'P_{\widehat F}(\tilde FH-\widehat F)H^{-1}\tilde\delta_{dt}\|_2=O_P(\Delta_{ud}+ \Delta_F^2+   \sqrt{\frac{\log p}{nT}})O_P(\Delta_F)   \sqrt{|J|_0}$.

 %Here $a_n$ in (vi) is 
%\begin{eqnarray}\label{eb.9a} a_n&=&(  \Delta_{fe}\sqrt{T\log (pT)}+ \Delta_{fe}\Delta_{fum}\sqrt{nT}+ \frac{\log p}{\sqrt{nT}}+  \sqrt{\frac{\log p}{n}})|J|_0 \cr
%&&+ (\Delta_{F}\sqrt{\log p}+\Delta_{F} \Delta_{fel}\sqrt{nT}+\Delta_{F}^2\sqrt{\log p}+ \Delta_{F}^2\Delta_{fel}\sqrt{nT})|J|_0
%\end{eqnarray}

\end{lemma} 
 

\proof (i) We have
\begin{align}
\nonumber
\left\|\frac{\sqrt{T}}{n\sqrt{n}} \right. & \left. \sum_{m=1}^p\widehat F'\frac{1}{T}\sum_t U_{t,m}\epsilon_t'\widehat F\gamma_{dm}\right\|_F \\
\label{G2.i1}
&\leq \left\|\frac{\sqrt{T}}{n\sqrt{n}}\sum_{m=1}^p(\widehat F-\tilde FH)'\frac{1}{T}\sum_t U_{t,m}\epsilon_t'(\widehat F-\tilde FH)\gamma_{dm}\right\|_F \\
\label{G2.i2}
&\qquad + O_P(1)\left\|\frac{\sqrt{T}}{n\sqrt{n}}\sum_{m=1}^p\tilde F'\frac{1}{T}\sum_t U_{t,m}\epsilon_t' \tilde F\gamma_{dm}\right\|_F \\
\label{G2.i3}
&\qquad + \left\|\frac{\sqrt{T}}{n\sqrt{n}}\sum_{m=1}^p(\widehat F-\tilde FH)'\frac{1}{T}\sum_t U_{t,m}\epsilon_t'\tilde F\gamma_{dm}\right\|_FO_P(1) \\
\label{G2.i4}
&\qquad +O_P(1)\left\|\frac{\sqrt{T}}{n\sqrt{n}}\sum_{m=1}^p\tilde F'\frac{1}{T}\sum_t U_{t,m}\epsilon_t'(\widehat F-\tilde FH)\gamma_{dm}\right\|_F.
\end{align}
Term (\ref{G2.i1}) is bounded by 
\begin{align*}
\sqrt{Tn}\left(\frac{1}{T}\sum_t\| \frac{1}{n}(\widehat F-\tilde FH)' U_{t}\gamma_{d}\|_2^2\right)^{1/2}\left(\frac{1}{T}\sum_t\|\frac{1}{n}\epsilon_t'(\widehat F-\tilde FH)\|_2^2\right)^{1/2}
&=O_P\left(|J|_0\Delta_F^2\sqrt{Tn}\right) \\
&=o_P(1)
\end{align*}
using the assumption that $|J|_0\Delta_F\Delta_{fe}\sqrt{Tn}\leq |J|_0\Delta_F^2\sqrt{Tn}=o(1)$. Term (\ref{G2.i2})
$$
\left\|\frac{\sqrt{T}}{n\sqrt{n}}\sum_{m=1}^p\tilde F'\frac{1}{T}\sum_t U_{t,m}\epsilon_t' \tilde F\gamma_{dm}\right\|_F=o_P(1).
$$
For term (\ref{G2.i3}),
\begin{align*}
& \left\|\frac{\sqrt{T}}{n\sqrt{n}}\sum_{m=1}^p(\widehat F-\tilde FH)'\frac{1}{T}\sum_t U_{t,m}\epsilon_t'\tilde F\gamma_{dm}\right\|_F
\leq \frac{\sqrt{T}}{\sqrt{n}}\|\widehat F'-\tilde FH\|_F\left\|\frac{1}{n }\sum_{m=1}^p\frac{1}{T}\sum_t U_{t,m}\epsilon_t'\tilde F\gamma_{dm}\right\|_F\\
&\qquad \leq O_P(\sqrt{T}\Delta_F)\left(\frac{1}{T}\sum_t\|\sum_{m=1}^p\frac{1}{\sqrt{n}}\gamma_{dm}U_{t,m}\|_2^2\right)^{1/2}
\left(\frac{1}{T}\sum_t\| \frac{1}{\sqrt{n}}\epsilon_t'\tilde F  \|_2^2\right)^{1/2}\\
&=O_P(|J|_0\Delta_F\sqrt{T})=o_P(1).
\end{align*}



 For (\ref{G2.i4}),  by Assumption \ref{ca.1} that 
$
\frac{1}{T} \sum_{t=1}^T\|\frac{1}{n}\sum_{i=1}^n (\widehat  f_i-H'\tilde f_i)  \epsilon_{it}\|^2_2
 =O_P(\Delta_{fe}^2)$, 
 \begin{align*}
& \left\|\frac{\sqrt{T}}{n\sqrt{n}}\sum_{m=1}^p\tilde F'\frac{1}{T}\sum_t U_{t,m}\epsilon_t'(\widehat F-\tilde FH)\gamma_{dm}\right\|_F\\
&\qquad \leq (\frac{1}{T}\sum_t\|\frac{1}{n}\epsilon_t'(\widehat F-\tilde FH)\|_2^2)^{1/2}  ( \frac{1}{T}\sum_t\|\frac{1}{\sqrt{n}}\sum_{m=1}^p\tilde F' U_{t,m}\gamma_{dm}\|^2_2)^{1/2}\sqrt{T}\\
&\qquad \leq O_P({\sqrt{T}} \Delta_{fe}|J|_0)\leq O_P({\sqrt{T}} \Delta_{F}|J|_0)=o_P(1).
 \end{align*}
 Combining these bounds yields the result.

 

(ii) Because $\frac{1}{n}\sum_i\left(\frac{1}{T}\sum_tU_{it}'\gamma_d\right)^2=O_P(|J|_0^2)$,
\begin{align*}
\left\|\frac{1}{nT}\sum_{i=1}^n\sum_t(\widehat f_i-H'\tilde f_i)U_{it}'\gamma_{d}\right\|_2^2 &\leq \frac{1}{n}\sum_i\left(\frac{1}{T}\sum_tU_{it}'\gamma_d\right)^2\frac{1}{n}\sum_i\left\|\widehat f_i-H'\tilde f_i\right\|_2^2 \\
&= O_P\left(\Delta_F^2|J|_0^2\right)
\end{align*}
follows by applying the Cauchy-Schwarz inequality.  
We also have  
$$
\|\frac{1}{nT}\sum_{j=1}^n\sum_t(\widehat f_j-H'\tilde f_j)
\epsilon_{jt}\|_2^2\leq \frac{1}{n}\sum_j(\frac{1}{T}\sum_t\epsilon_{jt})^2\frac{1}{n}\sum_j\|\widehat f_j  	-H'\tilde f_j\|_2^2=O_P(\Delta_F)$$ 
due to $ \frac{1}{n}\sum_j\left(\frac{1}{T}\sum_t\epsilon_{jt}\right)^2=O_P(1).$  
Hence,
\begin{align*}
\left\|\frac{1}{n}\sum_{j=1}^n\widehat f_j \bar U_{j\cdot}'\gamma_{d}\right\|_2 &\leq \left\|\frac{1}{nT}\sum_{j=1}^n\sum_t(\widehat f_j	-H'\tilde f_j) U_{jt}'\gamma_{d}\|_2+O_P(1) \|\frac{1}{nT}\sum_{j=1}^n\sum_t\tilde f_j 
 U_{jt}'\gamma_{d}\right\|_2 \\
&= O_P\left(\Delta_F|J|_0\right)+O_P(1) \left\|\frac{1}{nT}\sum_{j=1}^n\sum_t f_j  	
 U_{jt}'\gamma_{d}\right\|_2 + O_P(1) \left|\frac{1}{nT}\sum_{j=1}^n\sum_t 
 U_{jt}'\gamma_{d}\right\|_2  \\
&= O_P\left(\Delta_F|J|_0+\sqrt{\frac{|J|_0}{nT}}\right)
\end{align*}

The second bound follows similarly.


(iii)  Note that
\begin{align}\label{eb.7}
\begin{split}
\left\|\frac{\sqrt{nT}}{n} \right. & \left. \sum_{m=1}^p\sum_{j=1}^n \widehat f_j  	
\frac{1}{T} \sum_{t=1}^T U_{jt,m} \bar \epsilon_{\cdot t}
\gamma_{dm}\right\|_2 \\
&\leq 
\left\|\frac{\sqrt{nT}}{n} \sum_{j=1}^n (\widehat f_j  	-H'\tilde f_j)
\frac{1}{T} \sum_{t=1}^T U_{jt} '\gamma_d\bar \epsilon_{\cdot t}\right\|_2
+  \left\|\frac{\sqrt{nT}}{n}\sum_{j=1}^n H'  f_j
\frac{1}{T} \sum_{t=1}^T U_{jt}'\gamma_d \bar \epsilon_{\cdot t}\right\|_2 \\
& \qquad +  \left\|H'\bar f\frac{\sqrt{nT}}{n} \sum_{j=1}^n   
\frac{1}{T} \sum_{t=1}^T U_{jt}'\gamma_d \bar \epsilon_{\cdot t}\right\|_2\\
&\leq 
\left\|\frac{\sqrt{nT}}{n} \sum_{j=1}^n (\widehat f_j  	-H'\tilde f_j)
\frac{1}{T} \sum_{t=1}^T U_{jt} '\gamma_d\bar \epsilon_{\cdot t}\right\|_2
+ O_P(1)\left\|\frac{\sqrt{nT}}{n}\sum_{j=1}^n   f_j
\frac{1}{T} \sum_{t=1}^T U_{jt}'\gamma_d \bar \epsilon_{\cdot t}\right\|_2 \\
& \qquad + O_P(1)\left|\frac{\sqrt{nT}}{n} \sum_{j=1}^n   
\frac{1}{T} \sum_{t=1}^T U_{jt}'\gamma_d \bar \epsilon_{\cdot t}\right|
\end{split}
\end{align}
We also have that
\begin{align*}
E\left[\frac{1}{n}\sum_j\left(\frac{1}{T}\sum_tU_{jt}'\gamma_d\bar\epsilon_{\cdot t}\right)^2\right]
&= \frac{1}{n^3T^2}\sum_{j=1}^n\sum_{s,t\leq T} E\left[U_{jt}'\gamma_d\epsilon_{jt}
U_{js}'\gamma_d\epsilon_{js}\right] \\
&\qquad + \frac{1}{n^3T^2}\sum_j\sum_s\sum_{i\neq j}\sum_t E\left[U_{jt}'\gamma_d U_{js}'\gamma_d\right]E\left[\epsilon_{is}\epsilon_{it}\right] \\
&= O\left(\frac{|J|_0}{n^2} \right)\max_{j,s,t} E|\epsilon_{js}\epsilon_{jt}|\max_{j,t,s,k}\sum_{m=1}^p\left|E\left[U_{jt,m} U_{js,k}|\epsilon_{js}\epsilon_{jt}\right]\right| \\
&\qquad + O\left(\frac{|J|_0}{nT}\right) \max_{ist} \left|E\left[\epsilon_{is}\epsilon_{it}\right]\right|\max_{j,t,m} \sum_s\sum_m \left|E\left[U_{jt,m}U_{js,k}\right]\right| \\
&= O\left(\frac{|J|_0}{n^2} +\frac{|J|_0}{nT}\right),
\end{align*}
where we used the assumption that $\max_{j,t,s,k}\sum_{m=1}^p \left|E\left[U_{jt,m}
U_{js,k}|\epsilon_{js}\epsilon_{jt}\right]\right| =O(1)$ almost surely and $\max_{j,t,m} \sum_s\sum_m\left|E\left[U_{jt,m}U_{js,k}\right]\right|=O(1)$.
The first term of (\ref{eb.7}) is thus bounded by 
$$
\sqrt{nT}\Delta_F\left(\frac{1}{n}\sum_j\left(\frac{1}{T}\sum_tU_{jt}'\gamma_d\bar\epsilon_{\cdot t}\right)^2\right)^{1/2} = O_P\left(\sqrt{nT|J|_0}\Delta_F\right)\left(\frac{1}{n}+\frac{1}{\sqrt{nT}}\right) = o_P(1).
$$


For notational simplicity, suppose $\dim(f_j)=1$ in the following.  Note that otherwise we can do the analysis element-by-element as $\dim(f_j)$ is fixed.  We can bound the second term of (\ref{eb.7}) by
\begin{align*}
E\left|\frac{\sqrt{nT}}{n} \right. & \left. \sum_{j=1}^n f_j\frac{1}{T} \sum_{t=1}^T U_{jt}'\gamma_d \bar \epsilon_{\cdot t}\right|^2 \\
&= \frac{1}{n^3T} E\left[ \sum_{j=1}^n  \sum_{t=1}^T     \sum_{s=1}^T   f_j^2  U_{jt}'\gamma_d  \epsilon_{jt}    U_{js}'\gamma_d  \epsilon_{js} \right] \\
&\qquad + \frac{1}{n^3T} \sum_{j=1}^n  \sum_{t=1}^T \sum_{i\neq j}  \sum_{s=1}^T E\left[  f_j^2  U_{jt}'\gamma_d      U_{js}'\gamma_d \right] E\left[\epsilon_{is} \epsilon_{it}\right] \\
&\leq O\left(\frac{|J|_0}{n^3T}\right) \sum_{j=1}^n  \sum_{t=1}^T \sum_{s=1}^T   E\left|f_j^2      \epsilon_{js}  \epsilon_{jt} \right|   \sum_m \left|E(U_{js,k} U_{jt,m} |\epsilon_j, f_j)\right| \\
&\qquad + O\left(\frac{|J|_0}{n^3T} \right)\sum_{j=1}^n  \sum_{t=1}^T \sum_{i\neq j}    \left|E\left[\epsilon_{is} \epsilon_{it}\right]\right|
E\left[f_j^2\right] \max_{jtk} \sum_m \sum_{s=1}^T \left|E\left( U_{jt,m}     U_{js,k}    |f_j\right)\right| \\
&= O\left(\frac{|J|_0T}{n^2}+\frac{|J|_0}{n}\right)=o(1).
\end{align*} 
Note that we frequently used the fact that $\{f_i, U_i, \epsilon_i\}$ are independent across $i$ and that $E(\epsilon_i|f_i, U_i)=0$ in the above. 

It follows from a similar argument to that used to bound the second term of (\ref{eb.7}) that the third term of (\ref{eb.7}) is $o_P(1)$. Hence (\ref{eb.7}) is $o_P(1)$.


(iv) Note that
\begin{align*}
\left\|\frac{\sqrt{nT}}{n} \right. & \left. \sum_{i=1}^n  \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t}'\gamma_d\epsilon_{it}\widehat f_i'\right\|_2 \\
&\leq \left\|\frac{\sqrt{nT}}{n}  \sum_{i=1}^n  \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t}'\gamma_d\epsilon_{it}
(\widehat f_i-H'\tilde f_i)'\right\|_2 + \left\|\frac{\sqrt{nT}}{n}  \sum_{i=1}^n  \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t}'\gamma_d\epsilon_{it}\tilde  f_i'\right\|_2 \\
&\leq O_P\left(\sqrt{nT}\Delta_{fe}\right)\left(\frac{1}{T}\sum_t(\bar U_{\cdot t}'\gamma_d)^2\right)^{1/2}+ \left\|\frac{\sqrt{nT}}{n}  \sum_{i=1}^n  \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t}'\gamma_d\epsilon_{it}
\tilde  f_i'\right\|_2.
\end{align*} 
 Following similar calculations in part (iii), it can be shown that  $\frac{1}{T}\sum_t(\bar U_{\cdot t}'\gamma_d)^2=O_P\left(\frac{|J|_0}{n}\right)$.  Note that $\max_k\sum_m|EU_{it,m}U_{it,k}|=O(1)$, which enables us to remove   the impact of one of the $\sum_{m\leq p}$ terms. Hence, the first term is $O_P\left(\sqrt{T|J|_0\Delta^2_{F}}\right)=o_P(1)$. In addition, 
$$
\left\|\frac{\sqrt{nT}}{n}  \sum_{i=1}^n  \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t}'\gamma_d\epsilon_{it}
\tilde  f_i'\right\|_2 = O_P\left(\sqrt{\frac{|J|_0}{n}}\right)=o_P(1).
$$ 

The second conclusion that 
$$
\sqrt{nT}| \sum_{m=1}^p \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t,m} \bar\epsilon_{\cdot t}
\gamma_{dm}|=o_P(1)$$ 
follows from a similar calculation to that used to bound the first term given immediately above. 


(v) Note that (\ref{ea.7add}) holds when $\tilde\delta_{yt}$ is replaced with $\tilde\delta_{dt}$. Also, recall that $\Delta_F = o\left(\sqrt{\frac{\log p}{nT}}\right)$, so we have
$$
\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it} -\tilde U_{it} ) \tilde \delta_{dt}'H^{'-1}(H'\tilde f_i - \widehat f_i )   \right\|_{\infty} \leq
O_P(\Delta_{F})\left(\Delta_{ud}+\sqrt{\frac{\log p}{nT}}\right).
$$
Also, by the assumption $\max_m|\frac{1}{nT}\sum_{i}\sum_t\tilde U_{it, m}(\widehat f_i-H'\tilde f_i)'H^{-1}\tilde \delta_{dt}|=O_P(\Delta_{ud})$, we have
\begin{align*} 
\left\|\frac{1}{nT} \right. & \left. \sum_t\widehat U_{t,\widehat J}' (\tilde F H-\widehat F)H^{-1}\tilde\delta_{dt}\right\|_2
&\leq \max_m\left|\frac{1}{nT}\sum_{i}\sum_t\widehat U_{it, m}(\widehat f_i-H'\tilde f_i)'H^{-1}\tilde \delta_{dt}\right|O_P(\sqrt{|J|_0}) \\
&\leq O_P\left(\Delta_{ud}+   \Delta_F\sqrt{\frac{\log p}{nT}}\right)\sqrt{|J|_0}.
\end{align*} 

  
Note that Lemma \ref{lb.2} continues to hold when $\tilde\epsilon$ is replaced with $\tilde\eta$ and when $\gamma_d$ is replaced with $\theta$.   $\blacksquare$
 
\begin{lemma}\label{lb.2}
%$\frac{1}{\sqrt{nT}} \tilde \epsilon'(I_T\otimes P_{\widehat F}) P_{\widehat U_{\widehat J}}(I_T\otimes M_{\widehat F}(\tilde FH-\widehat F)H^{-1}) \tilde\Delta_d =o_P(1)$\\
%(iii)$\frac{1}{\sqrt{nT}}\tilde\epsilon'P_{\widehat U_{\widehat J}}(I_T\otimes P_{\widehat F})\tilde U\gamma_d=o_P(1)$.\\
%(v)$\frac{1}{\sqrt{nT}}\tilde\epsilon'P_{\widehat U_{\widehat J}}(I_T\otimes P_{\widehat F})\tilde \eta=o_P(1)$, 
%\\(vi) $\frac{1}{\sqrt{nT}} \tilde \epsilon'(I_T\otimes P_{\widehat F}) P_{\widehat U_{\widehat J}} ( \tilde U_{\widehat J}-\widehat U_{\widehat J})\gamma_d=o_P(1)$. 
(i)  $\frac{1}{\sqrt{nT}}\tilde\epsilon'M_{\widehat U_{\widehat J}}(I_T\otimes M_{\widehat F}(\tilde FH-\widehat F)H^{-1}) \tilde\Delta_d=o_P(1)$.\\ 
(ii) $\frac{1}{\sqrt{nT}}\tilde\epsilon' (I_T\otimes P_{\widehat F})\tilde U\gamma_d=o_P(1)$, \\
(iii)
$\frac{1}{\sqrt{nT}}\tilde\epsilon' (I_T\otimes P_{\widehat F})\tilde \eta=o_P(1)$, \\
 (iv) $ \frac{1}{\sqrt{nT}}\tilde\epsilon'M_{\widehat U_{\widehat J}} \tilde U\gamma_d=o_P(1)$.\\
(v) $\frac{1}{\sqrt{nT}}\tilde\epsilon'M_{\widehat U_{\widehat J}}(I_T\otimes M_{\widehat F})\tilde U\gamma_d=o_P(1)$.\\
(vi) $  \tilde\Xi'(I_T\otimes H^{'-1}(\tilde FH-\widehat F)'M_{\widehat F}) 
 \frac{1}{\sqrt{nT}}M_{\widehat U_{\widehat J}} \tilde U\gamma_d=o_P(1).$
\end{lemma}

 
\proof (i)
Note that 
\begin{align*} 
\left|\frac{1}{\sqrt{nT}}\sum_t\tilde \epsilon_t' P_{\widehat F}(\tilde FH-\widehat F)H^{-1}\tilde\delta_{dt}\right|^2 &= \frac{1}{nT}\tr^2\left(\sum_t\tilde\delta_{dt}\tilde\epsilon_t'P_{\widehat F}(\tilde FH-\widehat F)H^{-1}\right) \\
&\leq \frac{1}{nT}\left\|\sum_t\tilde\delta_{dt}\tilde\epsilon_t'\widehat F\right\|_F^2\left\|\frac{1}{n}\widehat F'(\tilde FH-\widehat F)\right\|_F^2  O_P(1) \\
&= O_P(\Delta_F^2)\left\|\frac{1}{\sqrt{nT}}\sum_t\tilde\delta_{dt}\tilde\epsilon_t'(\widehat F-FH)\right\|_F^2 \\ 
&\qquad +O_P(\Delta_F^2)\frac{1}{nT}\left\|\sum_t\tilde\delta_{dt}\tilde\epsilon_t'  F\right\|_F^2 =o_P(1).
\end{align*}

In addition, 
$$
\left\|\frac{1}{nT}\sum_t\widehat U_{t,\widehat J}' (\tilde FH-\widehat F)H^{-1}\tilde\delta_{dt}\right\|_2=O_P\left(\Delta_{ud} +\Delta_{F}\sqrt{\frac{\log p}{nT}}\right)\sqrt{|J|_0},
$$
by Lemma \ref{lb.2new}; and $\left\|\frac{1}{\sqrt{nT}}\sum_t\widehat U_{t,\widehat J}'P_{\widehat F}(\tilde FH-\widehat F)H^{-1}\delta_{dt}\right\|_2=0$ due to $\widehat U_t'\widehat F=0$. It follows that
\begin{align}\label{eb.9}
\begin{split}
\left\|P_{\widehat U_{\widehat J}} \right. & \left. (I_T\otimes M_{\widehat F}(\tilde FH-\widehat F)H^{-1}) \tilde\Delta_d\right\|_2 \\
&= \left\|\widehat U_{\widehat J}(\widehat U_{\widehat J}'\widehat U_{\widehat J})^{-1}\sum_t\widehat U_{t,\widehat J}'M_{\widehat F}(\tilde FH-\widehat F)H^{-1}\delta_{dt}\right\|_2 \\
&\leq \left\|\frac{1}{\sqrt{nT}}\sum_t\widehat U_{t,\widehat J}'M_{\widehat F}(\tilde FH-\widehat F)H^{-1}\delta_{dt}\right\|_2 \\
&\leq \left\|\frac{1}{\sqrt{nT}}\sum_t\widehat U_{t,\widehat J}' (\tilde FH-\widehat F)H^{-1}\delta_{dt}\right\|_2+\left\|\frac{1}{\sqrt{nT}}\sum_t\widehat U_{t,\widehat J}'P_{\widehat F}(\tilde FH-\widehat F)H^{-1}\delta_{dt}\right\|_2 \\
&= O_P\left(\Delta_{ud}+   \Delta_{F}\sqrt{\frac{\log p}{nT}}\right)\sqrt{|J|_0nT}.
\end{split}
\end{align}
By Assumption \ref{ca.1}, $\|\frac{1}{\sqrt{nT}}\sum_t\tilde\delta_{dt}\tilde \epsilon_t'  (\tilde FH-\widehat F)\|_F=o_P(1)$. By Lemma \ref{lb.1},  
$$
\|\frac{1}{\sqrt{nT}}P_{\widehat U_{\widehat J}} \tilde\epsilon\|_2^2=O_P\left(|J|_0\frac{\log p}{nT}\right).
$$   
Hence, using the assumption $ \left(\Delta_{ud}+  \Delta_{F}\sqrt{\frac{\log p}{nT}}\right)\sqrt{\log p|J|_0^2}=o(1)$,
\begin{align*}
\frac{1}{\sqrt{nT}} & \tilde\epsilon'M_{\widehat U_{\widehat J}}(I_T\otimes M_{\widehat F}(\tilde FH-\widehat F)H^{-1}) \tilde\Delta_d \\
&= -\frac{1}{\sqrt{nT}}\tilde\epsilon'P_{\widehat U_{\widehat J}}(I_T\otimes M_{\widehat F}(\tilde FH-\widehat F)H^{-1}) \tilde\Delta_d	+ \frac{1}{\sqrt{nT}}\tilde\epsilon' (I_T\otimes  (\tilde FH-\widehat F)H^{-1}) \tilde\Delta_d \\	
&\qquad -\frac{1}{\sqrt{nT}}\tilde\epsilon' (I_T\otimes P_{\widehat F}(\tilde FH-\widehat F)H^{-1}) \tilde\Delta_d \\
&\leq \left\|\frac{1}{\sqrt{nT}}\tilde\epsilon'P_{\widehat U_{\widehat J}}\right\|_2 \left\|P_{\widehat U_{\widehat J}}(I_T\otimes M_{\widehat F}(\tilde FH-\widehat F)H^{-1}) \tilde\Delta_d\right\|_2	\\
&\qquad +\left|\frac{1}{\sqrt{nT}}\sum_t\tilde \epsilon_t' P_{\widehat F}(\tilde FH-\widehat F)H^{-1}\tilde \delta_{dt}\right| + \left|\frac{1}{\sqrt{nT}}\sum_t\tilde \epsilon_t'  (\tilde FH-\widehat F)H^{-1}\tilde\delta_{dt}\right| \\
&= o_P(1)+O_P\left(\sqrt{|J|_0\frac{\log p}{nT}}\right)\left(\Delta_{ud}+   \Delta_{F}\sqrt{\frac{\log p}{nT}}\right)\sqrt{|J|_0nT} = o_P(1).
\end{align*}
 

(ii)  Note that  
\begin{align*}
\frac{1}{T} \sum_{t=1}^T\tilde U_{jt}\tilde\epsilon_{it}
&= \frac{1}{T}  \sum_{t=1}^T U_{jt}\epsilon_{it}-\bar U_{j\cdot}\bar \epsilon_{i\cdot}
- \frac{1}{T} \sum_{t=1}^T U_{jt} \bar \epsilon_{\cdot t}
+\bar U_{j\cdot}\bar{\bar\epsilon}
+\bar{\bar  U}\bar \epsilon_{i\cdot}
- \bar{\bar  U}\bar{\bar\epsilon} \\
&\qquad -  \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t}\epsilon_{it}
 + \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t} \bar\epsilon_{\cdot t}.
\end{align*}
Hence, 
\begin{align}\label{eadd}
\begin{split}
\frac{1}{\sqrt{nT}}\tilde\epsilon' (I_T\otimes P_{\widehat F})\tilde U\gamma_d &= \frac{1}{\sqrt{nT}}\sum_t\tilde\epsilon_t'P_{\widehat F}\tilde U_t\gamma_d \\
&\leq \left\|\frac{\sqrt{T}}{n\sqrt{n}}\sum_{m=1}^p\widehat F'\frac{1}{T}\sum_t\tilde U_{t,m}\tilde\epsilon_t'\widehat F\gamma_{dm}\right\|_F \\
&\leq \sum_{l=1}^8 a_l
\end{split}
\end{align}
where each $a_l$ is defined below and can be bounded employing Lemma \ref{lb.2new}.  Specifically,
\begin{align*}
a_1&= \left\|\frac{\sqrt{T}}{n\sqrt{n}}\sum_{m=1}^p\widehat F'\frac{1}{T}\sum_t U_{t,m}\epsilon_t'\widehat F\gamma_{dm}\right\|_F=o_P(1). \\
a_2&= \sqrt{nT} \left\|\frac{1}{n}\sum_{j=1}^n\widehat f_j  	
\bar U_{j\cdot}'\gamma_{d}\right\|_2 \left\|\frac{1}{n}\sum_{i=1}^n\bar \epsilon_{i\cdot}
\widehat f_i'\right\|_2 \\
&= O_P\left(\sqrt{nT}\right)\left(\Delta_{F}|J|_0+\sqrt{\frac{|J|_0}{nT}}\right) \left(\Delta_F+\frac{1}{\sqrt{nT}}\right)=o_P(1). \\
a_3& = O_P(1)\left\|\frac{\sqrt{nT}}{n}\sum_{m=1}^p\sum_{j=1}^n \widehat f_j  	
\frac{1}{T} \sum_{t=1}^T U_{jt,m} \bar \epsilon_{\cdot t}
\gamma_{dm}\right\|_F=o_P(1). \\
a_4&= O_P(1)\left\|\frac{\sqrt{nT}}{n}\sum_{j=1}^n\widehat f_j \bar U_{j\cdot}' \gamma_{d}\right\|_2|\bar{\bar\epsilon}| \\
&= O_P\left(\sqrt{nT}\right)\left(\Delta_F|J|_0+\sqrt{\frac{|J|_0}{nT}}\right) \frac{1}{\sqrt{nT}} = o_P(1). \\
a_5&= O_P(1)\left\|\frac{\sqrt{nT}}{n}  \sum_{i=1}^n \bar \epsilon_{i\cdot} \widehat f_i'\right\|_2\left|\bar{\bar  U}'\gamma_{d}\right| \\
&= O_P(\sqrt{nT}) \left(\Delta_F+\frac{1}{\sqrt{nT}}\right)\sqrt{\frac{|J|_0}{nT}} = o_P(1). \\
a_6 &= O_P(1) \sqrt{nT}\left|\bar{\bar\epsilon}\right|
  \left| \bar{\bar  U}'\gamma_{d}\right| = O_P\left(\sqrt{\frac{|J|_0}{nT}}\right)=o_P(1). \\
a_7 &= O_P(1)\left\|\frac{\sqrt{nT}}{n}  \sum_{i=1}^n \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t}'\gamma_d\epsilon_{it}\widehat f_i'\right\|_2 \leq o_P(1). \\
a_8 &= O_P(1)\sqrt{nT}\left| \sum_{m=1}^p \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t,m} \bar\epsilon_{\cdot t}
\gamma_{dm}\right| \leq o_P(1).
\end{align*}
where 
we used $\sqrt{nT}\Delta_F^2|J|_0=o(1)$.


(iii) The third conclusion of the lemma follows from an argument similar to that used to establish (ii).  We have 
$$
\frac{1}{\sqrt{nT}}  \tilde\epsilon' (I_T\otimes P_{\widehat F})\tilde\eta \leq 
\left\|\frac{\sqrt{T}}{n\sqrt{n}}\widehat F'\frac{1}{T}\sum_t\tilde \eta_{t}\tilde\epsilon_t'\widehat F \right\|_F \leq \sum_{l=1}^8 b_l = o_P(1)
$$
which follows from the bounds
\begin{align*}
b_1&= \left\|\sqrt{nT}   \frac{1}{n^2}\sum_i\sum_j\widehat f_i\frac{1}{T}\sum_t \eta_{it}\epsilon_{jt}\widehat f_j' \right\|_F = O_P\left(\Delta^2_{F}\sqrt{Tn}+\Delta_{F}\sqrt{T}\right) = o_P(1) \\
b_2&= \sqrt{nT} \left\|\frac{1}{n}\sum_{j=1}^n\widehat f_j  	
\bar \eta_{j\cdot}\|_2 \right\|\frac{1}{n}\sum_{i=1}^n\bar \epsilon_{i\cdot}
\widehat f_i'\|_2 = O_P\left(\sqrt{nT}\right)\left(\Delta_{F}+\frac{1}{\sqrt{nT}}\right)^2 = o_P(1) \\
b_3&= O_P(1)\left\|\frac{\sqrt{nT}}{n}\sum_{j=1}^n \widehat f_j \frac{1}{T} \sum_{t=1}^T \eta_{jt} \bar \epsilon_{\cdot t} \right\|_F = o_P(1) \\
b_4&= O_P(1)\left\|\frac{\sqrt{nT}}{n}\sum_{j=1}^n\widehat f_j \bar \eta_{j\cdot}\right\|_2 \left|\bar{\bar\epsilon}\right| = O_P\left(\sqrt{nT}\right)\left(\Delta_{F}+\sqrt{\frac{1}{nT}}\right) \frac{1}{\sqrt{nT}} = o_P(1) \\
b_5&= O_P(1)\left\|\frac{\sqrt{nT}}{n}  \sum_{i=1}^n \bar \epsilon_{i\cdot}\widehat f_i'\right\|_2 \left|\bar{\bar  \eta}\right| = O_P\left(\sqrt{nT}\right)  \left(\Delta_{F}+\frac{1}{\sqrt{nT}}\right)\sqrt{\frac{1}{nT}} = o_P(1) \\
b_6&= O_P(1) \sqrt{nT}\left|\bar{\bar\epsilon}\right| \left| \bar{\bar  \eta}\right| = O_P\left(\sqrt{\frac{1}{nT}}\right) = o_P(1) \\
b_7&= O_P(1)\left\|\frac{\sqrt{nT}}{n} \sum_{i=1}^n \frac{1}{T} \sum_{t=1}^T \bar \eta_{\cdot t}\epsilon_{it}
\widehat f_i'\right\|_2 \leq o_P(1) \\
b_8&= O_P(1)\sqrt{nT} \left|\frac{1}{T} \sum_{t=1}^T \bar \eta_{\cdot t} \bar\epsilon_{\cdot t}\right| \leq o_P(1)
\end{align*}
where we used that $\sqrt{nT}\Delta_{F}^2=o(1)$. 

 

(iv) By (\ref{eb.2}), $\widehat U\widehat m=P_{\widehat U_{\widehat J}}\widehat U\gamma_d$. We first bound $\|\gamma_d-\widehat m\|_1$. By Proposition \ref{p2.1},   
$$ \frac{1}{nT}\|\widehat  U(\gamma_d- \widehat m)\|_2^2\leq \frac{1}{nT}\sum_{t=1}^T\|\widehat  U_t(\gamma_d- \tilde\gamma)\|_2^2= O_P(\|R_y\|_1^2+\kappa_n^2|J|_0).
$$
We also have $\gamma_d-\widehat m$ is at most $|\widehat J|_0+|J|_0$-sparse. Hence, by Assumption \ref{a3.4} and using a proof similar to that used to show (\ref{eb.4add}), we have that
\begin{align*}
\frac{1}{n}\left\|\widehat U(\gamma_d-\widehat m)\right\|_2^2 \geq (\phi_{\min}(|\widehat J|_0+|J|_0)-o_P(1))\|\gamma_d-\widehat m\|_2^2 \\
&\geq C(|J|_0+|\widehat J|_0)^{-1}\|\gamma_d-\widehat m\|_1^2
\end{align*}
with probability approaching one.
Hence, 
$$
\|\gamma_d-\widehat m\|_1^2=O_P(\|R_y\|_1^2+\kappa_n^2|J|_0)(|J|_0+|\widehat J|_0)=O_P( \|R_y\|_1^2|J|_0+\kappa_n^2|J|_0^2).
$$

In addition, $\|\frac{1}{\sqrt{nT}}\tilde\epsilon'  \widehat U\|_{\infty} = O_P\left(\sqrt{\log p}\right)$ by an argument similar to that in the proof of Lemma \ref{la.5new}.
Hence,
\begin{align*} 
\left|\frac{1}{\sqrt{nT}}\tilde\epsilon'M_{\widehat U_{\widehat J}} \widehat U\gamma_d\right| &= \left|\frac{1}{\sqrt{nT}}\tilde\epsilon'(I-P_{\widehat U_{\widehat J}}) \widehat U\gamma_d \right| \\
&= \left|\frac{1}{\sqrt{nT}}\tilde\epsilon'  \widehat U(\gamma_d-\widehat m)\right| \\
&\leq \left\|\frac{1}{\sqrt{nT}}\tilde\epsilon'  \widehat U\right\|_{\infty} \left\|\gamma_d-\widehat m\right\|_1 \\
&= O_P\left(  \|R_y\|_1|J|_0^{1/2}+\kappa_n|J|_0 \right)\sqrt{\log p} = o_P(1)
\end{align*}
using $\|R_y\|_1^2|J|_0\log p=o(1)$ and $\kappa_n|J|_0\sqrt{\log p}=o(1)$.


Next,
\begin{align}\label{eb.11}
\begin{split}
\frac{1}{\sqrt{nT}}\tilde\epsilon'M_{\widehat U_{\widehat J}} (\tilde U-\widehat U)\gamma_d &= \frac{1}{\sqrt{nT}}\tilde\epsilon' (\tilde U-\widehat U)\gamma_d - \frac{1}{\sqrt{nT}}\tilde\epsilon'P_{\widehat U_{\widehat J}} (\tilde U-\widehat U)\gamma_d \\
&= \frac{1}{\sqrt{nT}}\sum_{i}\sum_t\sum_m\tilde\epsilon_{it}(\tilde U_{it,m}-\widehat U_{it,m})\gamma_{dm} \\
&\qquad -\frac{1}{\sqrt{nT}}\tilde\epsilon'  \widehat U_{\widehat J} ( \widehat U_{\widehat J}' \widehat U_{\widehat J})^{-1} \widehat U_{\widehat J}'(\tilde U-\widehat U)\gamma_d \\
&\leq \left\| \frac{1}{\sqrt{nT}}\sum_{i}\sum_m\sum_t\tilde\epsilon_{it}  \gamma_{dm} \tilde \lambda_{tm}\tilde f_i'  \right\|_F O_P(\Delta_F) \\
&\qquad + \left\| \frac{1}{\sqrt{nT}}\sum_{i}\sum_t\sum_m\tilde\epsilon_{it}  \widehat  f_i		 \frac{1}{n}\widehat F'\tilde U_{t, m}\gamma_{dm}\right\|_F O_P(1) \\
&\qquad + \left\| \frac{1}{\sqrt{nT}}\sum_{i}\sum_t\sum_m\tilde\epsilon_{it}\gamma_{dm}  \tilde\lambda_{tm} (\widehat  f_i -H'\tilde f_i)'\right\| O_P(1) \\
&\qquad + \left\|  \frac{1}{\sqrt{nT}}\tilde\epsilon'  \widehat U_{\widehat J} \|_2\|\frac{1}{nT}   \widehat U_{\widehat J}'(\tilde U-\widehat U)\gamma_d\right\|_2 O_P(1).
\end{split}
\end{align}
The first term in the last inequality in (\ref{eb.11}) is $O_P(|J|_0)(\Delta_F)=o_P(1)$.  By (\ref{eadd}),
the second term in the last inequality in (\ref{eb.11}) is $o_P(1)$.  The third term in the last inequality in (\ref{eb.11}) is $o_P(1)$ by assumption. For the last term in the last inequality in (\ref{eb.11}), we have  $  \|\frac{1}{\sqrt{nT}}{\widehat U_{\widehat J}}' \tilde\epsilon\|_2=O_P(\sqrt{|J|_0 \log p})$ by a proof similar to that of (\ref{eb.7new}).  Then, Lemma \ref{la.5new} (v) yields 
\begin{align*}  
\frac{1}{nT}\left\|\widehat U_{\widehat J}'( \tilde U-\widehat U )\gamma_d\right\|_2 &\leq \sqrt{|\widehat J|_0}\max_k\left|\frac{1}{nT}\sum_i\sum_t\sum_{m}\widehat U_{it, k}(\tilde U_{it,m}-\widehat U_{it,m})\gamma_{dm} \right| \\
&\leq O_P\left( |J|_0^{3/2} \right)
\max_{mk}\left|\frac{1}{nT}\sum_i\sum_t\sum_{m}\widehat U_{it, k}(\tilde U_{it,m}-\widehat U_{it,m})
\right| \\
&\leq O_P\left( |J|_0^{3/2} \right)  \left( \sqrt{\frac{\log p}{nT}}\Delta_{F}+ \Delta_{ud}+{\frac{\log (pT)}{n}}+\Delta_{fum}^2\right).  
\end{align*}
  
The last term of (\ref{eb.11}) is thus bounded by
$$
\left(|J|_0^2 \sqrt{\log p }\right) O_P\left( \sqrt{\frac{\log p}{nT}}\Delta_F+ \Delta_{ud}+{\frac{\log (pT)}{n}}+\Delta_{fum}^2\right) =o_P(1).
$$
Therefore, both $\frac{1}{\sqrt{nT}}\tilde\epsilon'M_{\widehat U_{\widehat J}} (\tilde U-\widehat U)\gamma_d$ and $\frac{1}{\sqrt{nT}}\tilde\epsilon'M_{\widehat U_{\widehat J}} \widehat U\gamma_d$ are $o_P(1)$.


 
(v) Since $\frac{1}{\sqrt{nT}}\tilde\epsilon'P_{\widehat U_{\widehat J}}(I_T\otimes P_{\widehat F})\tilde U\gamma_d=0$, it follows  from  part (ii) and Lemma \ref{lb.2}(ii) and (iii) that 
\begin{eqnarray*}
&& \frac{1}{\sqrt{nT}}\tilde\epsilon'M_{\widehat U_{\widehat J}}(I_T\otimes M_{\widehat F})\tilde U\gamma_d=
 \frac{1}{\sqrt{nT}}\tilde\epsilon'M_{\widehat U_{\widehat J}} \tilde U\gamma_d
 -\frac{1}{\sqrt{nT}}\tilde\epsilon' (I_T\otimes P_{\widehat F})\tilde U\gamma_d=o_P(1).
\end{eqnarray*}

   
(vi)
By Lemma \ref{lb.1}, 
$\left\|\frac{1}{\sqrt{nT}}M_{\widehat U_{\widehat J}} \tilde U\gamma\right\|_2^2 = O_P\left(\|R_y\|_1^2+\kappa_n^2|J|_0+|J|_0^2\Delta_F^2+\frac{|J|_0}{n}\right)$.  Thus, under $Tn\Delta_F^2\left(\kappa_n^2|J|_0+\|R_y\|_1^2+\Delta_F^2|J|_0^2 + \frac{|J|_0}{n}\right)=o(1)$,
\begin{align*} 
|\tilde\Xi'(I_T & \otimes H^{'-1}(\tilde FH-\widehat F)'M_{\widehat F}) 
 \frac{1}{\sqrt{nT}}M_{\widehat U_{\widehat J}} \tilde U\gamma_d|^2 \\
&\leq O_P({T}) \left\|\widehat F-\tilde FH\right\|_F^2 \left\|
 \frac{1}{\sqrt{nT}}M_{\widehat U_{\widehat J}} \tilde U\gamma_d\right\|_2^2 \\
&= O_P\left(Tn\Delta_F^2\right)\left(\kappa_n^2|J|_0+\|R_y\|_1^2+\Delta_F^2|J|_0^2+  \frac{|J|_0}{n}\right)
= o_P(1).
\end{align*}
     
  %Note that  
 %$
  %\sum_{s=1}^T   \tilde\epsilon_{is}    \tilde U_{js}'\gamma_s= \sum_{s=1}^T(\epsilon_{is}-\bar \epsilon_{i\cdot}-\bar \epsilon_{\cdot s}+\bar{\bar \epsilon})(U_{js}-\bar U_{\cdot s})\gamma_s, 
% $
 %hence   for each $k\leq K$, 
 %$
%E\sum_{i=1}^n |  \sum_{j=1}^n f_{jk}     \sum_{s=1}^T   \tilde\epsilon_{is}    \tilde U_{js}'\gamma_s|^2
 %$ can be divided into several terms. We bound them one by one:
 
  
% \begin{eqnarray*} 
 %&&E\sum_{i=1}^n |   \sum_{s=1}^T \sum_{j=1}^n f_{jk}     ( \epsilon_{is}  -\bar \epsilon_{i\cdot}) U_{js}'\gamma_s|^2
 %=
   % \sum_{t=1}^T \sum_{s=1}^T \sum_{i=1}^n\sum_{j=1}^n  \sum_{l=1}^nE f_{jk}    (  \epsilon_{is}    -\bar \epsilon_{i\cdot}) U_{js}'\gamma_s 
 %f_{lk}    (  \epsilon_{it}  -\bar \epsilon_{i\cdot})   U_{lt}'\gamma_t    \cr
 %&=&
    % \sum_{t=1}^T \sum_{s=1}^T \sum_{i=1}^n  E f_{ik}^2     ( \epsilon_{is} -\bar \epsilon_{i\cdot})    U_{is}'\gamma_s 
  %(  \epsilon_{it}  -\bar \epsilon_{i\cdot})   U_{it}'\gamma_t    
 %+  \sum_{t=1}^T \sum_{s=1}^T \sum_{i=1}^n\sum_{j\neq i}  E f_{jk} ^2       U_{js}'\gamma_s 
  % U_{jt}'\gamma_t  E    ( \epsilon_{is}  -\bar \epsilon_{i\cdot}) (\epsilon_{it}  -\bar \epsilon_{i\cdot}) \cr
   %&\leq&  \sum_{t=1}^T \sum_{s=1}^T \sum_{i=1}^n \|\gamma_s\|_1\|\gamma_t\|_1\max_{m, v\leq p}| E f_{ik}^2     ( \epsilon_{is} -\bar \epsilon_{i\cdot})    U_{is,m} 
  %(  \epsilon_{it}  -\bar \epsilon_{i\cdot})   U_{it, v} 
 %\end{eqnarray*}
 
    
        


The following technical lemma is used to prove the consistency of the asymptotic variance estimator.  $\blacksquare$
  
\begin{lemma}\label{lb.4}
(i) Both $\frac{1}{nT}\sum_{i=1}^n\left[\sum_{t=1}^T (\widehat\eta_{it}-\tilde\eta_{it})\tilde\epsilon_{it}\right]^2=o_P(1)$ and $\frac{1}{nT}\sum_{i=1}^n\left[\sum_{t=1}^T (\widehat\epsilon_{it}-\tilde\epsilon_{it})\tilde\eta_{it}\right] ^2 = o_P(1)$. \\
(ii) $\frac{1}{nT}\sum_{i=1}^n\left[\sum_{t=1}^T(\widehat\eta_{it}\widehat\epsilon_{it}- \tilde \eta_{it}\tilde\epsilon_{it})\right] ^2 =o_P(1) $.
 %(iii) $\frac{1}{n}\sum_{i=1}^n |\frac{1}{T}\sum_{t=1}^T(\widehat\eta_{it}\widehat\epsilon_{it}- \tilde \eta_{it}\tilde\epsilon_{it})|.$
\end{lemma}
 
 
\proof (i) We first bound  $\max_{i\leq n}\frac{1}{T}\sum_{t=1}^T\tilde\epsilon_{it}^2$.  We do this by bounding $\max_{i\leq n}\left|\frac{1}{T}\sum_{t=1}^T \epsilon_{it}^2-E\epsilon_{it}^2\right| $ and $\max_{i\leq n}  \left(\frac{1}{T}\sum_t\epsilon_{it}\right)^2$.

When $T$ is fixed,  $$
\max_{i\leq n}\left|\frac{1}{T}\sum_{t=1}^T \epsilon_{it}^2-E\epsilon_{it}^2\right|
\leq\max_{it}| \epsilon_{it}^2-E\epsilon_{it}^2| =O_P(\log n)=O_P\left(\frac{\log n}{\sqrt{T}}\right).\quad (T \text{ does not grow})
$$
 and $\max_{i\leq n}  \left(\frac{1}{T}\sum_t\epsilon_{it}\right)^2\leq \max_{it}\epsilon_{it}^2=O_P(\log n)=O_P(\frac{\log n}{T})$. 

When $T\to\infty$, recall that by assumption, $\log ^\gamma p=o(n)$ for some $\gamma>2$ and that strong mixing (Assumption \ref{a3.2}) holds for the process  $\{(\eta_t, \epsilon_t)\}_{t=-\infty}^{+\infty}$ with mixing coefficient bounded by $\exp(-CT^r)$, $r>1$.  Note that we also assume $\gamma r\geq 2$. 
Let $\bar r= \min\{r, 1\}$, $r_1=(0.5 +\bar r^{-1})^{-1}$,  $c=0.5(\gamma+1)$, then  $\gamma \bar r\geq 2$, $r_1<1$, $cr_1> 1$ and $2c\geq 1.$ Because $\bar r\leq r$, the strong mixing condition in Assumption \ref{a3.2} also holds with $\bar r$ in place of $r$. 
The     Bernstein inequality for weakly dependent data of \cite{merlevede2011bernstein} requires (a) exponential tails (fulfilled by the sub-Gaussian condition in Assumption \ref{a3.2}), and (b) a strong mixing condition (also assumed in Assumption \ref{a3.2}). 
    The introduction of $\bar r$ is to ensure that the so-defined $r_1<1$, another requirement of applying the     Bernstein inequality for strong mixing sequences. Then, by Theorem 1 of  \cite{merlevede2011bernstein} (proved using a ``coupling argument" of \cite{dedecker2004coupling} - see also \cite{wang2016homogeneity} for a similar argument),  
     for $y=M\frac{(\log n)^c}{\sqrt{T}}$, and sufficiently large $M>0$, we have
     $$
     P\left(\max_{i\leq n}\left|\frac{1}{T}\sum_{t=1}^T \epsilon_{it}^2-E\epsilon_{it}^2\right|>y\right) \leq  A_1+A_2+A_3
     $$
     where
     \begin{eqnarray*}
     A_1&=& nT\exp ( -C (Ty)^{r_1})=\exp(\log(nT) - CM^{r_1} T^{r_1/2} \log^{cr_1} n) =o(1),\quad (cr_1\geq 1)\cr
     A_2&=& n\exp\left(  -C\frac{(Ty)^2}{T} \exp\left(\frac{(Ty)^{r_1(1-r_1)}}{C\log^{r_1}(Ty)}\right)  \right)=o(1),\quad (r_1<1, 2c> 1, Ty\to\infty)\cr
     A_3&=& n \exp\left( -CTy^2 \right)= \exp(\log n-CM^2\log^{2c}n) =o(1).
     \end{eqnarray*}
Hence $\max_{i\leq n}\left|\frac{1}{T}\sum_{t=1}^T \epsilon_{it}^2-E\epsilon_{it}^2\right|=O_P\left(\frac{(\log n)^c}{\sqrt{T}}\right)
=O_P\left(\frac{(\log n)^{0.5+0.5\gamma}}{\sqrt{T}}\right)
$. Similarly, $\max_{i\leq n}|\frac{1}{T}\sum_{t=1}^T \epsilon_{it}|=O_P\left(\frac{(\log n)^{0.5+0.5\gamma}}{\sqrt{T}}\right)$.
Then  combining the two cases $T\nrightarrow \infty$ and $T\to \infty$,  we have
\begin{align*} 
\max_{i\leq n}\frac{1}{T}\sum_{t=1}^T \epsilon_{it}^2 &\leq \max_{i\leq n}E \epsilon_{it}^2+\max_{i\leq n}\left|\frac{1}{T}\sum_{t=1}^T \epsilon_{it}^2-E\epsilon_{it}^2\right| \\
&= O_P\left(1+\frac{(\log n)^{0.5+0.5\gamma}}{\sqrt{T}}\right), \\
\max_{i\leq n}  \bar\epsilon_{i\cdot}^2 &= \max_{i\leq n}  \left(\frac{1}{T}\sum_t\epsilon_{it}\right)^2 = O_P\left(\frac{\log^{1+\gamma} n}{T}\right), \\
\frac{1}{T}\sum_{t=1}^T\bar \epsilon_{\cdot t}^2 &= \frac{1}{T}\sum_{t=1}^T\left(\frac{1}{n}\sum_i\epsilon_{it}\right)^2
= O_P\left(\frac{1}{n}\right),
\end{align*}
and
\begin{align*}
\quad \bar{\bar \epsilon}^2=o_P(1).
\end{align*}
Hence, $\max_{i\leq n}\frac{1}{T}\sum_{t=1}^T\tilde\epsilon_{it}^2 = O_P\left(1+\frac{(\log n)^{0.5+0.5\gamma}}{\sqrt{T}}+  \frac{\log^{1+\gamma} n}{T}\right)$.  Thus, by the Cauchy-Schwarz inequality and with $\psi_n$ defined in (\ref{eb.3a}),
\begin{align*} 
\frac{1}{nT}\sum_{i=1}^n\left[\sum_{t=1}^T (\widehat\eta_{it}-\tilde\eta_{it})\tilde\epsilon_{it}\right]^2
&\leq \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T (\widehat\eta_{it}-\tilde\eta_{it})^2\sum_{s=1}^T\tilde\epsilon_{is}^2 \\
&\leq \frac{T}{nT}\|\widehat\eta-\tilde\eta\|_2^2\max_{i\leq n}\frac{1}{T}\sum_{t=1}^T\tilde\epsilon_{it}^2
=O_P(T\psi_n^2)\left(1+\frac{(\log n)^{0.5+0.5\gamma}}{\sqrt{T}}+  \frac{\log^{1+\gamma} n}{T}\right)
 \\
&= O_P\left(\psi_n^2T+\psi_n^2\log ^{1+\gamma}n +\psi_n^2 (T\log^{1+\gamma} n)^{1/2}\right) = o_P(1).
\end{align*}
where the last equality may be directly verified under our assumptions. 
 
 

The same arguments as used in the proof of Proposition \ref{pb.1} yield that 
$$
\frac{1}{nT}\|\widehat e-\tilde e\|_2^2=O_P(\psi_n^2).
$$
In addition, the first statement of Theorem \ref{t3.1} implies $|\alpha-\widehat\alpha|^2=O_P\left(\frac{1}{nT}\right)$. Thus, from $\widehat\epsilon=\widehat e-\widehat\alpha\widehat\eta$, we have
\begin{align*}
\frac{1}{nT} \|\widehat\epsilon-\tilde\epsilon\|_2^2 &\leq 2\frac{1}{nT}\|\widehat e-\tilde e\|_2^2+4\frac{1}{nT}\|(\alpha-\widehat\alpha)\tilde\eta\|_2^2+4\frac{1}{nT}\|\widehat\alpha(\tilde\eta-\widehat\eta)\|_2^2 \\
&=O_P( \psi_n^2)
\end{align*}
for $\psi_n$ defined in (\ref{eb.3a}) and $\frac{1}{nT}=O(\psi_n^2)$.	
In addition, $\max_{i\leq n}\frac{1}{T}\sum_{t=1}^T\tilde\eta_{it}^2=O_P\left(1+\frac{\log n}{T}\right)$ follows from the same argument. Hence, 
\begin{eqnarray*} 
\frac{1}{nT}\sum_{i=1}^n\left[\sum_{t=1}^T (\widehat\epsilon_{it}-\tilde\epsilon_{it})\tilde\eta_{it}\right]^2
&\leq \frac{T}{nT}\|\widehat\epsilon-\tilde\epsilon\|_2^2\max_{i\leq n}\frac{1}{T}\sum_{t=1}^T\tilde\eta_{it}^2 \\
&= O_P\left(\psi_n^2T+\psi_n^2\log n\right)=o_P(1).
\end{eqnarray*}
 
 
(ii) By part (i),  
\begin{align}
\nonumber 
\frac{1}{nT}\sum_{i=1}^n\left[\sum_{t=1}^T(\widehat\eta_{it}\widehat\epsilon_{it}- \tilde \eta_{it}\tilde\epsilon_{it})\right]^2  
&\leq \frac{2}{nT}\sum_{i=1}^n\left[\sum_{t=1}^T (\widehat\eta_{it}-\tilde\eta_{it})\tilde\epsilon_{it}\right] ^2+\frac{2}{nT}\sum_{i=1}^n\left[\sum_{t=1}^T\widehat\eta_{it}(\widehat\epsilon_{it}-\tilde\epsilon_{it})\right]^2 \\
\nonumber
&\leq o_P(1)+\frac{4T}{n}\sum_{i=1}^n\left[\frac{1}{T}\sum_{t=1}^T(\widehat\eta_{it}-\tilde\eta_{it})(\widehat\epsilon_{it}-\tilde\epsilon_{it})\right]^2 \\
\nonumber
&\leq o_P(1)+\frac{4}{nT}\|\widehat\epsilon-\tilde\epsilon\|_2^2\max_i\sum_t(\widehat\eta_{it}-\tilde\eta_{it})^2 \\
\label{eb.16}	
&= o_P(1)+O_P(T\psi_n^2)\max_i\frac{1}{T}\sum_t(\widehat\eta_{it}-\tilde\eta_{it})^2.
\end{align}
We now show the second term on the right of (\ref{eb.16}) is $o_P(1)$. 
Using the equalities
\begin{align*}
\tilde U_{it, m}-\widehat U_{it, m} &= \widehat  f_i' (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm}+\widehat f_i'(\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m} + \tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i)
\end{align*}
and
\begin{align*}
\widehat\delta_{dt}-H^{-1}\tilde\delta_{dt}=
(\widehat F'\widehat F)^{-1}\widehat F' (\tilde FH-\widehat F)H^{-1}\tilde\delta_{dt}
+(\widehat F'\widehat F)^{-1}\widehat F' \tilde U_t\gamma_d
+(\widehat F'\widehat F)^{-1}\widehat F' \tilde \eta_t,
\end{align*} 
we have
\begin{align}\label{eb.17}
\begin{split} 
\tilde\eta_{it}- \widehat\eta_{it} &= \widehat f_i'(\widehat \delta_{dt}-H^{-1}\tilde\delta_{dt}) + \tilde \delta_{dt}'H^{-1'}(\widehat f_i-H'\tilde f_i) \\
&\qquad + \sum_m (\widehat  U_{it,m}-\tilde U_{it,m})\widehat\gamma_{dm} + \tilde U_{it}'(\widehat\gamma_d-\gamma_d) \\
&= \widehat f_i'(\widehat F'\widehat F)^{-1}\widehat F' (\tilde FH-\widehat F)H^{-1}\tilde\delta_{dt}
+ \widehat f_i'(\widehat F'\widehat F)^{-1}\widehat F' \tilde U_t\gamma_d \\
&\qquad + \widehat f_i'(\widehat F'\widehat F)^{-1}\widehat F' \tilde \eta_t + \tilde \delta_{dt}'H^{-1'}(\widehat f_i-H'\tilde f_i) \\
&\qquad +\sum_m \widehat f_i' (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm} \widehat\gamma_{dm} + \sum_m \widehat f_i'(\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m}  \widehat\gamma_{dm} \\
&\qquad + \sum_m  \tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i)\widehat\gamma_{dm} + \tilde U_{it}'(\widehat\gamma_d-\gamma_d) \\
&:=\sum_{l=1}^8 C_{l,it}.
\end{split}
\end{align}

Since $f_i$ is sub-Gaussian, $\max_i\|\tilde f_i\|_2=O_P(\sqrt{\log n})$. 
Under $$\max_i\|\widehat f_i-H'f_i\|_2=O_P(1),$$ 
we have that $\max_i\|\widehat f_i\|_2=O_P(\sqrt{\log n})$.
By Lemma \ref{la.1}(iv), 
$$
\frac{1}{T}\sum_t\left\|\frac{1}{n}\widehat F'\tilde U_t\gamma\right\|_2^2=O_P\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^2\right),$$
and
$$
\frac{1}{T}\sum_t\left\|\frac{1}{n}\widehat F'\tilde \eta_t\right\|_2^2= O_P\left(\frac{1}{n}+\Delta_{fe}^2\right).
$$
Hence,
\begin{align*}
T\psi_n^2 \max_i\frac{1}{T}\sum_t C_{1,it}^2 &= T\psi_n^2 \max_i\frac{1}{T}\sum_t  (\widehat   f_i '(\widehat F'\widehat F)^{-1}\widehat F' (\tilde FH-\widehat F)H^{-1}\tilde\delta_{dt}  )^2 \\
&= O_P(T\psi_n^2\log n)O_P( \Delta_{F}^2) \\
T\psi_n^2 \max_i\frac{1}{T}\sum_t C_{2,it}^2 &= T\psi_n^2 \max_i\frac{1}{T}\sum_t  ( \widehat   f_i '(\widehat F'\widehat F)^{-1}\widehat F' \tilde U_t\gamma_d  )^2 \\
&= O_P(T\psi_n^2\log n)\frac{1}{T}\sum_t\|\frac{1}{n}\widehat F'\tilde U_t\gamma_d\|_2^2 \\
&= O_P(T\psi_n^2\log n) O_P\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^2\right) \\
T\psi_n^2 \max_i\frac{1}{T}\sum_t C_{3,it}^2 &= T\psi_n^2 \max_i\frac{1}{T}\sum_t ( \widehat f_i'(\widehat F'\widehat F)^{-1}\widehat F' \tilde \eta_t)^2 \\
&= O_P(T\psi_n^2\log n)\frac{1}{T}\sum_t\left\|\frac{1}{n}\widehat F'\tilde \eta\right\|_2^2 \\
&= O_P(T\psi_n^2\log n)O_P\left(\frac{1}{n}+\Delta_{fe}^2\right) \\
T\psi_n^2 \max_i\frac{1}{T}\sum_t C_{4,it}^2 &= T\psi_n^2 \max_i\frac{1}{T}\sum_t  ( \tilde \delta_{dt}'H^{-1'}(\widehat f_i-H'\tilde f_i)^2 \\
&= O_P(T\psi_n^2)\max_i\|\widehat f_i-H'f_i\|_2^2 \\ 
T\psi_n^2 \max_i\frac{1}{T}\sum_t C_{5,it}^2 &= T\psi_n^2 \max_i\frac{1}{T}\sum_t ( \sum_m   \widehat  f_i' (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm} \widehat\gamma_{dm})^2 \\
&= O_P(T\psi_n^2\log n)|J|_0^2O_P(\Delta_F^2) \\
T\psi_n^2 \max_i\frac{1}{T}\sum_t C_{6,it}^2 &= T\psi_n^2 \max_i\frac{1}{T}\sum_t  ( \sum_m \widehat  f_i' 		 (\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m}  \widehat\gamma_{dm}  )^2 \\
&= O_P(T\psi_n^2\log n) O_P\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^2\right) \\
T\psi_n^2 \max_i\frac{1}{T}\sum_t C_{7,it}^2 &= T\psi_n^2 \max_i\frac{1}{T}\sum_t  (   \sum_m  \tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i)\widehat\gamma_{dm} )^2 \\
&= O_P(T\psi_n^2)\max_i\|\widehat f_i-H'f_i\|_2^2 |J|_0^2 \\
T\psi_n^2 \max_i\frac{1}{T}\sum_t C_{8,it}^2 &= T\psi_n^2 \max_i\frac{1}{T}\sum_t  ( \tilde U_{it}'(\widehat\gamma_d-\gamma_d)  )^2  \\
&\leq T\psi_n^2\|\widehat\gamma_d-\gamma_d\|_1^2\max_{imk}\left|\frac{1}{T}\sum_t\tilde U_{it,m}\tilde U_{it,k}\right| \\
&= O_P( T\psi_n^2)(\kappa_n^2|J|_0^2+\|R_d\|_1^2)\left(1+\sqrt{\frac{\log (pn)}{T}}\right).
\end{align*}
 
Therefore, 
$T\psi_n^2\max_i\frac{1}{T}\sum_t(\widehat\eta_{it}-\tilde\eta_{it})^2=O_P(T\psi_n^2c_n)$
where
\begin{align*}
c_n &= \log n\left(\Delta_{F}^2|J|_0^2+\frac{|J|_0}{n}+|J|_0^2\Delta_F^2+\Delta_{fe}^2\right) \\
&\qquad + \max_i \|\widehat f_i-H'f_i\|_2^2|J|_0^2+\left(\kappa_n^2|J|_0^2+\|R_d\|_1^2\right)\left(1+\sqrt{\frac{\log (pn)}{T}}\right).
\end{align*}
Under our assumptions $T\psi_n^2 c_n = o(1)$. Then (\ref{eb.16}) implies 
$$ 
\frac{1}{nT}\sum_{i=1}^n\left[\sum_{t=1}^T(\widehat\eta_{it}\widehat\epsilon_{it}- \tilde \eta_{it}\tilde\epsilon_{it})\right] ^2  =o_P(1).
$$
 $\blacksquare$

\subsection{Technical Lemmas for Theorem  \ref{t6.1} }
 
 

  The following lemma presents  a version of the maximal inequality for a U-statistic.  While there are several inequalities of this type  available in the literature, e.g.,  \cite{nolan1987u, horvath1996limit}, we present and prove the following version, which is more directly related to the current context, and is useful for proofs in the bootstrap sampling space.
  \begin{lemma}\label{ld.1ad}
  Suppose the following conditions hold:\\
  (i)   $\{Z_{ijm}\}$ is a sequence of random variables in the original sampling space, satisfying 
$$
\max_{m\leq p, i\leq n}\frac{1}{n}\sum_{j=1}^nZ_{ijm}^2=O_P(a_n^2)
$$ 
for some deterministic sequence $a_n>0.$ \\
  (ii)  $\{X_i^*, Y_i^*\}_{i\leq n}$ is an i.i.d. sequence in the bootstrap sampling space such that $\{X_i^*\} $ is independent of  $\{Y_i^*\}$, $EX_i^*=EY_i^*=0$, and $\Var^*(X_i)<C$ and $\Var^*(Y_i)<C$ for a constant $C>0$ where $C$ is non-random in both the original and bootstrap sampling space.\\
  (iii) Both $X_i^*$ and $Y_i^*$ are sub-exponential random variables satisfying Assumption \ref{a3.2} (iv).  
  
  Then for any $\varepsilon_1, \varepsilon_2>0$, there is a $C_{\varepsilon_1,\varepsilon_2}>0$ such that
  $$
  P\left(P^*\left(\max_{m\leq p}\left|\frac{1}{n^2}\sum_{i,j\leq n}X_i^*Y_j^*Z_{ijm}\right|>\frac{2a_nC\sqrt{C_{\varepsilon_1, \varepsilon_2}\log p\log(pn)}}{n}\right)>\varepsilon_1\right)<\varepsilon_2.
  $$
  Thus $\max_{m\leq p}|\frac{1}{n^2}\sum_{i,j\leq n}X_i^*Y_j^*Z_{ijm}|=O_{P^*}\left(\frac{a_n\sqrt{\log p\log(np)}}{n}\right)$. 
  \end{lemma}
	
  \proof By condition (i), for any $\delta>0$, there is $C_{\delta}>0$ such that with probability at least $1-\delta$ the event $A_{\delta}:=\{\max_{m\leq p, i\leq n}\frac{1}{n}\sum_{j=1}^nZ_{ijm}^2<a_n^2C_{\delta}\}$ holds. 
  
Let $V^*= \max_{mi}|\frac{1}{n}\sum_jY_j^*Z_{ijm}|$. Define $W_{im}^*=X_i^*\frac{1}{n}\sum_jY_j^*Z_{ijm}$ and $Y^*=\{Y_i^*\}_{i\leq n}$.  Since $\{X_i^*\}$ and $\{Y_i^*\}$ are independent,   then on the event $A_{\delta}$,
\begin{align}\label{ed.2a}
\begin{split}
\max_{m,i}\frac{1}{n}\sum_j\Var^*(Y_j^*Z_{ijm}) &=\max_{m,i}\frac{1}{n}\sum_jZ_{ijm}^2\Var(Y_j^*) <a_n^2CC_{\delta} \\
\max_m\frac{1}{n}\sum_i\Var^*(W_{im}^*|Y^*)
&=\max_m\frac{1}{n}\sum_i\left(\frac{1}{n}\sum_jY_j^*Z_{ijm}\right)^2\Var^*(X_i^*) \leq   CV^{*2}.
\end{split}
\end{align}  
In the bootstrap sampling space (BSS), {$\{Y_j^*Z_{ijm}\}_{j \leq n}$ is independent across $j$} and $E^*Y_j^*Z_{ijm}=0$. By the Bernstein inequality,  for $y=(2a_n^2CC_{\delta}\log(pn)/n)^{1/2}$,
\begin{align}\label{ed.3a}
\begin{split}
P^*(V^*>y)1\{A_{\delta}\} &\leq pn\max_{m,i}P^*\left(|\frac{1}{n}\sum_jY_j^*Z_{ijm}|>y\right)1\{A_{\delta}\} \\
&\leq \exp\left(\log(pn)-\frac{ny^2}{\max_{m,i}\frac{1}{n}\sum_j\Var^*(Y_j^*Z_{ijm})}\right)1\{A_{\delta}\} \\
&\leq\exp\left(\log(pn)-\frac{ny^2}{ a_n^2CC_{\delta}}\right) = (pn)^{-1}.
\end{split}
\end{align}
In the BSS, $\{W_{im}^*\}_{i\leq n}$ is independent across $i$ conditional on $Y^*$.
 By (\ref{ed.2a}) and the Bernstein inequality, for $x=y\sqrt{\frac{2C\log p}{n}}=\frac{2a_nC\sqrt{C_{\delta}\log p\log(pn)}}{n}$,
\begin{align}\label{ed.4a}
\begin{split}
P^*(\max_{m\leq p}|\frac{1}{n}&\sum_{i}W_{im}^*|>x|Y^*)1\{V^*<y\} \\
&\leq p\max_mP^*\left(|\frac{1}{n}\sum_{i}W_{im}^*|>x |Y^*\right)1\{V^*<y\} \\
&\leq \exp\left(\log p-\frac{nx^2}{\max_m\frac{1}{n}\sum_i\Var^*(W_{im}^*|Y^*)}\right)1\{V^*<y\} \\
&\leq \exp\left(\log p-\frac{nx^2}{CV^{*2}}\right)1\{V^*<y\}  \exp\left(\log p-\frac{nx^2}{Cy^2}\right)=p^{-1}.
\end{split}
\end{align}
   
  
Let $E_{Y^*}$ denote the expectation operator with respect to the marginal distribution of $Y^*$ in the bootstrap sampling space; i.e., $E_{Y^*}$ is the conditional distribution of $Y^*$ given the original data. By the law of iterated expectations, $E_{Y^*}\left[P^*(\cdot|Y^*)\right]=P^*(\cdot)$. We then have
\begin{align*}
P^*&\left(\max_{m\leq p}|\frac{1}{n^2}\sum_{i,j\leq n}X_i^*Y_j^*Z_{ijm}|>x\right) \\
&\leq P^*\left(\max_{m\leq p}|\frac{1}{n^2}\sum_{i,j\leq n}X_i^*Y_j^*Z_{ijm}|>x\right)1\{A_{\delta}\}+1\{A_{\delta}^c\} \\
&= P^*\left(\max_{m\leq p}|\frac{1}{n}\sum_{i}W_{im}^*|>x\right)1\{A_{\delta}\}+1\{A_{\delta}^c\} \\
&= E_{Y^*}P^*\left(\max_{m\leq p}|\frac{1}{n}\sum_{i}W_{im}^*|>x|Y^*\right)1\{A_{\delta}\}+1\{A_{\delta}^c\} \\
&\leq^{(a)} E_{Y^*}P^*\left(\max_{m\leq p}|\frac{1}{n}\sum_{i}W_{im}^*|>x|Y^*\right)1\{V^*<y\}1\{A_{\delta}\}+E_{Y^*} 1\{V^*\geq y\}1\{A_{\delta}\}+1\{A_{\delta}^c\} \\
&\leq^{(b)} p^{-1}+P^*(V^*\geq y)1\{A_{\delta}\}+1\{A_{\delta}^c\} \\
&\leq^{(c)} p^{-1}+(pn)^{-1}+1\{A_{\delta}^c\},
\end{align*}
where we used $P^*(\cdot |Y^*)\leq P^*(\cdot |Y^*)1\{V^*<y\}+1\{V^*\geq y\}$ in (a), (\ref{ed.4a}) in (b), and (\ref{ed.3a}) in (c). Because $P(A_{\delta}^c)\leq \delta$, taking the expectation with respect to the distribution of the original data on both sides yields
    $$
  EP^*(\max_{m\leq p}|\frac{1}{n^2}\sum_{i,j\leq n}X_i^*Y_j^*Z_{ijm}|>x)\leq  p^{-1}+(pn)^{-1}+\delta.
  $$
For any $\varepsilon_1, \varepsilon_2>0$, let $\delta=\varepsilon_1\varepsilon_2/2$, and call $C_{\delta}$ in $x$ to  be $C_{\varepsilon_1,\varepsilon_2}$. By the Markov Inequality, 
\begin{align*}
 P\left(P^*(\max_{m\leq p}|\frac{1}{n^2}\sum_{i,j\leq n}X_i^*Y_j^*Z_{ijm}|>x)>\varepsilon_1\right)
 &\leq \frac{1}{\varepsilon_1}(p^{-1}+(pn)^{-1}+\delta)  \leq\frac{\varepsilon_1\varepsilon_2/2+\delta}{\varepsilon_1}=\varepsilon_2.
\end{align*}
   $\blacksquare$
\begin{lemma}\label{le.0}
(i) $ \frac{1}{T}\sum_t\|\frac{1}{n}\widehat F^{*'} \tilde U^{*'}_t\widehat\gamma\|_2^2=
O_{P^*}\left(\Delta_{F}^{*2}|J|_0^2+\frac{|J|_0}{n}\right).$\\
(ii)  $\frac{1}{T}\sum_t\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}^* \widehat f_i^*\|_2^2=O_{P^*}\left(\frac{1}{n}+\Delta_{F}^{*2}\right)$.\\
(iii) $ \|\frac{1}{nT}\sum_t\widehat U_{t,\widehat J}' (\tilde FH-\widehat F^{*})H^{*-1}\widehat\delta_{dt}\|_2=O_{P^*}(\Delta^*_{ud}+\Delta_F^{*2}+\Delta_F^*b_n^*)\sqrt{|J_0|},$ where \\
 $b_n^*=\sqrt{\frac{\log p}{n}\left( \frac{\log(n)\log(pT)}{n} +\Delta_{\max}^2+\frac{\log(pn)}{T} \right)}$.
\end{lemma}

 

\proof For notational simplicity, we assume $f_i$ is a scalar.

(i) We have $\frac{1}{T}\sum_t\|\frac{1}{n}\widehat F^{*'} \tilde U^{*'}_t\widehat\gamma\|_2^2\leq
 O_{P^*}(|J|_0^2\Delta_{F}^{*2})+ \frac{1}{T}\sum_t|\frac{1}{n}\sum_i\widehat f_i w_i^U\widehat U_{it}'\widehat\gamma|^2O_{P^*}(1).$  Hence, we need to bound $\frac{1}{T}\sum_t|\frac{1}{n}\sum_i\widehat f_i w_i^U\widehat U_{it}'\widehat\gamma|^2$.
\begin{align*}
E^*&\left[\frac{1}{T}\sum_t|\frac{1}{n}\sum_i\widehat f_i w_i^U\widehat U_{it}'\widehat\gamma|^2\right] \\
&= \frac{1}{T}\sum_t\frac{1}{n^2}\sum_i(\widehat f_i \widehat U_{it}'\widehat\gamma)^2 \\
&\leq \frac{2}{n^2T}\sum_t \sum_i(\widehat f_i \tilde U_{it}'\widehat\gamma)^2
+\frac{1}{n^2T}\sum_t \sum_i\widehat f_i^4\left( \sum_{m=1}^p   \tilde \lambda_{tm}  \widehat\gamma_m\right)^2O_P(\Delta_F^2) \\
&\qquad +\frac{1}{n^2T}\sum_t \sum_i\widehat f_i ^4\left(\frac{1}{n}\widehat F'\tilde U_{t}'\gamma\right)^2 +\frac{1}{nT}\sum_t  \left(  \sum_{m=1}^p \tilde\lambda_{tm}' \widehat\gamma_m\right)^2\max_i\widehat f_i^2O_P(\Delta_F^2) \\
&\leq \frac{4}{n^2T}\sum_t \sum_i(\widehat f_i \tilde U_{it}'(\widehat\gamma-\gamma))^2 
+ \frac{4}{n^2T}\sum_t \sum_i(\widehat f_i \tilde U_{it}'\gamma)^2 \\
&\qquad + \max_i|\widehat f_i|^2\frac{1}{n} O_P(\Delta_F^2|J|_0^2)
+ \max_i|\widehat f_i|^2\frac{1}{nT}\sum_t \left(\frac{1}{n}\widehat F'\tilde U_{t}'\gamma\right)^2 \\
&\leq \|\widehat\gamma-\gamma\|_1^2\max_i\widehat f_i^2\max_{mk}\frac{1}{n^2T}\sum_{it}|\tilde U_{it,m}\tilde U_{it,k}| + \frac{1}{n^2T}\sum_t \sum_i(\tilde f_i \tilde U_{it}'\gamma)^2 \\
&\qquad + \frac{8}{n^2T}\sum_t \sum_i(\widehat f_i -H'\tilde f_i)^2(\tilde U_{it}'\gamma)^2 
+ \max_i|\widehat f_i|^2\frac{1}{n} O_P(\Delta_F^2|J|_0^2) \\
&\qquad +\max_i|\widehat f_i|^2\frac{1}{nT}\sum_t \left(\frac{1}{n}\widehat F'\tilde U_{t}'\gamma\right)^2 \\
&\leq O_P\left(\frac{\log n}{n}\kappa_n^2|J|_0^2+\frac{|J|_0}{n}+\frac{\log n}{n}\Delta_F^2|J|_0^2+\frac{\log n|J|_0}{n^2}+\frac{\log n|J|_0^2}{n}\Delta_{F}^2 \right. \\
&\qquad\qquad \left. +\frac{|J|_0^2\Delta_F^2}{n}\left(\sqrt{\frac{\log n}{T}}+1\right)\right) \\
&= O_P\left(\frac{|J|_0}{n}\right),
\end{align*}
where we used the assumption $\max_k\sum_m|EU_{it,k}U_{it,m}|<C$ to bound $  \frac{1}{n^2T}\sum_t \sum_i(\tilde f_i \tilde U_{it}'\gamma)^2$, yielding its rate $O_P\left(\frac{|J|_0}{n}\right)$ instead of $O_P\left(\frac{|J|^2_0}{n}\right)$. Also,  
$$
\frac{1}{T}\sum_t\left\|\frac{1}{n}\widehat F'\tilde U_t\gamma\right\|_2^2= O_P\left(\frac{|J|_0}{n}+\Delta_F^2|J|_0^2\right)
$$
follows from Lemma \ref{la.1}. Finally,
$$
\frac{8}{n^2T}\sum_t \sum_i(\widehat f_i -H'\tilde f_i)^2(\tilde U_{it}'\gamma)^2=O_P\left(\frac{|J|_0^2\Delta_F^2}{n}\left(\sqrt{\frac{\log n}{T}}+1\right)\right)
$$
is due to  
\begin{align*}
\frac{1}{n^2T} \sum_i(\widehat f_i -H'\tilde f_i)^2\sum_t(\tilde U_{it}'\gamma)^2
&\leq O_P\left(\frac{1}{n}\Delta_F^2\right)\max_i\frac{1}{T}\sum_t(\tilde U_{it}'\gamma)^2 \\
&\leq O_P\left(\frac{1}{n}\Delta_F^2\right)\max_i\frac{1}{T}\sum_t(  U_{it}'\gamma)^2 \\
&\leq O_P\left(\frac{1}{n}\Delta_F^2\right)\max_i\left|\frac{1}{T}\sum_t(E(  U_{it}'\gamma)^2-(  U_{it}'\gamma)^2)\right| \\
&\qquad +O_P\left(\frac{1}{n}\Delta_F^2\right)\max_i\frac{1}{T}\sum_t E(  U_{it}'\gamma)^2 \\
&\leq O_P\left(\frac{|J|_0^2\Delta_F^2}{n}\left(\sqrt{\frac{\log n}{T}}+1\right)\right).
\end{align*}


(ii) By Lemma \ref{ld.1},  $\frac{1}{T}\sum_t\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}^* \widehat f_i\|_2^2=O_{P^*}(\frac{1}{n})$, which yields the result. 

 
(iii) By Assumption \ref{a6.2} and  (\ref{ed.18a}),
\begin{align*} 
\left\|\frac{1}{nT} \right. & \left. \sum_t\widehat U_{t,\widehat J^*}^{*'}(\tilde FH-\widehat F^{*})H^{*-1}\widehat\delta_{dt}\right\|_2 \\
&\leq O_{P^*}(\sqrt{|J|_0})\max_m \left|\frac{1}{n} \sum_{i=1}^n \frac{1}{T} \sum_{t=1}^T \tilde U_{it,m}^* \widehat\delta_{yt}'H^{*'-1}(\widehat f_i^*-H^{*'}\widehat f_i) \right| \\
&\qquad +O_{P^*}(\sqrt{|J|_0})\max_m\left|\frac{1}{n} \sum_{i=1}^n \frac{1}{T} \sum_{t=1}^T (\widehat U_{it,m}^* -\tilde U_{it,m}^*)\widehat\delta_{yt}'H^{*'-1}(\widehat f_i^*-H^{*'}\widehat f_i) \right| \\
&\leq O_{P^*}(\Delta^*_{ud}+\Delta_F^{*2}+\Delta_F^*b_n^*)\sqrt{|J_0|} 
\end{align*}
where $b_n^*=\sqrt{\frac{\log p}{n}\left( \frac{\log(n)\log(pT)}{n} +\Delta_{\max}^2+\frac{\log(pn)}{T} \right)}.$  
$\blacksquare$


In the following lemma, recall that $\tilde\gamma_y^*$ denotes the $k$-step lasso estimator, and $\widehat\gamma_y$ denotes the lasso estimator using the original data.


\begin{lemma}\label{le.1}   For $\widehat\gamma \in \{\widehat\gamma_y,\widehat\gamma_d,\theta\}$, \\
 (i)   $\left\|\frac{1}{\sqrt{nT}}M_{\widehat U^*_{\widehat J^*}} \tilde U^*\widehat\gamma\right\|_2^2=O_{P^*}\left(a_n+\kappa_n^2|J|_0+|J|_0^2\Delta_{F}^{*2}+\frac{|J|_0}{n}\right)$.
\\
 (ii) $\left\|\frac{1}{\sqrt{nT}} (I_T\otimes P_{\widehat F^*})\tilde U^*\widehat\gamma\right\|_2^2=O_{P^*}\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^{*2}\right).$
\\
(iii) $ \left\|  \frac{1}{\sqrt{nT}}  (I_T\otimes P_{\widehat F^*})\tilde\eta^*\right\|_2^2=O_{P^*}\left(\Delta_{F}^{*2}+\frac{1}{n}\right)$ and $\left\|  \frac{1}{\sqrt{nT}}  (I_T\otimes P_{\widehat F^*})\tilde\epsilon^*\right\|_2^2=O_{P^*}\left(\Delta_{F}^{*2}+\frac{1}{n}\right).$\\
(iv) $\left\|\frac{1}{\sqrt{nT}}P_{\widehat U^*_{\widehat J^*}} \tilde\eta^*\right\|_2^2=O_{P^*}\left(|J|_0\frac{\log p}{nT}\right)$ and $\left\|\frac{1}{\sqrt{nT}}P_{\widehat U^*_{\widehat J^*}} \tilde\epsilon^*\right\|_2^2=O_{P^*}\left(|J|_0\frac{\log p}{nT}\right)$.\\
(v) $\left\|\frac{1}{\sqrt{nT}}M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^*})\tilde U^*\widehat\gamma_d\right\|=O_{P^*}\left(a_n+\kappa_n^2|J|_0+|J|_0^2\Delta_{F}^{*2}+\frac{|J|_0}{n}\right)$.
%(v) $\frac{1}{T}\|P_{\widehat U_I}\eta\|_2^2=O_{P^*}(s^2\nu^2+s^2\frac{\Delta_F^4}{\nu^2})=\frac{1}{T}\|P_{\widehat U_I}\epsilon\|_2^2$.
\end{lemma}
 
\proof (i)  First, consider the following constrained problem:
\begin{equation*} 
\widehat m=\arg\min_m\|\widehat U^*(\widehat\gamma-m)\|_2^2, \quad m_j=0, j\notin\widehat J^*,
\end{equation*}
where $\widehat J^*$ is the support of the $k$-step lasso (instead of the support of the complete bootstrap lasso estimator $\tilde\gamma_{y,lasso}^*$).
The solution satisfies $\widehat U^*\widehat m=P_{\widehat U^*_{\widehat J^*}}\widehat U^*\widehat\gamma$.  For $\tilde\gamma^*$ being the $k$-step lasso (either $\tilde\gamma_y^*$ or $\tilde\gamma_d^*$), $\frac{1}{nT}\sum_{t=1}^T\|\widehat U_t ^* (\widehat\gamma-\tilde\gamma^*)\|_2^2=O_{P^*}(\kappa_n^2|J|_0+a_n)$ by Proposition \ref{pd.2}. Hence
$$\left\|\frac{1}{\sqrt{nT}}M_{\widehat U^*_{\widehat J^*}} \widehat U^*\widehat\gamma\right\|_2^2 =\left\|\frac{1}{\sqrt{nT}}\widehat  U^*(\widehat\gamma- \widehat m)\right\|_2^2\leq \left\|\frac{1}{\sqrt{nT}}\widehat  U^*(\widehat\gamma- \tilde\gamma^*)\right\|_2^2= O_{P^*}(a_n+\kappa_n^2|J|_0).
$$
Also, using $ \frac{1}{T}\sum_t\|\frac{1}{n}\widehat F^{*'} \tilde U^{*'}_t\widehat\gamma\|_2^2=
O_{P^*}\left(|J|_0^2\Delta_F^{*2}+\frac{|J|_0}{n}\right)$ from Lemma \ref{le.0} and
\begin{align*}
\tilde U^*_{it, m}-\widehat U_{it, m}^* &=  \widehat  f_i^{*'} (\widehat F^{*'}\widehat F^{*})^{-1}\widehat F^{*'}(\widehat FH^*-\widehat F^{*})H^{*-1}\widehat \lambda_{tm} + \widehat  f_i^{*'}(\widehat F^{*'}\widehat F^{*})^{-1}\widehat F^{*'} \tilde U^*_{t, m} \\
&\qquad +\widehat\lambda_{tm}'H^{*'-1}(\widehat  f_i^* -H^{*'}\widehat f_i),
\end{align*}
we have
\begin{align*} 
\left\|\frac{1}{\sqrt{nT}}M_{\widehat U^*_{\widehat J^*}} (\widehat U^*- \tilde U^*)\widehat\gamma\right\|_2^2 &\leq \frac{1}{nT}\left\|(\widehat U^*- \tilde U^*)\widehat\gamma\right\|_2^2\\ 
&= \frac{1}{nT}\sum_{t=1}^T\sum_i[\sum_m   (\widehat U_{it,m}^*- \tilde U^*_{it,m})   \widehat\gamma_m]^2 \\
&\leq \frac{1}{T}\sum_{t=1}^T \left[\sum_m \frac{1}{n}\widehat F^{*'} \tilde U^*_{t, m}  \widehat\gamma_m\right]^2O_{P^*}(1)+ \frac{1}{T}\sum_{t=1}^T[\sum_m  \widehat\lambda_{tm}'    \widehat\gamma_m]^2O_{P^*}(\Delta_F^{*2}) \\
&= O_{P^*}\left(|J|_0^2\Delta_F^{*2}+\frac{|J|_0}{n}\right).
\end{align*}  
 
Combining these yields $\left\|\frac{1}{\sqrt{nT}}M_{\widehat U^*_{\widehat J^*}}  \tilde U^*\widehat\gamma\right\|_2^2=O_{P^*}\left(a_n+\kappa_n^2|J|_0+|J|_0^2\Delta_{F}^{*2}+\frac{|J|_0}{n}\right)$.


(ii) Still by  by Lemma \ref{le.0}, 
\begin{align*}
\left\|\frac{1}{\sqrt{nT}} (I_T\otimes P_{\widehat F^*}) \tilde U^*\widehat\gamma\right\|_2^2 &= \frac{1}{nT}\sum_t\left\|P_{\widehat F^*} \tilde U^*_t\widehat\gamma\right\|_2^2 \\
&\leq \frac{1}{n^2T}\sum_t\left\|\widehat F^{*'} \tilde U^{*'}_t\widehat\gamma\right\|_2^2 \\
&=O_{P^*}\left(\frac{|J|_0}{n}+|J|_0^2\Delta_{F}^{*2}\right).
\end{align*}


(iii)  By Lemma \ref{le.0},  $\frac{1}{T}\sum_t\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}^* \widehat f_i\|_2^2=O_{P^*}(\frac{1}{n})$. Hence,
$$ \|  \frac{1}{\sqrt{nT}}  (I_T\otimes P_{\widehat F^*})\tilde\eta^*\|_2^2
=\frac{1}{nT}\sum_t\|P_{\widehat F^*}\tilde\eta^*_t\|_2^2\leq   \frac{1}{n^2T}\sum_t\|{\widehat F}^{*'}\tilde\eta^*_t\|_2^2=O_{P^*}\left(\Delta_{F}^{*2}+\frac{1}{n}\right).
$$
 
(iv) The same argument as used in the proof of Lemma \ref{ld.5new} yields
\begin{align*}
\max_m\left|\frac{1}{{nT}}\sum_t\sum_i \tilde U^*_{it,m}\tilde\eta^*_{it}\right| &=O_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right) \ \text{and} \\
\max_m\left|\frac{1}{{nT}}\sum_t\sum_i( \tilde U^*_{it,m}-\widehat U^*_{it,m})\tilde\eta^*_{it}\right| &=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right).
\end{align*}
Also, by Assumption \ref{a6.1}, $|\widehat J^*|=O_{P^*}(|J|_0)$.  Hence,
$$\left\|\frac{1}{{nT}}{\tilde U_{\widehat J^*}}^{*'}\tilde\eta^*\right\|_2=\left\|\frac{1}{{nT}}{\sum_t\tilde U^{*'}_{t,\widehat J^*}}\tilde\eta^*_t\right\|_2\leq  \max_m\left|\frac{1}{{nT}}\sum_t\sum_i \tilde U^*_{it,m}\tilde\eta^*_{it}\right|\sqrt{|\widehat J^*|_0}=O_{P^*}\left(\sqrt{|J|_0\frac{\log p}{nT}}\right)
$$
and 
$\|\frac{1}{{nT}}( \tilde U^*_{\widehat J^*}-\widehat U^*_{\widehat J^*})' \tilde\eta^*\|_2 
\leq \|\frac{1}{{nT}}\sum_t(\tilde U^{*'}_{t,\widehat J^*}-\widehat U^{*'}_{t,\widehat J^*})\tilde\eta^*_t\|_{\infty}\sqrt{|\widehat J|_0} =o_{P^*}\left(\sqrt{|J|_0\frac{\log p}{nT}}\right).$
We then have 
\begin{equation}\label{ee.6}
\left\|\frac{1}{{nT}}{\widehat U_{\widehat J^*}}^{*'} \tilde\eta^*\right\|_2=O_{P^*}\left(\sqrt{|J|_0\frac{\log p}{nT}}\right).
\end{equation}
Next, $\phi_{\min}(|\widehat J^*|)\leq\lambda_{\min}\left(\frac{1}{nT}\tilde U_{\widehat J^*}^{'}\tilde U_{\widehat J^*}\right)\leq \lambda_{\max}\left(\frac{1}{nT}\tilde U_{\widehat J^*}^{'}\tilde U_{\widehat J^*}\right) \leq \phi_{\max}(|\widehat J^*|)$. Also by  (\ref{ed.24}), 
$$
\left\|\frac{1}{nT}\widehat U_{\widehat J^*}^{*'}\widehat U_{\widehat J^*}^*- \frac{1}{nT}\tilde U_{\widehat J^*}^{'}\tilde U_{\widehat J^*}\right\|\leq
O_{P^*}(|J|_0)\max_{m,k\leq p}\left|\frac{1}{nT}\sum_{it}(\widehat U^*_{it,m}\widehat U^*_{it,k}-\tilde U_{it,m}\tilde U_{it,k})\right|
=o_{P^*}(1).
$$
This result implies
\begin{equation} \label{ee.5}
\phi_{\min}(|\widehat J^*|)-o_{P^*}(1)\leq\lambda_{\min}\left(\frac{1}{nT}\widehat U_{\widehat J^*}^{*'}\widehat U^*_{\widehat J^*}\right)\leq \lambda_{\max}\left(\frac{1}{nT}\widehat U_{\widehat J^*}^{*'}\widehat U^*_{\widehat J^*}\right)\leq\phi_{\max}(|\widehat J^*|)+o_{P^*}(1).
\end{equation}
Hence $\left\|\frac{1}{\sqrt{nT}}\widehat U^*_{\widehat J^*}\right\|^2=\lambda_{\max}\left(\frac{1}{nT}\widehat U_{\widehat J^*}^{*'}\widehat U^*_{\widehat J^*}\right)=O_{P^*}(1)$ and $\left\|(\frac{1}{nT}\widehat U_{\widehat J^*}^{*'}\widehat U_{\widehat J^*}^*)^{-1}\right\|=O_{P^*}(1)$.
% in the proof of (\ref{ea.0}), we showed
%$$\frac{1}{nT}\|\widehat U^*_{\widehat J^*}\|^2\leq 2\phi_{\max}(|\widehat J|_0)+ O_{P^*}(|\widehat J|_0\Delta_F^2+|\widehat J|_0\frac{\log (pT)}{n})=O_{P^*}(1),$$
%where the last equality follows from $|\widehat J|_0=O_{P^*}(|J|_0)$, $\phi_{\max}(| J|_0)=O_{P^*}(1)$ and the assumptions that $|J|_0\Delta_F^2$ and $|J|_0\frac{\log pT}{n}=O(1)$. Also, $\frac{1}{nT}\| \tilde U^*_{\widehat J^*}\|^2=O_{P^*}(1)$. Hence (\ref{ea.7}) implies
%\begin{eqnarray*}&&\|\frac{1}{nT}\widehat U_{\widehat J^*}^{*'}\widehat U_{\widehat J^*}^*-\frac{1}{nT}\tilde U_{\widehat J^*}^{*'} \tilde U^*_{\widehat J^*}\|\leq \|\frac{1}{\sqrt{nT}}(\widehat U_{\widehat J^*}^*- \tilde U^*_{\widehat J^*} )\|\frac{1}{\sqrt{nT}}(\|\widehat U_{\widehat J^*}^* \|+\| \tilde U^*_{\widehat J^*} \|)\cr&=&O_{P^*}(1)(|J|_0\max_m\frac{1}{{nT}}\sum_{it}( \tilde U^*_{it,m}-\widehat U_{it,m})^2)^{1/2} =o_{P^*}(1).\end{eqnarray*}
Finally,
$$
\left\|\frac{1}{\sqrt{nT}}P_{\widehat U_{\widehat J^*}^*} \tilde\eta^*\right\|_2 \leq \left\|\frac{1}{\sqrt{nT}}  \widehat U_{\widehat J^*}^* (\frac{1}{nT}\widehat U_{\widehat J^*} ^{*'}\widehat U_{\widehat J^*}^* )^{-1}\frac{1}{nT}\widehat U_{\widehat J^*} ^{*'}\tilde\eta^*\right\|_2
  =O_{P^*}\left(\sqrt{|J|_0\frac{\log p}{nT}}\right).
$$
  
(v) The result follows immediately from parts (i) and (ii) and the following inequality: 
$$
\left\|\frac{1}{\sqrt{nT}}M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^*})\tilde U^*\widehat\gamma_d\right\|_2 \leq \left\|\frac{1}{\sqrt{nT}}M_{\widehat U^*_{\widehat J^*}} \tilde U^*\widehat\gamma_d\right\|_2
+\left\|\frac{1}{\sqrt{nT}} (I_T\otimes P_{\widehat F^*})\tilde U^*\widehat\gamma_d\right\|_2.
$$ 

$\blacksquare$

\begin{lemma}\label{le.4}
%$\frac{1}{\sqrt{nT}} \tilde \epsilon'(I_T\otimes P_{\widehat F}) P_{\widehat U_{\widehat J}}(I_T\otimes M_{\widehat F}(\tilde FH-\widehat F)H^{-1}) \tilde\Delta_d =o_P(1)$\\
%(iii)$\frac{1}{\sqrt{nT}}\tilde\epsilon'P_{\widehat U_{\widehat J}}(I_T\otimes P_{\widehat F})\tilde U\gamma_d=o_P(1)$.\\
%(v)$\frac{1}{\sqrt{nT}}\tilde\epsilon'P_{\widehat U_{\widehat J}}(I_T\otimes P_{\widehat F})\tilde \eta=o_P(1)$, 
%\\(vi) $\frac{1}{\sqrt{nT}} \tilde \epsilon'(I_T\otimes P_{\widehat F}) P_{\widehat U_{\widehat J}} ( \tilde U_{\widehat J}-\widehat U_{\widehat J})\gamma_d=o_P(1)$. 
(i)  $\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^{*}}(\widehat FH^*-\widehat F^{*})H^{*-1}) \widehat\Delta_d=o_{P^*}(1)$.\\ 
(ii) $\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'} (I_T\otimes P_{\widehat F^{*}})\tilde U^{*}\widehat\gamma_d=o_{P^*}(1)$, \\
(iii)
$\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'} (I_T\otimes P_{\widehat F^{*}})\tilde \eta^*=o_{P^*}(1)$, \\
 (iv) $ \frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}M_{\widehat U^*_{\widehat J^*}} \tilde U^{*}\widehat\gamma_d=o_{P^*}(1)$.\\
(v) $\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^{*}})\tilde U^{*}\widehat\gamma_d=o_{P^*}(1)$.\\
(vi) $  \widehat\Xi'(I_T\otimes H^{*'-1}(\widehat FH^*-\widehat F^{*})'M_{\widehat F^{*}}) 
 \frac{1}{\sqrt{nT}}M_{\widehat U^*_{\widehat J^*}} \tilde U^{*}\widehat\gamma_d=o_{P^*}(1).$
\end{lemma}


\proof (i) First, by Lemma \ref{ld.1},  $\frac{1}{T}\sum_t\|\frac{1}{n}\sum_{i=1}^n \tilde \epsilon_{it}^* \widehat f_i\|_2^2=O_{P^*}\left(\frac{1}{n}\right)$. 
Thus,
\begin{align*} 
\left|\frac{1}{\sqrt{nT}} \right. & \left. \sum_t\tilde \epsilon_t^{*'} P_{\widehat F^{*}}(\widehat FH^*-\widehat F^{*})H^{*-1}\widehat\delta_{dt}\right|^2 \\
&=\frac{1}{nT}\tr^2\left(\sum_t\widehat\delta_{dt}\tilde \epsilon_t^{*'}P_{\widehat F^{*}}(\widehat FH^*-\widehat F^{*})H^{*-1}\right) \\
&\leq \frac{1}{nT}\left\|\sum_t\widehat\delta_{dt}\tilde \epsilon_t^{*'}\widehat F^{*}\right\|_F^2\left\|\frac{1}{n}\widehat F^{*'}(\widehat FH^*-\widehat F^{*})\right\|_F^2  O_{P^*}(1) \\
&=O_{P^*}( \Delta_{F}^{*2})\left\|\frac{1}{\sqrt{nT}}\sum_t\widehat\delta_{dt}\tilde \epsilon_t^{*'}(\widehat F^{*}-\widehat FH^*)\right\|_F^2 
+O_{P^*}( \Delta_{F}^{*2})\frac{1}{nT}\left\|\sum_t\widehat\delta_{dt}\tilde \epsilon_t^{*'}  \widehat F\right\|_F^2 \\
&=O_{P^*}( \Delta_{F}^{*4}nT)+O_{P^*}( \Delta_{F}^{*2}) nT\frac{1}{T}\sum_t\|   \frac{1}{n}\tilde \epsilon_t^{*'}  \widehat F\|_F^2 \\
&=o_{P^*}(1).
\end{align*}
By Lemma \ref{le.0}, 
$$ 
\left\|\frac{1}{nT}\sum_t\widehat U_{t,\widehat J}' (\tilde FH-\widehat F^{*})H^{*-1}\widehat\delta_{dt}\right\|_2=O_{P^*}\left(\Delta^*_{ud}+\Delta_F^{*2}+\Delta_F^*b_n^*\right)\sqrt{|J_0|},
$$
and 
$$ 
\left\|\frac{1}{\sqrt{nT}}\sum_t\widehat U_{t,\widehat J^*}^{*'}P_{\widehat F^{*}}(\widehat FH^*-\widehat F^{*})H^{*-1}\widehat\delta_{dt}\right\|_2=0
$$ 
due to $\widehat U_t'\widehat F^{*}=0$. Therefore,
\begin{align}
\nonumber
\left\|P_{\widehat U^*_{\widehat J^*}} \right. & \left. (I_T\otimes M_{\widehat F^{*}}(\widehat FH^*-\widehat F^{*})H^{*-1}) \widehat\Delta_d\right\|_2 \\
\nonumber
&= \left\|\widehat U^*_{\widehat J^*}(\widehat U_{\widehat J^*}^{*'}\widehat U^*_{\widehat J^*})^{-1}\sum_t\widehat U_{t,\widehat J^*}^{*'}M_{\widehat F^{*}}(\widehat FH^*-\widehat F^{*})H^{*-1}\widehat\delta_{dt}\right\|_2 \\
\nonumber
&\leq \left\|\frac{1}{\sqrt{nT}}\sum_t\widehat U_{t,\widehat J^*}^{*'}M_{\widehat F^{*}}(\widehat FH^*-\widehat F^{*})H^{*-1}\widehat\delta_{dt}\right\|_2 \\
\nonumber
&\leq \left\|\frac{1}{\sqrt{nT}}\sum_t\widehat U_{t,\widehat J^*}^{*'} (\widehat FH^*-\widehat F^{*})H^{*-1}\widehat\delta_{dt}\right\|_2 \\
%+\|\frac{1}{\sqrt{nT}}\sum_t\widehat U_{t,\widehat J^*}^{*'}P_{\widehat F^{*}}(\widehat FH^*-\widehat F^{*})H^{*-1}\widehat\delta_{dt}\|_2\cr
\label{ed.9a}
&=O_{P^*}\left(\Delta_{ud}^*+ \Delta_F^{*2}+ \Delta_{F}^{*} b_n^*\right)\sqrt{|J|_0nT}.
\end{align}
By Assumption \ref{a6.2}, $\|\frac{1}{\sqrt{nT}}\sum_t\widehat\delta_{dt}\tilde \epsilon_t^{*'}  (\widehat FH^*-\widehat F^{*})\|_F=o_{P^*}(1)$, and $\|\frac{1}{\sqrt{nT}}P_{\widehat U^*_{\widehat J^*}} \tilde\epsilon\|_2^2=O_{P^*}(|J|_0\frac{\log p}{nT})$ by Lemma \ref{le.1}.   Hence,
\begin{align*}
\frac{1}{\sqrt{nT}} & \tilde\epsilon^{*'}M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^{*}}(\widehat FH^*-\widehat F^{*})H^{*-1}) \widehat\Delta_d \\
&=-\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}P_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^{*}}(\widehat FH^*-\widehat F^{*})H^{*-1}) \widehat\Delta_d	\\
&\qquad + \frac{1}{\sqrt{nT}}\tilde\epsilon^{*'} (I_T\otimes  (\widehat FH^*-\widehat F^{*})H^{*-1}) \widehat\Delta_d	\\
&\qquad -\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'} (I_T\otimes P_{\widehat F^{*}}(\widehat FH^*-\widehat F^{*})H^{*-1}) \widehat\Delta_d \\
&\leq  \left\|\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}P_{\widehat U^*_{\widehat J^*}}\right\|_2 \left\|P_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^{*}}(\widehat FH^*-\widehat F^{*})H^{*-1}) \widehat\Delta_d\right\|_2	\\
&\qquad + \left|\frac{1}{\sqrt{nT}}\sum_t\tilde \epsilon_t^{*'} P_{\widehat F^{*}}(\widehat FH^*-\widehat F^{*})H^{-1}\tilde \delta_{dt}\right| \\
&\qquad + \left|\frac{1}{\sqrt{nT}}\sum_t\tilde \epsilon_t^{*'}  (\widehat FH^*-\widehat F^{*})H^{-1}\widehat\delta_{dt}\right| \\
&= o_{P^*}(1)+O_{P^*}\left(\sqrt{|J|_0\frac{\log p}{nT}}\right)\left(\Delta_{ud}^*+ \Delta_F^{*2}+ \Delta_{F}^{*} b_n^*\right)\sqrt{|J|_0nT} \\
&= o_{P^*}(1)
 \end{align*}
where we have used the assumption that $(\Delta_{ud}^*+ \Delta_F^{*2}+ \Delta_{F}^{*} b_n^*)\sqrt{\log p|J|_0^2}=o(1)$. 
 


(ii)  %Note that  $$
%\frac{1}{T} \sum_{t=1}^T\tilde U^{*}_{jt}\tilde\epsilon_{it}
 % =   \frac{1}{T}  \sum_{t=1}^T U_{jt}\epsilon_{it}
  %-\bar U_{j\cdot}\bar \epsilon_{i\cdot}
%- \frac{1}{T} \sum_{t=1}^T U_{jt} \bar \epsilon_{\cdot t}+\bar U_{j\cdot}\bar{\bar\epsilon}+\bar{\bar  U}\bar \epsilon_{i\cdot}- \bar{\bar  U}\bar{\bar\epsilon}-  \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t}\epsilon_{it} + \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t} \bar\epsilon_{\cdot t}.  $$
By the Cauchy-Schwarz inequality and Lemma \ref{le.0},
\begin{align*}%\label{eedd}
\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'} (I_T\otimes P_{\widehat F^{*}})\tilde U^{*}\widehat\gamma_d &=\frac{1}{\sqrt{nT}}\sum_t\tilde \epsilon_t^{*'}P_{\widehat F^{*}}\tilde U^{*}_t\widehat\gamma_d \\
&\leq \left\|\frac{\sqrt{nT}}{n^2}\sum_{m=1}^p\frac{1}{T}\sum_t\widehat F^{*'}\tilde U^{*}_{t,m}\tilde \epsilon_t^{*'}\widehat F^{*}\widehat\gamma_{dm}\right\|_F \\
&\leq \sqrt{nT}\left(\frac{1}{T}\sum_t(\frac{1}{n}\widehat F^{*'}\tilde U^{*'}_{t}\widehat\gamma_{d})^2\right)^{1/2}\left(\frac{1}{T}\sum_t\left(\frac{1}{n}\tilde \epsilon_t^{*'}\widehat F^{*}\right)^2\right)^{1/2} \\
&=O_{P^*}\left(\sqrt{nT}\left(\Delta_{F}^{*}|J|_0+\sqrt{\frac{|J|_0}{n}}\right)\right)\left(\frac{1}{\sqrt{n}}+\Delta_{F}^{*}\right) \\
&=o_{P^*}(1).
\end{align*}
%where  each $a_l$ is defined and   bounded below.

  
 
%\begin{eqnarray*}
%a_1&=&\|\frac{\sqrt{nT}}{n^2}\sum_{m=1}^p(\widehat F^{*}-\widehat FH^*)'\frac{1}{T}\sum_t\tilde U^{*}_{t,m}\tilde \epsilon_t^{*'}(\widehat F^{*}-\widehat FH^*)\widehat\gamma_{dm}\|_F\cr
%&\leq& O_{P^*}(\sqrt{nT}\Delta_{F}^{*2}) (\frac{1}{nT}\sum_t\|\tilde U_t^{*'}\widehat\gamma_d\|_2^2\frac{1}{nT}\sum_s\|\tilde\epsilon_s^*\|_2^2)^{1/2}=O_{P^*}(|J|_0\sqrt{nT}\Delta_{F}^{*2}) =o_{P^*}(1).
%\cr
%a_2&=&\|\frac{\sqrt{nT}}{n^2}\sum_{m=1}^p(\widehat F^{*}-\widehat FH^*)'\frac{1}{T}\sum_t\tilde U^{*}_{t,m}\tilde \epsilon_t^{*'}\widehat F\widehat\gamma_{dm}\|_FO_{P^*}(1)
%\leq \| \frac{1}{T}\sum_t\tilde U^{*'}_{t}\widehat\gamma_d\frac{1}{n}\tilde \epsilon_t^{*'}\widehat F\|_FO_{P^*}(\Delta_F^*\sqrt{T})\cr
%&\leq&(\frac{1}{T}\sum_t\|\tilde U^{*'}_{t}\widehat\gamma_d\|_2^2)^{1/2}(\frac{1}{T}\sum_t(\frac{1}{n}\tilde \epsilon_t^{*'}\widehat F)^2)^{1/2}O_{P^*}(\Delta_F^*\sqrt{T})
%=O_{P^*}(\frac{\Delta_F^*|J|_0\sqrt{nT}}{\sqrt{n}})=o_{P^*}(1) (\text{Lemma \ref{ld.1}})
%\cr
%a_3&=&\|\frac{\sqrt{nT}}{n^2}\sum_{m=1}^p\widehat F'\frac{1}{T}\sum_t\tilde U^{*}_{t,m}\tilde \epsilon_t^{*'}(\widehat F^*-\widehat FH^*)\widehat\gamma_{dm}\|_FO_{P^*}(1)
%\leq \|\frac{\sqrt{T}}{n}\widehat F'\frac{1}{T}\sum_t\tilde U^{*'}_{t}\widehat\gamma_{d}\tilde \epsilon_t^{*'} \|_FO_{P^*}(\Delta_F^*)\cr
%&\leq& 
 %(\frac{1}{T}\sum_t\|\frac{1}{n}\widehat F'\tilde U^{*'}_{t}\widehat\gamma_{d}\|_2^2)^{1/2}(\frac{1}{T}\sum_t\|\tilde \epsilon_t^*\|_2^2)^{1/2}O_{P^*}(\Delta_F^*\sqrt{T})
 %\leq O_{P^*}(\Delta_{F}^{*}|J|_0+\sqrt{\frac{|J|_0}{n}})\Delta_F^*\sqrt{Tn}\cr
% &=&o_{P^*}(1)\quad (\text{ Lemma \ref{le.0}})\cr
%a_4&=&\|\frac{\sqrt{nT}}{n^2}\sum_{m=1}^p\frac{1}{T}\sum_t\widehat F'\tilde U^{*}_{t,m}\tilde \epsilon_t^{*'}\widehat F\widehat\gamma_{dm}\|_FO_{P^*}(1)
%\leq \sqrt{nT} (\frac{1}{T}\sum_t(\widehat F'\tilde U^{*'}_{t}\widehat\gamma_{d})^2)^{1/2}(\frac{1}{T}\sum_t()^2)^{1/2}
 %\cr
%\end{eqnarray*}
 
(iii) It can be shown that
$$
\frac{1}{\sqrt{nT}}  \tilde\epsilon^{*'} (I_T\otimes P_{\widehat F^{*}})\tilde\eta^* \leq 
\left\|\frac{\sqrt{T}}{n\sqrt{n}}\widehat F^{*'}\frac{1}{T}\sum_t\tilde \eta^*_{t}\tilde \epsilon_t^{*'}\widehat F^{*} \right\|_F \leq O_{P^*}\left(\sqrt{nT} \left(\frac{1}{\sqrt{n}}+\Delta_{F}^{*}\right)^2\right) = o_{P^*}(1).
$$
  
   
  
(iv) Recall that there is $\widehat m$ such that $\widehat U^*\widehat m=P_{\widehat U^*_{\widehat J^*}}\widehat U^*\widehat\gamma_d$. We first bound $\|\widehat\gamma_d-\widehat m\|_1$. By Proposition \ref{pd.2},  
$$ 
\frac{1}{nT}\|\widehat U^*(\widehat\gamma_d- \widehat m)\|_2^2\leq \frac{1}{nT}\sum_{t=1}^T\|\widehat U^*_t(\widehat\gamma_d- \tilde\gamma^*)\|_2^2= O_{P^*}(a_n+\kappa_n^2|J|_0).
$$
Also, $\widehat\gamma_d-\widehat m$ is at most $|\widehat J|_0+|\widehat J^*|_0$-sparse.  Hence, by Assumption \ref{a3.4} and using arguments similar to those used in the proof of (\ref{ee.5}),  
$$
\frac{1}{nT}\|\widehat U^*(\widehat\gamma_d-\widehat m)\|_2^2\geq(\phi_{\min}(|\widehat J|_0+|\widehat J^*|_0)-o_{P^*}(1))\|\widehat\gamma_d-\widehat m\|_2^2\geq C|J|_0^{-1}\|\widehat\gamma_d-\widehat m\|_1^2.
$$
Hence, $\|\widehat\gamma_d-\widehat m\|_1^2=O_{P^*}(a_n+\kappa_n^2|J|_0)|J|_0.$
 
In addition, similar to the proof of Lemma \ref{ld.5new}, $\left\|\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}  \widehat U^*\right\|_{\infty} = O_{P^*}(\sqrt{\log p}).$
Hence,
\begin{align*} 
\left|\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}M_{\widehat U^*_{\widehat J^*}} \widehat U^*\widehat\gamma_d\right| 
&= \left|\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}  \widehat U^*(\widehat\gamma_d-\widehat m)\right| \\
&\leq \left\|\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}  \widehat U^*\|_{\infty}\|\widehat\gamma_d-\widehat m\right\|_1 \\
&=O_{P^*}\left(a_n^{1/2}|J|_0^{1/2}+\kappa_n|J|_0 \right)\sqrt{\log p}=o_{P^*}(1)
\end{align*}
under the assumption $a_n|J|_0\log p=o(1)$.

In addition,
\begin{align}
\nonumber
\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}M_{\widehat U^*_{\widehat J^*}} (\tilde U^{*}-\widehat U^*)\widehat\gamma_d &= \frac{1}{\sqrt{nT}}\tilde\epsilon^{*'} (\tilde U^{*}-\widehat U^*)\widehat\gamma_d- \frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}P_{\widehat U^*_{\widehat J^*}} (\tilde U^{*}-\widehat U^*)\widehat\gamma_d \\
\nonumber
&=\frac{1}{\sqrt{nT}}\sum_{i}\sum_t\sum_m\tilde\epsilon_{it}^*(\tilde U^{*}_{it,m}-\widehat U^*_{it,m})\widehat\gamma_{dm} \\
\nonumber
& \qquad -\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}  \widehat U^*_{\widehat J^*} ( \widehat U_{\widehat J^*}^{*'} \widehat U^*_{\widehat J^*})^{-1} \widehat U_{\widehat J^*}^{*'}(\tilde U^{*}-\widehat U^*)\widehat\gamma_d \\
\label{ee.7}
\begin{split}
&\leq \left\| \frac{1}{\sqrt{nT}}\sum_{i}\sum_m\sum_t\tilde\epsilon_{it}^* \widehat \gamma_{dm} \widehat\lambda_{tm}\widehat f_i'  \right\|_F  O_{P^*}(\Delta_{F}^{*}) \\
&\qquad + \left\| \frac{1}{\sqrt{nT}}\sum_{i}\sum_t\sum_m\tilde\epsilon_{it}^*  \widehat  f_i		 \frac{1}{n}\widehat F^{*'}\tilde U^{*}_{t, m}\widehat\gamma_{dm}\right\|_FO_{P^*}(1) \\
&\qquad + \left\| \frac{1}{\sqrt{nT}}\sum_{i}\sum_t\sum_m\tilde\epsilon_{it}^*\widehat\gamma_{dm}  \widehat\lambda_{tm} (\widehat  f_i ^*-H{^*}\widehat f_i)'\right\| O_{P^*}(1) \\
&\qquad + \left\|  \frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}  \widehat U^*_{\widehat J^*} \|_2\|\frac{1}{nT}   \widehat U_{\widehat J^*}^{*'}(\tilde U^{*}-\widehat U^*)\widehat\gamma_d\right\|_2 O_{P^*}(1).
\end{split}
\end{align}
The first term on the right of the inequality in (\ref{ee.7}) is $O_{P^*}(|J|_0)\Delta_{F}^{*}=o_{P^*}(1)$.  By part (ii), the second term on the right of the inequality in (\ref{ee.7}) is $ o_{P^*}(1)$, and the third term is $O_{P^*}(\Delta_{eg}^*\sqrt{nT})=o_{P^*}(1)$ by Assumption \ref{a6.2}.   For the last term on the right of the inequality in (\ref{ee.7}),  we have  $\left\|\frac{1}{\sqrt{nT}}{\widehat U_{\widehat J^*}}^{*'}\tilde\epsilon^*\right\|_2=O_{P^*}(\sqrt{|J|_0 \log p}).$  Also, by (\ref{ed.13})
\begin{align*}  
\frac{1}{nT}\|\widehat U_{\widehat J^*}^{*'}( \tilde U^{*}-\widehat U^* )\widehat\gamma_d\|_2 &\leq \sqrt{|\widehat J|_0}\max_k\left|\frac{1}{nT}\sum_i\sum_t\sum_{m}\widehat U^*_{it, k}(\tilde U^{*}_{it,m}-\widehat U^*_{it,m})\widehat\gamma_{dm} \right|  \\
&\leq  O_{P^*}(|J|_0^{1/2})\left( \Delta_F^{*2}|J|_0+\Delta_{ud}^*|J|_0+|J|_0\Delta_F^*b_n+|J|_0\frac{\log(pT)}{n}\right):=O_{P^*}(c_n),
\end{align*}
where $b_n=\left(\sqrt{\frac{\log p}{n}}\right)\sqrt{\frac{ \log(n)\log(pT)}{n}+ \Delta_{\max}^2+\frac{\log(pn)}{T}}$.
Therefore, the last term of (\ref{ee.7}) is bounded by
$O_{P^*}(\sqrt{|J|_0 \log p})c_n=o_{P^*}(1).$
Thus, both $  \frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}M_{\widehat U^*_{\widehat J^*}} (\tilde U^{*}-\widehat U^*)\widehat\gamma_d$
and $  \frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}M_{\widehat U^*_{\widehat J^*}} \widehat U^*\widehat\gamma_d$ are $o_{P^*}(1)$.



(v) Since $\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}P_{\widehat U^*_{\widehat J^*}}(I_T\otimes P_{\widehat F^{*}})\tilde U^{*}\widehat\gamma_d=0$, it follows from parts (ii) and (iv) that 
\begin{align*}
\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}M_{\widehat U^*_{\widehat J^*}}(I_T\otimes M_{\widehat F^{*}})\tilde U^{*}\widehat\gamma_d=
 \frac{1}{\sqrt{nT}}\tilde\epsilon^{*'}M_{\widehat U^*_{\widehat J^*}} \tilde U^{*}\widehat\gamma_d
 -\frac{1}{\sqrt{nT}}\tilde\epsilon^{*'} (I_T\otimes P_{\widehat F^{*}})\tilde U^{*}\widehat\gamma_d=o_{P^*}(1).
\end{align*}
  
   
(vi) By Lemma \ref{le.1}, 
$\left\|\frac{1}{\sqrt{nT}}M_{\widehat U^*_{\widehat J^*}} \tilde U^*\widehat\gamma\right\|_2^2=O_{P^*}\left(a_n+\kappa_n^2|J|_0+|J|_0^2\Delta_{F}^{*2}+\frac{|J|_0}{n}\right)$.
Thus, 
\begin{align*} 
\left|\widehat\Xi' \right. & \left. (I_T\otimes H^{*'-1}(\widehat FH^*-\widehat F^{*})'M_{\widehat F^{*}}) 
\frac{1}{\sqrt{nT}}M_{\widehat U^*_{\widehat J^*}} \tilde U^{*}\widehat\gamma_d\right|^2 \\
&\leq O_{P^*}({T}) \left\|\widehat F^{*}-\widehat FH^*\right\|_F^2 
\left\|\frac{1}{\sqrt{nT}}M_{\widehat U^*_{\widehat J^*}} \tilde U^{*}\widehat\gamma_d\right\|_2^2 \\
&= O_{P^*}\left(nT\Delta_F^{*2}\right)\left(\kappa_n^2|J|_0+ a_n+\Delta_F^{*2}|J|_0^2+  \frac{|J|_0}{n}\right)=o_{P^*}(1).
\end{align*}
$\blacksquare$




\subsection{Technical lemmas for Proposition \ref{p2.1}}
 
\begin{lemma}   
$\kappa_n^2|J|_0\sqrt{nT}=o(1).$
\end{lemma}
 
\proof 
Note that $\kappa_n^2=O(\log(pq_n^{-1})/(nT))$. To prove $ \kappa_n^2|J|_0\sqrt{nT}=o(1)$, it thus suffices to show $|J|_0^2\log ^2p=o(nT)$ and $|J|_0^2\log ^2\left(\frac{1}{q_n}\right)=o(nT)$, which is guaranteed by our assumptions. $\blacksquare$

 


\begin{lemma}\label{la0}
For all $t$, the $K\times p$ matrix $\widehat F'\widehat U_t$ satisfies $\widehat F'\widehat U_t=0$.  Also, $   P_{\widehat U_{\widehat J}}(I_T\otimes P_{\widehat F})=0$, and $\widehat U_{\widehat J}'(I_T\otimes P_{\widehat F})=0$.
\end{lemma}
  
  
\proof 
Note that the $n\times p$ matrix  $\widehat U_t=\tilde X_t- \widehat F(\widehat F'\widehat F)^{-1}\widehat F'\tilde X_t$. Thus, $\widehat F'\widehat U_t=\widehat F'M_{\widehat F}\tilde X_t=0$.
 Straightforward calculations yield $\widehat U_{\widehat J}'(I_T\otimes P_{\widehat F})=\sum_t\widehat U_{t,\widehat J}'\widehat F(\widehat F'\widehat F)^{-1}\widehat F'=0$ since $\widehat U_t'\widehat F=0$. It also follows that $ P_{\widehat U_{\widehat J}}(I_T\otimes P_{\widehat F})=0$. $\blacksquare$

\begin{lemma}\label{la.1} 
For $\gamma =\gamma_{y}$ or $\gamma = \gamma_{d}$,\\
(i) Both  
$$
\max_{t\leq T, m\leq p}\|\frac{1}{n}\sum_{i=1}^n\widehat f_i \tilde U_{it,m}\|_2 = O_P\left(\sqrt{\frac{\log (pT)}{n}}+\Delta_{fum}\right)
$$
and
$$
\max_{m\leq p} \left(\frac{1}{n^2T}\sum_t \|\tilde U_{t,m}'\widehat F   \|_2^2\right)^{1/2} = O_P\left(\sqrt{\frac{\log (pT)}{n}}+\Delta_{fum}\right).
$$
\\
(ii) $ \max_{m\leq p} \|\frac{1}{nT}\sum_{t=1}^T\widehat F'\tilde  U_t \gamma     \tilde\lambda_{tm}'\|_F = O_P\left(\sqrt{\frac{|J|_0\log p}{nT}}+\Delta_F|J|_0\right)$. 
\\
(iii) Both
$$  
\max_{m\leq p} \|\frac{1}{nT}\sum_{t=1}^T\widehat F'\tilde  e_t \tilde\lambda_{tm}'\|_F = O_P\left(\Delta_{eg}+\sqrt{\frac{\log p}{nT}}\right)
$$
and
$$
\max_m \|\frac{1}{T} \sum_{t=1}^T \frac{1}{n}\widehat F'\tilde U_{t, m} \tilde\delta_{yt}' \|_F =
 O_P\left(\Delta_{ud}+\sqrt{\frac{\log p}{nT}}\right).
$$
\\
(iv) We have 
$$
\frac{1}{T}\sum_t\|\frac{1}{n}\widehat F'\tilde e_t\|_2^2 = O_P\left(\frac{1}{n}+\Delta_{fe}^2\right),
$$ 
$$
\frac{1}{T}\sum_t\|\frac{1}{n}\widehat F'\tilde \eta_t\|_2^2 = O_P\left(\frac{1}{n}+\Delta_{fe}^2\right),
$$ 
and 
$$
\frac{1}{T}\sum_t\|\frac{1}{n}\widehat F'\tilde U_t\gamma\|_2^2 = O_P\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^2\right).
$$
\\
(v) $\max_{m\leq p} \|\frac{1}{nT}\sum_{t=1}^T\widehat F'\tilde  U_{t,m} \gamma' \tilde\Lambda_{t}\|_F = O_P\left(\Delta_F|J|_0+|J|_0\sqrt{\frac{\log p}{nT}}\right).$
\\
(vi) $\max_{m,l\leq p}|\frac{1}{nT} \sum_{it}\tilde U_{it,l}   \widehat  f_i'  \tilde \lambda_{tm} | = O_P\left(\Delta_{ud}+\sqrt{\frac{\log p}{nT}}\right)$.
\end{lemma}


\proof   For notational simplicity, we assume $f_i$ to be a scalar without loss of generality. 
 
(i) First, $\frac{1}{n}\sum_{i=1}^n \tilde f_i\tilde U_{it, m}=\frac{1}{n}\sum_{i=1}^n  f_i U_{it, m} - \bar f\bar U_{\cdot t, m}-\frac{1}{n}\sum_if_i\bar U_{i\cdot, m}+\bar f\bar{\bar U}_m$. Then 
$$
\max_{t\leq T, m\leq p}\|\frac{1}{n}\sum_{i=1}^n \tilde f_i\tilde U_{it, m}\|_2 = O_P\left( \sqrt{\frac{\log (pT)}{n}}\right).
$$
Thus, $\max_{t\leq T, m\leq p}\|\frac{1}{n}\sum_{i=1}^n \widehat f_i\tilde U_{it, m}\|_2 = O_P\left(\sqrt{\frac{\log (pT)}{n}}+\Delta_{fum}\right)$. Also,
$$
\max_m \frac{1}{n^2T}\sum_t \|\tilde U_{t,m}' \widehat F\|_2^2 = \max_m \frac{1}{T}\sum_t\|\frac{1}{n}\sum_i\tilde U_{it,m}\widehat f_i\|_2^2 = O_P\left( {\frac{\log (pT)}{n}}+\Delta^2_{fum}\right).
$$

(ii) We first bound $\max_m \|\frac{1}{nT}\sum_{t=1}^T\tilde F'\tilde  U_t \gamma     \tilde\lambda_{tm}'\|_F$. 
\begin{align}
\nonumber
\max_m  \left\|\frac{1}{nT}\sum_{t=1}^T\tilde F'\tilde  U_t \gamma  \tilde\lambda_{tm}' \right\|_F &=
\max_m  \left\|\frac{1}{nT}\sum_{t=1}^T\sum_i f_i\tilde  U_{it}' \gamma  \tilde\lambda_{tm}'\right\|_F \\
\label{la.1.ii.1}
&\leq |\bar{\bar U}'\gamma |O_P(1) \\
\label{la.1.ii.2}
& \qquad + \max_m \left\|\frac{1}{nT}\sum_{t=1}^T\sum_i f_i   U_{it} '\gamma \tilde\lambda_{tm}'\right\|_F \\
\label{la.1.ii.3}
& \qquad + O_P(1) \max_m \left\|\frac{1}{T}\sum_{t=1}^T\bar U_{\cdot t}' \gamma\tilde\lambda_{tm}'\right\|_2 \\
\label{la.1.ii.4}
& \qquad + O_P(1) \left\|\frac{1}{n}\sum_i f_i\bar U_{i\cdot} '\gamma \right\|_2.
\end{align}
For term (\ref{la.1.ii.1}), note
$$
E |\bar{\bar U}'\gamma    |^2 \leq \frac{1}{nT}\|\gamma\|_1    \max_{i,t,m} \sum_{s=1}^T\sum_{k=1}^p|EU_{it,m} U_{is,k}| = O\left(\frac{|J|_0}{nT}\right).
$$ 
Hence, $| \bar{\bar U}'\gamma | = O_P\left(\sqrt{\frac{|J|_0}{nT}}\right)$. 


For term (\ref{la.1.ii.2}), let $W_{im}=\frac{1}{T}\sum_t f_i  U_{it}'\gamma\tilde \lambda_{tm}$. Note that the $W_{im}$'s are independent across $i$ and $E(W_{im})=0$. By assumption, $\max_{l,t}\sum_{k=1}^p\sum_s|E( U_{it,k}\  U_{is,l}|F)|=O(1)$; so 
 $$
 \max_{im}\Var(W_{im})
\leq \max_{im}\frac{1}{T}   \|\gamma\|_1 Ef_i  ^2\max_{l,t}\sum_{k=1}^p\sum_s|E( U_{it,k}\  U_{is,l}|F)|
=O\left(\frac{|J|_0}{T}\right).
$$ 
Hence $\max_m|\frac{1}{n}\sum_iW_{im}| = O_P\left(\sqrt{\frac{|J|_0\log p}{nT}}\right)$.
 
For term (\ref{la.1.ii.3}), let  $Z_{im}=\frac{1}{T}\sum_tU_{it}'\gamma\tilde\lambda_{tm}$. Then $\max_{im}\Var(Z_{im})=O\left(\frac{|J|_0}{T}\right)$, and
$$
\max_m  \|  \frac{1}{T}\sum_{t=1}^T\bar U_{\cdot t}' \gamma     \tilde\lambda_{tm}'\|_2
  =\max_m\|\frac{1}{n}\sum_iZ_{im}\|_2=O_P(\sqrt{\frac{|J|_0\log p}{nT}}).
$$ 

Finally, 
$$ E  |  \frac{1}{n}\sum_i f_i\bar U_{i\cdot} '\gamma |^2 \leq \frac{1}{nT} \|\gamma\|_1 Ef_i ^2\max_{imt}\sum_s \sum_{k=1}^p|E(U_{is,k} U_{it,m}|F)|.
$$
Hence, $\| \frac{1}{n}\sum_i f_i\bar U_{i\cdot} '\gamma \|_2 = O_P\left(\sqrt{\frac{|J|_0}{nT}}\right)$. Combining the four terms above, we have 
$$
\max_m  \|  \frac{1}{nT}\sum_{t=1}^T\tilde F'\tilde  U_t \gamma \tilde\lambda_{tm}'\|_F = O_P\left(\sqrt{\frac{|J|_0\log p}{nT}}\right).
$$

Next, $\frac{1}{n}\sum_i|\frac{1}{T}\sum_t\tilde U_{it}'\gamma|^2\max_m\|\tilde\lambda_m\|_2^2=O_P(|J|_0^2)$, so
\begin{align*}
\max_m  \| \frac{1}{nT}\sum_{t=1}^T(\tilde FH-\widehat F)'\tilde  U_t \gamma \tilde\lambda_{tm}'\|^2_F
&\leq \frac{1}{n}\sum_i\|\widehat  f_i -H'\tilde f_i\|_2^2\frac{1}{n}\sum_i|\frac{1}{T}\sum_t\tilde U_{it}'\gamma|^2\max_m\|\tilde\lambda_m\|_2^2 \\
&= O_P(\Delta_F^2|J|_0^2).
\end{align*}
Hence, we  have $\max_m  \| \frac{1}{nT}\sum_{t=1}^T\widehat F'\tilde  U_t \gamma \tilde\lambda_{tm}'\|_F = O_P\left(\sqrt{\frac{|J|_0\log p}{nT}}+\Delta_F|J|_0\right).$


   
(iii)   First of all,  note that $\frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n \tilde  e_{it} \tilde\lambda_{tm}'=0$ and $\frac{1}{T}\sum_t\tilde\lambda_{tm}=0$, so
\begin{align*}
 \max_{m\leq p}& \|\frac{1}{nT}\sum_{t=1}^T\tilde F'\tilde  e_t \tilde\lambda_{tm}'\|_F= \max_{m\leq p} \|\frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n\tilde f_i\tilde  e_{it} \tilde\lambda_{tm}'\|_F= \max_{m\leq p} \|\frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n f_i\tilde  e_{it} \tilde\lambda_{tm}'\|_F
 \\
 &\leq  \max_{m\leq p} \|\frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n f_i  e_{it} \tilde\lambda_{tm}'\|_F
 + \max_{m\leq p} \|\frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n f_i\bar e_{i\cdot} \tilde\lambda_{tm}'\|_F\cr
 &\qquad + \max_{m\leq p} \|\frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n f_i\bar  e_{\cdot t} \tilde\lambda_{tm}'\|_F
 +  \max_{m\leq p} \|\frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n f_i  \bar {\bar e} \tilde\lambda_{tm}'\|_F\cr
 &\leq  \max_{m\leq p} \|\frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n f_i  e_{it} \tilde\lambda_{tm}'\|_F
+ \max_{m\leq p} \|\frac{1}{ T}\sum_{t=1}^T \bar  e_{\cdot t} \tilde\lambda_{tm}'\|_2O_P(1).
\end{align*}
Fix an element $k$ of $\tilde\lambda_{tm}$, let $W_{im,k}=\frac{1}{T}\sum_{t=1}^T  f_i  e_{it} \tilde\lambda_{tm,k}$. Then $EW_{im,k}=0$, and $\{W_{im,k}\}$'s are independent across $i\leq n$; $\max_{imk}\Var(W_{im,k})=O(\frac{1}{T})$. In addition, by Assumption \ref{a3.2}, $W_{im,k}$ satisfies the exponential-tail condition. Also, $K=O(1)$, hence we can ignore $\max_{k\leq K}$ in the follows by bounding it for each fixed $k$. By the Bernstein inequality for independent data, (e.g., \cite{bulmann}) we reach 
$$
\max_{m\leq p} \|\frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n f_i  e_{it} \tilde\lambda_{tm}'\|_F\leq K\max_{k\leq K}\max_{m\leq p}|\frac{1}{n}\sum_iW_{im,k}|=O_P(\sqrt{\frac{\log p}{nT}}).
$$
More specifically, by Assumption \ref{a3.2} (iv), $W_{im,k}$ has exponential tails, then we can apply the Bernstein inequality for independent data to reach  ($K=O(1)$), for $y=M\sqrt{\frac{\log p}{nT}}$ and sufficiently large $M>0$, 
\begin{align*}
P&(\max_{k\leq K}\max_{m\leq p}|\frac{1}{n}\sum_iW_{im,k}|>y)\leq Kp\max_{k\leq K}\max_{m\leq p}P(|\frac{1}{n}\sum_iW_{im,k}|>y)\cr
&\leq \exp\left(\log(Kp) -\frac{ny^2}{ \max_{imk}   \Var(W_{im,k})  }  \right)
\leq \exp\left(\log(Kp) -CTny^2 \right) =o(1).
\end{align*}
In addition, let $Z_{im,k}=\frac{1}{ T}\sum_{t=1}^T  e_{it} \tilde\lambda_{tm,k}$. Then $\max_{ikm}\Var(Z_{imk})=O(\frac{1}{T})$. Hence
$$
\max_{m\leq p} \|\frac{1}{ T}\sum_{t=1}^T \bar  e_{\cdot t} \tilde\lambda_{tm}'\|_2=
\max_{m\leq p} \|\frac{1}{ Tn}\sum_{t=1}^T\sum_{i=1}^n   e_{it} \tilde\lambda_{tm}'\|_2\leq \sqrt{K}\max_k\max_{m\leq p}|\frac{1}{n}\sum_{i=1}^nZ_{im,k}|=O_P(\sqrt{\frac{\log p}{nT}}).
$$
 This implies
 $$
 \max_{m\leq p}  \|\frac{1}{nT}\sum_{t=1}^T\tilde F'\tilde  e_t \tilde\lambda_{tm}'\|_F=O_P(\sqrt{\frac{\log p}{nT}}).
 $$
 Also, by Assumption \ref{ca.1}, $\max_{m\leq p}  \|\frac{1}{nT}\sum_{t=1}^T(\widehat F-\tilde FH)'\tilde  e_t \tilde\lambda_{tm}'\|_F=O_P(\Delta_{eg}),$ implying 
 $$
 \max_{m\leq p} \|\frac{1}{nT}\sum_{t=1}^T\widehat F'\tilde  e_t \tilde\lambda_{tm}'\|_F = O_P\left(\Delta_{eg}+\sqrt{\frac{\log p}{nT}}\right).
 $$
 
 On the other hand, similar calculations yield 
 \begin{align*}
\max_m \|\frac{1}{T} \sum_{t=1}^T \frac{1}{n}\tilde F'\tilde U_{t, m} \tilde\delta_{yt}' \|_F &\leq  \max_{m\leq p} \|\frac{1}{nT}\sum_{t=1}^T\sum_{i=1}^n f_i  U_{it,m} \tilde\delta_{yt}'\|_F
+ \max_{m\leq p} \|\frac{1}{ T}\sum_{t=1}^T \bar  U_{\cdot t,m} \tilde\delta_{yt}'\|_2O_P(1)\cr
&=O_P(\sqrt{\frac{\log p}{nT}}).
\end{align*}
Also, by Assumption \ref{ca.1}, $\max_m \|\frac{1}{T} \sum_{t=1}^T \frac{1}{n}(\widehat F-\tilde FH)'\tilde U_{t, m} \tilde\delta_{yt}' \|_F=O_P(\Delta_{ud})$. Hence
$$
\max_m \|\frac{1}{T} \sum_{t=1}^T \frac{1}{n}\widehat F'\tilde U_{t, m} \tilde\delta_{yt}' \|_F=O_P(\Delta_{ud}+\sqrt{\frac{\log p}{nT}}).
$$
  
(iv) Note that $\tilde e_{it} = e_{it}-\bar e_{\cdot t}-\bar e_{i\cdot}+\bar{\bar  e}$. Also,  $\frac{1}{n}\sum_i  \tilde  e_{it}=0$. Hence,
$$
\frac{1}{n}\sum_i\tilde f_i \tilde  e_{it}=  \frac{1}{n}\sum_if_i  e_{it}-\frac{1}{n}\sum_if_i \bar e_{i\cdot}- \bar f \bar e_{\cdot t}+\bar f \bar{\bar  e}.
$$
Therefore,
\begin{align*}
\frac{1}{T}\sum_t\|\frac{1}{n}\widehat F'\tilde e_t\|_2^2 &\leq \frac{2}{T}\sum_t\|\frac{1}{n}\sum_i(\widehat f_i-H'\tilde f_i)\tilde e_{it}\|_2^2 + O_P(1)\frac{1}{T}\sum_t\|\frac{1}{n}\sum_i\tilde f_i\tilde e_{it}\|_2^2 \\
&= O_P\left(\frac{1}{n}+\Delta_{fe}^2\right) \\
\frac{1}{T}\sum_t\|\frac{1}{n}\widehat F'\tilde U_t\gamma\|_2^2 &\leq 
\frac{2}{T}\sum_t\|\frac{1}{n}\sum_i (\widehat f_i-H'\tilde f_i)\tilde U_{it}'\gamma\|_2^2 + O_P(1)\frac{1}{T}\sum_t\|\frac{1}{n}\sum_i \tilde f_i\tilde U_{it}'\gamma\|_2^2 \\
&=O_P\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^2\right).
\end{align*}
 
(v) The proof is very similar to that of (ii).  
First,
\begin{align}\label{la.1.v.1}
\max_{m\leq p} \left\|\frac{1}{nT}\sum_{t=1}^T\tilde F'\tilde  U_{t,m} \gamma '    \tilde\Lambda_{t}\right\|_F &\leq   \max_m  \left\|\frac{1}{nT}\sum_{t=1}^T\sum_i f_i U_{it,m} \gamma'     \tilde\Lambda_{t} \right\|_F \\ 
\label{la.1.v.2}
& \qquad  + O_P(1)\max_m\left\|\frac{1}{T}\sum_t\bar U_{\cdot t,m} \gamma'\tilde\Lambda_t\right\|_2.
\end{align}
For the term (\ref{la.1.v.1}), let  $Z_{im}=\frac{1}{T}\sum_tf_iU_{it,m}\gamma'\tilde\Lambda_{t}$. Then 
$$
\max_{im}\Var(Z_{im})=O\left(\frac{|J|_0^2}{T}\right),
$$
and
$$
\max_m\|\frac{1}{n}\sum_iZ_{im}\|_2=O_P\left(|J|_0\sqrt{\frac{\log p}{nT}}\right),
$$ 
which gives the rate of convergence of (\ref{la.1.v.1}). 
For term (\ref{la.1.v.2}),  let $W_{im}=\frac{1}{T}\sum_t   U_{it,m}\gamma'\tilde \Lambda_{t}$. The $W_{im}$'s are independent across $i$, have $E(W_{im})=0$, and have $\max_{im}\Var(W_{im})=O(|J|_0^2/T)$.  Hence, $\max_m|\frac{1}{n}\sum_iW_{im}|=O_P\left(|J|_0\sqrt{\frac{\log p}{nT}}\right)$, which gives the rate of convergence for (\ref{la.1.v.2}).
Combining then yields 
$$
\max_{m\leq p} \|
  \frac{1}{nT}\sum_{t=1}^T\tilde F'\tilde  U_{t,m} \gamma '    \tilde\Lambda_{t}\|_F = O_P\left(|J|_0\sqrt{\frac{\log p}{nT}}\right).
$$
Finally, using $ \max_m\frac{1}{nT}\sum_{it}\tilde U_{it,m}^2\frac{1}{T}\sum_t\|\gamma_y'\tilde\Lambda_t\|_2^2=O_P(|J|_0^2)$, we have
\begin{align*}
\max_m & \left\|\frac{1}{nT} \sum_{i=1}^n   \sum_{t=1}^T(\widehat  f_i -H'\tilde f_i)\tilde U_{it,m}  \gamma_{y}'\tilde\Lambda_{t} \right\|_F^2 \\
&\leq \frac{1}{n}\sum_i\|\widehat  f_i -H'\tilde f_i\|_2^2 \max_m\frac{1}{nT}\sum_{it}\tilde U_{it,m}^2\frac{1}{T}\sum_t\|\gamma_y'\tilde\Lambda_t\|_2^2 = O_P\left(\Delta_F^2|J|_0^2\right).
\end{align*}
Hence, we obtain the desired  result. 

 (vi) A  proof similar to that of part (iii) yields 
 $$ \max_{m,l\leq p}|\frac{1}{nT} \sum_{it}\tilde U_{it,l}  \tilde  f_i'  \tilde \lambda_{tm} | = O_P\left( \sqrt{\frac{\log p}{nT}}\right).$$
 Then applying Assumption \ref{ca.1} yields $ \max_{m,l\leq p}|\frac{1}{nT} \sum_{it}\tilde U_{it,l}  (\widehat f_i-H'\tilde  f_i)'  \tilde \lambda_{tm} | = O_P(\Delta_{ud})$. Hence 
$$ \max_{m,l\leq p}|\frac{1}{nT} \sum_{it}\tilde U_{it,l}   \widehat  f_i'  \tilde \lambda_{tm} | = O_P\left(\Delta_{ud}+\sqrt{\frac{\log p}{nT}}\right).$$
 $\blacksquare$
\begin{lemma}\label{la.3june} 
(i)  $ \max_m\|  \frac{1}{n^2T}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n\tilde f_j \tilde U_{jt, m}  \tilde e_{it} \tilde f_i'\|_F=o_P(\sqrt{\frac{\log p}{nT}}).$\\
(ii)  $\max_{m\leq p} \|
  \frac{1}{n^2}\widehat F' \frac{1}{T}\sum_t\tilde  U_t \gamma_y   \tilde U_{t,m}'\widehat F   \|_F=o_P(\sqrt{\frac{\log p}{nT}}).$\\
(iii) $ \|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it} -\tilde U_{it} ) \tilde f_i'H
  (\widehat F'\widehat F )^{-1}\widehat F'\tilde  U_t \gamma_y\|_{\infty}=o_P(\sqrt{\frac{\log p}{nT}}).$
  \\
(iv) $ \|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it} -\tilde U_{it} ) \tilde f_i'H
  (\widehat F'\widehat F )^{-1}\widehat F'\tilde e_t\|_{\infty}=o_P(\sqrt{\frac{\log p}{nT}}).$
\end{lemma}
    

\proof  %Since the dimension of $f_i$ is fixed,   
For notational simplicity, we take $f_i$ to be a scalar without loss of generality. 
    

(i) We have 
\begin{align*}
\frac{1}{n^2T}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n\tilde f_j \tilde U_{jt, m}  \tilde e_{it} \tilde f_i'
&= \frac{1}{n^2T}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n\tilde f_j   U_{jt, m}   e_{it} \tilde f_i' \\
& \qquad - \frac{1}{n^2T}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n\tilde f_j \bar U_{j\cdot, m}  \bar e_{i\cdot } \tilde f_i'.
\end{align*}
Hence, as $n\to\infty$, 
\begin{align*}
\max_m &\left\|  \frac{1}{n^2T}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n\tilde f_j \tilde U_{jt, m}  \tilde e_{it} \tilde f_i' \right\|_F \\
&\leq \max_m \left\|  \frac{1}{n^2T}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n\tilde  f_j  U_{jt, m}   e_{it} \tilde f_i'\right\|_F+o_P\left(\sqrt{\frac{\log p}{nT}}\right).
\end{align*}
Define $W_{tm}= \frac{1}{n^2}\sum_{i=1}^n\sum_{j=1}^nU_{jt, m}   e_{it}\tilde  f_j\tilde f_i'$. We aim to bound $\max_m\|\frac{1}{T}\sum_tW_{tm}\|_F.$ We assume $\tilde f_j$ is one dimensional for simplicity, and the multivariate case follows from a similar argument.    By assumption, $E(U_{jt}e_{it}|F)=0$ almost surely. If $T=O(1)$, then  because   $\{(U_{it}, e_{it})\}_{i\leq n}$ are conditionally independent given $F$, 
\begin{align*}
\max_m|\frac{1}{T}\sum_tW_{tm}|&
\leq \max_{mt}|  \frac{1}{n^2}\sum_{i=1}^n\sum_{j=1}^nU_{jt, m}   e_{it}\tilde  f_j\tilde f_i|
\cr
&\leq \max_{mt}|\frac{1}{n}\sum_{i=1}^nU_{jt, m}  \tilde  f_j | \max_{t}| \frac{1}{n}\sum_{i=1}^ne_{it} \tilde f_i|\cr
&\leq O_P(\frac{\sqrt{\log p}}{n})=O_P(\sqrt{\frac{\log p}{Tn^2}}).\quad (T=O(1))
\end{align*}
If $T\to\infty$, using the assumption that $\{(U_{it}, e_{it})\}_{i\leq n}$ are conditionally independent given $F$, for  $E_f(\cdot)=E(\cdot|F)$,
\begin{align*}
\max_{tm} \Var(W_{tm}|F) &\leq  \frac{1}{n^4}  \sum_{l=1}^n \tilde  f_l^4   E_f U_{lt, m}^2   e_{lt}^2
 +\frac{1}{n^3}  \sum_{l=1}^n\tilde  f_l^2  E_f U_{lt, m} ^2  \frac{1}{n} \sum_{k=1}^n\tilde f_k^2E_fe_{kt}^2 \\
&\leq \frac{1}{n^3}  \left(\frac{1}{n}\sum_{l=1}^n \tilde  f_l^8\right)^{1/2} \left( \frac{1}{n}\sum_{l=1}^nE_f U_{lt, m}^4   e_{lt}^4\right)^{1/2} \\
& \qquad  + \frac{1}{n^2} \left(\frac{1}{n}\sum_{l=1}^n\tilde  f_l^4\right) \left(\frac{1}{n}\sum_l  E_f U_{lt, m} ^4\right)^{1/2}\left( \frac{1}{n} \sum_{k=1}^nE_fe_{kt}^4\right)^{1/2}.
\end{align*}
There is $C>0$ such that $\max_m \frac{1}{n}\sum_l  E_f U_{lt, m} ^8$ and $\sum_{k=1}^nE_fe_{kt}^8$ are upper bounded by $C$ almost surely (a.s.) in $F$. Thus a.s. in $F$, $\max_{tm} \Var(W_{tm}|F)\leq\frac{C}{n^2} (\frac{1}{n}\sum_{l=1}^n \tilde  f_l^8)^{1/2}$. Hence, for any $  x>0$ and a generic constant $C>0$,
\begin{align*}
P\left(\max_m|\frac{1}{T}\sum_tW_{tm}|>x\right) &\leq p\max_mEP\left( |\frac{1}{T}\sum_tW_{tm}|>x|F\right) \\
&\leq p\max_mEP\left( |\frac{1}{T}\sum_tW_{tm}|>x|F\right) 1\{\frac{1}{n}\sum_{l=1}^n \tilde  f_l^{8} <C\} \\
& \qquad +pP\left(|\frac{1}{n}\sum_{l=1}^n (   f_l^{16}- E  f_l^{16})|>C\right)+pP\left(  |\bar f-E\bar f|>C\right) \\
&\leq^{(a)} p\max_mEP\left( |\frac{1}{T}\sum_tW_{tm}|>x|F\right) 1\{  \max_{tm} \Var(W_{tm}|F)\leq\frac{C}{n^2} \} +o(1) 
\end{align*}
where $(a)$ follows from the Bernstein inequality for independent data  and $\log p=o(n)$, applied to 
$pP\left(|\frac{1}{n}\sum_{l=1}^n (   f_l^{16}- E  f_l^{16})|>C\right)$ and $pP\left(  |\bar f-E\bar f|>C\right)$.

We now bound $P\left( |\frac{1}{T}\sum_tW_{tm}|>x|F\right) $ conditioning on the event $\max_{tm} \Var(W_{tm}|F)\leq\frac{C}{n^2}$. We have $\min_{tm}[\Var(W_{tm}|F)^{-1/2}]\geq C_0n$ for a generic $C_0>0$.
Conditional on $F$,  $\{W_{tm}\}$ is a strong mixing sequence across $t$ with uniform mixing condition (uniform over $m\leq p$) that is bounded by $\exp(-CT^{r})$ (Assumption \ref{a3.2}). Write $\mathcal X_{tm}= W_{tm}\Var(W_{tm}|F)^{-1/2}$. 
Recall that by assumption, $\log ^\gamma p=o(n)$ for some $\gamma>2$ and the strong mixing (Assumption \ref{a3.2}) holds for the process  $\{(\eta_t, \epsilon_t)\}_{t=-\infty}^{+\infty}$, with mixing coefficient bounded by $\exp(-CT^r)$, $r>1$, and also we assume $\gamma r\geq 2$. 
Let $\bar r= \min\{r, 1\}$, $r_1=(0.5 +\bar r^{-1})^{-1}$,  $c=0.5(\gamma+1)$, then  $r_1<1$, $cr_1> 1$ and $2c\geq 1.$ Because $\bar r\leq r$, the strong mixing condition in Assumption \ref{a3.2} also holds with $\bar r$ in place of $r$. All these constants are independent of $F$. Then by the Bernstein inequality for strong mixing sequences \cite{merlevede2011bernstein},  for $y=M\frac{(\log p)^c}{\sqrt{T}}$, and sufficiently large $M>0$, for $C_0nx=y$,
\begin{align*}
 p\max_mP\left( |\frac{1}{T}\sum_tW_{tm}|>x|F\right)&\leq p\max_mP\left( |\frac{1}{T}\sum_t\mathcal X_{tm}|>x\min_{tm}[\Var(W_{tm}|F)^{-1/2}]|F\right)\cr
 &\leq p\max_mP\left( |\frac{1}{T}\sum_t\mathcal X_{tm}|>C_0nx |F\right)
 \leq  A_1+A_2+A_3
\end{align*}
     where for a generic constant $C$ that is independent of $F$, 
     \begin{eqnarray*}
     A_1&=& pT\exp ( -C (Ty)^{r_1})=\exp(\log(pT) - CM^{r_1} T^{r_1/2} \log^{cr_1} p) =o(1),\quad (cr_1\geq 1)\cr
     A_2&=& p\exp\left(  -C\frac{(Ty)^2}{T} \exp\left(\frac{(Ty)^{r_1(1-r_1)}}{C\log^{r_1}(Ty)}\right)  \right)=o(1),\quad (r_1<1, 2c> 1, Ty\to\infty)\cr
     A_3&=& p \exp\left( -CTy^2 \right)= \exp(\log p-CM^2\log^{2c}p) =o(1),
     \end{eqnarray*}
     and $o(1)$ in the above are non-stochastic converging sequences. (This is because the involved generic constant $C$ in \cite{merlevede2011bernstein} is independent of $F$, given that the conditional strong mixing and sub-Gaussian conditions (given $F$) in Assumption \ref{a3.2} hold almost surely in $F$.)
Hence $$
p\max_mEP\left( |\frac{1}{T}\sum_tW_{tm}|>x|F\right) 1\{  \max_{tm} \Var(W_{tm}|F)\leq\frac{C}{n^2} \}  =o(1).
$$
Consequently, $\max_m|\frac{1}{T}\sum_tW_{tm}|=O_P(x)=O_P(\frac{\log^{0.5\gamma+0.5} p}{\sqrt{Tn^2}})$ when $T\to\infty$. Combining with the case $T=O(1)$, we have 
$\max_m|\frac{1}{T}\sum_tW_{tm}|= O_P(\frac{\log^{0.5\gamma+0.5} p}{\sqrt{Tn^2}}).$
 Hence, 
\begin{align*} 
\max_m\|  \frac{1}{n^2T}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n\tilde  f_j  U_{jt, m}   e_{it} \tilde f_i'\|_F 
&= \max_m|\frac{1}{T}\sum_tW_{tm}|=O_P(\frac{\log^{0.5\gamma+0.5} p}{\sqrt{Tn^2}})=o_P(\sqrt{\frac{\log p}{nT}}).
\end{align*}
 where the last equality is due to $\log ^\gamma p=o(n)$.

(ii)    By Assumption \ref{ca.1},   $ \max_{mt}\|\frac{1}{n}\sum_{j=1}^n(\widehat f_j -H'\tilde f_j)\tilde U_{jt, m}\|_2=o_P\left(\sqrt{\frac{\log p}{T}}\right).$
Hence,
\begin{align}\label{ea.4addd}
\begin{split}
\max_{m\leq p} \|\frac{1}{n^2}\widehat F' &\frac{1}{T}\sum_t\tilde  U_t \gamma_y   \tilde U_{t,m}'\widehat F   \|_F \\
&\leq O_P(1) \max_{m\leq p} \|\frac{1}{n^2}
   \tilde F' \frac{1}{T}\sum_t\tilde  U_t \gamma_y   \tilde U_{t,m}'\tilde F   \|_F \\
& \qquad +\|\gamma_y\|_1
 \max_{mt}|\frac{1}{n}\sum_i(\widehat f_i' -H'\tilde f_i)\tilde  U_{it,m} |\max_{m\leq p}   \left(\frac{1}{n^2T}\sum_t \|\tilde U_{t,m}'\widehat F   \|_2^2\right)^{1/2}.
\end{split}
\end{align}
By Lemma \ref{la.1}, $\max_{m\leq p}   \left(\frac{1}{n^2T}\sum_t \|\tilde U_{t,m}'\widehat F   \|_2^2\right)^{1/2}=O_P\left(\sqrt{\frac{\log (pT)}{n}}+\Delta_{fum}\right)$. Hence, the second term on the right-hand-side of the inequality in (\ref{ea.4addd}) is $O_P\left(|J|_0\Delta_{fum}\right)\left(\sqrt{\frac{\log (pT)}{n}}+\Delta_{fum}\right)$. 

We now bound the first term on the right-hand-side of the inequality in (\ref{ea.4addd}). %$\max_{m\leq p}|\frac{1}{n^2}\tilde F' \frac{1}{T}\sum_t\tilde  U_t \gamma_y   \tilde U_{t,m}'\tilde  F   |  $. Without loss of generality we assume $\tilde f_i$ to be a scalar for notational simplicity. It is 
We have
\begin{align}\label{ea.4}
\begin{split}
\max_{m\leq p}&|\frac{1}{n^2}\tilde F' \frac{1}{T}\sum_t\tilde  U_t \gamma_y   \tilde U_{t,m}'\tilde  F   | \\
&= \max_{m\leq p}|\frac{1}{n^2T} \sum_t\sum_{j=1}^n \sum_{i=1}^n\sum_{l=1}^p\tilde f_i \tilde  f_j\tilde  U_{it,l}  \tilde U_{jt,m} \gamma_{yl}  | \\
&\leq O_P\left(\frac{\log p}{nT}|J|_0\right) \\
&\qquad + \max_{m\leq p}|\frac{1}{n^2T}\sum_l\gamma_{yl} \sum_t\sum_{j=1}^n \sum_{i=1}^n\tilde f_i \tilde  f_j(U_{it,l}-\bar U_{\cdot t, l})(U_{jt,m}-\bar U_{\cdot t,m})   |
\end{split}
\end{align}
where we obtain the $ O_P\left(|J|_0\frac{\log p}{nT}\right)$ term in the second inequality by bounding
\begin{align*} 
\max_m|\frac{1}{T}\sum_if_i\bar U_{i\cdot m}|^2 &= O_P\left(\frac{\log p}{n}\max_{im}\Var(f_i\bar U_{i,\cdot, m})\right) \\
&= O_P\left(\frac{\log p}{n}\right)  \max_{im}\frac{1}{T^2}\sum_{t}\sum_s Ef_i^2U_{it,m}U_{is,m} \\
&= O_P\left(\frac{\log p}{n}\right)  \max_{im,t}\frac{1}{T}Ef_i^2 |\sum_s (EU_{it,m}U_{is,m}|f_i)| \\
&= O_P\left(\frac{\log p}{nT}\right),
\end{align*}
using Assumption \ref{a3.3} that $\max_{im,t} \sum_s |E(U_{it,m}U_{is,m}|f_i)|<C$ almost surely in $f_i$.

It remains to bound the remaining term on the right-hand-side of the inequality in (\ref{ea.4}). This term is less than 

\begin{align*} 
&\underbrace{\max_{m\leq p}|\frac{1}{n^2T}\sum_l\gamma_{yl} \sum_t\sum_{j=1}^n \sum_{i=1}^n\tilde f_i \tilde  f_jU_{it,l}U_{jt,m} |}_{z_1} 
+ \underbrace{\max_{m\leq p}|\frac{1}{n^2T}\sum_l\gamma_{yl} \sum_t\sum_{j=1}^n \sum_{i=1}^n\tilde f_i \tilde  f_jU_{it,l}\bar U_{\cdot t,m}   |}_{z_2} \\
& \qquad + \underbrace{\max_{m\leq p}|\frac{1}{n^2T}\sum_l\gamma_{yl} \sum_t\sum_{j=1}^n \sum_{i=1}^n\tilde f_i \tilde  f_j\bar U_{\cdot t,l}U_{jt,m}   |}_{z_3} 
+ \underbrace{\max_{m\leq p}|\frac{1}{n^2T}\sum_l\gamma_{yl} \sum_t\sum_{j=1}^n \sum_{i=1}^n\tilde f_i \tilde  f_j\bar U_{\cdot t,l}\bar U_{\cdot t,m}   |}_{z_4}. 
\end{align*}
We now bound each of $z_1$-$z_4$.

$z_1$:  Define 
$$
Z_{1t,lm}=\frac{1}{n}  \sum_{i=1}^n\tilde f_i ^2(U_{it,l}U_{it,m} -E (U_{it,l}U_{it,m}  |F))
$$
and 
$$
Z_{2t,lm}=\frac{1}{n^2}\sum_{i=1}^n\sum_{j\neq i}\tilde f_i \tilde  f_jU_{it,l}U_{jt,m}.
$$ 
By the assumption that $E(U_{it,l}U_{jt,m}|F)=0$ for $i\neq j$, we have $E(Z_{1t,lm}|F)=0$.  Also note that  $\|\gamma_{y}\|_1\leq O(|J|_0)+\|R_y\|_1=O(|J|_0)+o(1)=O(|J|_0)$.

Now, consider
\begin{align*} 
z_1&\leq  \max_{m\leq p}|\frac{1}{n^2T}\sum_l\gamma_{yl} \sum_t  \sum_{i=1}^n\tilde f_i^2U_{it,l}U_{it,m} |
+\max_{m\leq p}|\frac{1}{n^2T}\sum_l\gamma_{yl} \sum_t \sum_{i=1}^n\sum_{j\neq i}\tilde f_i \tilde  f_jU_{it,l}U_{jt,m} |\\
&\leq  \max_{m\leq p} \left|\frac{1}{n^2T}\sum_l\gamma_{yl} \sum_t  \sum_{i=1}^n\tilde f_i ^2(U_{it,l}U_{it,m} - E( U_{it,l}U_{it,m} |F ))\right| \cr
&\qquad +  \max_{m\leq p}|\frac{1}{n^2T}\sum_l\gamma_{yl} \sum_t  \sum_{i=1}^n\tilde f_i^2E(U_{it,l}U_{it,m} |F)|   \\
&\qquad +\max_{m\leq p}|\frac{1}{n^2T}\sum_l\gamma_{yl} \sum_t \sum_{i=1}^n\sum_{j\neq i}\tilde f_i \tilde  f_jU_{it,l}U_{jt,m} |\\
&=  \max_{m\leq p} \left|\frac{1}{n}\sum_l\gamma_{yl}\frac{1}{T} \sum_t   Z_{1t,lm} \right| + \max_{m\leq p}|\frac{1}{n^2T}\sum_l\gamma_{yl} \sum_t  \sum_{i=1}^n\tilde f_i^2E(U_{it,l}U_{it,m} |F)| \\
&\qquad + \max_{m\leq p}|\frac{1}{T}\sum_l\gamma_{yl} \sum_t Z_{2t,lm} |\\
&\leq \|\gamma_{y}\|_1\frac{1}{n} \max_{m,l\leq p} \left| \frac{1}{T} \sum_t   Z_{1t,lm} \right| 
+\|\gamma_{y}\|_1   \max_{m,l\leq p}|\frac{1}{T} \sum_t Z_{2t,lm} |\\
&\qquad +\max_{l\leq p}|\gamma_{yl} | \max_{m\leq p}  \frac{1}{n^2T}\sum_t  \sum_{i=1}^n\tilde f_i^2\sum_{l=1}^p|E(U_{it,l}U_{it,m} |F)|
\\
&\leq O(\frac{|J|_0}{n}) \max_{m,l\leq p} \left| \frac{1}{T} \sum_t   Z_{1t,lm} \right| 
+O(|J|_0) \max_{l,m\leq p}\left|\frac{1}{T}  \sum_t Z_{2t,lm}  \right|\\  
& \qquad + O_P( \frac{1}{n})  \max_{m\leq p, i\leq n, t\leq T} \sum_{l=1}^p|E(U_{it,l}U_{it,m} |F)|  \\
&= O\left(\frac{|J|_0}{n}\right)\max_{m\leq p}  \left|\frac{1}{T}\sum_t Z_{1t,lm}\right|+O(|J|_0) \max_{l,m\leq p}\left|\frac{1}{T}  \sum_t Z_{2t,lm}  \right|+O\left(\frac{1}{n}\right),
\end{align*}
where in the last equality we used the assumption that $  \max_{m\leq p, i\leq n, t\leq T} \sum_{l=1}^p|E(U_{it,l}U_{it,m} |F)| =O(1)$ almost surely in $F$ (Assumption \ref{a3.2} (iii):  weak cross-sectional correlations).
%\textcolor{blue}{NEED TO CHECK:  Have we used arguments similar to those in this display previously?  What happened to $R_y$?  What happened to the $\tilde f_i^2$ and the $\gamma_y$ in the second term in the first inequality?  The simple argument looks like it would have the second term following the first inequality multiplied by $\max_{l}|\gamma_{y,l}|\max_i|\tilde f_i^2|$ which would not give $O\left(\frac{1}{n}\right)$.  I'm sure I'm missing something simple, but this argument needs to be clarified/fixed.}

To bound $\max_{m\leq p}  |\frac{1}{T}\sum_t Z_{1t,lm}|$, first note that 
\begin{align*}
\Var(Z_{1t,lm}|F) &= \Var\left(\frac{1}{n}  \sum_{i=1}^n\tilde f_i ^2U_{it,l}U_{it,m} |F\right) \\
&= \frac{1}{n^2}\sum_i\tilde f_i^4\Var(U_{it,l}U_{it,m}|F) \\
&\leq \frac{C}{n}\left(\frac{1}{n}\sum_i\tilde f_i^8\right)^{1/2}
\end{align*}
since $\max_{m,t}\frac{1}{n}\sum_i E_f U_{it,m}^8 < C$ a.s. in $F$.

Conditional on $F$, $\{Z_{1t,lm}\}_{t\leq T}$ is a strong mixing sequence. Hence, for $x=C\sqrt{\frac{\log p}{Tn}}$, 
\begin{align*}
P\left(\max_{lm}\right. & \left.\left|\frac{1}{T}\sum_tZ_{1t,lm}\right|>x\right) \\
&\leq p^2\max_{lm}E\left[P\left( \left|\frac{1}{T}\sum_tZ_{1t,lm}\right|>x|F\right)\right] \\
&\leq p^2\max_{lm} E\left[P\left( \left|\frac{1}{T}\sum_tZ_{1t,lm}\right|>x|F\right) 1\{\frac{1}{n}\sum_{l=1}^n \tilde  f_l^{8} <C\}\right] \\
&\qquad +p^2P\left(\left|\frac{1}{n}\sum_{l=1}^n (   f_l^{16}- E  f_l^{16})\right| > C\right) + p^2 P(  |\bar f-E\bar f|>C)\\
&\leq p^2\max_{lm}E\left[P\left( \left|\frac{1}{T}\sum_tZ_{1t,lm}\right|>x|F\right) 1\{  \max_{tm} \Var(Z_{1t,lm}|F) \leq \frac{C}{n} \}\right] + o(1) \\
&\leq E\left[\exp\left(2\log p -\frac{Tx^2}{\max_{tm}\Var(Z_{1t,lm}|F)}\right)1\{  \max_{tm} \Var(Z_{1t,lm}|F) \leq \frac{C}{n} \}\right] + o(1) \\
&\leq \exp(2\log p-CTnx^2)+o(1) = o(p^{-1}), \text{ for some large $C$.}
\end{align*}
This bound then implies $O(\frac{|J|_0}{n}) \max_{m\leq p}  \left|\frac{1}{T}\sum_t Z_{1t,lm}\right|=O_P\left(\frac{|J|_0}{n}\sqrt{\frac{\log p}{Tn}}\right)$. 
  
%\textcolor{blue}{NEED TO CHECK:  The previous version of the display above had some expectations where it was not obvious over which terms the expectations were taken and which seemed to drop prematurely.  Double check that the brackets are in the right places and that I didn't mess anything up when I tried to clarify.}


To bound $\max_{m\leq p}  |\frac{1}{T}\sum_t Z_{2t,lm}|$,  uniformly in $t,l,m$, (recall $E_f(.)=E(.|F)$) we note that 
\begin{align*}
\Var(Z_{2t,lm}|F) &=   \frac{1}{n^4}\sum_{i=1}^n\sum_{j\neq i}\sum_{k=1}^n\sum_{v\neq k} \tilde f_i \tilde  f_j\tilde f_k\tilde  f_v E_f U_{it,l}U_{jt,m}U_{kt,l}U_{vt,m} \\
&=\frac{1}{n^4}\sum_{i=1}^n\sum_{j\neq i}   \tilde f_i^2 \tilde  f_j^2E_fU_{it,l}^2E_fU_{jt,m} ^2
 +\frac{1}{n^4}\sum_{i=1}^n\sum_{j\neq i}   \tilde f_i^2 \tilde  f_j^2 E_fU_{it,l}U_{it,m}E_fU_{jt,m}U_{jt,l} \\
&\leq\frac{C}{n^3}\sum_{i=1}^n\tilde f_i^4.
\end{align*}
Hence, following an argument similar to that used to bound $\max_{m\leq p}  \left|\frac{1}{T}\sum_t Z_{1t,lm}\right|$, it can be shown that $\max_{m\leq p}  |\frac{1}{T}\sum_t Z_{2t,lm}|=O_P\left(\sqrt{\frac{\log p}{Tn^2}}\right)$. Therefore, 
$$
z_1=O_P\left(\frac{1}{n}+|J|_0\sqrt{\frac{\log p}{Tn^2}}\right).
$$
 


\noindent For $z_2, z_3$, note that
\begin{align*}
z_2 &\leq \max_{m\leq p}\left|\frac{1}{n^3T}\sum_l\gamma_{yl} \sum_t\sum_{j=1}^n \sum_{i=1}^n\sum_{k=1}^n\tilde f_i \tilde  f_jU_{it,l}U_{k t,m}   \right| \\
&\leq   \max_{m\leq p} \left|\frac{1}{n^2T}\sum_l\gamma_{yl} \sum_t \sum_{i=1}^n\sum_{k=1}^n\tilde f_i  U_{it,l}U_{k t,m}   \right| O_P(1)
\end{align*}
Following the same argument as we used to bound $z_1$, we reach $z_2=O_P\left(\frac{1}{n}+|J|_0\sqrt{\frac{\log p}{Tn^2}}\right).$ Similarly, we have $z_3=O_P\left(\frac{1}{n}+|J|_0\sqrt{\frac{\log p}{Tn^2}}\right).$  Finally,  as $\frac{1}{n}\sum_{j=1}^n\tilde f_i=\frac{1}{n}\sum_{j=1}^n  f_i-\bar f=0$, we have
$$z_4=\max_{m\leq p}|\frac{1}{n^2T}\sum_l\gamma_{yl} \sum_t\sum_{j=1}^n \sum_{i=1}^n\tilde f_i \tilde  f_j\bar U_{\cdot t,l}\bar U_{\cdot t,m}   |=0$$
 

 
 
Combining the above, we have that the second term of (\ref{ea.4}) is $O_P\left(\frac{1}{n}+|J|_0\sqrt{\frac{\log p}{Tn^2}}\right)$.  Therefore, 
$   \max_{m\leq p}|\frac{1}{n^2}\tilde F' \frac{1}{T}\sum_t\tilde  U_t \gamma_y   \tilde U_{t,m}'\tilde  F   | 
=O_P\left(\frac{\log p}{nT}|J|_0+|J|_0\sqrt{\frac{\log p}{Tn^2}}+\frac{1}{n}\right).$
Thus by (\ref{ea.4addd}), we have 
\begin{align*}
\max_{m\leq p} & \left\|
  \frac{1}{n^2}\widehat F' \frac{1}{T}\sum_t\tilde  U_t \gamma_y   \tilde U_{t,m}'\widehat F   \right\|_F \\
&= O_P\left(|J|_0\Delta_{fum}\right)\left(\sqrt{\frac{\log (pT)}{n}}+\Delta_{fum}\right)+O_P\left(\frac{\log p}{nT}|J|_0+|J|_0\sqrt{\frac{\log p}{Tn^2}}+\frac{1}{n}\right),
\end{align*}
which is $o_P\left(\sqrt{\frac{\log p}{nT}}\right)$ given that  $\Delta_{fum}^2=o\left(\frac{\log p}{T|J|^2\log(pT)}\right)$, $\Delta_{fum}^4=o\left(\frac{\log p}{|J|^2nT}\right)$, $|J|_0^2\log p=o(nT)$, and $|J|_0^2T=o(n)$. $\Delta_{fum}^2=o\left(\frac{\log p}{T|J|^2\log(pT)}\right)$ is guaranteed in Assumption \ref{ca.1}.  To establish $\Delta_{fum}^4=o\left(\frac{\log p}{|J|^2nT}\right)$, note that $$
\max_{mt}\|\frac{1}{n}\sum_{j=1}^n(\widehat f_j -H'\tilde f_j)\tilde U_{jt, m}\|_2^2\leq\frac{1}{n}\sum_i\|\widehat  f_i -H'\tilde f_i\|_2^2\max_{mt}\frac{1}{n}\sum_i\tilde U_{jt,m}^2.
$$ Because $\max_{mt}\frac{1}{n}\sum_i\tilde U_{it,m}^2\leq C\max_{imt} (E U_{it,m}^2+E\bar U_{i\cdot, m}^2)+o_P(1)=O_P(1)$, we have   $\Delta_{fum}\leq\Delta_F.$ 
We then have $\Delta_{fum}^4\leq\Delta_F^4=o(\frac{1}{nT|J|^4_0})=o(\frac{\log p}{|J|_0^2nT})$. 



(iii) Note that $
  \tilde U_{it, m}-\widehat U_{it, m}=(\widehat \lambda_{tm}'-\tilde\lambda_{tm}'H^{'-1})\widehat  f_i +\tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i).
 $ Also  note\\
 $
 \widehat\lambda_{tm}-H^{-1}\tilde \lambda_{tm}=(\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm}+(\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m}$.   Then up to an $O_P(1)$ term,     
 the object of interest is   
\begin{align*}
\max_{m\leq p} & \left|\frac{1}{n} \sum_{i=1}^n \tilde f_i'H
  (\widehat F'\widehat F )^{-1} \frac{1}{T}\sum_{t=1}^T\widehat F'\tilde  U_t \gamma_y(\widehat U_{it,m} -\tilde U_{it,m} ) \right| \\
&\leq \max_{m\leq p} \left\|\frac{1}{nT}\sum_{t=1}^T\widehat F'\tilde  U_t \gamma_y     \tilde\lambda_{tm}' \right\|_F O_P(\Delta_{F}) +\max_{m\leq p} \left\|\frac{1}{n^2}\widehat F' \frac{1}{T}\sum_t\tilde  U_t \gamma_y   \tilde U_{t,m}'\widehat F   \right\|_F.
\end{align*}
By part (ii), 
$$\max_{m\leq p} \left\|\frac{1}{n^2}\widehat F' \frac{1}{T}\sum_t\tilde  U_t \gamma_y   \tilde U_{t,m}'\widehat F   \right\|_F = o_P\left(\sqrt{\frac{\log p}{nT}}\right).
$$
By Lemma \ref{la.1},
\begin{align*}
\Delta_F \max_{m\leq p} \left\|\frac{1}{nT}\sum_{t=1}^T\widehat F'\tilde  U_t \gamma_y     \tilde\lambda_{tm}' \right\|_F &= O_P
\left(\Delta_F|J|_0+\sqrt{\frac{|J|_0\log p}{nT}}\right) \Delta_F \\ 
&=o_P\left(\sqrt{\frac{\log p}{nT}}\right),
\end{align*}
given the assumptions that  $\sqrt{ nT }\Delta_F^2|J|_0=o(1).$ The result follows. 
  


(iv) The object of interest is bounded by
\begin{align}\label{G4.iv}
\max_{m\leq p} \left\|\frac{1}{nT}\sum_{t=1}^T\widehat F'\tilde  e_t     \tilde\lambda_{tm}' \right\|_FO_P(\Delta_F) + \max_{m\leq p}  \left\|\frac{1}{n^2}\widehat F' \frac{1}{T}\sum_t\tilde  e_t  \tilde U_{t,m}'\widehat F   \right\|_F.
\end{align}
The first term in equation (\ref{G4.iv}) can be bounded similarly to the term 
$$
\max_{m\leq p} \|  \frac{1}{nT}\sum_{t=1}^T\widehat F'\tilde  U_t \gamma_y     \tilde\lambda_{tm}'\|_F
$$
in Lemma \ref{la.1}; and we have, using from Lemma \ref{la.1} that 
$$\max_{m\leq p} \|
\frac{1}{nT}\sum_{t=1}^T\widehat F'\tilde  e_t     \tilde\lambda_{tm}'\|_F = O_P\left(\Delta_{eg}+\sqrt{\frac{\log p}{nT}}\right),
$$
that 
$$  
O_P(\Delta_F)  \max_{m\leq p} \left\|\frac{1}{nT}\sum_{t=1}^T\widehat F'\tilde  e_t \tilde\lambda_{tm}'\right\|_F = O_P(\Delta_F)\left(\Delta_{eg}+\sqrt{\frac{\log p}{nT}}\right)=o_P\left(\sqrt{\frac{\log p}{nT}}\right),
$$
given Assumption \ref{ca.1} that $\Delta_{eg}=o\left(\frac{1}{\sqrt{nT}}\right).$
 
The second term in (\ref{G4.iv}) is bounded similarly to the term $\max_{m\leq p} \left\| \frac{1}{n^2}\widehat F' \frac{1}{T}\sum_t\tilde  U_t \gamma_y   \tilde U_{t,m}'\widehat F   \right\|_F$ in part (ii) using that $ \max_m \left\|  \frac{1}{n^2T}\sum_{i=1}^n\sum_{t=1}^T\sum_{j=1}^n\tilde  f_j  U_{jt, m}   e_{it} \tilde f_i'\right\|_F = O_P\left(\sqrt{\frac{\log p}{Tn^2}}\right)$ from part (i).  Specifically, we have
\begin{align}\label{ea.6add}
\begin{split}
\max_{m\leq p} \left\|\frac{1}{n^2}\widehat F' \frac{1}{T}\sum_t\tilde  e_t  \tilde U_{t,m}'\widehat F   \right\|_F &= O_P(\Delta_{fe})\left(\sqrt{\frac{\log (pT)}{n}}+\Delta_{fum}\right)+O_P\left(\frac{\log p}{nT}+ \sqrt{\frac{\log p}{Tn^2}}\right) \\
&= o_P\left(\sqrt{\frac{\log p}{nT}}\right),
\end{split}
\end{align}
due to $\Delta_{fe}\Delta_{fum} \leq \Delta_F^2 = o\left(\sqrt{\frac{\log p}{nT}}\right)$ and the assumption that  
$
\Delta_{fe}^2 = o\left(\frac{\log p}{T\log (pT)}\right).
$  
We omit the details for brevity.  $\blacksquare$
 
 
\begin{lemma}\label{la.4imp}
% (i) $\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it} -\tilde U_{it} ) \tilde f_i'H(H^{-1}\tilde\delta_{yt}-\widehat \delta_{yt} )   \|_{\infty}=o_P(\sqrt{\frac{\log p}{nT}}).$\\
 (i) $\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it} -\tilde U_{it} ) \tilde \delta_{yt}'H^{'-1}(H'\tilde f_i - \widehat f_i )   \|_{\infty}=o_P(\sqrt{\frac{\log p}{nT}})$.\\
 %(iii) $\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it} -\tilde U_{it} )(\widehat \delta_{yt}' -\tilde \delta_{yt}'H^{'-1})(H'\tilde f_i - \widehat f_i )   \|_{\infty}=o_P(\sqrt{\frac{\log p}{nT}})$\\
 %(iv) $ \|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T\tilde U_{it} \tilde f_i'H(H^{-1}\tilde\delta_{yt}-\widehat \delta_{yt} )   \|_{\infty}=o_P(\sqrt{\frac{\log p}{nT}}).$\\
%(v)  $\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it} -\tilde U_{it} ) (\delta_{yt}'\tilde f_i -\widehat\delta_{yt}'\widehat f_i )   \|_{\infty}=o_P(\sqrt{\frac{\log p}{nT}})$\\
  %(vi)$  \|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T\tilde U_{it} (\delta_{yt}'\tilde f_i -\widehat\delta_{yt}'\widehat f_i )   \|_{\infty}=o_P(\sqrt{\frac{\log p}{nT}})$\\
 (ii) $ \|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it}  -\tilde U_{it} ) (\widehat U_{it}  -\tilde U_{it} ) '\gamma_y \|_{\infty} =o_P(\sqrt{\frac{\log p}{nT}})$.\\
(iii) $ \|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T\tilde U_{it}    (\tilde U_{it} -\widehat U_{it} ) '\gamma_y \|_{\infty}=o_P(\sqrt{\frac{\log p}{nT}})$.
 
   \end{lemma}   
   
\proof   
   
   
  (i)  By Lemma \ref{la.1},   $\max_m \|\frac{1}{T}   \sum_{t=1}^T \frac{1}{n}\widehat F'\tilde U_{t, m} \tilde\delta_{yt}' \|_F=O_P\left(\Delta_{ud}+\sqrt{\frac{\log p}{nT}}\right).$ Hence,
\begin{align}\label{ea.7add}
\begin{split}
\|\frac{1}{nT} &\sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it} -\tilde U_{it} ) \tilde \delta_{yt}'H^{'-1}(H'\tilde f_i - \widehat f_i )   \|_{\infty} \\
&\leq \max_m \left\|\frac{1}{T}   \sum_{t=1}^T (\widehat \lambda_{tm}-H^{-1}\tilde\lambda_{tm})\tilde \delta_{yt}'\right\|_F  O_P(\Delta_F)+O_P(\Delta_F^2) \\
&\leq O_P(\Delta_F^2) + \max_m \left\|\frac{1}{T}   \sum_{t=1}^T  \frac{1}{n}\widehat F'\tilde U_{t, m}\tilde \delta_{yt}'\right\|_F  O_P(\Delta_F) \\
&= O_P(\Delta_F^2) + O_P(\Delta_F)\left(\Delta_{ud}+\sqrt{\frac{\log p}{nT}}\right) \\
&= o_P\left(\sqrt{\frac{\log p}{nT}}\right)
\end{split}
\end{align}
under the assumption  $\Delta_F^2 + \Delta_F\Delta_{ud}=o\left(\sqrt{\frac{\log p}{nT}}\right)$.


(ii) Using the equalities
\begin{align*}
\tilde U_{it, m}-\widehat U_{it, m} &= (\widehat \lambda_{tm}'-\tilde\lambda_{tm}'H^{'-1})\widehat  f_i +\tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i), \\
\widehat\lambda_{tm}-H^{-1}\tilde \lambda_{tm} &= (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm}+(\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m}, \text{and} \\ 
\widehat\delta_{yt}-H^{-1}\tilde\delta_{yt} &=
(\widehat F'\widehat F)^{-1}\widehat F' (\tilde FH-\widehat F)H^{-1}\tilde\delta_{yt}
+(\widehat F'\widehat F)^{-1}\widehat F' \tilde U_t\gamma_y
+(\widehat F'\widehat F)^{-1}\widehat F' \tilde e_t,
\end{align*}
we have
\begin{align*} 
\|\frac{1}{nT} &\sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it}  -\tilde U_{it} ) (\widehat U_{it}  -\tilde U_{it} ) '\gamma_y \|_{\infty} \\
&= \max_m|\frac{1}{nT} \sum_{i=1}^n\sum_{k=1}^p  \sum_{t=1}^T(\widehat U_{it,m}  -\tilde U_{it,m} ) (\widehat U_{it,k}  -\tilde U_{it,k} ) \gamma_{yk} | \\
&\leq \max_m\|\frac{1}{T}  \sum_{k=1}^p  \sum_{t=1}^T\tilde\lambda_{tm}(\widehat \lambda_{tk}- H^{-1}\tilde \lambda_{tk})' \gamma_{yk} \|_F O_P(\Delta_F) \\
&\qquad + \max_m\|\frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T(\widehat \lambda_{tm}-H^{-1}\tilde\lambda_{tm}) \tilde\lambda_{tk}'  \gamma_{yk} \|_FO_P(\Delta_F) \\
&\qquad +\max_m\|\frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T(\widehat \lambda_{tm}-H^{-1}\tilde\lambda_{tm}) (\widehat \lambda_{tk}'-\tilde\lambda_{tk}'H^{'-1})  \gamma_{yk} \|_F + O_P(\Delta_F^2|J|_0) \\
&\leq \left(\max_m\|\frac{1}{T}  \sum_{k=1}^p  \sum_{t=1}^T\tilde\lambda_{tm}      \frac{1}{n}\widehat F'\tilde U_{t, k}   \gamma_{yk} \|_F + \max_m\|\frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T\frac{1}{n}\widehat F'\tilde U_{t, m}\tilde\lambda_{tk}'  \gamma_{yk} \|_F
\right) O_P(\Delta_F) \\
&\qquad + \max_m\|\frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T\frac{1}{n}\widehat F'\tilde U_{t, m}\frac{1}{n}\widehat F'\tilde U_{t, k}  \gamma_{yk} \|_F +O_P(  \Delta_F^2|J|_0) \\
&= \left(|J|_0\sqrt{\frac{\log p}{nT}}+ \Delta_{F}|J|_0\right)O_P(\Delta_F)+o_P\left(\sqrt{\frac{\log p}{nT}}\right)+O_P( \Delta_F^2|J|_0) \\
&= o_P\left(\sqrt{\frac{\log p}{nT}}\right), 
\end{align*}
where the bound on the first term in the second to last equality follows from Lemma \ref{la.1} (ii)(v) and the bound on the second term in the second to last equality follows from Lemma \ref{la.3june}.  Finally, the last equality follows from the assumption that 
$
|J|_0\Delta_F^2=o\left(\sqrt{\frac{\log p}{nT}}\right).
$
   
   
 
  


     
 
(iii)  From the equality 
\begin{align*}
\tilde U_{it, m}-\widehat U_{it, m} &= \widehat  f_i' (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm} + \widehat  f_i'(\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m} \\
& \qquad  +\tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i),
\end{align*}
we have
\begin{align*} 
\|\frac{1}{nT} &\sum_{i=1}^n  \sum_{t=1}^T\tilde U_{it}    (\tilde U_{it} -\widehat U_{it} ) '\gamma_y \|_{\infty} \\
&\leq \max_m \left|\frac{1}{nT} \sum_{i=1}^n\sum_{k=1}^p  \sum_{t=1}^T\tilde U_{it,m} \tilde\lambda_{tk}'H^{'-1}(\widehat  f_i -H'\tilde f_i) \gamma_{yk} \right| \\
&\qquad +\max_m \left|\frac{1}{nT} \sum_{i=1}^n\sum_{k=1}^p  \sum_{t=1}^T\tilde U_{it,m}   \tilde \lambda_{tk}\widehat  f_i \gamma_{yk} \right|O_P(\Delta_F) \\
&\qquad +\max_m\left|\frac{1}{nT} \sum_{i=1}^n\sum_{k=1}^p  \sum_{t=1}^T\tilde U_{it,m}  \frac{1}{n}\widehat F'\tilde U_{t, k}\widehat  f_i \gamma_{yk} \right| \\
&= o_P\left(\sqrt{\frac{\log p}{nT}}\right).
\end{align*}
 $\blacksquare$

 
\subsection{Technical lemmas for Appendix \ref{sd}}
  

\begin{lemma}\label{ld.1}
(i) $  \max_{m} \|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}^*  \widehat  f_i \widehat \lambda_{tm} '  \|_F=O_{P^*}\left(\sqrt{\frac{ \log(p)\log(np)}{nT}}\right)$.\\
(ii) $\max_{mj}\frac{1}{n}\sum_{i}(\frac{1}{T}\sum_{t}   \widehat \eta_{it}	 \widehat f_j \widehat U_{jt, m}   \widehat  f_i	)^2=O_P\left(  \frac{\log ^2(n )\log(pn)}{T}  \right).$ \\
(iii) $\max_{m} \|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}^*  	 \frac{1}{n}\widehat F^{'}\tilde U_{t, m}^*   \widehat  f_i^{'}	\|_F=o_{P^*}\left(\sqrt{ \frac{\log p}{nT}}\right)$. \\
(iv) $\max_{mt}  \| \frac{1}{n}\widehat F^{'}\tilde U_{t, m}^*\|_2=O_{P^*}\left(\sqrt{ \frac{\log (pT)}{n}}\right)$,
  $\max_{m} \frac{1}{T}\sum_t  \| \frac{1}{n}\widehat F^{'}\tilde U_{t, m}^*\|_2^2=O_{P^*}\left(\sqrt{ \frac{\log (pT)}{n}}\right)$, \\
  and 
  $\frac{1}{T}\sum_t\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}^* \widehat f_i\|_2^2=O_{P^*}\left(\frac{1}{n}\right)$. \\
(v) $\max_{m} \|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}^*  	 \frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*   \widehat  f_i^{*'}	\|_F=o_{P^*}\left(\sqrt{ \frac{\log p}{nT}}\right).$
\end{lemma}
  
\proof For notational simplicity, we assume $f_i$ is one dimensional without loss of generality. 
  
  (i) Note $\tilde e^*_{it}=\tilde \epsilon_{it}^*+\widehat\alpha \tilde\eta_{it}^*=w_i^Y\widehat\epsilon_{it}+\widehat\alpha w_i^D\widehat\eta_{it}$. We bound  $\max_{m} |\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^Tw_i^D\widehat\eta_{it} \widehat  f_i \widehat \lambda_{tm} |.$ Let $\Gamma_{im}=\frac{1}{T}\sum_{t=1}^Tw_i^D\widehat\eta_{it} \widehat  f_i \widehat \lambda_{tm}$. Then, since $\Var(w_i^D)=1$, uniformly in $m\leq p$, 
\begin{align*}
\frac{1}{n}\sum_i\Var^*(\Gamma_{im}) &= \frac{1}{n}\sum_i\left(\frac{1}{T}\sum_t\widehat\eta_{it} \widehat  f_i \widehat \lambda_{tm} \right)^2 \\
&\leq O_P(1)\frac{1}{n}\sum_i \left(\frac{1}{T}\sum_t\widehat\eta_{it}\widehat\lambda_{tm}\right)^2 \\
&\leq O_P\left(\frac{1}{n}\right)\sum_i\left(\frac{1}{T}\sum_t\tilde\eta_{it}\tilde\lambda_{tm}\right)^2\\
&\qquad + O_P(1)\frac{1}{n}\sum_i\left(\frac{1}{T}\sum_t(\widehat\eta_{it}-\tilde\eta_{it})\widehat\lambda_{tm}\right)^2 \\
&\qquad + O_P(1)\frac{1}{n}\sum_i\left(\frac{1}{T}\sum_t\tilde\eta_{it}(\widehat\lambda_{tm}-H^{-1}\tilde\lambda_{tm})\right)^2 \\
&\leq^{(a)} O_P(\psi_n^2)\max_m\frac{1}{T}\sum_t\widehat\lambda_{tm}^2 \\
&\qquad + O_P(1)\max_m\frac{1}{n}\sum_i\left(\frac{1}{T}\sum_t\tilde\eta_{it}  (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm} \right)^2 \\
&\qquad + O_P(1)\max_m\frac{1}{n}\sum_i\left(\frac{1}{T}\sum_t\tilde\eta_{it} (\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m}\right)^2 \\
&\qquad + O_P(1)\max_m\frac{1}{n}\sum_i\left(\frac{1}{T}\sum_t\tilde\eta_{it}\tilde\lambda_{tm}\right)^2 \\
&\leq^{(b)} O_P(\psi_n^2) + \max_m\frac{1}{n}\sum_i\left(\frac{1}{T}\sum_t\tilde\eta_{it}  \tilde \lambda_{tm} \right)^2O_P(\Delta_F^2+1) \\
&\qquad +O_P(1)\max_m\frac{1}{n}\sum_i\left(\frac{1}{Tn}\sum_t\tilde\eta_{it}  \widehat F'\tilde U_{t, m}\right)^2 \\
&\leq O_P(\psi_n^2)+\max_{im} \left(\frac{1}{T}\sum_t\tilde\eta_{it}  \tilde \lambda_{tm} \right)^2O_P(1)+\frac{1}{n^2}\|\widehat F\|_F^2\max_{im} \left\|\frac{1}{T}\sum_t\tilde\eta_{it}\tilde U_{t,m}\right\|_2^2 \\
   %&&+\frac{1}{n^2}\|\widehat F-\tilde FH\|_F^2\max_m\frac{1}{n}\sum_i\|\frac{1}{T}\sum_t\tilde\eta_{it}\tilde U_{t,m}\|_2^2+O_P(1)\max_m\frac{1}{n}\sum_i(\frac{1}{Tn}\sum_t\tilde\eta_{it}  \tilde F'\tilde U_{t, m})^2\cr
&= O_P\left(\psi_n^2+\frac{\log(np)}{T}\right) \\
&=^{(c)} O_P\left(\frac{\log(np)}{T}\right) %\text{ (this is uniform in $m\leq p$)}
\end{align*}
where $\psi_n$ is defined in  (\ref{eb.3a}).
In (a), we used the equality 
$$
\widehat\lambda_{tm}-H^{-1}\tilde \lambda_{tm}=(\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm}+(\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m}
$$ 
and Proposition  \ref{pb.1}. In (b), we used
\begin{align}\label{ed.5ad}
\begin{split}
\max_{mt}|\widehat\lambda_{tm}| &\leq  O_P(1)\max_{mt}|\tilde\lambda_{tm}|+\max_{mt} | \widehat\lambda_{tm}-H^{-1}\tilde \lambda_{tm}| \\
&\leq O_P(1) +\max_{mt}|\frac{1}{n}\widehat F'\tilde U_{t, m}|=O_P(1).
\end{split}
\end{align}
which follows from Lemma \ref{la.1}.
Equality (c) follows from $\psi_n^2=O\left(\frac{1}{T}\right)$, given that $\Delta_F^2|J|_0^2=o\left(\frac{1}{\sqrt{nT}}\right)=o\left(\frac{1}{T}\right)$. 
 
 Recall that $\Gamma_{im}=\frac{1}{T}\sum_{t=1}^Tw_i^D\widehat\eta_{it} \widehat  f_i \widehat \lambda_{tm}$. 
It then follows that, for any arbitrarily small $\varepsilon>0$, there is a $C_{\varepsilon}>0$ which depends only on the true data generating process and not on any realized data such that the event $A=\left\{\max_m\frac{1}{n}\sum_i\Var^*(\Gamma_{im})\leq C_{\varepsilon}\frac{\log(np)}{T}\right\}$ holds with probability at least $1-\varepsilon$. On this event, by the Bernstein inequality for independent data,  for $x=\sqrt{\frac{2C_{\varepsilon}\log(p)\log(np)}{nT}}$,
$$
P^*\left(\max_{m}|\frac{1}{n}\sum_i\Gamma_{im}|>x\right)1\{A\} \leq \exp\left(\log p-\frac{nTx^2}{C_{\varepsilon}\log(np)}\right)=p^{-1}.
$$
Thus, $P^*(\max_{m}|\frac{1}{n}\sum_i\Gamma_{im}|>x)\leq p^{-1}+1\{A^c\}$ where $P^*$ is with respect to the bootstrap sampling space conditional on the original data. Taking expectations on both sides, we reach $P\left(\max_{m}\left|\frac{1}{n}\sum_i\Gamma_{im}\right|>x\right)\leq p^{-1}+\varepsilon$ where $P$ is with respect to the probability space of the joint distribution of $\{w_i^D, \text{original data}\}$.  Now for any $\varepsilon_1, \varepsilon_2>0$, take $\varepsilon=\varepsilon_1\varepsilon_2/2$.  Then, as $p\to\infty$,
$$
 P\left(P^*\left(\max_{m}|\frac{1}{n}\sum_i\Gamma_{im}|>x\right)>\varepsilon_1\right)<(p^{-1}+\varepsilon)/\varepsilon_1<\varepsilon_2.
$$
Since $C_{\varepsilon}$ is not random in either the bootstrap or the original sampling space, we have 
$$
\max_{m} \left|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^Tw_i^D\widehat\eta_{it} \widehat  f_i \widehat \lambda_{tm} \right|=O_{P^*}\left(\sqrt{\frac{ \log(p)\log(np)}{nT}}\right).
$$ 
The same argument applies to 
$$
\max_{m} \left|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^Tw_i^Y\widehat\epsilon_{it} \widehat  f_i \widehat \lambda_{tm} \right|
$$ 
and thus 
$$
\max_{m} \left|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}^* \widehat  f_i \widehat \lambda_{tm} \right|
$$ 
as well. The conclusion follows.
 

(ii) First,  by Lemma \ref{la.1} and Assumption \ref{ca.1},
$$ 
\max_{m\leq p}   \left(\frac{1}{n^2T}\sum_t \|\tilde U_{t,m}'\widehat F   \|_2^2\right)=O_P\left(\frac{\log(pT)}{n}+\Delta_{fum}^2\right)=O_P\left(\Delta_F^2+\frac{\log(pT)}{n}\right).
$$ 
Hence,
\begin{align}\label{ed.6}
\begin{split}
\max_{mi}\frac{1}{T}\sum_{t} ( \tilde U_{it, m}   -\widehat U_{it,m}	)^2 &\leq
\max_{mi}\frac{1}{T}\sum_{t} ( \widehat  f_i' 		 (\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m})^2 \\
&\qquad + \max_{mi}\frac{1}{T}\sum_{t} ( \widehat  f_i' (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm} )^2 \\
&\qquad + \max_{mi}\frac{1}{T}\sum_{t} ( \tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i)	)^2 \\
&\leq O_P(\log n)\max_m\frac{1}{T}\sum_{t} \left(\frac{1}{n}\widehat F'\tilde U_{t, m}\right)^2
 +O_P(\log (n)\Delta_{F}^2+\Delta_{\max}^2) \\
&= O_P\left( \log(n)\frac{\log(pT)}{n}+\log (n)\Delta_{F}^2+\Delta_{\max}^2\right)=O_P(\log n)
\end{split}
\end{align}
under the assumption $\Delta_{\max}=O\left(\sqrt{\log n}\right)$.
Then we have
\begin{align*}
\max_{mj}\frac{1}{n}&\sum_{i}\left(\frac{1}{T}\sum_{t}   \widehat \eta_{it}	 \widehat f_j \widehat U_{jt, m}   \widehat  f_i	\right)^2 \\
&\leq \max_{mj}\frac{1}{n}\sum_{i}\left(\frac{1}{T}\sum_{t}   \widehat \eta_{it}	 \widehat U_{jt, m}   	\right)^2\widehat  f_i^2 \max_j \widehat f_j^2 \\
&\leq O_P(\log n)\max_{mj}\frac{1}{n}\sum_{i}\left(\frac{1}{T}\sum_{t}   \widehat \eta_{it}	 \widehat U_{jt, m}   	\right)^2\widehat  f_i^2 \\
&\leq  O_P(\log n)\max_{mj}\frac{1}{n}\sum_{i}\left(\frac{1}{T}\sum_{t}   (\widehat \eta_{it}-\tilde\eta_{it})	 \widehat U_{jt, m}   	\right)^2\widehat  f_i^2 \\
&\qquad + O_P(\log n)\frac{1}{n}\sum_{i}\widehat  f_i^2\max_{mj}\left(\frac{1}{T}\sum_{t}   \tilde \eta_{it}	\tilde U_{jt, m}   	\right)^2 \\
&\qquad + O_P(\log n)\max_{mj}\frac{1}{n}\sum_{i}\left(\frac{1}{T}\sum_{t}   \tilde \eta_{it}	( \widehat U_{jt, m}   	-\tilde U_{jt,m})\right)^2\widehat  f_i^2 \\
&\leq O_P(\log ^2n)\frac{1}{nT}\sum_{it}   
  (\widehat \eta_{it}-\tilde\eta_{it})	^2  \max_{mj}\frac{1}{T}\sum_{s}  \widehat U_{js, m}^2 + O_P(\log n) \max_{mij}\left(\frac{1}{T}\sum_{t}   \tilde \eta_{it}	\tilde U_{jt, m} \right)^2\\
&\qquad + O_P(\log n)\max_{mj}\frac{1}{n}\sum_{i}\left(\frac{1}{T}\sum_{t}   \tilde \eta_{it}	
 \widehat  f_j' (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm}  \right)^2\widehat  f_i^2 \\
&\qquad + O_P(\log n)\max_{mj}\frac{1}{n}\sum_{i}\left(\frac{1}{T}\sum_{t}   \tilde \eta_{it}	
  \widehat  f_j' 		 (\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m}    \right)^2\widehat  f_i^2 \\
&\qquad + O_P(\log n)\max_{mj}\frac{1}{n}\sum_{i}\left(\frac{1}{T}\sum_{t}   \tilde \eta_{it}	
 \tilde\lambda_{tm}'H^{'-1}(\widehat  f_j -H'\tilde f_j) \right)^2\widehat  f_i^2 \\
&\leq O_P(\log ^2(n) \psi_n^2)\left( \max_{mj}\frac{1}{T}\sum_{s}  \tilde U_{js, m}   	^2+\log n\right) + O_P\left(\log (n)\frac{\log(pn)}{T}\right) \\
&\qquad + O_P(\log^2 (n)\Delta_F^2) \max_{mi}\left(\frac{1}{T}\sum_{t} \tilde\eta_{it}\tilde\lambda_{tm} \right)^2 \\
&\qquad + O_P(\log ^2n)\max_i \frac{1}{T}\sum_{t} |  \tilde \eta_{it}	|^2
  \max_{tm}\left|	\frac{1}{n}\widehat F'\tilde U_{t, m} \right|^2 \\
&\qquad + O_P(\log (n)\Delta_{\max}^2 )\max_{mi} \left(\frac{1}{T}\sum_{t}   \tilde \eta_{it}	
 \tilde\lambda_{tm}\right)^2 \\
 %&\leq& O_P(\log ^2(n )\psi_n^2)( 1+\sqrt{\frac{\log(pn)}{T}}+\log n)+ O_P(\log (n)\frac{\log(pn)}{T})  +O_P(\log^2 (n)\Delta_F^2+\log (n)\Delta_{\max}^2 ) \frac{\log(pn)}{T}\cr  && +O_P(\log ^2n)(1+\sqrt{\frac{\log n}{T}})(\frac{\log(pT)}{n}+\Delta_F^2)\cr
&\leq O_P\left(  \frac{\log ^2(n )\log(pn)}{T}\right),
\end{align*}
where we used 
$$
\max_{t\leq T, m\leq p} \left\|\frac{1}{n}\sum_{i=1}^n\widehat f_i \tilde U_{it,m}\right\|_2=O_P\left(\sqrt{\frac{\log (pT)}{n}}+\Delta_{fum}\right)
$$
and
$$
\max_i\frac{1}{T}\sum_t\tilde\eta_{it}^2=O_P\left(\max_{it}E\eta_{it}^2+\max_{i}\frac{1}{T}\sum_t(\eta_{it}^2-E\eta_{it}^2)\right)
$$ 
from Lemma \ref{la.1}, (\ref{eb.3a}) for the definition of $\psi_n$, $|J|^2_0\Delta_F^2=o\left((nT)^{-1/2}\right)$ from Assumption \ref{ca.1}, and $T=o(n)$.
  
 
(iii)  We have 
$$  
\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}^*  	 \frac{1}{n}\widehat F^{'}\tilde U_{t, m}^*   \widehat  f_i^{'}	 =\frac{1}{n^2T}\sum_{i, j\leq n}\sum_{t=1}^Tw_i^U (\tilde \epsilon_{it}^*+\widehat\alpha \tilde\eta_{it}^*)  	 \widehat f_j \widehat U_{jt, m}   \widehat  f_i^{'}.	
$$ 
We  bound 
$$
\max_m|\frac{1}{n^2T}\sum_{i, j\leq n}\sum_{t=1}^Tw_j^U   \tilde\eta_{it}^*	 \widehat f_j \widehat U_{jt, m}   \widehat  f_i	|=\max_m|\frac{1}{n^2T}\sum_{i, j\leq n}\sum_{t=1}^Tw_j^U w_i^D  \widehat \eta_{it}	 \widehat f_j \widehat U_{jt, m}   \widehat  f_i	|
$$ 
using Lemma \ref{ld.1ad}. 
%We note that the argument below can also be used to bound 
%$$
%\max_m|\frac{1}{n^2T}\sum_{i, j\leq n}\sum_{t=1}^Tw_j^U \tilde\epsilon_{it}^*	 \widehat f_j \widehat U_{jt, m}   \widehat  f_i	|,$$
%which we omit for brevity.

We first rewrite 
$$
\max_m|\frac{1}{n^2T}\sum_{i, j\leq n}\sum_{t=1}^Tw_j^U w_i^D  \widehat \eta_{it}	 \widehat f_j \widehat U_{jt, m}   \widehat  f_i	|
$$ 
as 
$$
\max_m|\frac{1}{n^2T}\sum_{i, j\leq n}\sum_{t=1}^Tw_i^U w_j^D  \widehat \eta_{jt}	 \widehat f_i\widehat U_{it, m}   \widehat  f_j	|$$ 
so that it looks more friendly for application of Lemma \ref{ld.1ad}.  Now define $Z_{ijm}=\frac{1}{T} \sum_t\widehat \eta_{jt}	 \widehat f_i\widehat U_{it, m}   \widehat  f_j$. Then by part (ii) of the present lemma,
\begin{align*}
\max_{m\leq p, i\leq n}\frac{1}{n}\sum_{j=1}^n Z_{ijm}^2 &= \max_{mi}\frac{1}{n}\sum_{j}\left(\frac{1}{T}\sum_{t}   \widehat \eta_{jt}	 \widehat f_i\widehat U_{it, m}   \widehat  f_j	\right)^2 \\
&=O_P\left(  \frac{\log ^2(n )\log(pn)}{T}  \right):=O_P(a_n^2).
\end{align*}
Now define $X_i^*=w_i^U$ and $Y_j^*=w_j^D$. Apply Lemma \ref{ld.1ad}, 
\begin{align*}
\max_m\left|\frac{1}{n^2T}\sum_{i, j\leq n}\sum_{t=1}^Tw_i^U w_j^D  \widehat \eta_{jt}	 \widehat f_i\widehat U_{it, m}   \widehat  f_j	\right| &=
\max_m\left|\frac{1}{n^2}\sum_{ij}X_i^*Y_j^*Z_{ijm}\right| \\
&=O_{P^*}\left(\frac{\log(n)\log(pn)}{\sqrt{n}}\sqrt{ \frac{\log p}{nT}}\right).
\end{align*}
Then, using the assumption that $\log ^{3}p=O(n)$, we have
$$
\max_m|\frac{1}{n^2T}\sum_{i, j\leq n}\sum_{t=1}^Tw_i^U w_j^D  \widehat \eta_{jt}	 \widehat f_i\widehat U_{it, m}   \widehat  f_j	|=o_{P^*}\left(\sqrt{ \frac{\log p}{nT}}\right).
$$
 
Similarly, 
$$
\max_m\left|\frac{1}{n^2T}\sum_{i, j\leq n}\sum_{t=1}^Tw_j^U \tilde\epsilon_{it}^*	 \widehat f_j \widehat U_{jt, m}   \widehat  f_i	\right|=o_{P^*}\left(\sqrt{ \frac{\log p}{nT}}\right).
$$  
Therefore, 
$$
\max_m \left|  \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}^*  	 \frac{1}{n}\widehat F^{'}\tilde U_{t, m}^*   \widehat  f_i^{'}	\right| =o_{P^*}\left(\sqrt{ \frac{\log p}{nT}}\right).
$$



(iv) First, we bound  $  \max_{tm}\frac{1}{n}\sum_i|\widehat U_{it,m}-\tilde U_{it,m}|^g$, for some $g\geq 2$.
Again, we use the following equality 
\begin{align*}
\tilde U_{it, m}-\widehat U_{it, m} &= 
\widehat  f_i' (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm} \\
&\qquad + \widehat f_i'(\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m} + \tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i).
\end{align*}
Note that 
$$
\frac{1}{n}\sum_i | \widehat  f_i	|^g\leq  \max_i|\widehat f_i|^{g-2}\frac{1}{n}\sum_i | \widehat  f_i	|^{2}=O_P((\log n)^{g-2}),
$$
$$ 
\frac{1}{n}\sum_i |  \widehat  f_i -H'\tilde f_i|^g\leq \max_i|\widehat f_i-H'\tilde f_i|^{g-2} \frac{1}{n}\sum_i |  \widehat  f_i -H'\tilde f_i|^2=O_P(\Delta_F^2\Delta_{\max}^{g-2}),
$$ 
and
$$ 
\max_{tm}|\frac{1}{n}\widehat F'\tilde U_{t, m}  |^g=(\max_{tm}|\frac{1}{n}\widehat F'\tilde U_{t, m}  |^2)^{g/2}=O_P((\frac{\log(pT)}{n})^{g/2}+\Delta_F^g).
$$
Thus,
\begin{align}\label{ed.5}
\begin{split}
\max_{tm}\frac{1}{n}\sum_i(\widehat U_{it,m}-\tilde U_{it,m})^g &\leq \max_{tm}| \tilde \lambda_{tm}    |^g\frac{1}{n}\sum_i|   \widehat  f_i|^g O_P( \Delta_F^g) \\
&\qquad + O_P(1)\frac{1}{n}\sum_i | \widehat  f_i	|^g	 \max_{tm}|\frac{1}{n}\widehat F'\tilde U_{t, m}  |^g \\
&\qquad + \max_{tm}| \tilde\lambda_{tm}|^g   \frac{1}{n}\sum_i |  \widehat  f_i -H'\tilde f_i|^g O_P(1) \\
&\leq O_P\left(\Delta_F^g(\log n)^{g-2}+(\frac{\log(pT)}{n})^{g/2}(\log n)^{g-2}+\Delta_F^2\Delta_{\max}^{g-2}\right).
\end{split}
\end{align} 
When $g=2$, $ \max_{tm}\frac{1}{n}\sum_i(\widehat U_{it,m}-\tilde U_{it,m})^2=o_P(1)$, yielding  $\max_{tm}\frac{1}{n}\sum_i\widehat U_{it,m}^2\leq O_P(1)$.  Hence, uniformly in $m\leq p, t\leq T$,
\begin{align*}
\frac{1}{n}\sum_i\Var^*(w_i^U\widehat f_i\widehat U_{it,m}) &= \frac{1}{n}\sum_i \widehat f_i^2\widehat U_{it,m}^2 \\
&\leq \frac{2}{n}\sum_i \widehat f_i^2\tilde  U_{it,m}^2 + \frac{2}{n}\sum_i \widehat f_i^2(\widehat U_{it,m}-\tilde U_{it,m})^2 \\
&\leq \frac{4}{n}\sum_i( \widehat f_i-H'\tilde f_i)^2\tilde  U_{it,m}^2 + O_P(1)\frac{1}{n}\sum_i \tilde f_i^2\tilde  U_{it,m}^2 \\
&\qquad + \frac{1}{n}\sum_i   \widehat  f_i^4 \tilde \lambda_{tm} ^2O_P(\Delta_F^2) + \frac{1}{n}\sum_i \widehat f_i^4 \left( \frac{1}{n}\widehat F'\tilde U_{t, m}\right)^2 \\
&\qquad + \frac{1}{n}\sum_i \widehat f_i^2(\tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i))^2 \\
&\leq O_P(\Delta_F^2)\max_{itm}\tilde U_{it,m}^2+O_P(1)+O_P(\Delta_F^2\log n) \\
&\qquad +O_P(\log n)\left(\Delta_{fum}^2+\frac{\log(pT)}{n}\right) \\
&= O_P\left(\Delta_F^2\log(nTp)+1\right) = O_P(1).
%  \leq O_P(\log n)\max_{tm}\frac{1}{n}\sum_i\widehat U_{it,m}^2=O_P(\log n).
\end{align*}

It then follows from the union bound and Bernstein inequality for independent data that
\begin{align*}
\max_{mt}  \| \frac{1}{n}\widehat F^{'}\tilde U_{t, m}^*\|_2 & =\max_{mt} \|\frac{1}{n}\sum_i w_i^U\widehat f_i\widehat U_{it,m}\|_2 \cr
&=O_{P^*}\left(\sqrt{\max_{tm}\frac{1}{n}\sum_i\Var^*(w_i^U\widehat f_i\widehat U_{it,m})\frac{\log (pT)}{n}}\right) \\
&=O_{P^*}\left(\sqrt{ \frac{\log (pT)}{n}}\right).
\end{align*}       
        
Next, note 
\begin{align*}
\frac{1}{T}\sum_t\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}^* \widehat f_i\|_2^2
= \frac{1}{T}\sum_t\|\frac{1}{n}\sum_{i=1}^n( \tilde \epsilon_{it}^*+\widehat\alpha\tilde\eta_{it}^*) \widehat f_i\|_2^2.
\end{align*}
Here we just bound $  \frac{1}{T}\sum_t\|\frac{1}{n}\sum_{i=1}^n \tilde \eta_{it}^* \widehat f_i\|_2^2$.  Note that bounding $\frac{1}{T}\sum_t\|\frac{1}{n}\sum_{i=1}^n \tilde \epsilon_{it}^* \widehat f_i\|_2^2$ follows from the same argument. Recall that we have taken $f_i$ to be a scalar for simplicity. We then have    
\begin{align*}
E^*\frac{1}{T}\sum_t\left(\frac{1}{n}\sum_{i=1}^n \tilde \eta_{it}^* \widehat f_i\right)^2 &= \frac{1}{T}\sum_t\Var^*(\frac{1}{n}\sum_{i=1}^n w_i^D\widehat\eta_{it} \widehat f_i) \\
&= \frac{1}{Tn^2}\sum_{it}(\widehat\eta_{it} \widehat f_i)^2 \\
&\leq \frac{2}{Tn^2}\sum_{it}(\tilde\eta_{it} \widehat f_i)^2+\frac{2}{Tn^2}\sum_{it}(\widehat\eta_{it}-\tilde\eta_{it})^2 \max_i\widehat f_i^2 \\
&\leq \frac{2}{Tn^2}\sum_{it}(\tilde\eta_{it} \tilde f_i)^2O_P(1)
+O_P\left(\frac{1}{n}\Delta_F^2\right)\max_i\frac{1}{T}\sum_{t}\tilde\eta_{it}^2  +O_P\left(\frac{\log n}{n}\psi_n^2\right) \\
&= O_P\left(\frac{1}{n}+\frac{\Delta_F^2}{n}(1+\sqrt{\frac{\log n}{T}})+\frac{\log n}{n}\psi_n^2\right) \\
&= O_P\left(\frac{1}{n}\right),
\end{align*}
where we used $\log n\psi_n^2=o(1)$ from (\ref{eb.3a}).
 
         
(v) We have
$\max_{m} \|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}^*  	 \frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*   \widehat  f_i^{*'}	\|_F\leq \sum_{l=1}^4 A_l$. Each $A_l$ is defined and bounded below.
\begin{align*}
A_1 &= \max_{m} \left\|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}^*  	 \frac{1}{n}(\widehat F^{*}-\widehat FH^*)'\tilde U_{t, m}^*  ( \widehat  f_i^{*}- H^{*'}\widehat f_i)'	\right\|_F \\
&\leq \frac{1}{T}\sum_{t=1}^T\left\|\frac{1}{n}\sum_{i=1}^n\tilde e_{it}^*  	 ( \widehat  f_i^{*}- H^{*'}\widehat f_i)\right\|_2  \max_{tm}\left\|\frac{1}{n}(\widehat F^{*}-\widehat FH^*)'\tilde U_{t, m}^* \right\|_2 \\
&= O_{P^*}(\Delta_{fe}^{*}\Delta_{fum}^*) =^{(a)} o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right). \\
A_2 &= \max_{m} \left\|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}^*  	 \frac{1}{n}\widehat F^{'}\tilde U_{t, m}^*   \widehat  f_i^{'}	\right\|_F \\
&=o_{P^*}\left(\sqrt{ \frac{\log p}{nT}}\right) \text{ by part (iii).} \\
A_3 &= \max_{m} \left\|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}^*  	 \frac{1}{n}\widehat F^{'}\tilde U_{t, m}^* (  \widehat  f_i^*-H^{*'}\widehat f_i	)'\right\|_F \\
&\leq \max_{mt}  \left\| \frac{1}{n}\widehat F^{'}\tilde U_{t, m}^*\right\|_2  \frac{1}{T}\sum_{t=1}^T \left\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}^*  	 (  \widehat  f_i^*-H^{*'}\widehat f_i	)'\right\|_F \\
&\leq \max_{mt}  \left\| \frac{1}{n}\widehat F^{'}\tilde U_{t, m}^*\right\|_2  O_{P^*}\left(\Delta_{fe}^*\right) \\
&= O_{P^*}\left(\sqrt{ \frac{\log (pT)}{n}}\Delta_{fe}^*\right) =^{(b)} o_{P^*}\left(\sqrt{ \frac{\log p}{nT}}\right) \\
A_4 &= \max_{m} \left\|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T 	 \frac{1}{n}(\widehat FH^*-\widehat F^*)'\tilde U_{t, m}^*   \widehat  f_i^{'}	\tilde e_{it}^* \right\|_F \\
&\leq O_{P^*}(\Delta_{fum}^*) \left(\frac{1}{T}\sum_t\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}^* \widehat f_i\|_2^2\right)^{1/2} \\
&\leq^{(c)} O_{P^*}\left(\Delta_{fum}^*\sqrt{\frac{1}{n}}\right)=^{(d)}o_{P^*}\left(\sqrt{ \frac{\log p}{nT}}\right).
\end{align*}
In the above derivations, $\Delta_{fe}^*$ is taken such that 
$$
\frac{1}{T}\sum_{t=1}^T\left\|\frac{1}{n}\sum_{i=1}^n\tilde e_{it}^*  	 ( \widehat  f_i^{*}- H^{*'}\widehat f_i)\right\|_2  ^2=O_{P^*}(\Delta_{fe}^{*2}).
$$
Also, the sequence $\Delta_{fum}^*$  is taken such that 
$$ 
\max_{tm}\left\|\frac{1}{n}(\widehat F^{*}-\widehat FH^*)'\tilde U_{t, m}^* \right\|_2^2=O_{P^*}(\Delta_{fum}^{*2}).
$$
Then (c) follows from part (iv) of the present lemma. We now prove (b) by showing that $\Delta_{fe}^*\leq \Delta_F^*$, then (b) follows from Assumption \ref{a6.2} that $\Delta_{F}^{*2}=o\left(\frac{\log p}{T (\log pT)}\right)$.  In addition, we prove (d) that $\Delta_{fum}^{*2}\leq \Delta_F^{*2}=o\left(\frac{\log p}{T  }\right)$ and (a) that $\Delta_{fe}^*\Delta_{fum}^*=o\left(\sqrt{\frac{\log p}{nT}}\right)$ in the following. 
 

\textbf{Show} $\mathbf{\Delta_{fe}^*\leq \Delta_F^*}$: By Cauchy-Schwarz,
$$
\frac{1}{T}\sum_{t=1}^T\left\|\frac{1}{n}\sum_{i=1}^n\tilde e_{it}^*( \widehat  f_i^{*}- H^{*'}\widehat f_i)\right\|_2^2 \leq \frac{1}{nT}\sum_{it}\tilde e_{it}^{*2}	\frac{1}{n}\sum_{i=1}^n \left\| \widehat  f_i^{*}- H^{*'}\widehat f_i\right\|_2^2 = O_{P^*}(\Delta_F^{*2}). 
$$
Hence, we can always take $\Delta_{fe}^*$ so that $\Delta_{fe}^*\leq\Delta_{F}^*$. 
 

\textbf{Show} $\mathbf{\Delta_{fum}^*\leq \Delta_F^*}$: Let $X_{i,tm}^*=\widehat U_{it,m}^{2}(w_i^{U})^2$. For a generic $C>0$, by (\ref{ed.5}) with $g=4$, we have
\begin{align*}
\max_{tm}\frac{1}{n}\sum_i\Var^*(X^*_{i,tm}) &\leq \max_{tm}C\frac{1}{n}\sum_i\widehat U_{it,m}^4 \\
&\leq C \max_{tm}\frac{1}{n}\sum_i(\widehat U_{it,m}-\tilde U_{it,m})^4+C \max_{tm}\frac{1}{n}\sum_i \tilde U_{it,m}^4 \\
&\leq O_P(1) + O_P\left(\Delta_F^4(\log n)^{2}+(\frac{\log(pT)}{n})^{2}(\log n)^{2}+\Delta_F^2\Delta_{\max}^{2}\right) = O_P(1).
\end{align*}
Applying a Bernstein inequality for independent data and union bounds and noting that $E^*X_{i,tm}^*=\widehat U_{it,m}^2$, we have
\begin{align*}
\max_{tm} \frac{1}{n} \sum_i X_{i,tm}^* &\leq \max_{tm}\left|\frac{1}{n}\sum_i(X_{i,tm}^*-E^*X_{i,tm}^*)\right|+\max_{tm}\frac{1}{n}\sum_i\widehat U_{it,m}^2 \\
&= O_{P^*}\left(\sqrt{\frac{\log(pT)}{n}}+1\right)=O_{P^*}(1).
\end{align*}
Now recall that $\Delta_{fum}^*$ is taken to satisfy
$$
\max_{tm}\|\frac{1}{n}\sum_i(\widehat f_i^{*}-H^{*'}\widehat f_i)\tilde U_{it, m}^* \|_2^2=O_{P^*}(\Delta_{fum}^{*2}).
$$       
By Cauchy-Schwarz, we have
\begin{align}\label{ec.24}
\max_{tm}\left\|\frac{1}{n}\sum_i(\widehat f_i^{*}-H^{*'}\widehat f_i)\tilde U_{it, m}^* \right\|_2^2 &\leq O_{P^*}(\Delta_F^{*2})\max_{tm}\frac{1}{n}\sum_i\tilde U_{it,m}^{*2} \\
&= O_{P^*}(\Delta_F^{*2})\max_{tm}\frac{1}{n}\sum_iX_{i,tm}^*=O_{P^*}(\Delta_F^{*2}).
\end{align}
Hence, we can take $\Delta_{fum}^*$ so that $\Delta_{fum}^*\leq\Delta_{F}^*$. 

Therefore, $\Delta_{fe}^{*}\Delta_{fum}^*\leq\Delta_F^{*2}=o\left(\frac{\log p}{nT}\right).$ $\blacksquare$
 

 
\begin{lemma}\label{ld.3} Recall that $\widehat\gamma_y$ is the post-lasso estimator based on the original data, used as the coefficient to generate the bootstrap data. 

(i) $\underset{m}{\max} \left|\frac{1}{T} \sum_{t=1}^T  \frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*   \widehat  \delta_{yt}'\right| = O_{P^*}\left(\sqrt{\frac{\log p}{n}\left( \frac{\log(n)\log(pT)}{n} +\Delta_{\max}^2+\frac{\log(pn)}{T} \right)}\right).$\\
(ii) $\underset{m,v\leq p}{\max} \left| \frac{1}{T}  \sum_{k=1}^p  \sum_{t=1}^T    \widehat \lambda_{tm}   		\frac{1}{n}\widehat F^{*'}\tilde U_{t, v}^* \widehat\gamma_{y} \right| = O_{P^*}\left(\sqrt{\frac{|J|_0^2\log(p)}{n}\left( \frac{\log(n)\log(pT)}{n}+ \Delta_{\max}^2+\frac{\log(pn)}{T}\right)}\right)$.\\
(iii) $\underset{m}{\max} \left| \frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T 	\frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*  	\frac{1}{n}\widehat F^{*'}\tilde U_{t, k}^* \widehat\gamma_{y} \right|=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$.\\
(iv) $\underset{m,k\leq p}{\max} \left|\frac{1}{nT} \sum_{it}\tilde U^*_{it,k}  \widehat  f_i  \widehat \lambda_{tm}  \right| = O_{P^*}\left(\sqrt{\frac{\log p}{n}}\right)\sqrt{\frac{ \log(n)\log(pT)}{n}+ \Delta_{\max}^2+\frac{\log(pn)}{T}}.$
\end{lemma}
  
  
\proof 
(i) \textbf{Bound} $\mathbf{\frac{1}{T}\sum_t (\widehat  \delta_{yt}-H^{-1}\tilde\delta_{yt})^2}$: We have the following equality:
\begin{align*}
\widehat\delta_{yt}-H^{-1}\tilde\delta_{yt}=
(\widehat F'\widehat F)^{-1}\widehat F' (\tilde FH-\widehat F)H^{-1}\tilde\delta_{yt}
+(\widehat F'\widehat F)^{-1}\widehat F' \tilde U_t\gamma_y
+(\widehat F'\widehat F)^{-1}\widehat F' \tilde e_t.
\end{align*}
Hence, using Lemma \ref{la.1}, we have
$$
\frac{1}{T}\sum_t (\widehat  \delta_{yt}-H^{-1}\tilde\delta_{yt})^2=O_P(\Delta_F^2+\frac{|J|_0}{n} +\Delta_{fe}^2)=O_P(|J|_0^2\Delta_F^2+\frac{|J|_0}{n} )
$$
where we used Cauchy-Schwarz to bound   $\Delta_{fe}^2\leq\Delta_F^2$.
  
 
\textbf{Bound} $\mathbf{\max_m|   \frac{1}{T} \sum_{t=1}^T  \frac{1}{n} \widehat F \tilde U_{t, m}^*   \widehat  \delta_{yt}'  |}$: Let $\mathcal{X}_{im}= w_i^U\frac{1}{T} \sum_{t=1}^T \widehat f_i \widehat U_{it, m}   \widehat  \delta_{yt}$.  
\begin{align*}
\frac{1}{n}\sum_i\Var^*(\mathcal{X}_{im}) &= \frac{1}{n}\sum_i\left(\frac{1}{T} \sum_{t=1}^T \widehat f_i \widehat U_{it, m}   \widehat  \delta_{yt}\right)^2 \\
&= \frac{1}{n}\sum_i\widehat f_i^2\left(\frac{1}{T} \sum_{t=1}^T  \widehat U_{it, m}   \widehat  \delta_{yt}\right)^2 \\
&\leq O_P(1) \left(\frac{1}{T} \sum_{t=1}^T  \widehat U_{it, m}   \widehat  \delta_{yt}\right)^2 \\
&\leq O_P(1) \left(\frac{1}{T} \sum_{t=1}^T ( \widehat U_{it, m} -\tilde U_{it,m})  \widehat  \delta_{yt}\right)^2 + O_P(1) \left(\frac{1}{T} \sum_{t=1}^T   \tilde U_{it, m}   (\widehat  \delta_{yt}-H^{-1}\tilde\delta_{yt})\right)^2 \\
&\qquad +O_P(1) \left(\frac{1}{T} \sum_{t=1}^T   \tilde U_{it, m}    \tilde \delta_{yt}\right)^2 \\
&\leq O_P(1)\max_{im}\frac{1}{T}\sum_t ( \widehat U_{it, m} -\tilde U_{it,m})^2 \\
&\qquad + O_P\left( 1+\sqrt{\frac{\log(pn)}{T}}\right)\frac{1}{T}\sum_t (\widehat  \delta_{yt}-H^{-1}\tilde\delta_{yt})^2 + O_P\left({\frac{\log(pn)}{T}}\right) \\
&=^{(a)} O_P\left( \log(n)\frac{\log(pT)}{n}+\log (n)\Delta_{F}^2+\Delta_{\max}^2+\frac{\log(pn)}{T}\right) \\
&\qquad + O_P\left( 1+\sqrt{\frac{\log(pn)}{T}}\right)\left(|J|_0^2\Delta_F^2+\frac{|J|_0}{n} \right)\\
&= O_P\left( \log(n)\frac{\log(pT)}{n} +\Delta_{\max}^2+\frac{\log(pn)}{T} \right)
\end{align*}
where (a) follows from (\ref{ed.6}) and the bound is uniform in $m\leq p$. Therefore, 
\begin{align*}
\max_m \left|   \frac{1}{T} \sum_{t=1}^T  \frac{1}{n} \widehat F \tilde U_{t, m}^*   \widehat  \delta_{yt}'  \right| &= \max_m \left|  \frac{1}{n}\sum_i  \mathcal{X}_{im}  \right| \\
&\leq O_{P^*}\left(\sqrt{\frac{\log p}{n}( \frac{\log(n)\log(pT)}{n} +\Delta_{\max}^2+\frac{\log(pn)}{T} )}\right).
\end{align*}


\textbf{Bound} $\mathbf{\max_m| \frac{1}{T} \sum_{t=1}^T  \frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*   \widehat\delta_{yt}'|}$:
\begin{align*}
\max_m & \left|\frac{1}{T} \sum_{t=1}^T  \frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*   \widehat  \delta_{yt}' \right| \\ 
&\leq \max_m \left|\frac{1}{T} \sum_{t=1}^T  \frac{1}{n}(\widehat F^{*}-\widehat FH^*)'\tilde U_{t, m}^*   \widehat  \delta_{yt}'\right| + \max_m\left|\frac{1}{T} \sum_{t=1}^T  \frac{1}{n} \widehat F \tilde U_{t, m}^*   \widehat  \delta_{yt}'\right| O_{P^*}(1).
\end{align*}
The second term is bounded from above. The first term is $O_{P^*}(\Delta_{ud}^*)=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$.
 

(ii) First, $\gamma_y=\gamma_{y}^0+R_y$ is the decomposition of exactly sparse and remainder terms for $\gamma_y$.  We know that $\|\gamma_y^0\|_1=O(|J|_0)$ and $\|R_y\|_1=o(1)$, so $\|\gamma_y\|_1=O(|J|_0)$.
$$\|\widehat\gamma_{y}\|_1\leq \|\widehat\gamma_{y}-\gamma_{y}\|_1+\|\gamma_{y}\|_1=o_P(1)+O(|J|_0)=O_P(|J|_0).
$$
Also, $\max_{tm}|\widehat\lambda_{tm}|=O_P(1)$ as shown in (\ref{ed.5ad}). So 
by Cauchy-Schwarz, \begin{align*}
\max_{m,v\leq p} \left| \frac{1}{T}  \sum_{k=1}^p  \sum_{t=1}^T    \widehat \lambda_{tm}   		\frac{1}{n}(\widehat F^{*}-\widehat FH^*)'\tilde U_{t, v}^* \widehat\gamma_{y} \right|
 &\leq \max_{m,v\leq p}\left( \frac{1}{nT}\sum_{it} 	 \tilde U_{it, v}^{*2}   \right)^{1/2}O_{P^*}(\Delta_F^*|J|_0) \\
&= O_{P^*}(\Delta_F^*|J|_0).
\end{align*}
 
Secondly, 
$$
\max_{m,v\leq p} \left| \frac{1}{T}  \sum_{k=1}^p  \sum_{t=1}^T \widehat \lambda_{tm} \frac{1}{n}\widehat F^{'}\tilde U_{t, v}^* \widehat\gamma_{y} \right| = \max_{m,v\leq p}\left| \frac{1}{Tn}  \sum_{k=1}^p  \sum_{t=1}^T  \sum_i  \widehat \lambda_{tm} \widehat f_i\widehat U_{it, v}w_i^U\widehat\gamma_{y} \right|.
$$ 
Let $\mathcal{X}_{i,mv}= \frac{1}{T}  \sum_{k=1}^p  \sum_{t=1}^T    \widehat \lambda_{tm}   	 \widehat f_i\widehat U_{it, v}w_i^U\widehat\gamma_{y}$. Then 
\begin{align}
\nonumber
\max_{mv}\frac{1}{n}\sum_i\Var^*(\mathcal{X}_{i,mv}) &= \max_{mv}\frac{1}{n}\sum_i\left( \frac{1}{T}  \sum_{k=1}^p  \sum_{t=1}^T\widehat \lambda_{tm}\widehat f_i\widehat U_{it, v}\widehat\gamma_{y}\right)^2 \\
\nonumber
&\leq \max_{mv} \frac{1}{n}\sum_i \widehat f_i^2 \left( \frac{1}{T}    \sum_{t=1}^T    \widehat \lambda_{tm}   	\widehat U_{it, v}\right)^2 O_P(|J|_0^2) \\
\nonumber
&\leq \max_{imv}\left( \frac{1}{T} \sum_{t=1}^T \widehat \lambda_{tm} \widehat U_{it, v}\right)^2 O_P(|J|_0^2) \\
\nonumber
&\leq O_P(|J|_0^2) \max_{imv} \left(\frac{1}{T} \sum_{t=1}^T ( \widehat U_{it, v} -\tilde U_{it,v})  \widehat  \lambda_{tm}\right)^2 \\
\nonumber
&\qquad + O_P(|J|_0^2) \max_{imv}\left(\frac{1}{T} \sum_{t=1}^T   \tilde U_{it, v}   (\widehat  \lambda_{tm}-H^{-1} \tilde \lambda_{tm})\right)^2 \\
\nonumber
&\qquad +O_P(|J|_0^2)\max_{imv} \left(\frac{1}{T} \sum_{t=1}^T   \tilde U_{it, v}    \tilde \lambda_{tm}\right)^2 \\
\nonumber
&\leq O_P\left({\frac{|J|_0^2\log(pn)}{T}}\right) + O_P(|J|_0^2)\max_{iv}\frac{1}{T}\sum_t ( \widehat U_{it, v} -\tilde U_{it,v})^2 \\
\nonumber
&\qquad + O_P\left( 1+\sqrt{\frac{\log(pn)}{T}}\right)\max_m\frac{|J|_0^2}{T}\sum_t (\widehat  \lambda_{tm}-H^{-1} \tilde \lambda_{tm})^2 \\
\nonumber
&=^{(a)} O_P\left( \log(n)\frac{\log(pT)}{n}+ \Delta_{\max}^2+\frac{\log(pn)}{T}\right)|J|_0^2 \\
\nonumber
&\qquad + O_P\left( 1+\sqrt{\frac{\log(pn)}{T}}\right) |J|_0^2\left(\Delta_F^2+\frac{\log (pT)}{n}\right)
\\
\label{ed.8}
&= O_P\left( \log(n)\frac{\log(pT)}{n}+ \Delta_{\max}^2+\frac{\log(pn)}{T}\right)|J|_0^2
\end{align}
where in (a) we used
$$
\widehat\lambda_{tm}-H^{-1}\tilde \lambda_{tm}=(\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm}+(\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m}
$$
and thus 
$$
\max_{tm}|  \widehat\lambda_{tm}-H^{-1}\tilde \lambda_{tm}|=O_P(\Delta_F +\sqrt{\frac{\log(pT)}{n}}).
$$ 
Therefore,
\begin{align*}
\max_{mv}\left|\frac{1}{T}\sum_{k=1}^p\sum_{t=1}^T \widehat \lambda_{tm} \frac{1}{n}\widehat F^{'}\tilde U_{t, v}^* \widehat\gamma_{y} \right| &= \max_{mv}\left|\frac{1}{n}\sum_i \mathcal{X}_{i,mv}\right| \\
&= O_{P^*}\left(\sqrt{\frac{|J|_0^2\log(p)}{n}( \frac{\log(n)\log(pT)}{n}+ \Delta_{\max}^2+\frac{\log(pn)}{T})}\right).
\end{align*}


(iii) Note that 
$$
\max_m| \frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T 	\frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*  	\frac{1}{n}\widehat F^{*'}\tilde U_{t, k}^* \widehat\gamma_{y} |
\leq \max_{m}  \frac{1}{T}  \sum_{t=1}^T| 	\frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*  	  | ^2O_P(|J|_0).
$$
By repeatedly using the triangle inequality, we have
\begin{align*}
\max_m\frac{1}{T} \sum_{t=1}^T \left|\frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^* \right|^2 &\leq 
\max_m\frac{1}{T} \sum_{t=1}^T \left|\frac{1}{n}(\widehat F^{*}-\widehat FH^*)'\tilde U_{t, m}^*\right|^2 & (A_1) \\
&\quad + O_{P^*}(1)\max_m \frac{1}{T} \sum_{t=1}^T \left|\frac{1}{n}\sum_iw_i^U(\widehat f_i-H'\tilde f_i)\widehat U_{it, m}\right|^2 & (A_2) \\
&\quad + O_{P^*}(1)\max_m \frac{1}{T} \sum_{t=1}^T \left|\frac{1}{n}\sum_iw_i^U \tilde f_i(\widehat U_{it, m} -\tilde U_{it,m})\right|^2 & (A_3) \\
%&&+O_{P^*}(1)\max_m \frac{1}{T}   \sum_{t=1}^T |	\frac{1}{n}\sum_iw_i^U \tilde f_i\tilde U_{it, m}  	\frac{1}{n}\sum_j\widehat f_j \widehat U_{jt, m} w_j^U  |\cr
&\quad +O_{P^*}(1) \left|\frac{1}{n}\sum_iw_i^U \tilde f_i\right|^2\max_m \frac{1}{T} \sum_{t=1}^T \left|\bar U_{\cdot t, m}\right|^2 & (A_4) \\
&\quad +O_{P^*}(1)\max_m \left|\frac{1}{n}\sum_iw_i^U \tilde f_i \bar U_{i\cdot, m}\right|^2 & (A_5) \\
&\quad +O_{P^*}(1) \left|\frac{1}{n}\sum_iw_i^U \tilde f_i \right|^2 \max_m \left| \bar{\bar U}_{ m} \right|^2 & (A_6) \\
&\quad +O_{P^*}(1)	\frac{1}{n}\sum_i	\frac{1}{n} (w_i^U)^2 \tilde f_i^2 \max_m \frac{1}{T} \sum_{t=1}^T EU_{it, m}^2 & (A_7) \\
&\quad +O_{P^*}(1)\max_m \frac{1}{T} \sum_{t=1}^T \frac{1}{n}\sum_i \frac{1}{n} (w_i^U)^2 \tilde f_i^2  (U_{it, m}^2-EU_{it, m}^2) & (A_8) \\
&\quad +O_{P^*}(1)\max_m \frac{1}{T} \sum_{t=1}^T \frac{1}{n}\sum_i \frac{1}{n}\sum_{j\neq i}w_i^U \tilde f_j w_j^U \tilde f_i  U_{it, m} U_{jt, m} & (A_9) \\
&:=\sum_{l=1}^{9} A_l.
%&&+O_{P^*}(1)\max_m| \frac{1}{T}   \sum_{t=1}^T 	\frac{1}{n}\sum_iw_i^U \tilde f_i\tilde U_{it, m}  	\frac{1}{n}\sum_j\tilde f_j \tilde U_{jt, m} w_j^U ( -\tilde\gamma_{ym})|\cr
\end{align*}
We shall show each $A_l=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$. Note that we shall show that the last two terms are $o_{PP^*}\left(\sqrt{\frac{\log p}{nT}}\right)$, where $o_{PP^*}(1)$ means convergence in probability with respect to the joint distribution of the original data and the bootstrap weights $\{w_i^U\}$. This convergence then implies by the Markov inequality that they are also $o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$, i.e., that they converge in probability with respect to the bootstrap sampling distribution (see Lemma 3 of \cite{cheng2010bootstrap}). 
 
\textbf{Term $\mathbf{A_1}$.}   It is clear that term $(A_1)$ is $O_{P^*}(\Delta_{fum}^{*2})$.
    %First, by the same proof of (\ref{ed.8}),  $	\frac{1}{n}\sum_i( \frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T  \widehat U_{it, k}\widehat\gamma_{y} \widehat f_i   )^2 =O_P( \log(n)\frac{\log(pT)}{n}+ \Delta_{\max}^2+\frac{\log(pn)}{T})|J|_0^2. $
 %Hence
%$$E^*   | \frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T 	\frac{1}{n}\widehat F^{'}\tilde U_{t, k}^* \widehat\gamma_{y} |^2= 	\frac{1}{n^2}\sum_i( \frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T  \widehat U_{it, k}\widehat\gamma_{y} \widehat f_i   )^2=O_P(\frac{ \log(n)\log(pT)}{n}+ \Delta_{\max}^2+\frac{\log(pn)}{T})\frac{|J|_0^2}{n}.$$ Therefore,
%\begin{eqnarray*}&&E^*    \frac{1}{T}\sum_t( \sum_{k=1}^p   	   	\frac{1}{n}\widehat F^{'}\tilde U_{t, k}^* \widehat\gamma_{y})^2=\frac{1}{Tn^2}\sum_t	 \sum_i ( \sum_{k=1}^p   	   \widehat f_i\widehat U_{it, k} \widehat\gamma_{y})^2\leq O_P(\frac{|J|_0^2\log n}{n}) \cr
%&&\max_m| \frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T 	\frac{1}{n}(\widehat F^{*}-\widehat FH^*)'\tilde U_{t, m}^*  	\frac{1}{n}\widehat F^{*'}\tilde U_{t, k}^* \widehat\gamma_{y} |\leq    (\frac{1}{T}\sum_t( \sum_{k=1}^p   	   	\frac{1}{n}\widehat F^{*'}\tilde U_{t, k}^* \widehat\gamma_{y})^2)^{1/2}O_{P^*}(\Delta_{fum}^*)\cr&\leq&   O_{P^*}(|J|_0\Delta_{fum}^{*2})+(\frac{1}{T}\sum_t( \sum_{k=1}^p   	   	\frac{1}{n}\widehat F^{'}\tilde U_{t, k}^* \widehat\gamma_{y})^2)^{1/2}O_{P^*}(\Delta_{fum}^*)
%=  O_{P^*}(|J|_0\Delta_{fum}^{*})(\Delta_{fum}^{*}+\sqrt{\frac{\log n}{n}}).
% | \frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T 	\frac{1}{n}\widehat F^{*'}\tilde U_{t, k}^* \widehat\gamma_{y} |  O_{P^*}(\Delta_{fum}^*)\cr
%&\leq&  O_{P^*}(|J|_0\Delta_{fum}^{*2})+| \frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T 	\frac{1}{n}\widehat F^{'}\tilde U_{t, k}^* \widehat\gamma_{y} |  O_{P^*}(\Delta_{fum}^*)\cr
%&\leq&  O_{P^*}(|J|_0\Delta_{F}^{*2})+| \frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T 	\frac{1}{n}\widehat F^{'}\tilde U_{t, k}^* \widehat\gamma_{y} |  O_{P^*}(\Delta_{F}^*)=o_{P^*}(\sqrt{\frac{\log p}{nT}}) .
%\end{eqnarray*}
%Below  we show $\Delta_{fum}^*\leq \Delta_F^*$. In fact,  by Cauchy-Schwarz,
%\begin{eqnarray*}
%&&   \max_{tm}\|\frac{1}{n}(\widehat F^{*}-\widehat FH^*)'\tilde U_{t, m}^* \|_2^2\leq O_{P^*}(  \Delta_{F}^{*2})\max_{tm}\frac{1}{n}\sum_i\tilde U_{it,m}^2w_i^{U2}
 %\cr
 %&\leq& O_{P^*}(  \Delta_{F}^{*2})\max_{tm}\frac{1}{n}\sum_i\tilde U_{it,m}^2+  O_{P^*}(  \Delta_{F}^{*2})\max_{tm}|\frac{1}{n}\sum_i\tilde U_{it,m}^2(w_i^{U2}-E^*w_i^{U2})|\cr
% &\leq& O_{P^*}(  \Delta_{F}^{*2}) + O_{P^*}(  \Delta_{F}^{*2}) \sqrt{\frac{\log(pT)}{n}\max_{tm}\frac{1}{n}\sum_iVar^*(\tilde U_{it,m}^2w_i^{U2})}\cr
% &\leq&O_{P^*}(  \Delta_{F}^{*2}) (1+ \sqrt{\frac{\log(pT)}{n}\max_{tm}\frac{1}{n}\sum_i\tilde U_{it,m}^4 })=O_{P^*}(  \Delta_{F}^{*2}).
% \end{eqnarray*}
By (\ref{ec.24}),  $\Delta_{fum}^*\leq \Delta_F^*$. Hence $ |J|_0\Delta_{fum}^{*2}\leq  |J|_0\Delta_{F}^{*2}=o\left(\sqrt{\frac{\log p}{nT}}\right)$.  
 
  
\textbf{Term $\mathbf{A_2}$.} Term (A.2) is bounded by $\frac{1}{n}\sum_i(w_i^U(\widehat f_i-H'\tilde f_i))^2 \max_m \frac{1}{T}   \sum_{t=1}^T \frac{1}{n}\sum_j\widehat U_{jt, m}^2$.
Note that $E^*\left[\frac{1}{n}\sum_i(w_i^U(\widehat f_i-H'\tilde f_i))^2\right]=O_P(\Delta_F^2)$, and $\max_m \frac{1}{T}   \sum_{t=1}^T \frac{1}{n}\sum_j\widehat U_{jt, m}^2=O_P(1)$. Hence, 
$|J|_0A_2=O_{P^*}(|J|_0\Delta_F^2)O_P(1)=O_{P^*}(|J|_0\Delta_F^2)=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$, where we used $O_{P^*}\times O_P=O_{P^*}$ using Lemma 3 of \cite{cheng2010bootstrap}. 
 

\textbf{Term $\mathbf{A_3\sim A_6}$.}  These terms can be shown to be $o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$ similarly to the other terms with provided details. We omit the details for brevity. 
 %  \textbf{Terms $A_{7}, A_8$.}  First note that $  |  \frac{1}{n}\sum_{j}\tilde f_j  w_j^U  |  =O_{P^*}(\frac{1}{\sqrt{n}})$ and  $\max_{kt}     |  \bar U_{\cdot t, k}|  =O_P(\sqrt{\frac{\log(pT)}{n}})$.
   %Also, 
  % $ \max_{mt} \frac{1}{n}\sum_i\Var^*(w_i^U \tilde f_i  U_{it, m})=\max_{mt} \frac{1}{n}\sum_i\tilde f_i ^2 U_{it, m}^2 =O_P(1)$. Hence \\ $ \max_{mt}  |\frac{1}{n}\sum_iw_i^U \tilde f_i  U_{it, m}  |=O_{P^*}(\sqrt{\frac{\log(pT)}{n}})$. Therefore,
%\begin{eqnarray*}
% &&|J|_0\max_{mk}| \frac{1}{T}   \sum_{t=1}^T 	\frac{1}{n}\sum_i\frac{1}{n}\sum_{j}\tilde f_j w_i^U \tilde f_i  U_{it, m}    \bar U_{\cdot t, k}w_j^U  |
 %\leq \max_{mk}| \frac{1}{T}   \sum_{t=1}^T 	\frac{1}{n}\sum_iw_i^U \tilde f_i  U_{it, m}    \bar U_{\cdot t, k}  | O_{P^*}(\frac{|J|_0}{\sqrt{n}})\cr
 %&\leq& O_{P^*}(\frac{|J|_0}{\sqrt{n}})  \max_{mt}  |\frac{1}{n}\sum_iw_i^U \tilde f_i  U_{it, m}  |
% \max_{kt}     |  \bar U_{\cdot t, k}|  = O_{P^*}(\frac{|J|_0}{\sqrt{n}}\sqrt{\frac{\log(pT)}{n}})O_P(\sqrt{\frac{\log(pT)}{n}})\cr
 %&=^{(a)}& O_{P^*}(\frac{|J|_0}{\sqrt{n}}{\frac{\log(pT)}{n}})=o_{P^*}(\sqrt{\frac{\log p}{nT}}),
 %\end{eqnarray*}
% where in (a) we used $O_{P^*}\times O_P=O_{P^*}$, given by Lemma 3 of \cite{cheng2010bootstrap}. Term $A_8=o_{P^*}(\sqrt{\frac{\log p}{nT}})$ follows very similarly.
 
 
\textbf{Term $\mathbf{A_{7}}$.}  Term $(A_{7})$ is bounded by
$\frac{1}{n}\sum_i\frac{1}{n} \tilde f_i^2  (w_i^U)^2   \max_{itm}| (EU_{it, m}^2)|.$ Also,
$$
E^*\left(\frac{1}{n}\sum_i\frac{1}{n} \tilde f_i^2  (w_i^U)^2 \right) = \frac{1}{n}\sum_i\frac{1}{n} \tilde f_i^2 = O_P\left(\frac{1}{n}\right).
$$ 
Hence, $|J|_0A_{7}=O_{P^*}(\frac{|J|_0}{n})=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$.


\textbf{Term $\mathbf{A_{8}}$.}
Let $W^U=\{w_i^*\}_{i\leq n}$.  Now let $P_{U, F, W^U}$,  $P_{U|F,W^U}$  and $P_{F,W^U}$ respectively  be the probability measures with respect to the joint distribution of $\{U, F, W^U\}$,  the conditional distribution $U|F, W^U$, and the joint distribution of $(F,W^U)$.  Let   $E_{F,W^U}$  be the expectation operator with respect to the distribution of $(F, W^U)$.

If $T=O(1)$, then for $Z_{itm}:=\tilde f_i^2 (U_{it, m}  ^2 -EU_{it, m}  ^2)(w_i^U)^2$,
$A_8\leq\max_{it}|\frac{1}{n}\sum_i\frac{1}{n}Z_{itm}|.$ 
Then conditional on $F, W^U$,
\begin{align*}
\max_{tm}\Var(Z_{itm}|F, W^U) &\leq 
 \tilde f_i^4(w_i^U)^4\max_{tm}\Var(     U_{it, m}  ^2  |F, W^U) \\
&\leq C \tilde f_i^4(w_i^U)^4,
\end{align*}
almost surely in $F,W^U$ for a generic constant $C$ that does not depend on $F,W^U.$   $\forall x>0,$ and $l\leq 8,$
\begin{align*}
P_{U, F, W^U}&\left(A_8>x\right) \\
&\leq P_{F,W^U} (\frac{1}{n} \sum_i(  |w_i^U|^l- E|w_i^U|^l)>C ) P_{F,W^U} (\frac{1}{n} \sum_i(  |f_i|^l- E|f_i|^l)>C) \\
&\qquad +E_{F,W^U}P_{U|F,W^U}\left(A_8>x\right)1\left\{\frac{1}{n} \sum_i\tilde f_i^4(w_i^U)^4<C\right\} \\
&= o(1) \\
&\qquad + E_{F,W^U}P_{U|F,W^U}\left(\max_{it}|\frac{1}{n}\sum_iZ_{itm}|>nx\right)1\{\max_{tm}\Var(Z_{itm}|F, W^U)<C\}.
\end{align*} 
Because $\{U_{i}\}_{i\leq n}$ are independent conditional on $\{w_i^U, F\}$, we can apply the Bernstein inequality for independent data to reach, for $x=M\frac{1}{n}\sqrt{\frac{\log (pT)}{n}}$, for sufficiently large $M>0,$
\begin{align*} 
&E_{F,W^U}P_{U|F,W^U}\left(\max_{it}|\frac{1}{n}\sum_iZ_{itm}|>nx\right)1\{\max_{tm}\Var(Z_{itm}|F, W^U)<C\}\cr
&\leq E_{F,W^U}\exp\left(\log(pT)-\frac{M^2\log(pT)}{ \max_{tm}\Var(Z_{itm}|F, W^U)}\right)1\left\{\max_{tm}\Var(Z_{itm}|F, W^U)<C\right\}=o(1).
\end{align*} 
Hence $A_8=O_P(x)=O_P( \sqrt{\frac{\log (p )}{n^3}})$ when $T=O(1)$. 

When $T\to\infty$,  define $\mathcal{X}_{t,m}=	\frac{1}{n}\sum_i\frac{1}{n} \tilde f_i^2 (U_{it, m}  ^2 -EU_{it, m}  ^2)(w_i^U)^2$, so $A_{8}=\max_m| \frac{1}{T}\sum_{t=1}^T\mathcal{X}_{t,m}|$.  
Then conditional on $F, W^U$,
\begin{align*}
\max_{tm}\Var(\mathcal{X}_{t,m}|F, W^U) &\leq 
\frac{1}{n^4} \sum_i\tilde f_i^4(w_i^U)^4\max_{tm}\Var(     U_{it, m}  ^2  |F, W^U) \\
&\leq C\frac{1}{n^4} \sum_i\tilde f_i^4(w_i^U)^4.
\end{align*}
almost surely in $F,W^U$ for a generic constant $C$ that does not depend on $F,W^U.$   
So
\begin{align*}
P_{U, F, W^U}&\left(\max_{m}| \frac{1}{T}   \sum_{t=1}^T\mathcal{X}_{t,m}|>x\right) \\
&\leq P_{F,W^U} (\frac{1}{n} \sum_i(  |w_i^U|^l- E|w_i^U|^l)>C ) P_{F,W^U} (\frac{1}{n} \sum_i(  |f_i|^l- E|f_i|^l)>C) \\
&\qquad +E_{F,W^U}P_{U|F,W^U}\left(\max_{m}| \frac{1}{T}   \sum_{t=1}^T\mathcal X_{t,m}|>x\right)1\left\{\frac{1}{n} \sum_i\tilde f_i^4(w_i^U)^4<C\right\} \\
&= o(1) \\
&\qquad + E_{F,W^U}P_{U|F,W^U}\left(\max_{m}| \frac{1}{T}   \sum_{t=1}^T\mathcal X_{t,m}|>x\right)1\{\max_{tm}\Var(\mathcal X_{t,m}|F, W^U)<\frac{C_0}{n^3}\}.
\end{align*} 

We now bound $P_{U|F,W^U}\left(\max_{m}| \frac{1}{T}   \sum_{t=1}^T\mathcal X_{t,m}|>x\right)$ conditional on $\max_{tm}\Var(\mathcal X_{t,m}|F, W^U)<\frac{C_0}{n^{3/2}}$. 
 We have $\min_{tm}[\Var(\mathcal X_{t,m}|F)^{-1/2}]\geq C_0n^{3/2}$ for a generic $C_0>0$.
Conditional on $F$,  $\{\mathcal X_{t,m}\}$ is a strong mixing sequence across $t$ with uniform mixing condition (uniform over $m\leq p$) that is bounded by $\exp(-CT^{r})$ (Assumption \ref{a3.2}). Write $\mathcal Z_{t,m}= \mathcal X_{t,m}\Var(\mathcal X_{t,m}|F)^{-1/2}$, whose conditional variance is one.
Recall that by assumption, $\log ^\gamma p=o(n)$ for some $\gamma>2$ and the conditional strong mixing (Assumption \ref{a3.2}) holds for the process  $\{(U_t)\}_{t=-\infty}^{+\infty}$ given $F, W^U$, with mixing coefficient bounded by $\exp(-CT^r)$, $r>1$, and also we assume $\gamma r\geq 2$. 
Let $\bar r= \min\{r, 1\}$, $r_1=(0.5 +\bar r^{-1})^{-1}$,  $c=0.5(\gamma+1)$, then  $r_1<1$, $cr_1> 1$ and $2c\geq 1.$ Because $\bar r\leq r$, the strong mixing condition in Assumption \ref{a3.2} also holds with $\bar r$ in place of $r$. All these constants are independent of $F$. Then apply the Bernstein inequality for strong mixing sequences \cite{merlevede2011bernstein} on $|\frac{1}{T}\sum_t\mathcal Z_{t,m}|$,  for $y=M\frac{(\log p)^c}{\sqrt{T}}$, and sufficiently large $M>0$, for $x=y/(C_0n^{3/2})$,
\begin{align*}
 P_{U|F,W^U}\left(\max_{m}| \frac{1}{T}   \sum_{t=1}^T\mathcal X_{t,m}|>x\right)&\leq p\max_mP_{U|F,W^U}\left( |\frac{1}{T}\sum_t\mathcal Z_{t,m}|>x\min_{tm}[\Var(\mathcal X_{t,m}|F)^{-1/2}]\right)\cr
 &\leq p\max_mP_{U|F,W^U}\left( |\frac{1}{T}\sum_t\mathcal Z_{t,m}|>C_0n^{3/2}x \right)
 \leq  A_1+A_2+A_3
\end{align*}
     where for a generic constant $C$ that is independent of $F$, 
     \begin{eqnarray*}
     A_1&=& pT\exp ( -C (Ty)^{r_1})=\exp(\log(pT) - CM^{r_1} T^{r_1/2} \log^{cr_1} p) =o(1),\quad (cr_1\geq 1)\cr
     A_2&=& p\exp\left(  -C\frac{(Ty)^2}{T} \exp\left(\frac{(Ty)^{r_1(1-r_1)}}{C\log^{r_1}(Ty)}\right)  \right)=o(1),\quad (r_1<1, 2c> 1, Ty\to\infty)\cr
     A_3&=& p \exp\left( -CTy^2 \right)= \exp(\log p-CM^2\log^{2c}p) =o(1),
     \end{eqnarray*}
     and $o(1)$ in the above are non-stochastic converging sequences. (This is because the involved generic constant $C$ in \cite{merlevede2011bernstein} is independent of $F, W^U$.
Hence $$
E_{F,W^U}P_{U|F,W^U}\left(\max_{m}| \frac{1}{T}   \sum_{t=1}^T\mathcal X_{t,m}|>x\right)1\{\max_{tm}\Var(\mathcal X_{t,m}|F, W^U)<\frac{C_0}{n^3}\}  =o(1).
$$
Consequently, $\max_m|\frac{1}{T}\sum_t\mathcal X_{t,m}|=O_P(x)=O_P(\frac{\log^{0.5\gamma+0.5} p}{\sqrt{Tn^3}})$ when $T\to\infty$.

 Combining with the case $T=O(1)$, we have 
$A_8\leq \max_m|\frac{1}{T}\sum_t\mathcal X_{t,m}|= O_P(\frac{\log^{0.5\gamma+0.5} p}{\sqrt{Tn^3}}).$
Because the convergence is with respect to the joint distribution of $(U,F, W^U)$, 
the above result implies, by Lemma 3 of \cite{cheng2010bootstrap},  $|J|_0A_{8}=O_{P^*}\left(|J|_0\frac{\log^{0.5\gamma+0.5} p}{\sqrt{Tn^3}}\right)$. Thus, $|J|_0A_{8}=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$, given that $|J|_0^2\log^{\gamma} p=o(n^2)$.


\textbf{Term $\mathbf{A_{9}}$.} If $T=O(1)$, then
$$A_9\leq 
\max_{mt} \|\  \frac{1}{n}\sum_iw_i^U \tilde f_i  U_{it, m} \|\max_{mt}  \|  \frac{1}{n}\sum_{j}\tilde f_j w_j^U U_{jt, m}\|.
$$
Using the same argument when we bound $A_9$ when $T=O(1)$, we conclude that $A_9=O_P(\frac{\log p}{n})$.

When $T\to\infty$,  define $G_{t,m}=\frac{1}{n}\sum_i\frac{1}{n}\sum_{j\neq i}\tilde f_j w_i^U \tilde f_i  U_{it, m}   	 U_{jt, m} w_j^U$. Then $E(	G_{t,m}|F, W^U)=0$, and almost surely in $F, W^U$, and $A_9\leq\max_{m}\|\frac{1}{T}\sum_tG_{t,m}\|_F$. We only focus on the case $\dim(f_i)=1$.
\begin{align*}
\max_{tm}&\Var(G_{t,m}|F, W^U) \\
&\leq \max_{tm}\sum_i \sum_{j\neq i} \frac{1}{n^4}  \tilde f_j ^2 w_i^U \tilde f_i ^2  w_j^U  w_i^U  w_j^UE (U_{it, m} ^2  	 U_{jt, m}  ^2   	|F, W^U) \\
&\qquad +\max_{tm}\sum_i \sum_{j\neq i} \frac{1}{n^4}  \tilde f_j w_i^U \tilde f_i   w_j^U \tilde f_i w_j^U \tilde f_j  w_i^UE (U_{jt, m}  ^2 	 U_{it, m}  ^2 	  |F, W^U) \\
&\leq \frac{C}{n^2}\frac{1}{n}\sum_i(w_i^U )^2 \tilde f_i ^2   \frac{1}{n}  \sum_{j\neq i} \tilde f_j ^2(  w_j^U)^2.
\end{align*} 
Hence, for $l\leq 4$ and any $x>0$,
\begin{align*}
P_{U, F, W^U} & \left(\max_{m} \left| \frac{1}{T}   \sum_{t=1}^TG_{t,m}\right|>x\right) \\
&\leq P_{F,W^U} \left(\frac{1}{n} \sum_i(  |w_i^U|^l- E|w_i^U|^l)>C \right)+ P_{F,W^U} \left(\frac{1}{n} \sum_i(  |f_i|^l- E|f_i|^l)>C\right)  \\
&\qquad + E_{F,W^U}P_{U|F,W^U}\left(\max_{m}| \frac{1}{T}   \sum_{t=1}^TG_{t,m}|>x\right)1\left\{\frac{1}{n}\sum_i(w_i^U )^2 \tilde f_i ^2   \frac{1}{n}  \sum_{j\neq i} \tilde f_j ^2(  w_j^U)^2<C\right\} \\
&\leq o(1) + E_{F,W^U}P_{U|F,W^U}\left(\max_{m}| \frac{1}{T}   \sum_{t=1}^TG_{t,m}|>x\right)1\left\{\max_{tm}\Var(G_{t,m}|F, W^U)<\frac{C_0}{n^2}\right\}.
\end{align*} 
We note that $\{G_{t,m}\}_{t\leq T}$ is conditionally strong mixing, given $F, W^U$, whose mixing coefficient is independent of $m\leq p, F, W^U$. Thus, we can use the same argument as in bounding term $A_8$ for the case of $T\to\infty$, to conclude  that  when $x=O_P\left(\frac{\log^{0.5\gamma+0.5} p}{\sqrt{Tn^2}}\right)$
$$
E_{F,W^U}P_{U|F,W^U}\left(\max_{m}| \frac{1}{T}   \sum_{t=1}^TG_{t,m}|>x\right)1\left\{\max_{tm}\Var(G_{t,m}|F, W^U)<\frac{C_0}{n^2}\right\}\leq o(1)
$$
 Combining with the case of $T=O(1)$, 
 $A_9=O_P(\frac{\log^{0.5\gamma+0.5} p}{\sqrt{Tn^2}})=O_{P^*}(\frac{\log^{0.5\gamma+0.5} p}{\sqrt{Tn^2}})$, where the second equality is due to Lemma 3 of \cite{cheng2010bootstrap}.
Thus $|J|_0A_{9}=O_{P^*}\left(|J|_0\frac{\log^{0.5\gamma+0.5} p}{\sqrt{Tn^2}}\right)=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$ given that $J^2\log^\gamma p=o(n)$.
Hence, we have proven that 
\begin{align}\label{ed.9}
\begin{split}
\max_m\left| \frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T 	\frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*  	\frac{1}{n}\widehat F^{*'}\tilde U_{t, k}^* \widehat\gamma_{y} \right| &\leq  \max_{m}  \frac{1}{T}  \sum_{t=1}^T \left| 	\frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^* \right| ^2O_P(|J|_0) \\
&=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right).
\end{split}
\end{align}
 
  
  
(iv) Let $G_{i,mk}=\frac{1}{T}\sum_t\widehat U_{it,k}w_i^U\widehat f_i\widehat\lambda_{tm}$.
Then by (\ref{ed.8}),
\begin{align*}
\max_{mk}\frac{1}{n}\sum_i\Var^*(G_{i,mk}) &\leq\max_{mk}
 \frac{1}{n}\sum_i \widehat f_i^2 \left( \frac{1}{T}    \sum_{t=1}^T    \widehat \lambda_{tm}   	\widehat U_{it, k}\right)^2 \\
&= O_P\left(\frac{ \log(n)\log(pT)}{n}+ \Delta_{\max}^2+\frac{\log(pn)}{T}\right).
\end{align*}
Hence, 
\begin{align*}
\max_{m,k\leq p}\left|\frac{1}{nT} \sum_{it}\tilde U^*_{it,k}  \widehat  f_i  \widehat \lambda_{tm}  \right| &=\max_{mk}  \left|\frac{1}{n}\sum_iG_{i,mk}\right| \\
&=O_{P^*}\left(\sqrt{\frac{\log p}{n}}\right)\sqrt{\frac{ \log(n)\log(pT)}{n}+ \Delta_{\max}^2+\frac{\log(pn)}{T}}.
\end{align*}
 
 $\blacksquare$

\begin{lemma}\label{ld.4} 
(i) $ \left\| \frac{1}{n} \sum_{i=1}^n \frac{1}{T} \sum_{t=1}^T(\widehat U_{it}^* -\tilde U_{it}^*)  \widehat  \delta_{yt}'H^{*'-1}(\widehat f_i^*-H^{*'}\widehat f_i) \right\|_{\infty}=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right).$\\
(ii) $ \left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it}^*  -\tilde U_{it}^* )  (\tilde U_{it}^* -\widehat U_{it}^* ) '\widehat\gamma_y \right\|_{\infty} =o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$\\
(iii)
$ \left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T\tilde U_{it}^*    (\tilde U_{it}^* -\widehat U_{it}^* ) '\widehat\gamma_y \right\|_{\infty}=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$.
\end{lemma}
 
 
\proof We shall repeatedly use the following equality: 
\begin{align}\label{ed.5a}
\begin{split}
\tilde U_{it, m}^*-\widehat U_{it, m}^* &= \widehat  f_i^{*'} (\widehat F^{*'}\widehat F^{*})^{-1}\widehat F^{*'}(\widehat FH^*-\widehat F^{*})H^{*-1}\widehat \lambda_{tm} \\
& \qquad + \widehat  f_i^{*'}(\widehat F^{*'}\widehat F^{*})^{-1}\widehat F^{*'}\tilde U_{t, m}^*
  +\widehat\lambda_{tm}'H^{*'-1}(\widehat  f_i^* -H^{*'}\widehat f_i),
\end{split}
\end{align}
where $\widehat F^*$  and $\widehat U^*$ represent the estimators using bootstrapped data.  


(i) First, by (\ref{ed.5ad}), $\max_{tm}|\widehat\lambda_{tm}|=O_P(1)$. Hence, 
$$ 
\max_m \left| \frac{1}{T} \sum_{t=1}^T   \widehat \lambda_{tm}    \widehat  \delta_{yt} \right|=O_P(1)\left(\frac{1}{T}\sum_t\widehat\delta_{yt}^2\right)^{1/2}=O_P(1).
$$
Also, 
\begin{align}
\nonumber
\max_m & \left|\frac{1}{n} \sum_{i=1}^n \frac{1}{T} \sum_{t=1}^T   (\widehat U_{it,m}^* -\tilde U_{it,m}^*)      \widehat  \delta_{yt}'H^{*'-1}(\widehat f_i^*-H^{*'}\widehat f_i) \right| \\
\nonumber
&= \max_m \left|\frac{1}{n} \sum_{i=1}^n    \frac{1}{T} \sum_{t=1}^T   \widehat  f_i^{*'} (\widehat F^{*'}\widehat F^{*})^{-1}\widehat F^{*'}(\widehat FH^*-\widehat F^{*})H^{*-1}\widehat \lambda_{tm}   \widehat  \delta_{yt}'H^{*'-1}(\widehat f_i^*-H^{*'}\widehat f_i) \right| \\
\nonumber
&\qquad + \max_m\left|\frac{1}{n} \sum_{i=1}^n   \frac{1}{T} \sum_{t=1}^T \widehat  f_i^{*'}		 (\widehat F^{*'}\widehat F^{*})^{-1}\widehat F^{*'}\tilde U_{t, m}^*   \widehat  \delta_{yt}'H^{*'-1}(\widehat f_i^*-H^{*'}\widehat f_i) \right| \\
\nonumber
&\qquad + \max_m  \left|\frac{1}{n} \sum_{i=1}^n    \frac{1}{T} \sum_{t=1}^T \widehat\lambda_{tm}'H^{*'-1}(\widehat  f_i^* -H^{*'}\widehat f_i)     \widehat  \delta_{yt}'H^{*'-1}(\widehat f_i^*-H^{*'}\widehat f_i) \right| \\
\nonumber
&\leq \max_m \left| \frac{1}{T} \sum_{t=1}^T   \widehat \lambda_{tm} \widehat\delta_{yt}\right| O_{P^*}(\Delta_F^{*2})+ 
\max_m \left|   \frac{1}{T} \sum_{t=1}^T  \frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*   \widehat  \delta_{yt}' \right| O_{P^*}(\Delta_F^{*}) \\
\label{ed.18a}
\begin{split}
&=^{(a)} O_{P^*}(\Delta_{F}^{*2}) + O_{P^*}\left(\sqrt{\frac{\log p}{n}( \frac{\log(n)\log(pT)}{n} +\Delta_{\max}^2+\frac{\log(pn)}{T} )}\right)\Delta_F^* \\
&=^{(b)} o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)
\end{split}
\end{align}
where (a) follows from Lemma \ref{ld.3}(i) and (b) follows from Assumption \ref{ca.1}.
 

(ii) We have   
\begin{align*}
\max_m & \left| \frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it,m}^*  -\tilde U_{it,m}^* )  (\tilde U_{it}^* -\widehat U_{it}^* ) '\widehat\gamma_y \right| \\
&\leq \max_m  \left| \frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T \widehat  f_i^{*'}  \widehat \lambda_{tm}  (\tilde U_{it}^* -\widehat U_{it}^* ) '\widehat\gamma_y \right|O_{P^*}(\Delta_F^*) \\
&\qquad +\max_m \left| \frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T \widehat  f_i^{*'}		\frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^* (\tilde U_{it}^* -\widehat U_{it}^* ) '\widehat\gamma_y \right|O_{P^*}(1) \\
&\qquad +\max_m \left| \frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T\widehat\lambda_{tm}' (\widehat  f_i^* -H^{*'}\widehat f_i)  (\tilde U_{it}^* -\widehat U_{it}^* ) '\widehat\gamma_y \right|O_{P^*}(1) \\
&\leq \max_m \left| \frac{1}{T}  \sum_{k=1}^p  \sum_{t=1}^T   \widehat \lambda_{tm}    \widehat \lambda_{tk}   \widehat\gamma_{y} \right|O_{P^*}(\Delta_F^{*2}) \\
&\qquad + \max_m \left| \frac{1}{T}  \sum_{k=1}^p  \sum_{t=1}^T    \widehat \lambda_{tm}   		\frac{1}{n}\widehat F^{*'}\tilde U_{t, k}^* \widehat\gamma_{y} \right|O_{P^*}(\Delta_F^*) \\
&\qquad + \max_m \left| \frac{1}{T} \sum_{k=1}^p   \sum_{t=1}^T 		\frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*    \widehat \lambda_{tk} \widehat\gamma_{y} \right|O_{P^*}(\Delta_F^*) \\
&\qquad +  \max_m\left| \frac{1}{T} \sum_{k=1}^p  \sum_{t=1}^T 	\frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*  	\frac{1}{n}\widehat F^{*'}\tilde U_{t, k}^* \widehat\gamma_{y} \right|O_{P^*}(1) \\
&\leq^{(a)} O_{P^*}(|J|_0\Delta_F^{*2}) + O_{P^*}\left(\sqrt{\frac{|J|_0^2\log(p)}{n}\left( \frac{\log(n)\log(pT)}{n}+ \Delta_{\max}^2+\frac{\log(pn)}{T}\right)}\right)\Delta_F^* \\
&\qquad +o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right) \\
&=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right),
\end{align*}
where we used  Lemma \ref{ld.3} in (a) and the assumption that $\Delta_{\max}^2|J|_0^2T\Delta_{F}^{*2}=o(1)$.
 
 
(iii) We have
\begin{align*}
\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T\tilde U_{it}^*    (\tilde U_{it}^* -\widehat U_{it}^* ) '\widehat\gamma_y \right\|_{\infty} &= \max_{k\leq p} \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T\tilde U_{it,k}^*  \sum_{m=1}^p\widehat\gamma_{ym}  (\tilde U_{it,m}^* -\widehat U_{it,m}^* )  \right| \\
&\leq \max_{k\leq p} \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T\tilde U_{it,k}^* \widehat  f_i^{*}   \widehat\gamma_{y}' \widehat \Lambda_{t}   \right|O_{P^*}(\Delta_F^*) \\
&\qquad +\max_{k\leq p}\left|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T\tilde U_{it,k}^* \widehat  f_i^{*}	 \sum_{m=1}^p\widehat\gamma_{ym}  	\frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^* \right|O_{P^*}(1) \\
&\qquad +\max_{k\leq p}\left|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T\tilde U_{it,k}^* (\widehat  f_i^* -H^{*'}\widehat f_i)   \widehat\gamma_{y}' \widehat\Lambda_{t}   \right|O_{P^*}(1) \\
&=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right).
\end{align*}
The first term following the inequality is $O_{P^*}\left(\sqrt{\frac{|J|_0^2\log(p)}{n}\left( \frac{\log(n)\log(pT)}{n}+ \Delta_{\max}^2+\frac{\log(pn)}{T}\right)}\right)\Delta_F^*=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$ follows from Lemma \ref{ld.3}. 
The second term following the inequality is bounded using Lemma \ref{ld.3}(iii). By Assumption \ref{a6.2}, the third term is $o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$.

  $\blacksquare$
 
\begin{lemma}\label{ld.5new}  
(i)
  $ \|\widehat\Psi^{y-1}\frac{1}{nT}\sum_{t=1}^T\tilde U_t ^{*'}\tilde e_t^* \|_{\infty}
\leq   \frac{1}{\sqrt{nT}}\Phi^{-1}(1-\frac{q_n}{2p})(1+o_{P^*}(1))$.\\
(ii) $
 \|\frac{1}{nT}\sum_{t=1}^T(\tilde U_t^* -\widehat U_t^*)'\tilde e_t^* \|_{\infty}= o_{P^*}(\sqrt{\frac{\log p}{nT}}).\\
$
(iii) $\|\frac{1}{nT}\sum_{t=1}^T M_t^{*'}\widehat U_t ^*    \|_{\infty}=o_P(\sqrt{\frac{\log p}{nT}})$.\\
(iv) $|J|_0\max_{m,k\leq p}|\frac{1}{nT}\sum_{it}(\widehat U_{it,m}^*\widehat U_{it,k}^*-\tilde U_{it,m}^*\tilde U_{it,k}^*)|=o_{P^*}(1)$.\\
(v) 
$\frac{1}{nT}\sum_{t=1}^T \|\widehat U_t^* \beta\|_2^2\geq 
\frac{1}{nT}\sum_{t=1}^T \|\tilde U_t \beta\|_2^2-\|\beta\|_2^2o_{P^*}(1),
$
for any vector $p\times 1$ vector   
 $\|\beta\|_1^2\leq   C\|\beta\|_2^2|J|_0.$     
\end{lemma}
 
\proof  (i)  Note that $\tilde e^*_{it}=w_i^Y\widehat\epsilon_{it}+\widehat\alpha w_i^D\widehat\eta_{it}$. Define $a_{im}=\frac{1}{\sqrt{T}}\sum_t\widehat U_{it,m}\widehat\epsilon_{it}$, $b_{im}=\frac{1}{\sqrt{T}}\sum_t\widehat U_{it,m}\widehat\alpha\widehat\eta_{it}$, and $W_{im}=\frac{1}{\sqrt{T}}\sum_t\widehat U_{it,m}^*\tilde e^*_{it}     =w_i^Uw_i^Ya_{im}+w_i^Uw_i^Db_{im}$. Note that 
\begin{align*}
E^*\left|w_i^Uw_i^Ya_{im}\right|^3 &\leq |a_{im}|^3E^*|w_i^U|^3E^*|w_i^Y|^3 \\
&\leq C \left|\frac{1}{\sqrt{T}}\sum_t\widehat U_{it,m}\widehat\epsilon_{it}\right|^3 \\
&=  \left|\frac{C}{\sqrt{T}}\sum_t\tilde  U_{it,m}\tilde \epsilon_{it}\right|^3+o_P(1)=O_P(1). \\
E^*|w_i^Uw_i^Db_{im}|^3 &\leq |b_{im}|^3E^*|w_i^U|^3E^*|w_i^D|^3 \\
&\leq C\left|\frac{1}{\sqrt{T}}\sum_t\widehat U_{it,m}\widehat\alpha\widehat\eta_{it}\right|^3 \\
&=\left|\frac{C}{\sqrt{T}}\sum_t\tilde U_{it,m} \alpha\tilde \eta_{it}\right|^3+o_P(1)=O_P(1). \\
E^*|w_i^Uw_i^Db_{im}|^2|w_i^Uw_i^Ya_{im}| &\leq C|b_{im}|^2|a_{im}|=O_P(1).\cr
E^*|w_i^Uw_i^Ya_{im}|^2|w_i^Uw_i^Db_{im}| &\leq C|a_{im}|^2|b_{im}|=O_P(1).
\end{align*}
Therefore, $ E^*|W_{im}|^3=O_P(1)$. Also, 
\begin{align*}
E^*W_{im}^2 &= E^*(w_i^Uw_i^Y)^2a_{im}^2+(w_i^Uw_i^D)^2b_{im}^2+2(w_i^U)^2w_i^Yw_i^Da_{im}b_{im} \\
&=a_{im}^2+b_{im}^2 \\
&\geq \left(\frac{1}{\sqrt{T}}\sum_t\tilde U_{it,m}\tilde\epsilon_{it}\right)^2+\left(\frac{1}{\sqrt{T}}\sum_t\tilde U_{it,m}\tilde\alpha\tilde\eta_{it}\right)^2-o_P(1).
\end{align*}
Hence, $E^*W_{im}^2\geq C-o_P(1)$ uniformly in $i$ which implies 
$$
\max_{i\leq n, m\leq p}\frac{(E^*|W_{im}|^3)^{1/3}  }{  (E^*W_{im}^2)^{1/2}}\leq C+o_P(1).
$$
Also note that  $E^*W_{im}=0$. 
It follows from Lemma 5 of \cite{belloni2012sparse}, 
$$
\max_{m\leq p}\frac{|\frac{1}{n}\sum_iW_{im}|}{(\frac{1}{n}\sum_iW_{im}^2)^{1/2}}<\frac{1}{\sqrt{n}}\Phi^{-1}\left(1-\frac{q_n}{2p}\right)
$$ 
with probability approaching one.
  
Since 
$$   
\frac{1}{nT}\sum_i\sum_{t=1}^T\tilde U_{it,m} ^{*}\tilde e_{it}^*= \frac{1}{nT}\sum_i\sum_{t=1}^T\widehat U_{it,m}w_i^U(w_i^Y\widehat\epsilon_{it}+\widehat\alpha w_i^D\widehat\eta_{it})= \frac{1}{n\sqrt{T}}\sum_iW_{im},
$$    
by the triangle inequality, for 
$$
G=\max_{m}|[\frac{1}{n}\sum_i(\frac{1}{\sqrt{T}}\sum_t\widehat U_{it,m}\widehat e_{it})^2]^{-1/2}-[\frac{1}{n}\sum_iW_{im}^2]^{-1/2}|,$$
we have
\begin{align}\label{ed.2}
\begin{split}
\left\|\widehat\Psi^{y-1} \right. & \left.\frac{1}{nT}\sum_{t=1}^T\tilde U_t ^{*'}\tilde e_t^* \right\|_{\infty} \\
&\leq G\max_m\left|\frac{1}{nT}\sum_{it}\tilde U_{it,m}^*\tilde e^*_{it}\right|+\max_m\left|\left[\frac{1}{n}\sum_iW_{im}^2\right]^{-1/2}\frac{1}{nT}\sum_{it}\tilde U_{it,m}^*\tilde e^*_{it}\right| \\
&\leq G\max_m\left[\frac{1}{n}\sum_iW_{im}^2\right]^{1/2}\max_m\left|\left[\frac{1}{n}\sum_iW_{im}^2\right]^{-1/2}\frac{1}{\sqrt{T}}\frac{1}{n}\sum_iW_{im}\right| \\
&\qquad + \max_m\left|\left[\frac{1}{n}\sum_iW_{im}^2\right]^{-1/2}\frac{1}{\sqrt{T}}\frac{1}{n}\sum_iW_{im}\right| \\
&\leq \left(G\max_m[\frac{1}{n}\sum_iW_{im}^2]^{1/2}+1\right)\frac{1}{\sqrt{nT}}\Phi^{-1}\left(1-\frac{q_n}{2p}\right).
\end{split}
\end{align}
We now show $G=o_{P^*}(1)$ for which it suffices to show $\max_{m}|\frac{1}{n}\sum_i(\frac{1}{\sqrt{T}}\sum_t\widehat U_{it,m}\widehat e_{it})^2-\frac{1}{n}\sum_iW_{im}^2|=o_{P^*}(1)$. In the bootstrap sampling space, $\{W_{im}\}_{i\leq n}$ is i.i.d. with exponential tails. Hence, by the Bernstein inequality for i.i.d. data and union bounds, it can be shown that 
$$
\max_m|\frac{1}{n}\sum_iW_{im}^2-E^*\frac{1}{n}\sum_iW_{im}^2|=o_{P^*}(1)
$$
where $E^*\frac{1}{n}\sum_iW_{im}^2=\frac{1}{n}\sum_i (a_{im}^2+b_{im}^2)$. Also  since $\widehat e_{it}=\widehat \epsilon_{it}+\widehat\alpha\widehat\eta_{it},$ we have 
$$
\frac{1}{n}\sum_i(\frac{1}{\sqrt{T}}\sum_t\widehat U_{it,m}\widehat e_{it})^2=\frac{1}{n}\sum_ia_{im}^2+b_{im}^2+2a_{im}b_{im}.
$$ 
Thus,
\begin{equation}\label{ed.3}
\max_{m}\left|\frac{1}{n}\sum_i(\frac{1}{\sqrt{T}}\sum_t\widehat U_{it,m}\widehat e_{it})^2-\frac{1}{n}\sum_iW_{im}^2\right|\leq o_{P^*}(1)+\max_m\left|\frac{2}{n}\sum_ia_{im}b_{im}\right|
\end{equation}
It remains to show the second term is $o_P(1)$. It follows from the Cauchy-Schwarz inequality, 
$$
\max_m\frac{1}{n}\sum_i(\frac{1}{\sqrt{T}}\sum_t\widehat U_{it,m}\widehat\epsilon_{it}-\tilde U_{it,m}\tilde\epsilon_{it})^2=o_P(1),
$$
and 
$$ 
\max_m\frac{1}{n}\sum_i(\frac{1}{\sqrt{T}}\sum_t\widehat U_{it,m}\widehat\alpha\widehat\eta_{it}-\tilde U_{it,m}\alpha\tilde\eta_{it})^2=o_P(1)
$$ 
that 
$$
\max_m\left|\frac{1}{n}\sum_ia_{im}b_{im}-\frac{1}{n}\sum_i(\frac{1}{\sqrt{T}}\sum_t\tilde U_{it,m}\tilde \epsilon_{it})(\frac{1}{\sqrt{T}}\sum_t\tilde U_{it,m}\alpha\tilde \eta_{it})\right|=o_P(1).
$$
Also,
$$
\max_m\left|\frac{1}{n}\sum_i(\frac{1}{\sqrt{T}}\sum_t\tilde U_{it,m}\tilde \epsilon_{it})(\frac{1}{\sqrt{T}}\sum_t\tilde U_{it,m}\alpha\tilde \eta_{it})\right| = o_P(1)+\max_m\left|\frac{1}{n}\sum_iX_{im}\right|,
$$
where $X_{im}=\left(\frac{1}{\sqrt{T}}\sum_tU_{it,m}\epsilon_{it}\right)\left(\frac{1}{\sqrt{T}}\sum_t U_{it,m}\alpha \eta_{it}\right)$. Now $EX_{im}=0$ since $E(\epsilon|U,\eta)=0$, and $\max_{im}\Var(X_{im})<C$. Thus by the Bernstein inequality for independent data and union bounds, it follows that $$\max_m\left|\frac{1}{n}\sum_iX_{im}\right|=o_P(1).$$ This proves that $\max_i\frac{1}{n}\sum_ia_{im}b_{im}=o_P(1)$, and therefore the left hand side of (\ref{ed.3})) is $o_{P^*}(1)$, which also implies $G=o_{P^*}(1).$  

In view of (\ref{ed.2}), we have,
$
\|\widehat\Psi^{y-1}\frac{1}{nT}\sum_{t=1}^T\tilde U_t ^{*'}\tilde e_t^* \|_{\infty}\leq (1+o_{P^*}(1))\frac{1}{\sqrt{nT}}\Phi^{-1}\left(1-\frac{q_n}{2p}\right).
$

(ii) We make use of equality (\ref{ed.5a}). For any $w_{it,k}$, 
\begin{align}
\nonumber
\max_{m,k} &\left|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T w_{it,k}  (\tilde U_{it,m}^*-\widehat U_{it,m}^*)\right| \\
\nonumber
&\leq \max_{m,k} \left|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T w_{it,k}  \widehat  f_i^{*'} (\widehat F^{*'}\widehat F^{*})^{-1}\widehat F^{*'}(\widehat FH^*-\widehat F^{*})H^{*-1}\widehat \lambda_{tm}   \right|\\
\nonumber
&\qquad + \max_{m,k} \left|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T w_{it,k}   \widehat  f_i^{*'}		 (\widehat F^{*'}\widehat F^{*})^{-1}\widehat F^{*'}\tilde U_{t, m}^*  \right|  \\
&\qquad + \max_{m,k} \left|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T w_{it,k}\widehat\lambda_{tm}'H^{*'-1}(\widehat  f_i^* -H^{*'}\widehat f_i)   \right| \\
\label{ed.4}
\begin{split}
&\leq \max_{m,k} \left\|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T w_{it,k}  \widehat  f_i^{*}  \widehat \lambda_{tm} '  \right\|_FO_{P^*}(\Delta_F^*) + \max_{m,k} \left\|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T w_{it,k}  	 \frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*   \widehat  f_i^{*'}	\right\|_F \\
&\qquad  + \max_{m,k} \left|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T w_{it,k}\widehat\lambda_{tm}'H^{*'-1}(\widehat  f_i^* -H^{*'}\widehat f_i)\right|. 
\end{split}
\end{align} 
Let $w_{it,k}=\tilde e_{it}^*$.  Then (\ref{ed.4}) implies $\max_{m} |\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T \tilde e_{it}^*  (\tilde U_{it,m}^*-\widehat U_{it,m}^*)   |$ is bounded by
$$   \max_{m} \|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}^*  \widehat  f_i \widehat \lambda_{tm} '  \|_FO_{P^*}(\Delta_F^*) +\max_{m} \|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}^*  	 \frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*   \widehat  f_i^{*'}	\|_F+O_{P^*}(\Delta_{eg}^*).
$$
By Assumption \ref{a6.2},  $\Delta_{eg}^*=o(\sqrt{\frac{\log p}{nT}})$. By Lemma \ref{ld.1}, the first term is $ O_{P^*}\left(\sqrt{\frac{ \log(p)\log(np)}{nT}}\Delta_F^*\right)$, which is $o_{P^*}\left(\sqrt{\frac{\log(p)}{nT}}\right)$ given that $\log(np)\Delta_F^{*2}=o(1)$.  By Lemma \ref{ld.1}, the second term satisfies
$$
\max_{m} \left\|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\tilde e_{it}^*  	 \frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*   \widehat  f_i^{*'}	\right\|_F=o_{P^*}\left(\sqrt{ \frac{\log p}{nT}}\right).
$$ 
Hence 
$$
\max_{m} \left|\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T \tilde e_{it}^*  (\tilde U_{it,m}^*-\widehat U_{it,m}^*)   \right|=o_{P^*}\left(\sqrt{ \frac{\log p}{nT}}\right).
$$


(iii) Note that $M_t^*=\widehat F\widehat\delta_{yt}-\widehat F^*\widehat\delta_{yt}^*+(\tilde U_t^*-\widehat U_t^*) \widehat\gamma_y$. Thus,
\begin{align*}
\left\|\frac{1}{nT} \right. & \left. \sum_{t=1}^T M_t^{*'}\widehat U_t^*\right\|_{\infty} \\
&\leq \left\| \frac{1}{n} \sum_{i=1}^n \frac{1}{T} \sum_{t=1}^T\widehat U_{it}^*   (\widehat \delta_{yt}^*-H^{*-1}\widehat \delta_{yt})'\widehat f_i^*\right\|_{\infty}  \\
&\qquad + \left\| \frac{1}{n} \sum_{i=1}^n \frac{1}{T} \sum_{t=1}^T  \widehat  \delta_{yt}'H^{*'-1}(\widehat f_i^*-H^{*'}\widehat f_i)\tilde U_{it}^* \right\|_{\infty}  \\
&\qquad + \left\| \frac{1}{n} \sum_{i=1}^n\frac{1}{T} \sum_{t=1}^T(\widehat U_{it}^* -\tilde U_{it}^*)   \widehat  \delta_{yt}'H^{*'-1}(\widehat f_i^*-H^{*'}\widehat f_i) \right\|_{\infty} \\
&\qquad + \left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat U_{it}^*  -\tilde U_{it}^* )  (\tilde U_{it}^* -\widehat U_{it}^* ) '\widehat\gamma_y \right\|_{\infty} \\ 
&\qquad + \left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T\tilde U_{it}^*    (\tilde U_{it}^* -\widehat U_{it}^* ) '\widehat\gamma_y \right\|_{\infty}.
\end{align*}
The first term after the inequality in the preceding display is zero since $\widehat F^{*'}\widehat U_t^*=0$. The second term is $O_{P^*}(\Delta_{ud}^*)=o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$ due to Assumption \ref{a6.2}. That the other three terms following the inequality are also $o_{P^*}\left(\sqrt{\frac{\log p}{nT}}\right)$ follows from  Lemma \ref{ld.4}.
 
 
(iv) By Lemma \ref{ld.3}, 
$$
\max_{m,k\leq p}|\frac{1}{nT} \sum_{it}\tilde U^*_{it,k}  \widehat  f_i  \widehat \lambda_{tm}  |=O_{P^*}\left(\sqrt{\frac{\log p}{n}}\right)\sqrt{\frac{ \log(n)\log(pT)}{n}+ \Delta_{\max}^2+\frac{\log(pn)}{T}}.
$$ 
By Lemma \ref{ld.1}, $\max_{m}\frac{1}{T}\sum_t  \| \frac{1}{n}\widehat F^{'}\tilde U_{t, m}^*\|_2^2=O_{P^*}\left({ \frac{\log (pT)}{n}}\right)$.
Thus, we have
\begin{align*}
|J|_0\max_{m,k\leq p}|\frac{1}{nT}\sum_{it}(\widehat U^*_{it,m}\widehat U^*_{it,k}-\tilde U^*_{it,m}\tilde U^*_{it,k})| &\leq A+B,
\end{align*}
where
\begin{align}\label{ed.13}
\begin{split}
A &:=  |J|_0\max_{m,k\leq p}\left|\frac{2}{nT} \sum_{it}\tilde U^*_{it,m}(\widehat U^*_{it,k} - \tilde U^*_{it,k})\right| \\
&\leq |J|_0\max_{m,k\leq p}\left|\frac{2}{nT} \sum_{it}\tilde U^*_{it,k}  \widehat  f_i  \widehat \lambda_{tm}  \right|  O_{P^*}(\Delta_F^*) \\ 
&\qquad +O_{P^*}|J|_0\max_{m}	\frac{1}{T}\sum_t \left|\frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^* \right|^2 + O_{P^*}(\Delta_{ud}^*|J|_0) \\
&=o_{P^*}(1)  \\
B&:= |J|_0\max_{m,k\leq p}\left|\frac{1}{nT}\sum_{it}(\widehat U^*_{it,m}   -  \tilde U^*_{it,m})(\widehat U^*_{it,k}-\tilde U^*_{it,k})\right| \\
&\leq |J|_0 O_{P^*}(\Delta_F^{*2})
+  \max_{m} \frac{1}{T}  \sum_{t=1}^T 	\left|\frac{1}{n}\widehat F^{'}\tilde U_{t, m}^*   \right|^2O_{P^*}( |J|_0 ) \\ 
&=O_{P^*}\left(|J|_0\Delta_F^{*2}+|J|_0\frac{\log(pT)}{n}\right)
= o_{P^*}(1).
\end{split}
\end{align}
 

(v) First note that for any $\beta$, by (\ref{ed.13}),
\begin{align}\label{ed.15}
\begin{split}
\frac{1}{nT}\sum_{t=1}^T \|(\widehat U_t^* -\tilde U_t^*)\beta\|_2^2 &= \frac{1}{nT}\sum_{it} ((\widehat U_{it}^* -\tilde U_{it}^*)'\beta)^2 \\
&=\beta' \frac{1}{nT}\sum_{it} (\widehat U_{it}^* -\tilde U_{it}^*)(\widehat U_{it}^* -\tilde U_{it}^*)'\beta \\
&\leq \|\beta\|_1^2\max_{mk}\left|\frac{1}{nT}\sum_{it} (\widehat U_{it,m}^* -\tilde U_{it,m}^*)(\widehat U_{it,k}^* -\tilde U_{it,k}^*)\right| \\
&\leq \|\beta\|_1^2o_{P^*}(|J|_0^{-1})
\leq o_{P^*}(1)\|\beta\|_2^2.
\end{split}
\end{align}
Note that the condition  $\|\beta\|_1^2\leq   C\|\beta\|_2^2|J|_0$ is only used in the last inequality. 


Secondly, define $X_{i,mk}=\frac{1}{T}\sum_{t}\widehat U_{it,m}\widehat U_{it,k} ((w_i^U)^2-E^*(w_i^U)^2)$. Then
$$\max_{mk}\frac{1}{n}\sum_i\Var^*(X_{i,mk})\leq C\max_{mk}\frac{1}{n}\sum_i(\frac{1}{T}\sum_{t}\widehat U_{it,m}\widehat U_{it,k})^2 =O_P(1).
$$
Hence $\max_{mk}|\frac{1}{n}\sum_{i}X_{i,mk}|=  O_{P^*}(\sqrt{\frac{\log(p)}{n} })$ and 
\begin{align*}
\left|\beta' \right. & \left. \frac{1}{nT}\sum_{it}\widehat U_{it}\widehat U_{it}' ((w_i^U)^2-E^*(w_i^U)^2)\beta\right| \\
&\leq \|\beta\|_1^2 \max_{mk}\left|
\frac{1}{nT}\sum_{it}\widehat U_{it,m}\widehat U_{it,k} ((w_i^U)^2-E^*(w_i^U)^2)\right| \\
&=\|\beta\|_1^2 \max_{mk}\left|\frac{1}{n}\sum_{i}X_{i,mk}\right| \\
&=\|\beta\|_1^2 O_{P^*}\left(\sqrt{\frac{\log(p)}{n} }\right).
\end{align*}
Combining the above result with (\ref{ed.15}) and noting that 
$$
\frac{1}{nT}\sum_{t=1}^T \|\tilde U_t^* \beta\|_2^2=\beta'\frac{1}{nT}\sum_{it}\widehat U_{it}\widehat U_{it}' (w_i^U)^2\beta,
$$
we obtain, for any $\beta$,
\begin{align}\label{ed.18}
\begin{split}
\frac{1}{nT}\sum_{t=1}^T \|\widehat U_t^* \beta\|_2^2 &\geq \frac{1}{nT}\sum_{t=1}^T \|\tilde U_t^* \beta\|_2^2-o_{P^*}( |J|_0^{-1})\|\beta\|_1^2 \\
&\geq \beta'\frac{1}{nT}\sum_{it}\widehat U_{it}\widehat U_{it}'   \beta-\|\beta\|_1^2 \left(o_{P^*}( |J|_0^{-1})+O_{P^*}\left(\sqrt{\frac{\log(p)}{n} }\right)\right) \\
&\geq^{(a.1)}  \frac{1}{nT}\sum_{t=1}^T \|\tilde U_t \beta\|_2^2
-\|\beta\|_1^2 \left(o_{P^*}(  |J|_0^{-1}  )+
o_P(|J|_0^{-1})\right) \\
&\geq^{(a.2)} \frac{1}{nT}\sum_{t=1}^T \|\tilde U_t \beta\|_2^2
-\|\beta\|_2^2o_{P^*}(1),
\end{split}
\end{align}
where (a.1) follows from  Lemma \ref{la.5new} (v)  that 
$$
\max_{m,k\leq p}|\frac{1}{nT}\sum_{it}(\widehat U_{it,m}\widehat U_{it,k}-\tilde U_{it,m}\tilde U_{it,k})|=o_P(|J|_0^{-1})
$$ 
and (a.2) follows from  $\|\beta\|_1^2\leq   C\|\beta\|_2^2|J|_0.$ 
%Note  that this condition is not needed to achieve inequality (a).    
%  In addition, $o_P(|J|_0^{-1})$ is also $o_{P^*}(|J|_0^{-1})$ by Lemma 3 of \cite{cheng2010bootstrap}.
Note that we have shown
\begin{equation}\label{ed.24}
|J|_0\max_{m,k\leq p}\left|\frac{1}{nT}\sum_{it}(\widehat U^*_{it,m}\widehat U^*_{it,k}-\tilde U_{it,m}\tilde U_{it,k})\right|
=o_{P^*}(1).
\end{equation}
 
 $\blacksquare$


\section{Technical lemmas for the estimated factors }





\subsection{Proof of Proposition \ref{p4.1} (for \texorpdfstring{$\widehat F$}{F-hat} using the original data)} \label{sf}
This section verifies Assumption  \ref{ca.1}  when factors are estimated using PCA.
%We first prove a less sharp rate: $ \frac{1}{n}\sum_{i=1}^n\|\widehat f_i-H' f_i\|_2^2=O_P(\frac{1}{pT}+\frac{1}{n})$, that is, $n^{-2}$ is replaced with $n^{-1}$. Then we derive the shaper rate.
 
 %\begin{lemma}
 %$ \frac{1}{n}\sum_{i=1}^n\|\widehat f_i-H' \tilde f_i\|_2^2=O_P(\frac{1}{pT}+\frac{1}{n})$
 %\end{lemma}
 
(i) By Assumption \ref{pervasive},     %the pervasive condition, 
it can be shown that $\|H\|=O_P(1)=\|V^{-1}\|$.
In addition, we have the following identity:
 $$
 \widehat f_i-H'\tilde f_i=V^{-1} \sum_{l=1}^5A_{il},
 $$
 where 
\begin{align}\label{ec.1}
\begin{split}
 A_{i1} &= \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T (\tilde U_{it}'\tilde U_{jt}-U_{it}' U_{jt}), \\ 
 A_{i2} &= \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T (U_{it}' U_{jt}-EU_{it}' U_{jt}), \\
 A_{i3} &= \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T EU_{it}'U_{jt}, \\
 A_{i4} &= \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_j'\tilde \Lambda_t'\tilde U_{it}, \\
 A_{i5} &= \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_i'\tilde \Lambda_t'\tilde U_{jt}.
\end{split}
\end{align}
Each term can be written in the form $A_{il}=\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^TB_{ijt, l}$. By Cauchy-Schwarz,
\begin{align}\label{ee.2}
\begin{split}
 \frac{1}{n}\sum_{i=1}^n\| \widehat f_i-H'\tilde f_i\|_2^2 &= O_P(1)  \sum_{l=1}^5\frac{1}{n}\sum_{i=1}^n\|A_{il}\|_2^2  \\
&\leq  O_P(1)\sum_{l=1}^5 \frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n\left(\frac{1}{pT}\sum_{t=1}^TB_{ijt,l}\right)^2.
\end{split}
\end{align}
Note that we used $\frac{1}{n}\sum_{i=1}^n\|\widehat f_i\|^2_2=K=O(1)$ because the columns of $n^{-1/2}(\widehat f_1,...,\widehat f_K)'$ are defined as the eigenvectors of $\tilde X'\tilde X$.
We bound the terms in (\ref{ee.2}) in Lemmas \ref{lc.1} and \ref{lc.2} below.  Then, applying the bounds in Lemmas \ref{lc.1} and \ref{lc.2} and using $T=o(n)$, we have 
$$
\frac{1}{n}\sum_{i=1}^n\|\widehat f_i-H'\tilde f_i\|_2^2=O_P(\Delta_F^2),\quad \Delta_F^2=\frac{1}{n^2}+ 
\frac{1}{nT^2}+\frac{1}{pT}.
$$
It  is then straightforward to verify that $|J|^2_0\Delta_F^2=o\left(\sqrt{\frac{1}{nT}}\right)$
  holds when $|J|_0^4=o(nT^3)$, $|J|_0^4n=o(p^2T)$, $|J|_0^2 \sqrt{\log p }  \log(pT)=o(n)$, and  $|J|_0^2T=o(n)$.  For example, to show $|J|_0^2 \sqrt{\log p }\log(T)=o(n)$, note that $|J|_0 =o(\sqrt{n/(\log p)})$ and $|J|_0=o(\sqrt{n/T})$ implies 
$$
|J|_0^2 \sqrt{\log p }  \log(T)=o(n\log T\sqrt{\log p}/\sqrt{T\log p})=o(n).
$$ 
  
%$|J|_0^2=o(n)$,  $|J|_0^2\log p=o(nT)$,   


(ii) We now verify that we can produce sequences $\Delta_{eg}$ so that $\Delta_{eg}=o\left(\frac{1}{\sqrt{nT}}\right)$. First, note that we can set $g_{tm} \in \{\gamma_d'\tilde\Lambda_t, \tilde\lambda_{tm}, \tilde\delta_t\}$ in applying Lemma \ref{lc.4}, each of which yields $\omega_n=O(|J|_0^2)$ for $\omega_n$ defined in Lemma \ref{lc.4}.  It then follows from Lemma \ref{lc.4} that we can take $\Delta_{eg}=(\frac{1}{\sqrt{npT}}+\frac{1}{n}) |J|_0$ so that $\Delta_{eg}=o\left(\sqrt{\frac{1}{nT}}\right),$ given $T|J|_0^2=o(n)$ and $|J|_0^2=o(p)$.  Note that $|J|_0^2=o(p)$ is implied by the assumption that $|J|_0^4n=o(p^2T)$.  
   
    
 
    
% Then $\omega_n$ in Lemma \ref{lc.4} is $\omega_n=O(1)$, and hence $\Delta_{fel}=(\frac{1}{\sqrt{npT}}+\frac{1}{n})$ satisfies $\Delta_{fel}=o(\sqrt{\frac{\log p}{nT}})$.

 
(iii)
By Lemma \ref{lc.6}, we take $\Delta_{fum}=\frac{1}{n} + \frac{1}{T\sqrt{n}}+\frac{1}{T\sqrt{p}} +\sqrt{\frac{\log (pT)}{npT}}$, and $\Delta_{fe}^2=\frac{1}{n^2}+ \frac{1}{Tpn}$. We then have 
$$ 
\max_{s\leq T, m\leq p}|\frac{1}{n}  \sum_{i=1}^n (H^{'}\tilde f_i-\widehat f_i)\tilde U_{is,m} |=
O_P(  \Delta_{fum})
$$ 
and 
$$
\frac{1}{T} \sum_{t=1}^T\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}(\widehat  f_i-H'\tilde f_i)\|^2_2
 =O_P(\Delta_{fe}^2).  
$$    
Then, it is straightforward to check    $\Delta_{fum}^2=o\left(\frac{\log p}{T|J|^2\log(pT)}\right)$ and $\Delta_{fe}^2=o\left(\frac{\log p}{T\log(pT)}\right).$ 
 
(iv) By Lemma \ref{lc.7}, we can define $\Delta_{ud}=   \frac{1}{\sqrt{n}} \sqrt{\frac{\log(pT)}{nT}}  
 +\sqrt{\frac{\log p}{nT}}(\frac{1}{\sqrt{pT}}+   \frac{1}{T\sqrt{n}} )+\frac{1}{pT}$.  Given $|J|_0^4n=o(p^2T)$ and $|J|_0^4=o(nT^3)$, it is straightforward to verify that  $\Delta_{ud}=o\left(\sqrt{\frac{\log p}{nT}}\right) $ and 
 $|J|_0^2 \sqrt{\log p }      \Delta_{ud}=o(1)$.  This result follows by verifying $\sqrt{\frac{\log p}{nT}}\frac{1}{\sqrt{pT}}|J|_0^2\sqrt{\log p}=o(1)$ which can be shown by noting that $|J|_0^4n=o(p^2T)$ implies $|J_0|^2=o(p\sqrt{T/n})$ and that $|J|_0^2\log (p)=o(n)$. Thus, because $\log^2 p=o(n)$,
 $$
\left( \sqrt{\frac{\log p}{nT}}\frac{|J|_0^2\sqrt{\log p}}{\sqrt{pT}}\right)^2=\frac{\log^2 p|J|_0^4}{npT^2}=o\left(\frac{\log(p)pn\sqrt{T}}{npT^2\sqrt{n}}\right)=o\left(\frac{\log(p)\sqrt{T}}{T^2\sqrt{n}}\right)=o(1).
 $$

(v) First, by Lemma \ref{lc.8}, we can take $\Delta_{\max}=\frac{1}{\sqrt{n}}  +\sqrt{\frac{\log n}{Tp}}$.
 %Secondly, by Lemma \ref{lc.6}, we have
    % $ \frac{1}{T}\sum_t\|\frac{1}{n}\sum_i (\widehat f_i-H'\tilde f_i)\tilde U_{it}'\gamma\|_2^2\leq \max_{ms}|\frac{1}{n}  \sum_{i=1}^n (H^{'}\tilde f_i-\widehat f_i)\tilde U_{is,m} |^2\|\gamma\|_1^2= O_P(\Delta_{fu}^2),$where  $\Delta_{fu}^2=(\frac{1}{n^2}+   \frac{1}{T^2{n}}+\frac{1}{T^2{p}} +{\frac{\log (pT)}{npT}}  )|J|_0^2$.
%Also by parts (i)(iii)$\Delta_{fe}^2=\frac{1}{n^2}+ \frac{1}{Tpn}$ and 
Also $\Delta_F^2=\frac{1}{n^2}+ 
 \frac{1}{nT^2}+\frac{1}{pT}$. This implies 
 $
 \Delta_F^2|J|^2_0+ {\frac{|J|_0}{n}}=O\left( \frac{|J|_0}{n}+  \frac{|J|_0^2}{nT^2}+\frac{|J|_0^2}{pT}\right).
 $
 In addition,  $ \kappa_n^2|J|_0\sqrt{nT}=o(1)$  and $\|R_y\|_1^2=o\left(\frac{\log p}{nT}\right)$ imply 
  $\lambda^2_n|J|_0+\|R_y\|_1^2=O\left(\frac{1}{\sqrt{nT}}\right)$. Hence with the conditions  $|J|_0^4=o(nT^3)$, $|J|_0^4n=o(p^2T)$, we have
  $$
  \lambda^2_n|J|_0+\|R_y\|_1^2+\Delta_F^2|J|^2_0+ {\frac{|J|_0}{n}}=O\left(\frac{1}{\sqrt{nT}}\right).
  $$
Thus, in order to verify 
$\Delta_{\max}^2|J|_0^2T(\lambda^2_n|J|_0+\|R_y\|_1^2+\Delta_F^2|J|^2_0+ {\frac{|J|_0}{n}})=o(1),$
it suffices to verify 
$$
\left(\frac{1}{n}+\frac{\log n}{Tp}\right)|J|_0^2T \frac{1}{\sqrt{nT}}=o(1),
$$
which holds given the conditions $|J|_0^2T=o(n)$ and $|J|_0^2=o(p)$.  Note that $|J|_0^2=o(p)$ is implied by $|J|_0^4n=o(p^2T)$.
 $\blacksquare$
    

\begin{lemma}\label{lc.8}    
$$\max_i\|\widehat f_i-H'\tilde f_i\|_2=O_P \left(  \frac{1}{\sqrt{n}}  +\sqrt{\frac{\log n}{Tp}}\right):=O_P(\Delta_{\max}^2).$$
\end{lemma}
    
  
\proof   
By Assumption \ref{ad.3ad},
$
\max_i\frac{1}{pT}\sum_{tm} ( U_{it,m}^2-E     U_{it,m}^2 )=O_P(1).
$
Also,  $\max_{it,m}E     U_{it,m}^2=O(1)$. Hence $\max_i\frac{1}{pT}\sum_{tm}  U_{it,m}^2=O_P(1)$.
So  $\max_i\|\widehat f_i-H'\tilde f_i\|_2\leq\sum_{l=1}^3G_l$ where each $G_l$ is defined and bounded below.  Specifically, 
\begin{align*}
G_1 &= \max_i\|\frac{1}{pTn}\sum_{t=1}^T \tilde U_{it}'\sum_{j=1}^n\tilde U_{jt}\widehat f_j \|_2 \\
    &\leq \max_i\|\frac{1}{pTn}\sum_{t=1}^T   U_{it}'\sum_{j=1}^n\tilde U_{jt}\widehat f_j \|_2+ \|\frac{1}{pTn}\sum_{t=1}^T   \bar U_{\cdot t}'\sum_{j=1}^n\tilde U_{jt}\widehat f_j \|_2 \\
%\leq\max_i\|\frac{1}{pTn}\sum_{t}\sum_j\sum_m\tilde U_{it,m}\tilde U_{jt,m}\widehat f_j\|_2\cr
    &\leq \left(2\max_i\frac{1}{pT}\sum_{tm}  U_{it,m}^2+  2\frac{1}{pT}\sum_{tm}  \bar U_{\cdot t,m}^2\right)^{1/2}\left(\frac{1}{pT}\sum_{tm} \| \frac{1}{n}\sum_j\tilde U_{jt,m}\widehat f_j\|_2^2\right)^{1/2} \\
    &= O_P\left(O_P\left(1\right)\left( \frac{1}{T\sqrt{p}}  + \frac{1}{\sqrt{n}}\right)\right)  
\end{align*}
by equation (\ref{ec.4}) given below in Lemma \ref{lc.4}.  
\begin{align*}
G_2 &= \max_i\| \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_j'\tilde\Lambda_t'\tilde U_{it}\|_2 \\
    &\leq O_P(1)\max_i\| \frac{1}{pT}\sum_{t=1}^T\tilde \Lambda_t'U_{it}\|_2+O_P(1)\| \frac{1}{pT}\sum_{t=1}^T\tilde \Lambda_t' \bar U_{\cdot t}\|_2 \\
    &= O_P\left(\sqrt{\frac{\log n}{Tp}}\right)
\end{align*}
by Assumption \ref{ad.3ad}  that   
$\max_{i\leq n}\|\frac{1}{pT}\sum_{t=1}^T\sum_{m=1}^p\tilde\lambda_{t,m}U_{it,m}\|_2=O_P(\sqrt{\frac{\log n}{pT}}).$    Finally, recall that $\|A\|_F^2=\tr(A'A)$ and $\|A\|_2\leq \|A\|_F$ for any matrix $A$, 
\begin{align*}
G_3 &= \max_i\|\frac{1}{pTn}\sum_{t=1}^T\tilde f_i'\tilde\Lambda_t'\sum_{j=1}^n\tilde U_{jt}\widehat f_j'\|_2 \\
    &\leq \max_i\|\tilde f_i\|_2\|  \frac{1}{pTn}\sum_{t=1}^T\tilde\Lambda_t'\sum_{j=1}^n\tilde U_{jt}\widehat f_j'\|_F \\
    &\leq O_P(\sqrt{\log n})\| \frac{1}{pTn}\sum_{t=1}^T\tilde\Lambda_t'\sum_{j=1}^n\tilde U_{jt} \tilde f_j'\|_F
\\
	  & \qquad + O_P(\sqrt{\log n})\| \frac{1}{pTn}\sum_{t=1}^T\tilde\Lambda_t'\sum_{j=1}^n\tilde U_{jt}(\widehat f_j-H'\tilde f_j)'\|_F \\
    &= O_P\left(\sqrt{\frac{\log n}{pTn}} \right) + O_P(\sqrt{\log n}\Delta_F)\left(\frac{1}{n}\sum_j\|\frac{1}{pT}\sum_t\tilde\Lambda_t'\tilde U_{jt}\|_2^2\right)^{1/2} \\
		&= O_P\left(\sqrt{\frac{\log n}{pTn}} +\Delta_F\sqrt{\frac{\log n}{pT}}\right).
\end{align*}
Hence, $\max_i\|\widehat f_i-H'\tilde f_i\|_2=O_P \left(  \frac{1}{\sqrt{n}}  +\sqrt{\frac{\log n}{Tp}}\right).$
 $\blacksquare$
 
 
\begin{lemma}\label{lc.4}
Let $\{z_{it}\}$ be a random sequence with $E(z_{it}|f_t, U_t)=0$ and $\Var(z_{it})>0$.  In addition, let $\{g_{tm}\}$ be a deterministic sequence of vectors with a  fixed dimension, $m\leq p$.  Then for $\tilde z_{it}=z_{it}-\bar z_{i\cdot}-\bar z_{\cdot t}+\bar{\bar z}$,  and $\omega_n=\max_{m\leq p}\frac{1}{T}\sum_{t=1}^T \|g_{tm}\|_2^2,$
$$
\max_{m\leq p}\|  \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^Tg_{tm}\tilde z_{it}(\widehat f_i-H'\tilde f_i)'\|_F=
O_P\left(\frac{1}{\sqrt{npT}}+\frac{1}{n}\right)\omega_n^{1/2}.
$$
\end{lemma}
  
\proof It follows from equation (\ref{ec.1}) that 
$$
\max_{m\leq p}\|  \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^Tg_{tm}\tilde z_{it}(\widehat f_i-H'\tilde f_i)'\|_F\leq \sum_{l=1}^3\bar C_lO_P(1),
$$
where each term $\bar C_l$ is defined and bounded in  below.
\begin{align*}
\bar C_1 &= \max_{m\leq p}\|  \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{s=1}^T \tilde U_{is}'\tilde U_{js}\tilde z_{it}g_{tm}'\|_F \\
&\leq \max_{m\leq p}\|  \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T \frac{1}{pTn}\sum_{j=1}^n(\widehat f_j-H'\tilde f_j)\sum_{s=1}^T \tilde U_{is}'\tilde U_{js}\tilde z_{it}g_{tm}'\|_F \\
& \qquad + O_P(1)\max_{m\leq p}\|  \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T \frac{1}{pTn}\sum_{j=1}^n\tilde f_j\sum_{s=1}^T \tilde U_{is}'\tilde U_{js}\tilde z_{it}g_{tm}'\|_F \\
 %&\leq&\left(  \frac{1}{pT^2}\sum_{t=1}^T \sum_{s=1}^T\|\frac{1}{n}\sum_{i=1}^n\tilde z_{it}\tilde U_{is}\|_2^2 \right)^{1/2}\left( \max_{m\leq p}\frac{1}{T}\sum_{t=1}^T \|g_{tm}\|_2^2\right)^{1/2}\left( \frac{1}{npT}\|\tilde U\|_F^2   \right)^{1/2}O_P(\Delta_F) \cr
% &&+\left(\frac{1}{pT^2}\sum_s  \sum_{t=1}^T   \|\frac{1}{n}\sum_{i=1}^n     \tilde z_{it}  \tilde  U_{is}  \|_2 ^2\right)^{1/2}  \left( \max_{m\leq p} \frac{1}{T} \sum_{t=1}^T\| g_{tm}\|_2^2\right)^{1/2}  O_P(1)(\frac{1}{pT}\sum_s  \| \frac{1}{n} \sum_{j=1}^n\tilde f_j \tilde U_{js}' \|_F^2  )^{1/2} \cr
&\leq (x_n\omega_n)^{1/2}\left( \left( \frac{1}{npT}\|\tilde U\|_F^2   \right)^{1/2}O_P(\Delta_F)  
 +O_P(1)(\frac{1}{pT}\sum_s  \| \frac{1}{n} \sum_{j=1}^n\tilde f_j \tilde U_{js}' \|_F^2  )^{1/2}
 \right) \\
&= (x_n\omega_n)^{1/2}O_P(\Delta_F+\frac{1}{\sqrt{n}}) \\
&=\omega_n^{1/2} O_P\left(\frac{1}{\sqrt{npT}}+\frac{1}{n}\right).
\end{align*}
where $x_n= \frac{1}{pT^2}\sum_{t=1}^T \sum_{s=1}^T\|\frac{1}{n}\sum_{i=1}^n\tilde z_{it}\tilde U_{is}\|_2^2=O_P(\frac{1}{{n}})$ and $\omega_n=\max_{m\leq p}\frac{1}{T}\sum_{t=1}^T \|g_{tm}\|_2^2$.  Next,
\begin{align*}
\bar C_2 &= \max_{m\leq p}\|  \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T  \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{s=1}^T\tilde f_j'\tilde \Lambda_s'\tilde U_{is} \tilde z_{it}g_{tm}'\|_F \\
&\leq O_P(\omega_n^{1/2})  \left( \frac{1}{T}\sum_{t=1}^T \| \frac{1}{pT} \frac{1}{n}  \sum_{i=1}^n\sum_{s=1}^T \tilde \Lambda_s'\tilde U_{is} \tilde z_{it}\|_2^2\right)^{1/2} \\
&= O_P\left(\omega_n^{1/2}\frac{1}{\sqrt{npT}}\right) 
\end{align*}
by Lemma \ref{lc.3}).  Finally,
\begin{align*}
\bar C_3 &= \max_{m\leq p}\|  \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{s=1}^T\tilde f_i'\tilde \Lambda_s'\tilde U_{js} \tilde z_{it}g_{tm}'\|_F \cr
&\leq  \max_{m\leq p}\|  \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\frac{1}{pTn}\sum_{j=1}^n(\widehat f_j-H'\tilde f_j)\sum_{s=1}^T\tilde f_i'\tilde \Lambda_s'\tilde U_{js} \tilde z_{it}g_{tm}'\|_F\cr
& \qquad + O_P(1) \max_{m\leq p}\|  \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\frac{1}{pTn}\sum_{j=1}^n \tilde f_j\sum_{s=1}^T\tilde f_i'\tilde \Lambda_s'\tilde U_{js} \tilde z_{it}g_{tm}'\|_F\cr
&\leq  (\omega_nc_n)^{1/2}\left( O_P(\Delta_F)( \frac{1}{n}\sum_j \|\frac{1}{pT}  \sum_{s=1}^T\tilde \Lambda_s'\tilde U_{js}\|_2^2 )^{1/2}   +O_P(1)
 \max_{m\leq p}\| \frac{1}{pTn}  \sum_{j=1}^n\sum_{s=1}^T    \tilde \Lambda_s'\tilde U_{js}\tilde f_j '  \|_F\right) \\
 &= \omega_n^{1/2} O_P\left(\frac{1}{\sqrt{npT}}+\frac{1}{n}\right)
\end{align*}
where $ c_n=\frac{1}{T}\sum_{t=1}^T\|\frac{1}{n}\sum_{i=1}^n\tilde f_i\tilde z_{it}\|_2^2 =O_P(\frac{1}{n})$.
Therefore, 
$$
\max_{m\leq p}\|  \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^Tg_{tm}\tilde z_{it}(\widehat f_i-H'\tilde f_i)'\|_F=O_P\left(\frac{1}{\sqrt{npT}}+\frac{1}{n}\right)\omega_n^{1/2}.
$$
  $\blacksquare$
 
\begin{lemma}\label{lc.7}
Let $\{g_{tk}: k\leq p\}$ be a deterministic sequence of vectors of fixed dimension with $\max_{tk}\|g_{tk}\|_2=O(1)$, and let $\tilde g_{tk}=g_{tk}-\frac{1}{T}\sum_t g_{tk}$. Then
\begin{align*}
\max_{m,k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat  f_i -H'\tilde f_i) \tilde U_{it,m} \tilde g_{tk}'  \|_F &= O_P\left(   \frac{1}{\sqrt{n}} \sqrt{\frac{\log(pT)}{nT}}  
 +\sqrt{\frac{\log p}{nT}}(\frac{1}{\sqrt{pT}}+   \frac{1}{T\sqrt{n}} )+\frac{1}{pT}\right) \\
&:=O_P(\Delta_{ud}).
\end{align*}
\end{lemma}
\proof  



First, note that $\sum_t\tilde g_{tk}=0$. Hence,
\begin{align}
\nonumber 
\max_{m,k\leq p}\|\frac{1}{nT} &\sum_{i=1}^n  \sum_{t=1}^T(\widehat  f_i -H'\tilde f_i) \tilde U_{it,m} \tilde g_{tk}'  \|_F \\
\label{lemmaE3.i}
&\leq \max_{m,k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat  f_i -H'\tilde f_i)  U_{it,m} \tilde g_{tk}'  \|_F \\
\label{lemmaE3.ii}
& \qquad +\|\frac{1}{n} \sum_{i=1}^n (\widehat  f_i -H'\tilde f_i) \|_2\max_{m,k\leq p}\| \frac{1}{T} \sum_{t=1}^T\bar U_{\cdot t,m} \tilde g_{tk}'  \|_F.
\end{align}
Term (\ref{lemmaE3.ii}) is $O_P\left(\Delta_F\sqrt{\frac{\log p}{nT}}\right)$. Term (\ref{lemmaE3.i}) is bounded by $\sum_{l=1}^7C_l$, where each $C_l$ is defined and bounded blow.
 
First, note that applying Lemma \ref{lc.6} gives     
\begin{align}\label{ec.4}
\begin{split}
\frac{1}{\sqrt{p}}\left(\frac{1}{T}\sum_t  \|   \frac{1}{n}\sum_j\widehat f_j\tilde U_{jt}'\|_F^2  \right)^{1/2}&\leq \frac{2}{\sqrt{p}}\left(\frac{1}{T}\sum_t  \|   \frac{1}{n}\sum_j(\widehat f_j-H'\tilde f_j)\tilde U_{jt}'\|_F^2  \right)^{1/2} \\
& \qquad +\frac{2}{\sqrt{p}}\left(\frac{1}{T}\sum_t  \|   \frac{1}{n}\sum_jH'\tilde f_j\tilde U_{jt}'\|_F^2  \right)^{1/2} \\
&= O_P\left(\frac{1}{n}+   \frac{1}{T\sqrt{n}}+\frac{1}{T\sqrt{p}} +\sqrt{\frac{\log (pT)}{npT}} + \frac{1}{\sqrt{n}} \right) \\
&= O_P( \frac{1}{T\sqrt{p}}  + \frac{1}{\sqrt{n}} ) \ \text{and} \\
\max_{tl}\|\frac{1}{n}\sum_{j=1}^n\widehat f_j \tilde U_{jt,l}\|_2 &= O_P\left(\sqrt{\frac{\log(pT)}{n}}+ \frac{1}{T\sqrt{p}} \right).
\end{split}
\end{align}
We then have, up to an $\|V^{-1}\|=O_P(1)$ term, 
\begin{align*}
C_1 &= \max_{m,k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T \tilde U_{jt}'(\tilde U_{it}   -U_{it}) U_{is,m} \tilde g_{sk}'  \|_F \\
%&\leq&\max_{m,k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T \tilde U_{jt}'\bar U_{i\cdot }    U_{is,m} \tilde g_{sk}'  \|_F\cr
&= \max_{m,k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T \tilde U_{jt}'\bar U_{\cdot t }    U_{is,m} \tilde g_{sk}'  \|_F  \text{ (because $\frac{1}{T}\sum_t\tilde U_{jt}=0$)} \\
&\leq \left(\frac{1}{T}\sum_t  \|   \frac{1}{n}\sum_j\widehat f_j\tilde U_{jt}'\|_F^2  \right)^{1/2} \left(\frac{1}{T}\sum_t\|\bar U_{\cdot t }  \|_2^2\right)^{1/2}\max_{mk} \|\frac{1}{nTp}\sum_{i=1}^n\sum_{s=1}^T  U_{is,m} \tilde g_{sk}'  \|_2 \\
&= O_P\left( \frac{1}{T\sqrt{p}}  + \frac{1}{\sqrt{n}} \right)
\frac{1}{\sqrt{n}} \sqrt{\frac{\log p}{nT}} \\
&= O_P\left( \frac{1}{T\sqrt{pn}}  + \frac{1}{n} \right)
 \sqrt{\frac{\log p}{nT}}.
\end{align*}
%&&+\max_{m,k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T \tilde U_{jt}'\bar {\bar U}    U_{is,m} \tilde g_{sk}'  \|_F\cr
\begin{align}
\label{ee.9a2}
\begin{split}
C_2 &= \max_{m,k\leq p}\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T \tilde U_{jt}'(U_{it}    U_{is,m}-EU_{it}U_{is,m}) \tilde g_{sk}'  \right\|_F \\
%&\leq &(\frac{1}{T}\sum_t  \|   \frac{1}{n}\sum_j\widehat f_j\tilde U_{jt}'\|_F^2  )^{1/2}(\max_{mkt}\frac{1}{T}\sum_t  \|   \frac{1}{nTp}\sum_{i=1}^n\sum_{s=1}^T  (U_{it}    U_{is,m}-EU_{it}U_{is,m}) \tilde g_{sk}'   \|_F^2  )^{1/2}
&\leq \left(\frac{1}{T}\sum_t  \|   \frac{1}{n}\sum_j\widehat f_j\tilde U_{jt}'\|_F^2  \right)^{1/2} \max_{mkt}  \|   \frac{1}{nTp}\sum_{i=1}^n\sum_{s=1}^T  (U_{it}    U_{is,m}-EU_{it}U_{is,m}) \tilde g_{sk}'   \|_F
\\
&= O_P\left( \frac{1}{T\sqrt{p}}  + \frac{1}{\sqrt{n}} \right) \left(\sqrt{\frac{\log(pT)}{nT}}\right)
\end{split}
\end{align}
using Lemma \ref{lc.5}.
\begin{align*}
C_3 &= \max_{m,k\leq p}\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T \tilde U_{jt}'(EU_{it}    U_{is,m}) \tilde g_{sk}'  \right\|_F \\
&\leq \max_{itm}\sum_{s=1}^T\sum_{l=1}^p |(EU_{it,l}    U_{is,m})   |   \max_{sk}\|\tilde g_{sk}\|_2   \frac{1}{pT}  \max_{tl}\|\frac{1}{n}\sum_{j=1}^n\widehat f_j \tilde U_{jt,l}\|_2 \\
&= O_P\left(\frac{1}{pT} \right) O_P\left(\sqrt{\frac{\log(pT)}{n}}+ \frac{1}{T\sqrt{p}} \right).
\end{align*}
 



\begin{align*}
C_4 &= \max_{m,k\leq p}\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_j'\tilde \Lambda_t'(\tilde U_{it} -U_{it}) U_{is,m} \tilde g_{sk}'  \right\|_F \\
&= \max_{m,k\leq p}\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_j'\tilde \Lambda_t' \bar U_{\cdot t} U_{is,m} \tilde g_{sk}'  \right\|_F \text{ (because $\sum_t\tilde\Lambda_t=0$)} \\
&\leq O_P(1)  \left\|   \frac{1}{pT}  \sum_{t=1}^T \tilde \Lambda_t' \bar U_{\cdot t}\right\|_2\max_{m,k\leq p}  \left\|\frac{1}{nT} \sum_{i=1}^n\sum_{s=1}^T U_{is,m} \tilde g_{sk}'  \right\|_2 \\
&= O_P\left(\frac{1}{\sqrt{npT}}\right)\sqrt{\frac{\log p}{nT}}.
\end{align*}
\begin{align}
\label{ee.9a5}
\begin{split}
C_5 &= \max_{m,k\leq p}\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_j'\tilde \Lambda_t'  (U_{it}  U_{is,m}-EU_{it}  U_{is,m}) \tilde g_{sk}'  \right\|_F \\
&\leq O_P(1)\max_{m,k\leq p}\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}  \sum_{t=1}^T \tilde \Lambda_t'  (U_{it}  U_{is,m}-EU_{it}  U_{is,m}) \tilde g_{sk}'  \right\|_F \\
&= O_P\left(\sqrt{\frac{\log p}{npT^2}}\right) 
\end{split}
\end{align}
using the same proof as that of Lemma  \ref{lc.5} (ii).
\begin{align*}
C_6 &= \max_{m,k\leq p}\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_j'\tilde \Lambda_t'  (EU_{it}  U_{is,m}) \tilde g_{sk}'  \right\|_F \\
&\leq O_P(1)
\max_{m,k\leq p}\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT} \sum_{t=1}^T\tilde \Lambda_t'  (EU_{it}  U_{is,m}) \tilde g_{sk}'  \right\|_F \\
&=O_P\left(\frac{1}{pT}\right).
\end{align*}
\begin{align*}
 C_7 &= \max_{m,k\leq p}\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_i'\tilde \Lambda_t'\tilde U_{jt}  U_{is,m} \tilde g_{sk}'  \right\|_F \\
 &\leq \left\| \frac{1}{pTn}\sum_{j=1}^n\sum_{t=1}^T  \widehat f_j\tilde U_{jt}    '    \tilde \Lambda_t\right\|_2 \max_{m,k\leq p}\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \tilde f_i      U_{is,m} \tilde g_{sk}'  \right\|_F \\
 &\leq O_P\left(\sqrt{\frac{\log p}{nT}}\right)\left( \| \frac{1}{pTn}\sum_{j=1}^n\sum_{t=1}^T \tilde  f_j\tilde U_{jt}    '    \tilde \Lambda_t\|_2+ \| \frac{1}{n}\sum_{j=1}^n   ( \widehat f_j-H'\tilde f_j)\tilde U_{jt,m} \|_2  \right) \\
 &= O_P\left(\sqrt{\frac{\log p}{nT}}\right)\left( \frac{1}{n}+   \frac{1}{T\sqrt{n}}+\frac{1}{T\sqrt{p}} +\sqrt{\frac{\log (pT)}{npT}} \right).
\end{align*} 
 Combining the above, we reach
 \begin{align*}
 \max_{m,k\leq p}&\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat  f_i -H'\tilde f_i) \tilde U_{it,m} \tilde g_{tk}'  \right\|_F \\
&= O_P\left(   \frac{1}{\sqrt{n}} \sqrt{\frac{\log(pT)}{nT}}  
 +\sqrt{\frac{\log p}{nT}}(\frac{1}{\sqrt{pT}}+   \frac{1}{T\sqrt{n}} +\Delta_F)+\frac{1}{pT}\right) \\
&= O_P\left(   \frac{1}{\sqrt{n}} \sqrt{\frac{\log(pT)}{nT}}  
 +\sqrt{\frac{\log p}{nT}}(\frac{1}{\sqrt{pT}}+   \frac{1}{T\sqrt{n}} )+\frac{1}{pT}\right).
 \end{align*} 
 $\blacksquare$

\subsection{Proof of Proposition \ref{p4.1} (for \texorpdfstring{$\widehat F^*$}{F-hat} using the bootstrap data)} 
This section verifies Assumption    \ref{a6.2} when factors are estimated using PCA.


 (i) Similar to (\ref{ec.1}), it  can be proven that there is $\|H^*\|=O_{P^*}(1)$ and $\|V^{*-1}\|=O_{P^*}(1)$such that 
 $$
 \widehat f_i^*-H^{*'}\widehat f_i=V^{*-1} \sum_{l=1}^4 A^*_{il},
 $$
 where 
\begin{equation}\label{ee.9}
\begin{aligned}
A^*_{i1} &= \frac{1}{pTn}\sum_{j=1}^n\widehat f_j^*\sum_{t=1}^T (\tilde U_{it}^{*'}\tilde U_{jt}^*- E^*\tilde U_{it}^{*'}\tilde U_{jt}^*),  
& A^*_{i2} &= \frac{1}{pTn}\sum_{j=1}^n\widehat f_j^*\sum_{t=1}^T E^*\tilde U_{it}^{*'}\tilde U_{jt}^*, \\
A^*_{i3} &= \frac{1}{pTn}\sum_{j=1}^n\widehat f_j^*\sum_{t=1}^T\widehat f_j'\widehat \Lambda_t'\tilde U_{it}^*,
& A^*_{i4} &= \frac{1}{pTn}\sum_{j=1}^n\widehat f_j^*\sum_{t=1}^T\widehat f_i'\widehat\Lambda_t'\tilde U_{jt}^*.
\end{aligned}
\end{equation}
% A^*_{i2}=\frac{1}{pTn}\sum_{j=1}^n\widehat f_j^*\sum_{t=1}^T (U_{it}' U_{jt}-EU_{it}' U_{jt}),

We first treat $A^*_{i2} - A_{i4}^*$. 
Because $\tilde U_{it}^*$ and $\tilde U_{jt}^*$ are independent if $i\neq j$, we have
\begin{align*}
\frac{1}{n}\sum_i\|A_{i2}^*\|_2^2 &=  \frac{1}{n}\sum_i  \|\frac{1}{pTn}\sum_{j=1}^n\widehat f_j^*\sum_{t=1}^T E^*\tilde U_{it}^{*'}\tilde U_{jt}^*\|_2^2 \\
&=  \frac{1}{n}\sum_i  \|\frac{1}{pTn} \widehat f_i^*\sum_{t=1}^T E^*\tilde U_{it}^{*'}\tilde U_{it}^*\|_2^2 \\
&= O_{P^*}\left(\frac{1}{n^2}\right).
\end{align*}
  By Lemma \ref{le.10}, $  \frac{1}{n}\sum_i\|A_{i3}^*\|_2^2+  \frac{1}{n}\sum_i\|A_{i4}^*\|_2^2=O_{P^*}(\Delta_F^2)$.  Hence 
  \begin{equation}\label{ee.10a}
  \sum_{l=2}^4 \frac{1}{n}\sum_i\|A_{il}^*\|_2^2=O_{P^*}(\Delta_F^2).
  \end{equation}
  
  
Now we bound $\frac{1}{n}\sum_{i=1}^n\|A^*_{i1}\|_2^2$. A preliminary rate is provided by Lemma \ref{le.9} where we have that 
$$
\frac{1}{n}\sum_{i=1}^n\|A^*_{i1}\|_2^2=O_{P^*}\left(\Delta_F^2+\frac{1}{n}\right).
$$
However, this rate is not sharp due to the $O_{P^*}(n^{-1})$ term and can be improved.  Specifically, the proof of Lemma \ref{le.9} (iii) uses a Cauchy-Schwarz inequality and is not sharp for  terms involving $E\left[U_{it}'U_{jt}\right]$. To see this intuitively, consider a simple example where we bound 
$$
\frac{1}{n}\sum_{i=1}^n\|\frac{1}{nTp}\sum_{j=1}^n\sum_{t=1}^Tf_jEU_{it}'U_{jt}\|^2_2.
$$
Since $U_{it}$ and $U_{jt}$ are independent when $i\neq j$, this term may be simplified to 
$$
\frac{1}{n}\sum_{i=1}^n\|\frac{1}{nTp}\sum_{t=1}^Tf_iEU_{it}'U_{it}\|^2_2 = O_P\left(\frac{1}{n^2}\right).
$$ 
In contrast, using the Cauchy-Schwarz inequality gives
\begin{align*}
\frac{1}{n}\sum_{i=1}^n\|\frac{1}{nTp}\sum_{j=1}^n\sum_{t=1}^Tf_jEU_{it}'U_{jt}\|^2_2 &\leq \frac{1}{n}\sum_j \|\widehat f_j\|_2^2 \frac{1}{n^2}\sum_j\sum_{i=1}^n\left(\frac{1}{Tp}\sum_{t=1}^TEU_{it}'U_{jt}\right)^2 \\
&=O_P\left(\frac{1}{n}\right).
\end{align*}
  	
  
Lemma \ref{le.9} (iii) does provide a useful preliminary rate to build upon.  Applying Lemma \ref{le.9} (iii) and (\ref{ee.10a}), we obtain a preliminary rate 
$$
\frac{1}{n}\sum_{i=1}^n\|\widehat f_i^*-H^{*'}\widehat f_i\|_2^2=O_{P^*}\left(\Delta_F^2+\frac{1}{n}\right).
$$
Our goal is to remove the term $\frac{1}{n}$ through improving the bound for $\frac{1}{n}\sum_{i=1}^n\|A^*_{i1}\|_2^2$. By the triangle inequality,
\begin{align*}
\frac{1}{n}\sum_{i=1}^n\|A^*_{i1}\|_2^2 &\leq \frac{2}{n}\sum_{i=1}^n\left\|\frac{1}{pTn}\sum_{j=1}^nH^{*'}\widehat f_j\sum_{t=1}^T (\tilde U_{it}^{*'}\tilde U_{jt}^*- E^*\tilde U_{it}^{*'}\tilde U_{jt}^*)\right\|_2^2 \\
& \qquad + \frac{2}{n}\sum_{i=1}^n\left\|\frac{1}{n}\sum_{j=1}^n(\widehat f_j^*-H^{*'}\widehat f_j)\frac{1}{pT}\sum_{t=1}^T (\tilde U_{it}^{*'}\tilde U_{jt}^*- E^*\tilde U_{it}^{*'}\tilde U_{jt}^*)\right\|_2^2 \\
&\leq^{(a)} O_{P^*}(\Delta_F^2) + O_{P^*}(\Delta_F^2)\frac{1}{n^2}\sum_{ij}\left(\frac{1}{pT}\sum_{t=1}^T (\tilde U_{it}^{*'}\tilde U_{jt}^*- E^*\tilde U_{it}^{*'}\tilde U_{jt}^*)\right)^2 \\
&=O_{P^*}(\Delta_F^2) 
\end{align*}
 where in (a) we used Lemma \ref{le.11} and the last equality follows from (\ref{ee.13}). Hence combining with (\ref{ee.10a}), we have $ \frac{1}{n}\sum_{i=1}^n\|\widehat f_i^*-H^{*'}\widehat f_i\|_2^2=O_{P^*}(\Delta_F^2).$
Thus, we have $\Delta_F^*=\Delta_F.$
 

(ii)  We now verify the conditions in Assumption \ref{a6.2}:
 $\sqrt{nT}|J|^2_0\Delta_F^{*2}=o(1)$,     $\Delta_{eg}^*=o(\frac{1}{\sqrt{nT}})$,  $ \Delta_{ud}^*=o(\sqrt{\frac{\log p}{nT}}) $,   $|J|_0^2 \sqrt{\log p }      \Delta_{ud}^*=o(1)$,    $  \Delta_{F}^{*2}=o(\frac{\log p}{T \log (pT)}), $     and     $\Delta_{\max}^2|J|_0^2T\Delta_{F}^{*2}=o(1)$.
 
$\Delta_F^{*}$:  We have previously proven that $\Delta_F^2=\frac{1}{n^2}+ 
 \frac{1}{nT^2}+\frac{1}{pT}.$  In addition, Lemma \ref{lc.8} gives $ \Delta_{\max}=\frac{1}{\sqrt{n}}  +\sqrt{\frac{\log n}{Tp}}$. Hence it is straightforward to verify the required conditions involving $\Delta_F^*$, given the assumption that $|J|_0^2\log n=o(p)$.
 

$\Delta^*_{ud}$:  $\Delta_{ud}^*$ is defined in Lemma \ref{le.13} which gives $\Delta_{ud}^*=  b_n+\Delta_{ud}$ for  
$$  
b_n=  \frac{1}{T}\sqrt{\frac{\log p\log (np)}{np}} + \sqrt{\frac{\log n}{npT}}  +\frac{\log p}{n\sqrt{T}} +\frac{1}{pT}+\sqrt{\frac{\log p}{npT}}+\frac{\sqrt{\log p}}{n}+\frac{\sqrt{|J|_0}}{nT}+\sqrt{\frac{|J|_0}{npT}}.
$$   
In the proof of Proposition \ref{p4.1}, we verified  $ \Delta_{ud}=o\left(\sqrt{\frac{\log p}{nT}}\right)$ and $|J|_0^2 \sqrt{\log p }\Delta_{ud}=o(1)$.
 It is also straightforward to verify that $b_n=o\left(\sqrt{\frac{\log p}{nT}}\right) $ and that $|J|_0^2 \sqrt{\log p } b_n=o(1)$ given that $\log n=o(p)$, $|J|^4_0n=o(p^2T)$, $|J|_0^2\log^3p=o(n)$, and $|J|_0^4=o(nT^3)$. In particular, we need to  verify $|J|_0^5\log p\left(\frac{1}{n^2T^2}+\frac{1}{npT}\right)=o(1)$. To verify this condition, we use $|J|^4_0n=o(p^2T)$ and $|J|_0^4=o(nT^3)$ to show
\begin{align*}
|J|_0^5\log p\left(\frac{1}{n^2T^2}+\frac{1}{npT}\right) &= |J|_0^3\log p\left(\frac{|J|_0^2}{n^2T^2}+\frac{|J|_0^2n^{1/2}}{n^{3/2}pT}\right) \\
&=o(1)|J|_0^3\log p\left(\frac{n^{1/2}T^{3/2}}{n^2T^2}+\frac{ pT^{1/2}}{n^{3/2}pT}\right) \\
&=o(1) \frac{|J|_0^3\log p}{n^{3/2}T^{1/2}} \\
&=o(1)(\frac{|J|_0\log p}{n^{1/2}})^3=o(1).
\end{align*}
 
 
$\Delta_{eg}^*$:  Note that for $\widehat g_{tm} \in \{\widehat\Lambda_t'\widehat\gamma_d, \widehat\Lambda_t'\widehat\gamma_y, \widehat\delta_{dt},\widehat\delta_{yt} , \widehat\lambda_{tm}\}$, we have 
$$
\omega_n^*=\max_{m\leq p}\frac{1}{T}\sum_{t=1}^T \|\widehat g_{tm}\|_2^2=O_P(|J|_0^2).
$$ 
Hence by Lemma \ref{le.13a}, $\Delta_{eg}^{*2}=\left(\frac{1}{n^2}+\frac{\log n}{npT}+\frac{\log n}{n^2T^2}+\frac{\log^{1/2}n}{n^2T^{1/2}}\right)|J|_0^2.$  Given $|J|_0^2\log n=o(p)$ and $|J|_0^2T=o(n)$, it is then straightforward to verify $ \Delta_{eg}^{*2}=o\left(\frac{1}{nT}\right)$ which follows by verifying $\frac{|J|_0^2\log n}{nT}=o(1)$. To see $\frac{|J|_0^2\log n}{nT}=o(1)$, note that we have, by $|J|_0^{4/3}=o(n^{1/3}T)$,
$$
\frac{|J|_0^2\log n}{nT}=\frac{|J|_0^{2/3}|J|_0^{4/3}\log n}{nT}=o(1)\frac{|J|_0^{2/3}n^{1/3}T\log n}{nT}=o(1)\frac{|J|_0^{2/3}\log n}{n^{2/3}}=o(1).
$$
$\blacksquare$


\begin{lemma}\label{le.13a}
In the bootstrap sampling space, let  $\tilde z_{it}^*=\widehat z_{it}w_i^Z$ where $\{w_i^Z\}_{i=1}^n$ are  i.i.d. with mean zero and bounded variance and independent of $\{w_i^U\}$ and $\widehat z_{it}=\widehat\eta_{it}$ or $\widehat z_{it}=\widehat\epsilon_{it}$.  In addition, let $\{\widehat g_{tm}\}$ be a deterministic sequence (in the bootstrap sampling space) of vectors with a  fixed dimension, $m\leq p$.  Then for  $\omega_n^*=\max_{m\leq p}\frac{1}{T}\sum_{t=1}^T \|\widehat g_{tm}\|_2^2,$
$$
\max_{m\leq p}\left\|  \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\widehat g_{tm}\tilde z_{it}^*(\widehat f^*_i-H^{*'}\widehat f_i)'\right\|_F^2=
O_{P^*}\left(\frac{1}{n^2}+\frac{\log n}{npT}+\frac{\log n}{n^2T^2}+\frac{\log^{1/2}n}{n^2T^{1/2}}\right)\omega_n^{*}
$$
where the term $O_{P^*}\left(\frac{1}{n^2}+\frac{\log n}{npT}+\frac{\log n}{n^2T^2}+\frac{\log^{1/2}n}{n^2T^{1/2}}\right)\omega_n^{*}$ defines $\Delta_{eg}^{*2}$.
\end{lemma}

 
\proof It follows from (\ref{ee.9}) that 
$$
\max_{m\leq p}\|  \frac{1}{nT}\sum_{i=1}^n\sum_{s=1}^T\tilde z_{is}^*(\widehat f^*_i-H^{*'}\widehat f_i)\widehat g_{sm}'\|_F\leq \sum_{l=1}^3\bar C_l O_P(1),
$$
where each term $\bar C_l$ is defined and bounded below.

First, we have 
\begin{eqnarray*}
\bar C_1 &=& \max_{m\leq p} \left\|  \frac{1}{nT}\sum_{i=1}^n\sum_{s=1}^T\tilde z_{is}^*\frac{1}{pTn}\sum_{j=1}^n\widehat f_j^*\sum_{t=1}^T \tilde U_{it}^{*'}\tilde U_{jt}^*\widehat g_{sm}' \right\|_F\cr
         &\leq&   \left(\frac{1}{pT}\sum_t\|\frac{1}{n}\sum_{j=1}^n\widehat f_j^* \tilde U_{jt}^{*'}\|_F^2\right)^{1/2}\left(\frac{1}{T^2p} \sum_{s, t}\|\frac{1}{n}\sum_{i=1}^n\tilde z_{is}^*\tilde U_{it}^*\|_2^2 \right)^{1/2}\omega_n^{*1/2}\cr
         &=&O_{P^*}\left(\Delta_F+\frac{1}{\sqrt{n}}\right)\frac{1}{\sqrt{n}}\omega_n^{*1/2}\cr
				&=&\omega_n^{1/2} O_{P^*}\left(\frac{1}{\sqrt{npT}}+\frac{1}{n}\right),
              \end{eqnarray*}
where we used  
\begin{align}\label{ee.15}
\begin{split}
\frac{1}{pT}\sum_t\|\frac{1}{n}\sum_{j=1}^n\widehat f_j^* \tilde U_{jt}^{*'}\|_F^2 &\leq O_{P^*}(\Delta_F^2) + O_{P^*}(1)\frac{1}{pT}\sum_t\|\frac{1}{n}\sum_{j=1}^n\widehat f_j \tilde U_{jt}^{*'}\|_F^2 \\
&= O_{P^*}\left(\Delta_F^2+\frac{1}{n}\right), \\
\frac{1}{T^2p} \sum_{s, t}\|\frac{1}{n}\sum_{i=1}^n\tilde z_{is}^*\tilde U_{it}^*\|_2^2 &=
\frac{1}{T^2p} \sum_{s, t}\|\frac{1}{n}\sum_{i=1}^n\widehat  z_{is}w_i^Zw_i^U\widehat  U_{it}\|_2^2 \\
&=O_{P^*}\left(\frac{1}{n}\right),
\end{split}
\end{align}
and $\Delta_F=\Delta_F^*$.

For the second term, we have by Lemma \ref{le.11} that
\begin{align*}
\bar C_2 &= \max_{m\leq p}\left\|  \frac{1}{nT}\sum_{i=1}^n\sum_{s=1}^T\tilde z_{is}^*
\frac{1}{pTn}\sum_{j=1}^n\widehat f_j^*\sum_{t=1}^T\widehat f_j'\widehat \Lambda_t'\tilde U_{it}^*  \widehat g_{sm}'\right\|_F \\
&\leq O_{P^*}(\omega_n^{*1/2})  \left( \frac{1}{T}\sum_{s=1}^T \| \frac{1}{pTn}    \sum_{i=1}^n\sum_{t=1}^T \widehat \Lambda_t'\tilde U_{it}^* \tilde z^*_{is}\|_2^2\right)^{1/2} \\
&= O_{P^*}\left(\sqrt{\frac{\log n}{npT}}+\frac{\sqrt{\log n}}{nT}+ \frac{1}{n}+\frac{1}{n}(\frac{\log n}{T})^{1/4}\right)\omega_n^{*1/2}.
\end{align*}
Finally, 
\begin{align*}
\bar C_3 &= \max_{m\leq p}\left\|  \frac{1}{nT}\sum_{i=1}^n\sum_{s=1}^T\tilde z_{is}^*\frac{1}{pTn}\sum_{j=1}^n\widehat f_j^*\sum_{t=1}^T\widehat f_i'\widehat \Lambda_t'\tilde U_{jt}^*\widehat g_{sm}'\right\|_F \\
&\leq \left(\frac{1}{T}\sum_{s=1}^T( \frac{1}{n}\sum_{i=1}^n\tilde z_{is}^* \widehat f_i'  )^2\right)^{1/2}\omega_n^{*1/2} \left\|\frac{1}{nTp}\sum_{j=1}^n\sum_{t=1}^T\widehat f_j^*\widehat \Lambda_t'\tilde U_{jt}^*\right\|_2 \\
&= O_{P^*}\left(\omega_n^{*1/2}\frac{1}{\sqrt{n}}\right)\left(\Delta_F^2+\sqrt{\frac{\Delta_F^2\log n}{n}}\right)
\end{align*}
where we used Lemma \ref{ld.1} to obtain
$\frac{1}{T}\sum_t\|\frac{1}{n}\sum_{i=1}^n \tilde z_{it}^* \widehat f_i\|_2^2=O_{P^*}\left(\frac{1}{n}\right)$.
 
Combining the above, we have
$$
\max_{m\leq p}\|  \frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^T\widehat g_{tm}\tilde z_{it}^*(\widehat f^*_i-H^{*'}\widehat f_i)'\|_F^2=
O_{P^*}\left(\frac{1}{n^2}+\frac{\log n}{npT}+\frac{\log n}{n^2T^2}+\frac{\log^{1/2}n}{n^2T^{1/2}}\right)\omega_n^{*}.
$$
 $\blacksquare$

\begin{lemma}\label{le.13}
For $\widehat h_{tk} \in \{\widehat \delta_{yt},\widehat \delta_{dt}, \widehat\lambda_{tk}\}$, we have 
\begin{align*}
\max_{m, k\leq p}&\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat f_i^*-H^{*'}\widehat f_i) \tilde U_{it,m}^*  \widehat h_{tk}' \right\|_F \\
&\leq O_{P^*}\left(   \frac{1}{T}\sqrt{\frac{\log p\log (np)}{np}} + \sqrt{\frac{\log n}{npT}}  +\frac{\log p}{n\sqrt{T}} +\frac{1}{pT}+\sqrt{\frac{\log p}{npT}}+\frac{\sqrt{\log p}}{n}+\frac{\sqrt{|J|_0}}{nT}+\sqrt{\frac{|J|_0}{npT}}\right) \\
&\qquad + O_{P^*}(\Delta_{ud})
\end{align*}
for $\Delta_{ud}$ defined as in Lemma \ref{lc.7}.  The term on the right-hand-side of the inequality defines $\Delta_{ud}^{*}$.
%$ \max_{m, k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{t=1}^T(\widehat f_i^*-H^{*'}\widehat f_i) \tilde U_{it,m}^*  \widehat h_{tk}'  \|_F$ is  upper bounded by ($\Delta_{ud}$ is defined as in Lemma \ref{lc.7}): $$  O_{P^*}(   \frac{1}{T}\sqrt{\frac{\log p\log (np)}{np}} + \sqrt{\frac{\log n}{npT}}  +\frac{\log p}{n\sqrt{T}} +\frac{1}{pT}+\sqrt{\frac{\log p}{npT}}+\frac{\sqrt{\log p}}{n}+\frac{\sqrt{|J|_0}}{nT}+\sqrt{\frac{|J|_0}{npT}})+O_{P^*}(\Delta_{ud}).$$   This gives $\Delta_{ud}^*$.
\end{lemma}


\proof We have $ \max_{m, k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T(\widehat f_i^*-H^{*'}\widehat f_i) \tilde U_{is,m}^*  \widehat h_{sk}'  \|_F\leq \sum_{l=1}^5D_l$, where each $D_l$ is defined and bounded below. 
\begin{align*}
D_1 &= \max_{m, k\leq p} \left\| \frac{1}{T}\sum_{t=1}^T\frac{1}{n}\sum_{j=1}^n\widehat f_j^* \tilde U_{jt}^{*'} \frac{1}{nTp} \sum_{i=1}^n\sum_{s=1}^T(\tilde U_{it}^* \tilde U_{is,m}^*-E^*\tilde U_{it}^* \tilde U_{is,m}^*)  \widehat h_{sk}'  \right\|_F \\
&\leq \left(\frac{1}{T}\sum_{t=1}^T\|\frac{1}{n}\sum_{j=1}^n\widehat f_j^* \tilde U_{jt}^{*'}\|_2^2\right)^{1/2}\\
&\qquad \times \max_{m, k\leq p} \left(\frac{1}{T}\sum_{t=1}^T\|\frac{1}{nTp} \sum_{i=1}^n\sum_{s=1}^T(\tilde U_{it}^* \tilde U_{is,m}^*-E^*\tilde U_{it}^* \tilde U_{is,m}^*)  \widehat h_{sk}'\|_F^2\right)^{1/2} \\
&= O_{P^*}\left(\sqrt{\frac{\log (pT)}{n}(\frac{\log n\log (pT)}{n}+\frac{\log(np)}{T})}\right)\left(\Delta_F+\frac{1}{\sqrt{n}} \right)
\end{align*}
where we used Lemma \ref{le.11} and equation (\ref{ee.15}) from the proof of Lemma \ref{le.13a} which gives
$$
\frac{1}{pT}\sum_t\|\frac{1}{n}\sum_{j=1}^n\widehat f_j^* \tilde U_{jt}^{*'}\|_F^2\leq  O_{P^*}\left(\Delta_F^2+\frac{1}{n}\right).$$
Next, 
\begin{eqnarray*}    
D_2 &=&    \max_{m, k\leq p}\left\|\sum_{l=1}^p\frac{1}{T}\sum_{t=1}^T\frac{1}{pn}\sum_{j=1}^n\widehat f_j^* \tilde U_{jt,l}^{*'}\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^TE^*(\tilde U_{it,l}^* \tilde U_{is,m}^*)  \widehat h_{sk}'  \right\|_F\cr
&\leq & \max_{tl}\left\|\frac{1}{n}\sum_{j=1}^n\widehat f_j^* \tilde U_{jt,l}^{*'}\right\|_2
       \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^TE^*(\tilde U_{it,l}^* \tilde U_{is,m}^*)  \widehat h_{sk}'  \right\|_F \cr
&=& O_{P^*}\left(\sqrt{ \frac{\log (pT)}{n}}+\Delta_F\right)\left(\Delta_F+\frac{\sqrt{\log(pT)\log p}}{n}+ \sqrt{\frac{\log (pT)}{nT}}\right)
\end{eqnarray*}
where we used Lemma \ref{le.11} and $\max_{mt} \| \frac{1}{n}\widehat F^{*'}\tilde U_{t, m}^*\|_2 = O_{P^*}\left(\sqrt{ \frac{\log (pT)}{n}}+\Delta_F\right)$ due to Lemma \ref{ld.1}.    
We then have
\begin{eqnarray*}    
D_3&=&    \max_{m, k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pTn}\sum_{j=1}^n\widehat f_j^*\sum_{t=1}^T\widehat f_j'\widehat \Lambda_t'(\tilde U_{it}^* \tilde U_{is,m}^* -E^*\tilde U_{it}^* \tilde U_{is,m}^*) \widehat h_{sk}'  \|_F\cr
&\leq& O_{P^*} (1) \max_{m, k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'(\tilde U_{it}^* \tilde U_{is,m}^* -E^*\tilde U_{it}^* \tilde U_{is,m}^*) \widehat h_{sk}'  \|_F\cr
&=& O_{P^*}\left(\sqrt{\frac{\log p}{n}(\frac{\log n\log (pT)}{n}+\frac{\log(np)}{T})\Delta_F^2}\right)
\end{eqnarray*}
by Lemma \ref{le.11}.  We also have
\begin{eqnarray*}
D_4 &=& \max_{m, k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pTn}\sum_{j=1}^n\widehat f_j^*\sum_{t=1}^T\widehat f_j'\widehat \Lambda_t'E^*\tilde U_{it}^* \tilde U_{is,m}^*  \widehat h_{sk}'  \|_F\cr
&\leq& O_{P^*} (1) \max_{m, k\leq p}\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'E^*\tilde U_{it}^* \tilde U_{is,m}^*  \widehat h_{sk}'  \|_F\cr
&=& O_{P^*} \left(\frac{1}{pT}+\sqrt{\frac{\log p}{npT}}+\frac{\sqrt{\log p}}{n}+\frac{\sqrt{|J|_0}}{nT}+\sqrt{\frac{|J|_0}{npT}}\right)+ O_{P^*}(\Delta_{ud})
\end{eqnarray*}
where the inequality follows from Lemma \ref{le.12} (iii). 
Finally,
\begin{align*}    
D_5 &= \max_{m, k\leq p}\left\|  \frac{1}{pTn}\sum_{j=1}^n\widehat f_j^*\sum_{t=1}^T\widehat\Lambda_t'\tilde U_{jt}^* \frac{1}{nT}\sum_{s=1}^T \sum_{i=1}^n\widehat f_i\tilde U_{is,m}^*  \widehat h_{sk}'  \right\|_F \\
&\leq \left\|  \frac{1}{pTn}\sum_{j=1}^n\widehat f_j^*\sum_{t=1}^T\widehat\Lambda_t'\tilde U_{jt}^*\right\|_2\max_{m, k\leq p}\left\| \frac{1}{nT}\sum_{s=1}^T \sum_{i=1}^n\widehat f_i\tilde U_{is,m}^*  \widehat h_{sk}'  \right\|_F \\
&=^{(a)} O_{P^*}\left(\Delta_F^2+\sqrt{\frac{\Delta_F^2\log n}{n}}\right)\max_{m, k\leq p} \left\| \frac{1}{nT}\sum_{s=1}^T \sum_{i=1}^n\widehat f_i\tilde U_{is,m}^*  \widehat h_{sk}'  \right\|_F \\
&= O_{P^*}\left(\Delta_F^2+\sqrt{\frac{\Delta_F^2\log n}{n}}\right)
\end{align*}
where equality (a) results by applying Lemma \ref{le.11} (iii).  Note that the upper bound achieved in the last equality is not sharp but is sufficient to verify Assumptions about $\Delta_{ud}^*$.
 
Combining the above terms, we reach 
\begin{align*}   
\max_{m, k\leq p}&\left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T(\widehat f_i^*-H^{*'}\widehat f_i) \tilde U_{is,m}^*  \widehat h_{sk}' \right\|_F \\
&= O_{P^*}\left(\sqrt{\frac{\log (pT)}{n}(\frac{\log n\log (pT)}{n}+\frac{\log(np)}{T})}\right)\left(\Delta_F+\frac{1}{\sqrt{n}} \right) \\
& \qquad + O_{P^*}\left(\sqrt{ \frac{\log (pT)}{n}}+\Delta_F\right)\left(\Delta_F+\frac{\sqrt{\log(pT)\log p}}{n}+ \sqrt{\frac{\log (pT)}{nT}}\right) \\
& \qquad + O_{P^*}(\Delta_{ud}+\Delta_F\sqrt{\frac{\log n}{n}}) +
  O_{P^*}\left(\Delta_F\sqrt{\frac{\log p}{n}(\frac{\log n\log (pT)}{n}+\frac{\log(np)}{T})}\right) \\
& \qquad +O_{P^*}\left(\frac{1}{pT}+\sqrt{\frac{\log p}{npT}}+\frac{\sqrt{\log p}}{n}+\frac{\sqrt{|J|_0}}{nT}+\sqrt{\frac{|J|_0}{npT}}\right) \\
&= O_{P^*}\left(   \frac{1}{T}\sqrt{\frac{\log p\log (np)}{np}} + \sqrt{\frac{\log n}{npT}}  +\frac{\log p}{n\sqrt{T}} +\frac{1}{pT}+\sqrt{\frac{\log p}{npT}}+\frac{\sqrt{\log p}}{n}+\frac{\sqrt{|J|_0}}{nT}+\sqrt{\frac{|J|_0}{npT}}\right) \\
& \qquad +O_{P^*}(\Delta_{ud})
\end{align*}
where $\Delta_{ud}$ is defined as in Lemma \ref{lc.7}. $\blacksquare$
 
   




\subsection{Technical lemmas for Proposition \ref{p4.1}: the original data  }


 

\begin{lemma}\label{lc.1} Let $A_{i1}$ be as defined in (\ref{ec.1}).  Then
$$
\frac{1}{n}\sum_{i=1}^n\|A_{i1}\|_2^2= O_P\left(\frac{1}{n^2}+ 
\frac{1}{npT }+\frac{1}{nT^2}+\frac{1}{pT^2}\right).
$$
\end{lemma}

\proof   First, we have the following equality:
$$
\frac{1}{T} \sum_{t=1}^T\tilde U_{it}'\tilde U_{jt}
  =   \frac{1}{T}  \sum_{t=1}^T U_{it} 'U_{jt}
  -\bar U_{i\cdot}'\bar  U_{j\cdot}
- \frac{1}{T} \sum_{t=1}^T U_{it}' \bar  U_{\cdot t}
+\bar U_{i\cdot}'\bar{\bar U}
+\bar{\bar  U}'\bar  U_{j\cdot}
- \bar{\bar  U}'\bar{\bar U}
 -  \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t} 'U_{jt}
 + \frac{1}{T} \sum_{t=1}^T \bar U_{\cdot t} '\bar U_{\cdot t}.
$$
Therefore,  $  \frac{1}{n}\sum_{i=1}^n\|A_{i1}\|_2^2\leq  \frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n(\frac{1}{pT}\sum_{t=1}^T   \tilde U_{it}'\tilde U_{jt}-U_{it}' U_{jt})^2 O_P(1) $, and 
\begin{align*}
E\left[\frac{1}{n^2} \right. & \left. \sum_{i=1}^n \sum_{j=1}^n(\frac{1}{pT}\sum_{t=1}^T  (\tilde U_{it}'\tilde U_{jt}-U_{it}' U_{jt}))^2\right] \\
&\leq C E\left[\frac{1}{n}   \sum_{i=1}^n(\frac{1}{pT}     \sum_{t=1}^T \bar U_{\cdot t} 'U_{it}   )^2 \right]  
 +CE\left[  (\frac{1}{pT}\sum_{t=1}^T \bar U_{\cdot t}'\bar U_{\cdot t})^2\right] \\
&\qquad + CE\left[  \frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n(\frac{1}{p}  \bar U_{i\cdot}'\bar  U_{j\cdot} )^2\right]  
+CE\left[  \frac{1}{n }\sum_{i=1}^n (\frac{1}{p}   \bar U_{i\cdot}'\bar{\bar U}  )^2  \right]
+ CE\left[   (\frac{1}{p}   \bar{\bar  U}'\bar{\bar U}    )^2 \right] \\
&:=\sum_{l=1}^5\Delta_l.
%+   \frac{C}{n}\sum_{i=1}^n E (\frac{1}{pT}\sum_{t=1}^T  U_{it}'\bar U_{\cdot t})^2
% +CE  \frac{1}{n}     \sum_{j=1}^n(\frac{1}{p}   \bar{\bar  U}'\bar  U_{j\cdot}    )^2  
\end{align*}
We now separately bound $\Delta_1-\Delta_5$. 
\begin{align*}%\label{ec.2}
\Delta_1 =& E\left[  \frac{1}{n}   \sum_{i=1}^n(\frac{1}{pT}     \sum_{t=1}^T \bar U_{\cdot t} 'U_{it}   )^2\right] \\  
&= \frac{1}{n^3}\sum_{i=1}^n \sum_{j=1}^n \sum_{l=1}^n  \frac{1}{p^2T^2}\sum_{t=1}^T 
 \sum_{s=1}^T E\left[  U_{is}'  U_{js}   U_{it}'  U_{l t}\right] \\
&= \frac{1}{n^3}\sum_{i=1}^n   \frac{1}{p^2T^2}\sum_{t=1}^T 
 \sum_{s=1}^T E\left[  U_{is}'  U_{is}   U_{it}'  U_{i t}\right]
 +\frac{1}{n^3}\sum_{i=1}^n \sum_{j\neq i}  \frac{1}{p^2T^2}\sum_{t=1}^T 
 \sum_{s=1}^T E\left[  U_{is}'  U_{js}   U_{it}'  U_{j t}\right] \\
&= \frac{1}{n^3}\sum_{i=1}^n \sum_{j=1}^n    \frac{1}{p^2T^2}
E\left[ \left( \sum_{s=1}^TU_{is}'  U_{js}\right)^2\right] \\
&= \frac{1}{n^3}\sum_{i=1}^n \sum_{j=1}^n    \frac{1}{p^2T^2}
\Var\left(  \sum_{s=1}^TU_{is}'  U_{js}\right) 
+\frac{1}{n^3}\sum_{i=1}^n \sum_{j=1}^n    \frac{1}{p^2T^2}
 \left( E\left[ \sum_{s=1}^TU_{is}'  U_{js}\right]\right)^2 \\
&= \frac{1}{n^3}\sum_{i=1}^n \sum_{j=1}^n    \frac{1}{p^2T^2}
  \sum_{t=1}^T  \sum_{s=1}^T\Cov (U_{is}'U_{js}, U_{it}'U_{jt}) 
+\frac{1}{n^3}\sum_{i=1}^n    \frac{1}{p^2T^2}
 \left(  \sum_{s=1}^TE\left[U_{is}'  U_{is}\right]\right)^2 \\
&= \frac{1}{n^3}\sum_{i=1}^n      \frac{1}{p^2T^2}
  \sum_{t=1}^T  \sum_{s=1}^T\Cov (U_{is}'U_{is}, U_{it}'U_{it}) \\
&\qquad +\frac{1}{n^3}\sum_{i=1}^n \sum_{j\neq i}    \frac{1}{p^2T^2}
  \sum_{t=1}^T  \sum_{s=1}^T\sum_{m=1}^p\sum_{v=1}^p E \left[U_{is,m} U_{it,v} EU_{js,m}U_{jt,v}\right] \\
&\qquad +\frac{1}{n^3}\sum_{i=1}^n    \frac{1}{p^2T^2}
 \left(  \sum_{s=1}^T\sum_{m=1}^p E\left[U_{is,m}^2\right]\right)^2 \\
&\leq \frac{1}{n^2T}     
 \max_{i\leq n, t\leq T}\sum_{s=1}^T\left|\Cov \left(\frac{1}{p}U_{is}'U_{is},\frac{1}{p} U_{it}'U_{it}\right) \right| \\
&\qquad + \frac{1}{n^3}\sum_{i=1}^n \sum_{j\neq i}    \frac{1}{p^2T^2}
  \sum_{t=1}^T \sum_{m=1}^p\max_{istmv}\left|E\left[ U_{is,m} U_{it,v} \right]\right|\sum_{v=1}^p \sum_{s=1}^T \left|E\left[U_{js,m}U_{jt,v}\right]\right| \\
&\qquad +\frac{1}{n^2}  
   \max_{i,s,m}\Var^2(U_{is,m} ) \\
&= O\left(\frac{1}{n^2}+\frac{1}{npT}\right),
\end{align*}
where we used the condition that $\max_{i\leq n}\max_{m\leq p,t\leq T} \sum_{s=1}^T\sum_{v=1}^p\left|\Cov(U_{it,m}, U_{is,v})\right|=O(1)$.
    
\begin{align*}%\label{ec.3}
\Delta_2 &= E\left[\left(\frac{1}{pT}\sum_{t=1}^T \bar U_{\cdot t}'\bar U_{\cdot t}\right)^2\right] \\
&=E\left[ \frac{1}{p^2T^2}\sum_{t=1}^T  \sum_{s=1}^T \bar U_{\cdot t}'\bar U_{\cdot t} 
 \bar U_{\cdot s}'\bar U_{\cdot s}\right] \\
&= \frac{1}{p^2T^2}\sum_{t=1}^T  \sum_{s=1}^T \frac{1}{n^4} \sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n\sum_{h=1}^n E\left[U_{i t}'U_{jt}  U_{k s}' U_{h s}\right] \\
&= \frac{1}{p^2T^2} \frac{1}{n^4} \sum_{i=1}^n  E\left[\left(\sum_{t=1}^T U_{i t}'U_{it}\right)^2\right]
+ \frac{1}{p^2T^2}  \frac{1}{n^4} \sum_{i=1}^n\sum_{j\neq i}  E\left[\left(\sum_{t=1}^TU_{i t}'U_{jt} \right)^2\right] \\  
&\qquad + \frac{1}{p^2T^2}\frac{1}{n^4} \sum_{i=1}^n\sum_{k\neq i}\sum_{t=1}^T E\left[U_{i t}'U_{it}\right]   \sum_{s=1}^T  E\left[U_{k s}' U_{k s}\right]  
+ \frac{1}{p^2T^2}\sum_{t=1}^T  \sum_{s=1}^T \frac{1}{n^4} \sum_{i=1}^n\sum_{j\neq i} E\left[U_{i t}'U_{jt}  U_{j s}' U_{i s}\right] \\
&=O\left(\frac{1}{n^2}\right).
\end{align*}
\begin{align*}
\Delta_3 &= \frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n E\left[\left(\frac{1}{p}  \bar U_{i\cdot}'\bar  U_{j\cdot} \right)^2\right] \\ 
&= \frac{1}{n^2}\sum_{i=1}^n E\left[\left(\frac{1}{p}  \sum_{m=1}^p\bar U_{i\cdot, m}^2\right)^2\right]  %\frac{1}{n^2p^2T^4}\sum_{i=1}^n \sum_{j=1}^n     \sum_{k=1}^T\sum_{v=1}^T\sum_{s=1}^T\sum_{t=1}^T \sum_{m=1}^p\sum_{l=1}^pEU_{it,m}U_{jsm}      U_{iv,l} U_{jk,l}    
%=\frac{1}{n^2p^2}\sum_{i=1}^n \sum_{j=1}^n\Var( \bar U_{i\cdot}'\bar  U_{j\cdot} )   
\\
&\qquad + \frac{1}{n^2p^2T^4}\sum_{i=1}^n \sum_{j\neq i}     \sum_{k=1}^T\sum_{v=1}^T\sum_{s=1}^T\sum_{t=1}^T \sum_{m=1}^p\sum_{l=1}^p E\left[U_{it,m}   U_{iv,l} E U_{jk,l}   U_{js,m}   \right]
\\
&\leq \frac{1}{n^2}\sum_{i=1}^n \Var\left(\frac{1}{p}  \sum_{m=1}^p\bar U_{i\cdot, m}^2\right)
+\frac{1}{n^2}\sum_{i=1}^n \left(\frac{1}{p}  \sum_{m=1}^pE\left[\bar U_{i\cdot, m}^2\right]\right)^2 \\
&\qquad + \frac{1}{n^2p^2T^4}\sum_{i=1}^n \sum_{j\neq i}  \sum_{s=1}^T\sum_{t=1}^T \sum_{m=1}^p       \max_{l\leq p}    \sum_{k=1}^T \left| E \left[U_{jk,l}   U_{js,m}\right] \right|  \max_{i\leq n, m\leq p, t\leq T} \sum_{l=1}^p \sum_{v=1}^T  \left|E\left[U_{it,m}   U_{iv,l}\right] \right|  \\
&\leq \frac{1}{n^2}\sum_{i=1}^n \Var\left(\frac{1}{p}  \sum_{m=1}^p\bar U_{i\cdot, m}^2\right)
+\frac{1}{n} \left[\max_{i\leq n, m\leq p}\Var\left(\bar U_{i\cdot, m}\right)\right]^2+\frac{C}{pT^2} \\
&= O\left(\frac{1}{npT^2 }\right)+O\left(\frac{1}{nT^2}\right)+O\left(\frac{1}{pT^2}\right)%=O(\frac{1}{nT^2})+O(\frac{1}{pT^2}).
%+\frac{1}{n^2p^2}\sum_{i=1}^n \sum_{j=1}^n (E \bar U_{i\cdot}'\bar  U_{j\cdot} )^2\cr
%&=&\frac{1}{n^2 }\sum_{i=1}^n  \Var(\frac{1}{p} \bar U_{i\cdot}'\bar  U_{i\cdot} )   +\frac{1}{n^2p^2}\sum_{i=1}^n \sum_{j\neq i}\Cov( \bar U_{i\cdot}'\bar  U_{j\cdot}, \bar U_{i\cdot}'\bar  U_{j\cdot} )  +\frac{1}{n^2p^2}\sum_{i=1}^n   (E \bar U_{i\cdot}'\bar  U_{i\cdot} )^2
\end{align*}
where we assumed $\frac{1}{n}\sum_i  \Var\left(\frac{1}{p}  \sum_{m=1}^p\bar U_{i\cdot, m}^2\right)=O\left(\frac{1}{pT^2}\right)$.
\begin{align*}
\Delta_4&= E\left[  \frac{1}{n }\sum_{i=1}^n \left(\frac{1}{p}   \bar U_{i\cdot}'\bar{\bar U}  \right)^2\right] \\ 
&=\frac{1}{np^2}\frac{1}{n^2T^4}\sum_{i=1}^n \sum_{t=1}^T\sum_{s=1}^T
 \sum_{k=1}^T\sum_{v=1}^T\sum_{m=1}^p\sum_{l=1}^p  E\left[U_{it,m}{ U}_{is,m} U_{ik,l}{ U}_{iv,l}\right] \\
&\qquad +\frac{1}{np^2}\frac{1}{n^2T^4}\sum_{i=1}^n\sum_{j\neq i}\sum_{t=1}^T\sum_{s=1}^T
 \sum_{k=1}^T\sum_{m=1}^p\sum_{l=1}^p\sum_{v=1}^T  E\left[U_{it,m} U_{ik,l}E{ U}_{jv,l}  { U}_{js,m}\right] \\
&\leq 
 \frac{1}{n^3}\sum_{i=1}^n   
   E\left[\left(\frac{1}{p} \sum_{m=1}^p\bar U_{i\cdot ,m}^2  \right)^2\right] \\
&\qquad + 
 \frac{1}{n^3T^4p^2}\sum_{i=1}^n\sum_{j\neq i}\sum_{t=1}^T\sum_{s=1}^T
\sum_{m=1}^p  \max_{itml}\sum_{k=1}^T \left|E\left[U_{it,m} U_{ik,l}\right]\right| \max_{jsm}\sum_{l=1}^p \sum_{v=1}^T\left|E\left[{ U}_{jv,l}  { U}_{js,m} \right] \right| \\
&=O\left(\frac{1}{n^2p}+\frac{1}{n^2T^2}\right)+O\left(\frac{1}{npT^2}\right). 
\end{align*}
\begin{align*}
\Delta_5 &=E\left[\left(\frac{1}{p}   \bar{\bar  U}'\bar{\bar U}    \right)^2\right] \\
&=\frac{1}{p^2}E\left[\left(  \sum_{m=1}^p\left(\frac{1}{nT}\sum_{i=1}^n\sum_{t=1}^TU_{it,m}\right)^2  \right)^2\right] \\
&= \frac{1}{n^2}\frac{1}{n^2}\sum_{i=1}^n 
    E\left[\left(\frac{1}{p}\sum_{m=1}^p   \bar U_{i\cdot ,m}^2\right)^2\right] \\
&\qquad +\frac{1}{p^2}\frac{1}{n^2T^2}\frac{1}{n^2T^2}\sum_{i=1}^n \sum_{t=1}^T \sum_{j\neq i}\sum_{s=1}^T \sum_{l=1}^p\sum_{m=1}^p\sum_{v=1}^T E\left[U_{iv,l}U_{it,m}\right] \sum_{b=1}^T E\left[U_{jb,l}U_{js,m}\right] \\
&\qquad +\frac{1}{p^2}\frac{1}{n^2T^2}\frac{1}{n^2T^2}\sum_{i=1}^n
\sum_{j\neq i}\sum_{v=1}^T\sum_{b=1}^T \sum_{l=1}^p E\left[U_{jv,l}U_{jb,l}\right] \sum_{t=1}^T  \sum_{s=1}^T\sum_{m=1}^p E\left[U_{it,m}U_{is,m}\right] \\
&\qquad +\frac{1}{p^2}\frac{1}{n^2T^2}\frac{1}{n^2T^2}\sum_{i=1}^n \sum_{m=1}^p\sum_{t=1}^T \sum_{j\neq i}\sum_{s=1}^T \sum_{l=1}^p\sum_{v=1}^T E\left[U_{jv,l}U_{js,m}\right] \sum_{b=1}^T E\left[U_{ib,l}U_{it,m}\right] \\
%=\frac{1}{p^2}E(\|\bar{\bar U}\|_2^2)^2=\frac{1}{p^2}(E\|\bar{\bar U}\|_2^2)^2+\frac{1}{p^2}\Var(\|\bar{\bar U}\|_2^2)
&\leq O\left(\frac{1}{n^3p}+\frac{1}{n^3T^2}\right)+O\left(\frac{1}{n^2pT^2}\right)+O\left(\frac{1}{n^2T^2}\right)+O\left(\frac{1}{n^2pT^2}\right).
\end{align*}
Hence,
$
\frac{1}{n}\sum_{i=1}^n\|A_{i1}\|_2^2=O_P(1)\sum_{l=1}^5\Delta_l=O_P\left(\frac{1}{n^2}+ 
\frac{1}{npT }+\frac{1}{nT^2}+\frac{1}{pT^2}\right).
$
$\blacksquare$
 
\begin{lemma} \label{lc.2} Let $A_{i2},..., A_{i5}$ be as defined in (\ref{ec.1}). Then
$$
\frac{1}{n}\sum_{i=1}^n\|A_{i2}+...+A_{i5}\|_2^2= O_P\left(\frac{1}{pT}+ 
\frac{1}{n^2 }\right).
$$
\end{lemma}
  

\proof 
$A_{i2}$: We prove $\frac{1}{n}\sum_{i=1}^n\|A_{i2}\|_2^2=O_P\left(\frac{1}{Tp}\right)$. By (\ref{ee.2}),
\begin{align}
\nonumber
\frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n & E\left[\left(\frac{1}{pT}\sum_{t=1}^TU_{it}' U_{jt}-EU_{it}' U_{jt}\right)^2\right] \\
\nonumber
&=\frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n\Var\left(\frac{1}{pT}\sum_{t=1}^TU_{it}' U_{jt}\right) \\
\nonumber
&= \frac{1}{n^2}\sum_{i=1}^n \frac{1}{p^2T^2} \sum_{s=1}^T\sum_{t=1}^T\sum_{m=1}^p\sum_{v=1}^p\Cov(U_{it,m}^2,U_{is,v}^2) \\
\nonumber
&\qquad + \frac{1}{n^2}\sum_{i=1}^n \sum_{j\neq i}\frac{1}{p^2T^2} \sum_{s=1}^T\sum_{t=1}^T\sum_{m=1}^p\sum_{v=1}^p E\left[U_{it,m}U_{is,v}EU_{js,v}U_{jt,m}\right] \\
\label{ee.6a}
\begin{split}
&\leq \frac{1}{np}    \max_{i\leq n, m\leq p}\max_{t,s\leq T}\sum_{v=1}^p\left|\Cov(U_{it,m}^2,U_{is,v}^2)\right| \\
&\qquad + \max_{j,s,v,t,m}\left|EU_{js,v}U_{jt,m}\right| \frac{1}{pT} \max_{i\leq n}\max_{m\leq p,t\leq T} \sum_{s=1}^T\sum_{v=1}^p\left|\Cov(U_{it,m}, U_{is,v})\right| \\
&=O\left(\frac{1}{pT}\right),
\end{split}
\end{align}
where we used the conditions 
$$
\max_{i\leq n}\max_{m\leq p,t\leq T} \sum_{s=1}^T\sum_{v=1}^p|\Cov(U_{it,m}, U_{is,v})|=O(1),
$$
$$ 
\max_{i\leq n, m\leq p}\max_{t,s\leq T}\sum_{v=1}^p|\Cov(U_{it,m}^2,U_{is,v}^2)|=O(1),
$$
and $T=o(n)$.
 

$A_{i3}:$ We prove $\frac{1}{n}\sum_{i=1}^n\|A_{i3}\|_2^2=O_P\left(\frac{1}{n^2}\right)$.
$$
\frac{1}{n}\sum_{i=1}^n\|A_{i3}\|_2^2= \frac{1}{n}\sum_{i=1}^n\left\|\frac{1}{pTn} \widehat f_i\sum_{t=1}^T EU_{it}'U_{it}\right\|_2^2=O\left(\frac{1}{n^2}\right).
$$
 

$A_{i4}:$ We prove $\frac{1}{n}\sum_{i=1}^n\|A_{i4}\|_2^2=O_P\left(\frac{1}{pT}\right)$. Note that 
$$
\frac{1}{T}\sum_{t=1}^T\tilde \Lambda_t'\bar U_{i\cdot} =\frac{1}{pT}\sum_{t=1}^T\tilde \Lambda_t'\bar {\bar U} =0.
$$
Then
\begin{align*}
\frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n&\left(\frac{1}{pT}\sum_{t=1}^T\tilde f_j'\tilde \Lambda_t'\tilde U_{it}\right)^2 \\
&\leq \frac{2}{n^2}\sum_{i=1}^n \sum_{j=1}^n(\tilde f_j'\frac{1}{pT}\sum_{t=1}^T\tilde \Lambda_t'U_{it} )^2 +\frac{2}{n^2}\sum_{i=1}^n \sum_{j=1}^n\left(\frac{1}{pT}\sum_{t=1}^T\tilde f_j'\tilde \Lambda_t'\bar U_{\cdot t}\right)^2 \\
&\leq \frac{2}{n} \sum_{j=1}^n\left\|\tilde f_j\right\|_2^2\frac{1}{n}\sum_{i=1}^n \left\|\frac{1}{pT}\sum_{t=1}^T\tilde \Lambda_t'U_{it} \right\|_2^2 
+ \frac{2}{n} \sum_{j=1}^n\left\|\tilde f_j\right\|_2^2  \left\|\frac{1}{pT}\sum_{t=1}^T\tilde \Lambda_t'\bar U_{\cdot t} \right\|_2^2. 
\end{align*}
The first term following the inequality is $O_P\left(\frac{1}{pT}\right)$ given the condition  
$$
\max_{i\leq n}\max_{m\leq p,t\leq T} \sum_{s=1}^T\sum_{v=1}^p \left|\Cov(U_{it,m}, U_{is,v})\right|=O(1).
$$
The second term is $O_P\left(\frac{1}{np}\right)$ and is negligible since $T=o(n)$.
    
Finally, $\frac{1}{n}\sum_{i=1}^n\|A_{i5}\|_2^2=O_P\left(\frac{1}{pT}\right)$ is bounded similarly to $\frac{1}{n}\sum_{i=1}^n\|A_{i4}\|_2^2$.
  $\blacksquare$

\begin{lemma}\label{lc.3} Let $\{z_{it}\}$ be a random sequence with $E(z_{it}|f_t, U_t)=0$, $E(z_{it}^2)=O(1)$, and $$
\max_{ism}\sum_{h=1}^T \sum_{k=1}^p  |E[   U_{is,m}
 U_{ih,k} |z_{i1},...,z_{iT}]|<C
$$ 
almost surely in $\{z_{it}\}$. 
Then for $\tilde z_{it}=z_{it}-\bar z_{i\cdot}-\bar z_{\cdot t}+\bar{\bar z}$,
$$\frac{1}{T}\sum_{t=1}^T \left\| \frac{1}{pT} \frac{1}{n}  \sum_{i=1}^n\sum_{s=1}^T \tilde \Lambda_s'\tilde U_{is} \tilde z_{it}\right\|_2^2=O\left(\frac{1}{pTn}\right).$$
\end{lemma}

%\textcolor{blue}{What do $E[\cdot|\{z_i\}]$ and $E[\cdot|\{z_{it}\}]$ denote?  Are these different from $E[\cdot|z_i]$ and $E[\cdot|z_{it}]$?}

\proof Since $\tilde\Lambda_s'$ has $K$ number of rows with fixed $K$, for notational simplicity, we assume that $\tilde\Lambda_s'$ is a row vector without loss of generality as we can always look at each row of  $\tilde\Lambda_s'$ separately. Then 
$$
E\left[ \frac{1}{T}\sum_{t=1}^T \left| \frac{1}{pT} \frac{1}{n}  \sum_{i=1}^n\sum_{s=1}^T \tilde \Lambda_s'\tilde U_{is} \tilde z_{it}\right|^2\right] \leq C \sum_{l=1}^7 \bar A_l
$$
where each $\bar A_l$ is defined and bounded below.

\begin{align*}
\bar  A_1 &=  E\left[\frac{1}{T}\sum_{t=1}^T \left| \frac{1}{pT} \frac{1}{n}  \sum_{i=1}^n\sum_{s=1}^T \tilde \Lambda_s'  U_{is}   z_{it}\right|^2\right] \\
 &=  \frac{1}{T}\sum_{t=1}^T \frac{1}{p^2T^2n^2}  \sum_{m=	1}^p\sum_{k=1}^p\sum_{h=1}^T  \sum_{i=1}^n\sum_{s=1}^T \tilde \lambda_{sm} \tilde \lambda_{hk}  E\left[   z_{it}^2U_{is,m}
 U_{ih,k}\right]   \\
 &\leq O(1)\frac{1}{T}\sum_{t=1}^T \frac{1}{p^2T^2n^2}  \sum_{m=	1}^p\sum_{i=1}^n\sum_{s=1}^T   {E\left[z_{it}^2\sum_{h=1}^T \sum_{k=1}^p  \left| E\left[  U_{is,m}
 U_{ih,k} |z_{it}\right]\right|\right]}=O\left(\frac{1}{pTn}\right). \\
\bar A_2 &= E\left[\frac{1}{T}\sum_{t=1}^T \left| \frac{1}{pT} \frac{1}{n}  \sum_{i=1}^n\sum_{s=1}^T \tilde \Lambda_s'  U_{is}   \bar z_{i\cdot}\right|^2\right] \\
 &\leq O(1)\frac{1}{T}\sum_{t=1}^T \frac{1}{pT} \frac{1}{n^2}  \frac{1}{T^2}\sum_{h=1}^T  \frac{1}{pT}  \sum_{i=1}^n \sum_{v=1}^T\sum_{s=1}^T\sum_{m=1}^p
 {E\left[\left|z_{ih}z_{iv}\right|  \sum_{w=1}^T  \sum_{k=1}^p\left|E\left[U_{is,m} U_{iw,k}    |z_i\right]\right|\right]} \\
 &= O\left(\frac{1}{pTn}\right). \\
 \bar A_3 &= E\left[\frac{1}{T}\sum_{t=1}^T \left| \frac{1}{pT} \frac{1}{n}  \sum_{i=1}^n\sum_{s=1}^T \tilde \Lambda_s'  U_{is}   \bar z_{\cdot t}\right|^2\right] \\
&\leq O(1) \frac{1}{T}\sum_{t=1}^T  \frac{1}{pT} \frac{1}{n^4}  \sum_{i=1}^n\sum_{s=1}^T   \frac{1}{pT}   \sum_{m=1}^p      {E\left[z_{i t}^2\sum_{k=1}^p\sum_{v=1}^T\left|E\left[U_{is,m}   U_{iv,k} |z_i\right]\right|\right]}   \\
 &\qquad +O(1)\frac{1}{T}\sum_{t=1}^T  \frac{1}{pT} \frac{1}{n^4}  \sum_{i=1}^n\sum_{s=1}^T  \sum_{j\neq i} 
 \frac{1}{pT} \sum_{m=1}^p     {E\left[z_{j t}^2\right] \sum_{v=1}^T\sum_{k=1}^p |E\left[U_{is,m}   U_{iv,k}\right]|} \\
&= O\left(\frac{1}{Tpn^2}\right). \\
 \bar A_4&= E\left[\frac{1}{T}\sum_{t=1}^T | \frac{1}{pT} \frac{1}{n}  \sum_{i=1}^n\sum_{s=1}^T \tilde \Lambda_s'  U_{is}   \bar {\bar z}|^2\right] =O\left(\frac{1}{n^2pT}\right). \\
 \bar A_5&=E\left[\frac{1}{T}\sum_{t=1}^T | \frac{1}{pT} \frac{1}{n}  \sum_{i=1}^n\sum_{s=1}^T \tilde \Lambda_s'  \bar U_{i\cdot}   \tilde z_{it}|^2\right] =0  \quad (\text{because } \frac{1}{T}\sum_s\tilde\Lambda_s=0). \\
 \bar A_6&= E\left[\frac{1}{T}\sum_{t=1}^T | \frac{1}{pT} \frac{1}{n}  \sum_{i=1}^n\sum_{s=1}^T \tilde \Lambda_s'  \bar U_{\cdot s}   \tilde z_{it}|^2\right] =0  \quad (\text{because } \frac{1}{n}\sum_i\tilde z_{it}=0). \\
  \bar A_7&= E\left[\frac{1}{T}\sum_{t=1}^T | \frac{1}{pT} \frac{1}{n}  \sum_{i=1}^n\sum_{s=1}^T \tilde \Lambda_s'  \bar {\bar{U}}   \tilde z_{it}|^2\right] =0.
\end{align*}
     $\blacksquare$

%\textcolor{blue}{DOUBLE CHECK all the expressions in blue in the previous display.}
 
       
The following lemmas are used to bound $ \frac{1}{T} \sum_{t=1}^T\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}(\widehat  f_i-H'\tilde f_i)\|^2_2$ and \\ $ \max_{s\leq T, m\leq p}|\frac{1}{n}  \sum_{i=1}^n (H^{'}\tilde f_i-\widehat f_i)\tilde U_{is,m} |.$   


\begin{lemma}\label{lc.5}
 (i) $ \max_{m,s}  \left\|   \frac{1}{pTn^2} \sum_{i,j} \widehat f_j\sum_{t=1}^T (U_{it}' U_{jt} -EU_{it}' U_{jt} )\tilde U_{is,m}\right\|_2=O_P\left(\frac{1}{n}+\frac{1}{pT}\right).$\\
 (ii) $\max_{m}\left\|\frac{1}{pT} \sum_{t=1}^T\tilde\Lambda_t'
 \frac{1}{n}  \sum_{i=1}^nU_{it}\bar U_{i\cdot, m}    \right\|_2 =O_P\left(\sqrt{\frac{\log p}{nT^2p}}+\frac{1}{pT}\right).$\\
 (iii) $ \max_{m,s}\left\|\frac{1}{pTn^2}\sum_{i, j}\widehat f_j\sum_{t=1}^T\tilde f_j'\tilde\Lambda_t'(\tilde U_{it} \tilde U_{is,m}- U_{it}  U_{is,m}) \right\|_2=O_P\left(\sqrt{\frac{\log p}{nT^2p}}+\frac{1}{pT}+\sqrt{\frac{\log(pT)}{n^2pT}}\right).$\\
 (iv) $ \max_{mktl}  \left\|   \frac{1}{nT}\sum_{i=1}^n\sum_{s=1}^T  (U_{it,l}    U_{is,m}-EU_{it,l}U_{is,m}) \tilde g_{sk}'   \right\|_F=O_P\left(\sqrt{\frac{\log(pT)}{nT}}\right)$,
  Here  $\{g_{sk}: k\leq p\}$, as   in Lemma \ref{lc.7},   is a deterministic sequence of vectors of fixed dimension with $\max_{sk}\|g_{sk}\|_2=O(1)$, and $\tilde g_{sk}=g_{sk}-\frac{1}{T}\sum_s g_{sk}$.  
  
% (iv) $\max_{ms}\|\frac{1}{n}  \sum_{i=1}^n\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T (\tilde IU_{it}'\tilde U_{jt} -U_{it}'U_{jt})\tilde U_{is,m}\|_2=$
\end{lemma}

 

\proof  
(i)  Let $\tilde W_{i}=\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T (U_{it}' U_{jt} -EU_{it}' U_{jt} )$.  We first bound  $\frac{1}{n}\sum_i\|\tilde W_{i}\|_2^2$.
\begin{align*}
\frac{1}{n}\sum_i\|\tilde W_{i}\|_2^2 &\leq \frac{2}{n}\sum_i \left\|  \frac{1}{pTn}\sum_{j=1}^n(\widehat f_j-H'\tilde f_j)\sum_{t=1}^T (U_{it}' U_{jt} -EU_{it}' U_{jt} )\right\|_2^2 \\
&\qquad + \frac{2}{n}\sum_i\left\|  \frac{1}{pTn}\sum_{j=1}^nH'\tilde f_j\sum_{t=1}^T (U_{it}' U_{jt} -EU_{it}' U_{jt} )\right\|_2^2 \\
&\leq \frac{2}{p^2T^2n^3}\sum_{l=1}^n\|\widehat f_l-H'\tilde f_l\|_2^2\sum_i\sum_{j=1}^n(\sum_{t=1}^T U_{it}' U_{jt} -EU_{it}' U_{jt} )^2 \\
&\qquad +O_P\left(\frac{1}{p^2T^2n^3}\right)\sum_i \left\| \sum_{j=1}^n f_j\sum_{t=1}^T (U_{it}' U_{jt} -EU_{it}' U_{jt} )\right\|_2^2 \\
&\qquad +O_P\left(\frac{\|\bar f\|_2^2}{p^2T^2n^3}\right)\sum_i\left| \sum_{j=1}^n\sum_{t=1}^T (U_{it}' U_{jt} -EU_{it}' U_{jt} )\right|^2 \\
&= O_P\left(  \frac{\Delta_F^2}{p^2T^2n^2}\right) \sum_i\sum_{j=1}^n\Var\left(\sum_{t=1}^T U_{it}' U_{jt}   \right) + O_P\left(\frac{1}{p^2T^2n^3}\right)\sum_i\Var\left( \sum_{j=1}^n\sum_{t=1}^T U_{it}' U_{jt}  \right) \\
&\qquad +O_P\left(\frac{1}{p^2T^2n^3}\right)\sum_i E\left\| \sum_{j=1}^n\sum_{t=1}^T  f_j(U_{it}' U_{jt} -EU_{it}' U_{jt} )\right\|_2^2.
\end{align*} 
Below we bound each of the three terms on the right of the equal sign in the preceding display. First,     
\begin{align*} 
\sum_i\sum_{j=1}^n\Var\left(\sum_{t=1}^T U_{it}' U_{jt}   \right) &=   \sum_i\sum_{j=1}^n\sum_{t=1}^T \sum_{m=1}^p\sum_{s=1}^T \sum_{v=1}^p\Cov(U_{it,m} U_{jt,m}, U_{is,v} U_{js,v}   ) \\
&\leq pnT\max_{mti}    \sum_{s=1}^T \sum_{v=1}^p\Cov(U_{it,m} ^2, U_{is,v} ^2 )+ O(n^2Tp)   \max_{tm}\sum_{s=1}^T \sum_{v=1}^p|EU_{it,m} U_{is,v} | \\
&=O(n^2Tp).
\end{align*}
Hence, $O_P\left(  \frac{\Delta_F^2}{p^2T^2n^2}\right) \sum_i\sum_{j=1}^n\Var\left(\sum_{t=1}^T U_{it}' U_{jt}   \right)=O_P\left(\frac{\Delta_F^2}{pT}\right).$

Second,
\begin{align*}   
\sum_i\Var\left( \sum_{j=1}^n\sum_{t=1}^T U_{it}' U_{jt}  \right) &=     \sum_i \sum_{j=1}^n\sum_{m=1}^p\sum_{t=1}^T   \sum_{l=1}^n\sum_{v=1}^p\sum_{s=1}^T\Cov(  U_{it,m}U_{jt,m}, U_{is,v}U_{ls,v}    ) \\
&= \sum_i  \sum_{m=1}^p\sum_{t=1}^T   \sum_{v=1}^p\sum_{s=1}^T\Cov(  U_{it,m}^2, U_{is,v} ^2 ) \\
&\qquad +  \sum_i \sum_{j\neq i}\sum_{m=1}^p\sum_{t=1}^T  \sum_{v=1}^p\sum_{s=1}^TE  U_{it,m}U_{is,v}EU_{js,v}  U_{jt,m}   \\
&\leq O(Tpn)    \max_{tmi}   \sum_{v=1}^p\sum_{s=1}^T\Cov(  U_{it,m}^2, U_{is,v} ^2 ) \\
&\qquad + O(n^2Tp)\max_{tmi}   \sum_{v=1}^p\sum_{s=1}^T|E  U_{it,m}U_{is,v}| \\
&= O(n^2Tp).
\end{align*} 
Hence, $O_P\left(\frac{1}{p^2T^2n^3}\right)\sum_i\Var\left( \sum_{j=1}^n\sum_{t=1}^T U_{it}' U_{jt}  \right)=O\left(\frac{1}{pTn}\right).$ 

Third, for each $k\leq K$,
\begin{align*}   
\sum_i E & \left| \sum_{j=1}^n\sum_{t=1}^T  f_{jk}(U_{it}' U_{jt} -EU_{it}' U_{jt} )\right|^2 \\
&\leq 2 \sum_i E\left| \sum_{t=1}^T  f_{ik}(\|U_{it}\|_2^2-E\|U_{it}\|_2^2 )\right|^2 
+2\sum_i\Var\left( \sum_{j\neq i}\sum_{t=1}^T  f_{jk}U_{it}' U_{jt}   \right) \\
&= O(np^2T^2)+2\sum_i \sum_{l\neq i}\sum_{s=1}^T \sum_{v=1}^p \sum_{t=1}^T \sum_{m=1}^p E\left[ U_{it,m} U_{is,v} \right]  E\left[ f_{lk}^2U_{lt,m}U_{ls,v}\right] \\
&= O(np^2T^2)+O(n^2Tp)\max_{tmi} \sum_{s=1}^T \sum_{v=1}^p   |E U_{it,m} U_{is,v} |   \\
&=O(np^2T^2+n^2Tp).
\end{align*}       
Hence, $O_P\left(\frac{1}{p^2T^2n^3}\right)\sum_i E\left\| \sum_{j=1}^n\sum_{t=1}^T  f_j(U_{it}' U_{jt} -EU_{it}' U_{jt} )\right\|_2^2 = O_P\left(\frac{1}{n^2}+\frac{1}{pTn}\right)$.       
Therefore, 
\begin{equation} \label{ee.7a}            
\frac{1}{n}\sum_i \left\|  \frac{1}{pTn}\sum_{j=1}^n\tilde f_j\sum_{t=1}^T (U_{it}' U_{jt} -EU_{it}' U_{jt} )\right\|_2^2 \leq O_P\left(\frac{1}{pTn}+\frac{1}{n^2}\right).
\end{equation}             
By $\Delta_F^2=\frac{1}{n^2}+\frac{1}{nT^2}+\frac{1}{pT}$,
$\frac{1}{n}\sum_i\|\tilde W_{i}\|_2^2 = O_P\left(\frac{\Delta_F^2}{pT}+\frac{1}{pTn}+ \frac{1}{n^2}\right) = O_P\left(\frac{1}{n^2}+\frac{1}{p^2T^2}\right).$ 
Hence, the object we aim to bound is $\max_{ms}\left\| \frac{1}{n}  \sum_{i=1}^n\tilde W_{i}\tilde U_{is,m}\right\|_2 \leq \left(\frac{1}{n}\sum_i\|\tilde W_{i}\|_2^2\right)^{1/2}\max_{ms}\left(\frac{1}{n}\sum_i\tilde U_{is,m}^2\right)^{1/2}$.  We now show  $\max_{ms}\left(\frac{1}{n}\sum_i\tilde U_{is,m}^2\right)^{1/2}=O_P(1)$. Once this is done, then $\max_{ms}\left\| \frac{1}{n}  \sum_{i=1}^n\tilde W_{i}\tilde U_{is,m}\right\|_2=O_P\left(\frac{1}{n}+\frac{1}{pT}\right)$ as desired.  Note that 
\begin{align*}
\max_{ms}\frac{1}{n}\sum_i\tilde U_{is,m}^2&\leq \max_{ms}\frac{2}{n}\sum_i(U_{is,m}-\bar U_{i\cdot, m})^2+ 2\max_{ms}(\bar U_{\cdot s,m}- \bar{\bar U}_{ m})^2
\end{align*}   
Let $\mathcal{U}_{i,ms}=2(U_{ms,i}-\bar U_{i\cdot, m})^2-2E(U_{ms,i}-\bar U_{i\cdot, m})^2$. Then $E\mathcal{U}_{i,ms}=0$ and $\max_{i,ms}\Var(\mathcal{U}_{i,ms})<C$. Hence by the Bernstein inequality for independent data, we have
$$
\max_{ms}\frac{1}{n}\sum_i\mathcal{U}_{i,ms}=O_P(\sqrt{\frac{\log(Tp)}{n}})=o_P(1).
$$
This implies $\max_{ms}\frac{2}{n}\sum_i(U_{is,m}-\bar U_{i\cdot, m})^2\leq o_P(1)+ \max_{ms}\frac{2}{n}\sum_iE(U_{is,m}-\bar U_{i\cdot, m})^2=O_P(1)$. On the other hand, 
\begin{align*}
\max_{ms}(\bar U_{\cdot s,m}- \bar{\bar U}_{ m})^2
&\leq 2(\max_{ms}|\bar U_{\cdot s,m}|)^2+2(\max_{ms}|\bar{\bar U}_{ m}|)^2\\
&\leq 2(\max_{ms}|\bar U_{\cdot s,m}|)^2+2(\max_{ms}|\frac{1}{T}\sum_s\bar U_{\cdot s,m}|)^2\leq 
4(\max_{ms}|\bar U_{\cdot s,m}|)^2\\
&\leq 4(\max_{ms}|\frac{1}{n}\sum_i U_{is,m}|)^2=O_P\left(\frac{\log (Tp)}{n}\right).
\end{align*} 
This shows $\max_{ms}\frac{1}{n}\sum_i\tilde U_{is,m}^2=O_P(1)$.
 
     
(ii) Define $\Gamma_{im}= \frac{1}{pT^2} \sum_{t=1}^T  \sum_{s=1}^T\tilde\Lambda_t'(U_{it}U_{is, m}  -EU_{it}U_{is, m}) $. Without loss of generality, we assume $\tilde\Lambda_t'$ has a single row, as we can always conduct the analysis for each fixed row of $\tilde\Lambda_t'$. Then $EX_{im}=0$, and 
\begin{align*}
\max_{im}\Var(X_{im}) &\leq \max_{im}\frac{1}{T^4p^2}\sum_{t=1}^T \sum_{k=1}^p  \sum_{h=1}^T \sum_{l=1}^p \sum_{v=1}^T\sum_{s=1}^T |\Cov(U_{it,k}U_{is, m}, U_{ih,l}U_{iv, m})| \\
&= O\left(\frac{1}{T^2p}\right).
\end{align*}    
By the Bernstein inequality for independent data,          
$ \max_{m} \|\frac{1}{n} \sum_{i=1}^nX_{im} \|_2 = O_P\left(\sqrt{\frac{\log p}{nT^2p}}\right)$. Thus, 
\begin{align*}       
\max_{m} \left\|\frac{1}{pT} \sum_{t=1}^T\tilde\Lambda_t'
 \frac{1}{n}  \sum_{i=1}^nU_{it}\bar U_{i\cdot, m}    \right\|_2  &\leq   \max_{m} \left\|
 \frac{1}{n} \sum_{i=1}^nX_{im} \right\|_2 + \max_{m} \left\|\frac{1}{pT^2} \sum_{t=1}^T
 \frac{1}{n} \sum_{s=1}^T \sum_{i=1}^n\tilde\Lambda_t'EU_{it}U_{is, m}    \right\|_2 \\
 &\leq O_P\left(\sqrt{\frac{\log p}{nT^2p}}+\frac{1}{pT}\right).
\end{align*}   
                  
  
(iii) With  $\left\|\frac{1}{n}\sum_i\widehat f_j\tilde f_j'\right\|=O_P(1)$ and the equalities $\frac{1}{T}\sum_t\tilde\Lambda_t=0$ and
\begin{align*}
\frac{1}{n}  \sum_{i=1}^n(\tilde U_{it} \tilde U_{is,m}- U_{it}  U_{is,m}) &= 
-\frac{1}{n}  \sum_{i=1}^nU_{it}\bar U_{i\cdot, m}
-\bar U_{\cdot t}\bar U_{\cdot s,m}
+\bar U_{\cdot t}\bar{\bar U}_{m}
-\frac{1}{n}\sum_{i=1}^n\bar U_{i\cdot}U_{is,m} \\
&\qquad +\bar{\bar U}\bar U_{\cdot s,m}
+ \frac{1}{n}\sum_{i=1}^n\bar U_{i\cdot} \bar U_{i\cdot, m}-\bar{\bar U}\bar{\bar U}_m,
\end{align*}
we have that $ \max_{m,s}|\frac{1}{n}  \sum_{i=1}^n\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_j'\tilde\Lambda_t'(\tilde U_{it} \tilde U_{is,m}- U_{it}  U_{is,m}) |$ is bounded by 
\begin{align*}
O_P(1) & \max_{ms}\left\|\frac{1}{pT} \sum_{t=1}^T\tilde\Lambda_t'\frac{1}{n}  \sum_{i=1}^n(\tilde U_{it} \tilde U_{is,m}- U_{it}  U_{is,m})\right\|_2 \\
&\leq O_P(1)\max_{m}\left\|\frac{1}{pT} \sum_{t=1}^T\tilde\Lambda_t'
 \frac{1}{n}  \sum_{i=1}^nU_{it}\bar U_{i\cdot, m} \right\|_2 + O_P(1)\max_{ms}\left\|\frac{1}{pT} \sum_{t=1}^T\tilde\Lambda_t' \bar U_{\cdot t}\bar U_{\cdot s,m}   \right\|_2 \\
&\qquad + O_P(1)\max_{m}\left\|\frac{1}{pT} \sum_{t=1}^T\tilde\Lambda_t'
   \bar U_{\cdot t}\bar{\bar U}_{m}     \right\|_2 \\
&\leq O_P(1)\max_{m}\left\|\frac{1}{pT} \sum_{t=1}^T\tilde\Lambda_t'
 \frac{1}{n}  \sum_{i=1}^nU_{it}\bar U_{i\cdot, m}    \right\|_2 
 + O_P(1)\left\|\frac{1}{pT} \sum_{t=1}^T\tilde\Lambda_t'
 \bar U_{\cdot t}   \right\|_2\max_{ms} \left(  |\bar U_{\cdot s,m} |+ |\bar{\bar U}_{m}  |\right) \\
 &= O_P\left(\sqrt{\frac{\log p}{nT^2p}}+\frac{1}{pT}+\sqrt{\frac{\log(pT)}{n^2pT}}\right),
 \end{align*} 
 where the last equality follows from part (ii) that 
$$
\max_{m} \left\|\frac{1}{pT} \sum_{t=1}^T\tilde\Lambda_t'
 \frac{1}{n}  \sum_{i=1}^nU_{it}\bar U_{i\cdot, m}    \right\|_2 =O_P\left(\sqrt{\frac{\log p}{nT^2p}}+\frac{1}{pT}\right),
$$
that  $\max_{ms} (  |\bar U_{\cdot s,m} |+ |\bar{\bar U}_{m}  |)=O_P\left(\sqrt{\frac{\log pT}{n}}\right)$, and that $\left\| \frac{1}{pT} \sum_{t=1}^T\tilde\Lambda_t'\bar U_{\cdot t}\right\|_2=O\left(\frac{1}{\sqrt{pTn}}\right)$. 
 
% (iv)  $\max_{ms}\|\frac{1}{n}  \sum_{i=1}^n\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T (\tilde U_{it}'\tilde U_{jt} -U_{it}'U_{jt})\tilde U_{is,m}\|_2\leq \sum_{l=1}^d G_l$, where
 %   \begin{eqnarray*}
   % G_1&=&\max_{ms}\|\frac{1}{n}  \sum_{i=1}^n\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T  U_{it}' \bar  U_{\cdot t}\tilde U_{is,m}\|_2\leq (\frac{1}{n}  \sum_{i=1}^n   (\frac{1}{pT} \sum_{t=1}^T  U_{it}' \bar  U_{\cdot t})^2)^{1/2}\max_{ms}(\frac{1}{n}  \sum_{i=1}^n \tilde U_{is,m}^2)^{1/2}O_P(1)\cr
    %&=&O_P(\frac{1}{n}+\frac{1}{\sqrt{npT}})\quad (\text{ by (\ref{ec.2})})\cr
    %G_2&=&\max_{ms}\|\frac{1}{n}  \sum_{i=1}^n\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T \bar U_{\cdot t} '\bar U_{\cdot t}    \tilde U_{is,m}\|_2\leq O_P(1)\max_{ms}|\frac{1}{n}\sum_i\bar U_{is,m}|\frac{1}{Tp}\sum_t\|\bar U_{\cdot t}\|^2=o_P(\frac{1}{n}),\cr
 %  G_3 &=&\max_{ms}\|\frac{1}{n}  \sum_{i=1}^n\frac{1}{pn}\sum_{j=1}^n\widehat f_j   \bar U_{i\cdot}'\bar  U_{j\cdot}   \tilde U_{is,m}\|_2\leq O_P(1)  (  \frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n(\frac{1}{p}  \bar U_{i\cdot}'\bar  U_{j\cdot} )^2 )^{1/2}=O(\frac{1}{\sqrt{nT^2}})+O(\frac{1}{\sqrt{pT^2}}).
    %  \end{eqnarray*} 
     
(iv)  As the dimension of $g_{sk}$ does not grow, without loss of generality, we assume it is a scalar. Let
$X_{i, mktl}= \frac{1}{T}\sum_{s=1}^T  (U_{it,l}    U_{is,m}-EU_{it,l}U_{is,m}) \tilde g_{sk}. $ Then $EX_{i, mktl}=0$ and  we are calculating the rate of $\max_{mktl}  |  \frac{1}{n}\sum_{i=1}^nX_{i,mktl}  |$. Note that by Assumption \ref{adep}, 
$$
\max_{imktl}\Var(X_{i, mktl})\leq \max_{imt}\frac{1}{T^2}\sum_{s=1}^T\sum_{v=1}^T |\Cov (  U_{it,l}    U_{is,m},  U_{it,l}    U_{iv,m}  )|=O\left(\frac{1}{T}\right).
$$
 Hence by the Bernstein inequality with independent data, 
$$
\max_{mktl}  |  \frac{1}{n}\sum_{i=1}^nX_{i,mktl}  |=O_P\left(\sqrt{\frac{\log(pT)}{nT}}\right).
$$
$\blacksquare$

\begin{lemma}\label{lc.6}
(i) $ \underset{s\leq T, m\leq p}{\max} \left|\frac{1}{n}  \sum_{i=1}^n (H^{'}\tilde f_i-\widehat f_i)\tilde U_{is,m} \right| = O_P\left(\frac{1}{n}+   \frac{1}{T\sqrt{n}}+\frac{1}{T\sqrt{p}} +\sqrt{\frac{\log (pT)}{npT}}\right).$   \\
(ii) $\frac{1}{T} \sum_{t=1}^T \left\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}(\widehat  f_i-H'\tilde f_i)\right\|^2_2 = O_P\left(\frac{1}{n^2}+ \frac{1}{Tpn}\right).$
\end{lemma}
 
Note that the rate of convergence in (i) is better than the rate obtained from the simple inequality 
$$ 
\max_{s\leq T, m\leq p}|\frac{1}{n}  \sum_{i=1}^n (H^{'}\tilde f_i-\widehat f_i)\tilde U_{is,m} |\leq O_P(\Delta_F)\max_{ms}\sqrt{\frac{1}{n}\sum_i\tilde U_{is,m}^2}=O_P\left(\frac{1}{n}+   \frac{1}{T\sqrt{n}}+\frac{1}{\sqrt{pT}}\right)
$$  
in some regimes such as when $p/n\ll T\ll n^2/p$.

 
\proof (i) We have $ \frac{1}{n}  \sum_{i=1}^n (\widehat f_i-H^{'}\tilde f_i)\tilde U_{is,m}  =V^{-1}\sum_{l=1}^7 D_{l,ms}$ where 
\begin{align*}
D_{1,ms} &=  \frac{1}{n}  \sum_{i=1}^n\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T (U_{it}' U_{jt} -EU_{it}' U_{jt} )\tilde U_{is,m}, \\
D_{2,ms} &= \frac{1}{n}  \sum_{i=1}^n\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_j'\tilde\Lambda_t'(\tilde U_{it} \tilde U_{is,m}- U_{it}  U_{is,m})  \\
D_{3,ms} &= \frac{1}{n}  \sum_{i=1}^n\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T(E U_{it}' U_{jt}) \tilde U_{is,m}, \\
D_{4,ms} &= \frac{1}{n}  \sum_{i=1}^n\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_j'\tilde\Lambda_t' (EU_{it}  U_{is,m}),  \\
D_{5,ms} &= \frac{1}{n}  \sum_{i=1}^n\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T (\tilde U_{it}'\tilde U_{jt} -U_{it}'U_{jt})\tilde U_{is,m}, \\ 
D_{6,ms} &= \frac{1}{n}  \sum_{i=1}^n\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_i'\tilde\Lambda_t'\tilde U_{jt} \tilde U_{is,m}, \\
D_{7,ms} &= \frac{1}{n}  \sum_{i=1}^n\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_j'\tilde\Lambda_t'( U_{it}  U_{is,m}-EU_{it}  U_{is,m}).  
\end{align*} 
We respectively bound these terms. 
   
   
\textbf{Bounding $\mathbf{D_{1,ms}}$ and $\mathbf{D_{2,ms}}$:}  The bounds 
$$
\max_{ms}\|D_{1,ms}\|_2=O_P\left(\frac{1}{n}+\frac{1}{pT}\right)
$$ 
and 
$$
\max_{ms}\|D_{2,ms}\|_2 = O_P\left(\sqrt{\frac{\log p}{nT^2p}}+\frac{1}{pT}+\sqrt{\frac{\log(pT)}{n^2pT}}\right)
$$ 
directly follow from Lemma \ref{lc.5}.
   

\textbf{Bounding $\mathbf{D_{3,ms}}$ and $\mathbf{D_{4,ms}}$:}   
\begin{eqnarray*}
&& \max_{ms}\|D_{3,ms}\|_2=\max_{ms}\left\|\frac{1}{n}  \sum_{i=1}^n\frac{1}{n}\widehat f_i\tilde U_{is,m}\frac{1}{Tp}\sum_{t=1}^T E \|U_{it}\|_2^2\right\|_2 = O_P\left(\frac{1}{n}\right). \cr
&& \max_{ms}\|D_{4,ms}\|_2=O_P(1)\max_{ms} \frac{1}{pTn}\sum_{i=1}^n\sum_{t=1}^T\sum_{v=1}^p | EU_{it,v}  U_{is,m}| = O_P\left(\frac{1}{pT}\right).
\end{eqnarray*} 
   
	
\textbf{Bounding $\mathbf{D_{5,ms}}$:}    By the Cauchy-Schwarz inequality and Lemma \ref{lc.1}, we have
\begin{align*}
\max_{ms}\|D_{5,ms}\|_2 &\leq \max_{ms}\left(\frac{1}{n}\sum_i\tilde U_{is,m}^2\right)^{1/2} \left(\frac{1}{n}\sum_{i=1}^n\|A_{i1}\|_2^2\right)^{1/2} \\
&=O_P\left(\frac{1}{n}+\frac{1}{\sqrt{npT}}+\frac{1}{T\sqrt{n}}+\frac{1}{T\sqrt{p}}\right).
\end{align*}
 

\textbf{Bounding $\mathbf{D_{6, ms}}$:} First note that $\left\| \frac{1}{npT}\sum_{j=1}^n\sum_{t=1}^T  \tilde\Lambda_t'  \tilde U_{jt} \widehat  f_{j}' \right\|_F \leq O_P\left(\frac{\Delta_F}{\sqrt{pT}} +\frac{1}{\sqrt{pTn}}\right).$  Then
\begin{align*}
\max_{ms} \|D_{6,ms}\|_2 &\leq O(1)\max_{ms} \left\|\frac{1}{n}   \sum_{i=1}^n\tilde U_{is,m}\tilde f_i\right\|_2 \left\| \frac{1}{npT}\sum_{j=1}^n\sum_{t=1}^T  \tilde\Lambda_t'  \tilde U_{jt} \widehat  f_{j}' \right\|_F \\
&= O_P\left(\frac{\Delta_F}{\sqrt{pT}} +\frac{1}{\sqrt{pTn}}\right)\sqrt{\frac{\log(pT)}{n}}.
\end{align*} 

 
\textbf{Bounding $\mathbf{D_{7, ms}}$:}  Let $X_{i,ms}=\frac{1}{pT}\sum_{t=1}^T\tilde\Lambda_t'( U_{it} U_{is,m}- EU_{it}  U_{is,m})$. Then $EX_{i,ms}=0$; and for each $k\leq K$ where $K$ denotes the number of factors, 
\begin{align*}
\max_{ims}\Var(X_{i,ms}) &= \max_{ims}\frac{1}{p^2T^2}\Var\left(\sum_{t=1}^T\sum_{v=1}^p\lambda_{tk,v} U_{it,v}  U_{is,m}\right) \\
&= \max_{ims}\frac{1}{p^2T^2}\sum_{j=1}^T\sum_{l=1}^p\sum_{t=1}^T\sum_{v=1}^p \lambda_{jk,l} \lambda_{tk,v}\Cov( U_{it,v}  U_{is,m}, U_{ij,l}  U_{is,m}) \\
&\leq O\left(\frac{1}{pT}\right)\max_{imstv}\sum_{k=1}^T\sum_{l=1}^p   |\Cov( U_{it,v}  U_{is,m}, U_{ik,l}  U_{is,m})|=O\left(\frac{1}{pT}\right).
\end{align*} 
Hence, by the Bernstein inequality for independent data, 
\begin{eqnarray*}
\max_{ms}\|D_{7,ms}\|_2  &\leq& O_P(1) \max_{ms}\|\frac{1}{n}\sum_{i=1}^nX_{i,ms}\|_2= O_P\left( \sqrt{\frac{\log (pT)}{npT}}\right).
\end{eqnarray*} 
Combining the above results, we reach %(since $J^4n=o(p^2T)$)
$$
\max_{s\leq T, m\leq p}|\frac{1}{n}  \sum_{i=1}^n (H^{'}\tilde f_i-\widehat f_i)\tilde U_{is,m} |=
O_P\left(\frac{1}{n}+   \frac{1}{T\sqrt{n}}+\frac{1}{T\sqrt{p}} +\sqrt{\frac{\log (pT)}{npT}}\right).
$$
    
    
(ii) $\frac{1}{T} \sum_{t=1}^T\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}(\widehat  f_i-H'\tilde f_i)\|^2_2\leq \sum_{l=1}^3\bar B_l$ where each $\bar B_l$ is defined and bounded below making use of (\ref{ec.1}).
\begin{align*} 
\bar B_1 &= \frac{1}{T} \sum_{s=1}^T \left\|\frac{1}{n}\sum_{i=1}^n \tilde e_{is} \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T \tilde U_{it}'\tilde U_{jt} \right\|^2_2 \\
&\leq \frac{1}{T} \sum_{s=1}^T \left\|\frac{1}{n}\sum_{i=1}^n \tilde e_{is} \frac{1}{pTn}\sum_{j=1}^n\tilde f_j\sum_{t=1}^T \tilde U_{it}'\tilde U_{jt} \right\|^2_2O_P(1) \\
&\qquad +\frac{1}{T}\sum_{t=1}^T\left\|\frac{1}{n}\sum_{j=1}^n(\widehat f_j-H'\tilde f_j) \tilde U_{jt}'\right\|_F^2\frac{1}{T^2} \sum_{s=1}^T\sum_{t=1}^T \left\|\frac{1}{pn}\sum_{i=1}^n   \tilde U_{it}     \tilde e_{is} \right\|_2^2 \\
&\leq \frac{1}{T} \sum_{s=1}^T\frac{1}{T}\sum_t \left\|\frac{1}{n}\sum_{i=1}^n \tilde e_{is}\tilde U_{it} \right\|_2^2 \frac{1}{T}\sum_t\left\|\frac{1}{pn}\sum_{j=1}^n\tilde f_j  \tilde U_{jt}\right\|^2O_P(1) \\
&\qquad + \frac{1}{pT^2} \sum_{s=1}^T\sum_{t=1}^T \left\|\frac{1}{n}\sum_{i=1}^n   \tilde U_{it}     \tilde e_{is}   \right\|_2^2\max_{tm}\left\|\frac{1}{n}\sum_{j=1}^n(\widehat f_j-H'\tilde f_j) \tilde U_{jt,m}\right\|_2^2 \\
&\leq O_P\left(\frac{1}{n^2}\right)+\frac{1}{n}O_P\left(\frac{1}{n^2}+\frac{1}{T^2n}+\frac{1}{T^2p}+\frac{\log (pT)}{npT}\right) \\
&= O_P\left(\frac{1}{n^2}+ \frac{1}{T^2pn}\right). \text{ (Lemma \ref{lc.6})} \\
\bar B_2 &= \frac{1}{T} \sum_{s=1}^T\left\|\frac{1}{n}\sum_{i=1}^n \tilde e_{is}   \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_j'\tilde \Lambda_t'\tilde U_{it} \right\|^2_2 \\
&\leq O_P(1) \frac{1}{T} \sum_{s=1}^T \left\|\frac{1}{n}\sum_{i=1}^n \tilde e_{is}   \frac{1}{pT} \sum_{t=1}^T \tilde \Lambda_t'\tilde U_{it} \right\|^2_2 \\
&= O\left(\frac{1}{pTn}\right).  \text{ (Lemma \ref{lc.3})} \\
\bar B_3 &= \frac{1}{T} \sum_{s=1}^T\left\|\frac{1}{n}\sum_{i=1}^n \tilde e_{is}  \frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\tilde f_i'\tilde \Lambda_t'\tilde U_{jt} \right\|^2_2 \\
&\leq \frac{1}{T} \sum_{s=1}^T\left\|\frac{1}{n}\sum_{i=1}^n \tilde e_{is} \frac{1}{pTn}\sum_{j=1}^n\tilde f_j\sum_{t=1}^T \tilde f_i'\tilde\Lambda_t'\tilde U_{jt} \right\|^2_2O_P(1) \\
&\qquad +\frac{1}{T}\sum_{t=1}^T\left\|\frac{1}{n}\sum_{j=1}^n(\widehat f_j-H'\tilde f_j) \tilde U_{jt}'\right\|_F^2\frac{1}{T} \sum_{s=1}^T\frac{1}{p} \left\|\frac{1}{n}\sum_{i=1}^n   \tilde f_i'  \tilde e_{is}   \right\|_2^2 \\
&= O_P\left(\frac{1}{n^2pT}\right)+\frac{1}{n}O_P\left(\frac{1}{n^2}+\frac{1}{T^2n}+\frac{1}{T^2p}+\frac{\log (pT)}{npT}\right) \\
&= O_P\left(\frac{1}{n^3}+\frac{1}{T^2n^2}+\frac{1}{T^2pn}+\frac{\log (pT)}{n^2pT}\right).
\end{align*}
Hence,
$$
\frac{1}{T} \sum_{t=1}^T \left\|\frac{1}{n}\sum_{i=1}^n \tilde e_{it}(\widehat  f_i-H'\tilde f_i)\right\|^2_2 = O_P\left(\frac{1}{n^2}+ \frac{1}{Tpn}\right).
$$
 $\blacksquare$


\subsection{Technical lemmas for Proposition \ref{p4.1}: the bootstrap data  }

\begin{lemma}\label{le.9}
(i) $\frac{1}{n} \sum_{j=1}^n \left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T
 	\frac{1}{n}\tilde F'\tilde U_{t, m}  \tilde  U_{jt,m} \right)^2 = O_P\left(\frac{1}{n^2}+\frac{1}{nT^2}+\frac{1}{pTn}\right).$\\
	(ii) $ \frac{1}{n}\sum_{i=1}^n \left(\frac{1}{pT}\sum_{t=1}^T \sum_{m=1}^p  \frac{1}{n}\tilde F'\tilde U_{t, m}\widehat  U_{it,m}\right)^2=O_P\left(\frac{1}{n^2}+\frac{1}{nT^2}+\frac{1}{pTn}\right).$\\
(iii) $\frac{1}{n}\sum_{i=1}^n\|A^*_{i1}\|_2^2=O_{P^*}\left(\Delta_F^2+\frac{1}{n}\right)$ {for $A^*_{i1}$ defined in (\ref{ee.9}).}
\end{lemma}
 
%\textcolor{blue}{DOUBLE CHECK reference for $A^*_{i1}$ is correct.}

\proof (i) Note that $\sum \tilde f_i=\sum_i\tilde U_{it}=\sum_t \tilde U_{it}=0$.  We may then bound 
$$
\frac{1}{n} \sum_{j=1}^n \left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T \frac{1}{n}\tilde F'\tilde U_{t, m} \tilde U_{jt,m} \right)^2
$$ 
from above with
\begin{align*} 
\frac{C}{n} & \sum_{j=1}^n(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T \frac{1}{n}\sum_{i=1}^n\tilde f_i \tilde U_{it, m} U_{jt,m} )^2 + \frac{C}{n} \sum_{j=1}^n(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T \frac{1}{n} \sum_{i=1}^n\tilde f_i\tilde U_{it, m} \bar U_{\cdot t,m} )^2 \\
&\leq O_P\left(\frac{1}{n^2}\right) + \frac{C}{n} \sum_{j=1}^n\left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T
 	\frac{1}{n}\sum_{i=1}^n\tilde f_i  U_{it, m}     U_{jt,m} \right)^2 \\
&\qquad + \frac{C}{n} \sum_{j=1}^n\left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T
 	\frac{1}{n}\sum_{i=1}^n\tilde f_i \bar U_{i\cdot , m}     U_{jt,m} \right)^2 \\
&\leq O_P\left(\frac{1}{n^2}+\frac{1}{nT^2}\right) + \frac{C}{n} \sum_{j=1}^n \left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T\frac{1}{n}\sum_{i=1}^n\tilde f_i  (U_{it, m} U_{jt,m} -EU_{it, m} U_{jt,m})\right)^2 \\
&\qquad +\frac{C}{n} \sum_{j=1}^n \left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T
 	\frac{1}{n}\sum_{i=1}^n\tilde f_i  EU_{it, m}     U_{jt,m} \right)^2 \\
&= O_P\left(\frac{1}{n^2}+\frac{1}{nT^2}+\frac{1}{pTn}\right),
\end{align*}
where we used (\ref{ee.7a}) that 
$$        
\frac{1}{n}\sum_i \left\|  \frac{1}{pTn}\sum_{j=1}^n\tilde f_j\sum_{t=1}^T (U_{it}' U_{jt} -EU_{it}' U_{jt} )\right\|_2^2 \leq O_P\left(\frac{1}{pTn}+\frac{1}{n^2}\right).
$$

(ii) The conclusion follows from part (i) and the following inequality:
\begin{align*} 
\frac{1}{n}\sum_{i=1}^n & \left(\frac{1}{pT}\sum_{t=1}^T \sum_{m=1}^p  \frac{1}{n}\tilde F'\tilde U_{t, m}(\widehat  U_{it,m}-\tilde U_{it,m}) \right)^2 \\
&\leq \left(\frac{1}{pT}\sum_{t=1}^T \sum_{m=1}^p  \frac{1}{n}\tilde F'\tilde U_{t, m}
  \tilde \lambda_{tm} \right)^2 O_P(\Delta_F^2) + \left(\frac{1}{pT}\sum_{t=1}^T \sum_{m=1}^p \left\|\frac{1}{n}\tilde F'\tilde U_{t, m} \right\|^2\right)^2 \\
&= O_P\left(\frac{\Delta_F^2}{nTp}+\frac{1}{n^2}\right).
\end{align*}
 

(iii)  Note that $\max_{ij}\Var^*(w_i^Uw_j^U)<C$. %Let $\tau_{ij}=\Var^*(w_i^Uw_j^U)$.
  %Note that $E^*\tilde U_{it}^{*'} \tilde U_{jt}^*=E^*w_i^Uw_j^U\widehat U_{it}'\widehat U_{jt}$
\begin{align*}
\frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n & E^*\left(\frac{1}{pT}\sum_{t=1}^T\tilde U_{it}^{*'} \tilde U_{jt}^*-E^*\tilde U_{it}^{*'}\tilde U_{jt}^*\right)^2 \\
&= \frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n\Var^*\left(\frac{1}{pT}\sum_{t=1}^T\tilde U_{it}^{*'} \tilde U_{jt}^*\right) \\
&= \frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n\left(\frac{1}{pT}\sum_{t=1}^T\widehat  U_{it}^{'} \widehat  U_{jt} \right)^2\Var^*(w_i^Uw_j^U) \\
&\leq  \underbrace{ \frac{8}{n^2}\sum_{i=1}^n \sum_{j=1}^n\left(\frac{1}{pT}\sum_{t=1}^T(\tilde U_{it}^{'} \tilde U_{jt} -EU_{it}'U_{jt})\right)^2C}_{a_1} 
+ \underbrace{ \frac{8}{n^2}\sum_{i=1}^n \sum_{j=1}^n\left(\frac{1}{pT}\sum_{t=1}^TE U_{it}^{'}  U_{jt} \right)^2C}_{a_2} \\
&\qquad + \underbrace{\frac{8}{n^2}\sum_{i=1}^n \sum_{j=1}^n\left(\frac{1}{pT}\sum_{t=1}^T\widehat  U_{it}^{'} \widehat  U_{jt} -\tilde U_{it}'\tilde U_{jt}\right)^2C}_{a_3}.
%&=&\frac{1}{(npT)^2}\sum_{i=1}^n \sum_{m=1}^p \sum_{v=1}^p \sum_{s=1}^T \sum_{t=1}^T\Cov^*(\tilde U_{it,m}^{*2}, \tilde U_{is,v}^{*2}  )
%+\frac{1}{(npT)^2}\sum_{i=1}^n \sum_{m=1}^p\sum_{v=1}^p\sum_{j\neq i} \sum_{s=1}^T \sum_{t=1}^TE^*\tilde U_{it,m}^{*} \tilde U_{is,v}^{*}E^*\tilde U_{jt,m}^*  \tilde U_{js,v}^*\cr
%&=&\frac{1}{(npT)^2}\sum_{i=1}^n \sum_{m=1}^p \sum_{v=1}^p \sum_{s=1}^T \sum_{t=1}^T\widehat U_{is,v}^{2}\widehat U_{it,m}^{2}\Cov^*(w_i^{U2},  w_i^{U2} )
%+\frac{1}{(npT)^2}\sum_{i=1}^n \sum_{m=1}^p\sum_{v=1}^p\sum_{j\neq i} \sum_{s=1}^T \sum_{t=1}^TE^*\tilde U_{it,m}^{*} \tilde U_{is,v}^{*}E^*\tilde U_{jt,m}^*  \tilde U_{js,v}^*
\end{align*}
Note that 
$$a_1\leq  \frac{C}{n^2}\sum_{i=1}^n \sum_{j=1}^n(\frac{1}{pT}\sum_{t=1}^T(\tilde U_{it}^{'} \tilde U_{jt} -U_{it}'U_{jt}))^2+ \frac{C}{n^2}\sum_{i=1}^n \sum_{j=1}^n(\frac{1}{pT}\sum_{t=1}^T(  U_{it}^{'}  U_{jt} -EU_{it}'U_{jt}))^2.
$$
The first term in the upper bound for $a_1$ is $O_P\left(\frac{1}{n^2}+\frac{1}{npT }+\frac{1}{nT^2}+\frac{1}{pT^2}\right)$ by Lemma \ref{lc.1}. The second term in the bound for $a_1$ is $O_P\left(\frac{1}{pT}\right)$ by (\ref{ee.6a}). Therefore, 
\begin{equation}\label{ee.10}
a_1=O_P\left(\frac{1}{n^2}+ \frac{1}{nT^2}+\frac{1}{pT}\right)=O_P(\Delta_F^2).
\end{equation}
Next we have $a_2 = \frac{C}{n^2}\sum_{i=1}^n \left(\frac{1}{pT}\sum_{t=1}^TE U_{it}^{'}  U_{it} \right)^2=O\left(\frac{1}{n}\right)$. Finally, 
\begin{align}
\nonumber
a_3 &= \frac{C}{n^2}\sum_{i=1}^n \sum_{j=1}^n\left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T\widehat  U_{it,m}\widehat  U_{jt,m} -\tilde U_{it,m}\tilde U_{jt,m}\right)^2 \\
\nonumber
&\leq \frac{C}{n^2}\sum_{i=1}^n \sum_{j=1}^n\left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T(\widehat  U_{it,m}-\tilde U_{it,m})\widehat  U_{jt,m} \right)^2 \\
\nonumber
&\qquad + \frac{C}{n^2}\sum_{i=1}^n \sum_{j=1}^n \left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T(\widehat  U_{it,m}-\tilde U_{it,m})\tilde  U_{jt,m} \right)^2 \\
\nonumber
&\leq \frac{C}{n}\sum_{j=1}^n \left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T
 \tilde \lambda_{tm}    \widehat  U_{jt,m} \right)^2 O_P(\Delta_F^2)
 + \frac{C}{n} \sum_{j=1}^n \left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T
\frac{1}{n}\widehat F'\tilde U_{t, m}   \widehat  U_{jt,m} \right)^2 \\
\nonumber
&\qquad + \frac{C}{n}\sum_{j=1}^n \left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T
 \tilde \lambda_{tm}   \tilde  U_{jt,m} \right)^2 O_P(\Delta_F^2)
 + \frac{C}{n} \sum_{j=1}^n \left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T
 	\frac{1}{n}\widehat F'\tilde U_{t, m}  \tilde  U_{jt,m} \right)^2 \\
\label{ee.12}
\begin{split}	
&= O_P(\Delta_F^2)+ \frac{C}{n} \sum_{j=1}^n \left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T
 	\frac{1}{n}\tilde F'\tilde U_{t, m}  \tilde  U_{jt,m} \right)^2 \\
&\qquad +\frac{C}{n}\sum_{i=1}^n \left(\frac{1}{pT}\sum_{t=1}^T \sum_{m=1}^p  \frac{1}{n}\tilde F'\tilde U_{t, m}(\widehat  U_{it,m}-\tilde U_{it,m})\right)^2 \\
&= O_P(\Delta_F^2).
\end{split}
\end{align}
where $\frac{1}{n} \sum_{j=1}^n \left(\frac{1}{pT}\sum_{m=1}^p\sum_{t=1}^T
 	\frac{1}{n}\tilde F'\tilde U_{t, m}  \tilde  U_{jt,m} \right)^2=O_P\left(\frac{1}{n^2}+\frac{1}{nT^2}+\frac{1}{pTn}\right)$ follows from part (i). 
Hence, we have proved 
\begin{equation}\label{ee.13}
\frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n E^*\left(\frac{1}{pT}\sum_{t=1}^T\tilde U_{it}^{*'} \tilde U_{jt}^*-E^*\tilde U_{it}^{*'}\tilde U_{jt}^*\right)^2=O_{P^*}\left(\Delta_F^2+\frac{1}{n}\right).
\end{equation}
The Cauchy-Schwarz inequality then implies
\begin{align*}
\frac{1}{n}\sum_{i=1}^n \|A^*_{i1}\|_2^2 &\leq O_{P^*}(1) \frac{1}{n^2}\sum_{i=1}^n \sum_{j=1}^n \left(\frac{1}{pT}\sum_{t=1}^T\tilde U_{it}^{*'} \tilde U_{jt}^*-E^*\tilde U_{it}^{*'}\tilde U_{jt}^*\right)^2 \\
&=O_{P^*}\left(\Delta_F^2+\frac{1}{n}\right).
\end{align*}
 $\blacksquare$

\begin{lemma}\label{le.10}
$\frac{1}{n}\sum_i\|A^*_{i3}\|_2^2=O_{P^*}(\Delta_F^2)$ and  $\frac{1}{n}\sum_i\|A^*_{i4}\|_2^2=O_{P^*}(\Delta_F^2)$ for $A^*_{i3}$ and $A^*_{i4}$ defined in (\ref{ee.9}).
\end{lemma}
 
\proof By the Cauchy-Schwarz inequality,
$\frac{1}{n}\sum_{i=1}^n\|A^*_{i3}\|_2^2\leq O_{P^*}(1)\frac{1}{n}\sum_{i=1}^n \left(\frac{1}{pT}\sum_{t=1}^T \widehat \Lambda_t'\tilde U_{it}^*\right)^2$.  Now 
\begin{align}
\nonumber
E^*\left[\frac{1}{n}\sum_{i=1}^n\left(\frac{1}{pT}\sum_{t=1}^T \widehat \Lambda_t'\tilde U_{it}^*\right)^2\right] 
\nonumber
&= \frac{1}{n}\sum_{i=1}^n\Var^*\left(\frac{1}{pT}\sum_{t=1}^T \widehat \Lambda_t'\widehat  U_{it} w_i^U\right) \\
\nonumber
&= \frac{1}{n}\sum_{i=1}^n \left(\frac{1}{pT}\sum_{t=1}^T \widehat \Lambda_t'\widehat  U_{it}\right)^2 \\
\nonumber
&\leq O_P(1)\frac{1}{n}\sum_{i=1}^n \left(\frac{1}{pT}\sum_{t=1}^T  \tilde\Lambda_t'\widehat  U_{it}\right)^2 \\
\nonumber
&\qquad + \frac{1}{n}\sum_{i=1}^n \left(\frac{1}{pT}\sum_{t=1}^T (\widehat \Lambda_t-\tilde\Lambda_tH^{'-1})'\widehat  U_{it}\right)^2 \\
\nonumber
&\leq O_P(1)\frac{1}{n}\sum_{i=1}^n \left(\frac{1}{pT}\sum_{t=1}^T  \tilde\Lambda_t'\widehat  U_{it}\right)^2 + O_P(\Delta_F^2) \\
\nonumber
&\qquad + \frac{1}{n}\sum_{i=1}^n \left(\frac{1}{pT}\sum_{t=1}^T \sum_{m=1}^p  \frac{1}{n}\tilde F'\tilde U_{t, m}\widehat  U_{it,m}\right)^2 O_P(1) \\
\nonumber
&\leq O_P(\Delta_F^2) + O_P(1)\frac{1}{n}\sum_{i=1}^n \left(\frac{1}{pT}\sum_{t=1}^T  \tilde\Lambda_t'\tilde  U_{it}\right)^2 \\
\nonumber
&\qquad + O_P(1)\frac{1}{n}\sum_{i=1}^n \left(\frac{1}{pT}\sum_{t=1}^T \sum_{m=1}^p \tilde\lambda_{tm}(\widehat  U_{it,m}-\tilde U_{it,m})\right)^2 \\
\nonumber
&\leq O_P(\Delta_F^2) + O_{P}(1) \left(\frac{1}{pT}\sum_{t=1}^T \sum_{m=1}^p \frac{1}{n}\tilde F'\tilde U_{t, m}  \tilde\lambda_{tm}\right)^2 \\
\label{ee.14}
&= O_P\left(\Delta_F^2+\frac{1}{nTp}\right) = O_P(\Delta_F^2).
\end{align}
where we used Lemma \ref{le.9} (ii) to bound $\frac{1}{n}\sum_{i=1}^n (\frac{1}{pT}\sum_{t=1}^T \sum_{m=1}^p  \frac{1}{n}\tilde F'\tilde U_{t, m}\widehat  U_{it,m})^2=O_P(\Delta_F^2)$
and the equalities   $ \widehat\lambda_{tm}-H^{-1}\tilde \lambda_{tm}
=  (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm}
+  (\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m}   $,
$$
  \tilde U_{it, m}-\widehat U_{it, m}=
   \widehat  f_i' (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm}
    +   \widehat  f_i' 		 (\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m}
    +  \tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i)   .$$
 
Hence, $\frac{1}{n}\sum_i\|A^*_{i3}\|_2^2=O_{P^*}(\Delta_F^2)$. $\frac{1}{n}\sum_i\|A^*_{i4}\|_2^2$ is bounded similarly. 
$\blacksquare$

\begin{lemma}\label{le.11}
(i) $\frac{1}{n}\sum_i\|\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T (\tilde U_{it}^{*'}\tilde U_{jt}^*- E^*\tilde U_{it}^{*'}\tilde U_{jt}^*)\|_2^2=O_{P^*}(\Delta_F^2).$\\
(ii) $ \frac{1}{T}\sum_s \left\| \frac{1}{pTn} \sum_{i=1}^n\sum_t \widehat \Lambda_t'\tilde U_{it}^* \tilde z^*_{is}\right\|_2^2=O_{P^*}\left(\frac{\log n}{npT}+\frac{\log n}{n^2T^2}+ \frac{1}{n^2}(1+\sqrt{\frac{\log n}{T}})\right)$ where $\tilde z^*_{is} \in \{\tilde\eta^*_{is}, \tilde\epsilon^*_{is}\}$.\\
(iii) $\left\|\frac{1}{nTp}\sum_{j=1}^n\sum_{t=1}^T\widehat f_j^*\widehat \Lambda_t'\tilde U_{jt}^*\right\|_2=O_{P^*}\left(\Delta_F^2+\sqrt{\frac{\Delta_F^2\log n}{n}}\right)$.\\
(iv) 
\begin{align*}
\underset{m,k,l,t}{\max} & \left\|\frac{1}{nT} \sum_{i=1}^n\sum_{s=1}^T(\tilde U_{it,l}^* \tilde U_{is,m}^*-E^*\tilde U_{it,l}^* \tilde U_{is,m}^*)  \widehat h_{sk}'\right\|_F \\
&=O_{P^*}\left(\sqrt{\frac{\log (pT)}{n}\left(\frac{\log n\log (pT)}{n}+\frac{\log(np)}{T}\right)}\right)
\end{align*}
(v) 
\begin{align*}
\underset{m, k\leq p}{\max} & \frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left\|\frac{1}{nT} \sum_{i s}E^*(\tilde U_{it,l}^* \tilde U_{is,m}^*)  \widehat h_{sk}'  \right\|_F \\
&=O_P\left(\Delta_F+\frac{\sqrt{\log(pT)\log p}}{n}+ \sqrt{\frac{\log (pT)}{nT}}\right),
\end{align*}
(vi) 
\begin{align*}
\underset{m, k\leq p}{\max} & \left\|\frac{1}{nT} \sum_{i=1}^n   \frac{1}{pT}    \sum_{s,t}  \widehat \Lambda_t'(\tilde U_{it}^* \tilde U_{is,m}^* -E^*\tilde U_{it}^* \tilde U_{is,m}^*) \widehat h_{sk}'  \right\|_F \\
&=O_{P^*}\left(\sqrt{\frac{\log p}{n}\left(\frac{\log n\log (pT)}{n}+\frac{\log(np)}{T}\right)\Delta_F^2}\right),
\end{align*}
where $\widehat h_{sk} \in \{\widehat\delta_{ys}, \widehat\delta_{ds},\widehat\lambda_{sk}\}$, $s\leq T, k\leq p$.
 \end{lemma}

 
\proof  We shall assume $f_j$ to be a scalar without loss of generality.   

(i) 
\begin{align*} 
\frac{1}{n}\sum_i & E^*\left|\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T (\tilde U_{it}^{*'}\tilde U_{jt}^*- E^*\tilde U_{it}^{*'}\tilde U_{jt}^*)\right|^2 \\
&= \frac{1}{n}\sum_i \Var^*\left(\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T \tilde U_{it}^{*'}\tilde U_{jt}^*\right) \\
&= \frac{1}{n}\sum_i\left(\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T\widehat U_{it}^{'}\widehat U_{jt})^2\Var^*(w_i^Uw_j^U \right) \\
&\leq \frac{C}{n}\sum_i\left(\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T(\widehat U_{it}^{'}\widehat U_{jt}-\tilde U_{it}'\tilde U_{jt})\right)^2 \\
&\qquad + \frac{C}{n}\sum_i\left(\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^T(\tilde U_{it}^{'}\tilde U_{jt}-EU_{it}'U_{jt})\right)^2+\frac{C}{n}\sum_i\left(\frac{1}{pTn}\sum_{j=1}^n\widehat f_j\sum_{t=1}^TEU_{it}^{'} U_{jt}\right)^2 \\
&\leq \underbrace{\frac{C}{n^2}\sum_{i=1}^n\sum_{j=1}^n\left(\frac{1}{pT}\sum_{t=1}^T(\widehat U_{it}^{'}\widehat U_{jt}-\tilde U_{it}'\tilde U_{jt})\right)^2}_{a_3}+ \underbrace{ \frac{C}{n^2}\sum_{i=1}^n\sum_{j=1}^n\left(\frac{1}{pT}\sum_{t=1}^T( \tilde U_{it}'\tilde U_{jt}-EU_{it}'U_{jt})\right)^2}_{a_1} \\
&\qquad +\frac{C}{n}\sum_i\left(\frac{1}{pTn} \widehat f_j\sum_{t=1}^TEU_{it}^{'} U_{it}\right)^2 \\
&=a_1+a_3+O_P\left(\frac{1}{n^2}\right) = O_P(\Delta_F^2)
\end{align*}
where $a_1$ and $a_3$ are as defined in Lemma \ref{le.9}.  We also have $a_1=O_P(\Delta_F^2)$ and $a_3=O_P(\Delta_F^2)$  by (\ref{ee.10}) and (\ref{ee.12}).

 
(ii)  We prove the result for $\tilde z_{is}^*=\tilde\eta^*_{is}$ only.  The other results follow similarly. Let $a_n:=\frac{\log n}{npT}+\frac{\Delta_F^2\log n}{n}+\frac{ \Delta_{\max}^2}{n}$.
\begin{align*}  
E^*&\left[\frac{1}{T}\sum_{s=1}^T \left| \frac{1}{pTn}\sum_{i=1}^n\sum_{t=1}^T \widehat \Lambda_t'\tilde U_{it}^* \tilde z^*_{is}\right|^2\right] \\
&= \frac{1}{T}\sum_{s=1}^T \Var^*\left(\frac{1}{pTn}\sum_{i=1}^n\sum_{t=1}^T \widehat \Lambda_t'\widehat U_{it}\widehat  \eta_{is}w_i^Uw_i^Z\right) \\
&= \frac{1}{T}\sum_{s=1}^T \frac{1}{n^2}\sum_{i=1}^n \left( \frac{1}{pT}    \sum_{t=1}^T \widehat \Lambda_t'\widehat U_{it}\widehat  \eta_{is}\right)^2\Var^*(w_i^Uw_i^Z) \\
& \leq \frac{1}{T}\sum_{s=1}^T \frac{1}{n^2}\sum_{i=1}^n\widehat\eta_{is}^2\left( \frac{1}{pT}    \sum_{t=1}^T \widehat \Lambda_t'\widehat U_{it}\right)^2C \\
&\leq O_P\left(\frac{1}{n}\right)\max_i\left( \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T \widehat \lambda_{tm}(\widehat U_{it,m}-\tilde U_{it,m})\right)^2 \\
&\qquad + O_P\left(\frac{1}{n}\right)\max_i\left( \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T (\widehat \lambda_{tm}-H^{'-1}\tilde\lambda_{tm})\tilde U_{it,m}\right)^2 \\
&\qquad +O_P\left(\frac{1}{n}\right)\max_i\left( \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T\tilde  \lambda_{tm}\tilde U_{it,m}\right)^2 \\
&\leq O_P\left(\frac{\log n}{npT}+\frac{\Delta_F^2\log n}{n}+\frac{ \Delta_{\max}^2}{n} \right) 
+ O_P\left(\frac{\log n}{n}\right)\left( \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T \widehat \lambda_{tm}
 	\frac{1}{n}\widehat F'\tilde U_{t, m}\right)^2 \\ 
&\qquad +O_P\left(\frac{1}{n}\right)\max_i\left( \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T (\widehat \lambda_{tm}-H^{'-1}\tilde\lambda_{tm})\tilde U_{it,m}\right)^2 \\
&\leq O_P(a_n)+O_P\left(\frac{\log n}{n}\right)\left( \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T \widehat \lambda_{tm}\frac{1}{n}\widehat F'\tilde U_{t, m}\right)^2 \\
&\qquad + O_P\left(\frac{1}{n}\right)\max_i\left( \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T
	\frac{1}{n}\widehat F'\tilde U_{t, m}  \tilde U_{it,m}\right)^2 
+ O_P\left(\frac{\Delta_F^2}{n}\right)\max_i\left( \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T
	 \tilde \lambda_{tm} \tilde U_{it,m}\right)^2 \\
&\leq O_P(a_n)+O_P\left(\frac{1}{n}\right)\max_i\left( \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T
	\frac{1}{n}\tilde F'\tilde U_{t, m}  \tilde U_{it,m}\right)^2 \\
&\qquad +	O_P\left(\frac{\log n}{n}\right)\left( \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T \widehat \lambda_{tm}
 	\frac{1}{n}\tilde F'\tilde U_{t, m}\right)^2 \\
&\leq O_P(a_n) + O_P\left(\frac{1}{n^2}\right)\max_i \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T \tilde U_{it,m}^2 + O_P\left(\frac{\log n}{n}\right)\left( \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T \tilde \lambda_{tm}\frac{1}{n}\tilde F'\tilde U_{t, m}\right)^2 \\
&\qquad + O_P\left(\frac{\log n}{n}\right)\left( \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T( \widehat \lambda_{tm}-H^{'-1}\tilde\lambda_{tm})
 	\frac{1}{n}\tilde F'\tilde U_{t, m} \right)^2 \\
&\leq O_P\left(a_n+\frac{1}{n^2}\left(1+\sqrt{\frac{\log n}{T}}\right)\right)+
 O_P\left(\frac{\log n}{n}\right)\left( \frac{1}{pT}  \sum_{m=1}^p  \sum_{t=1}^T( 
 	\frac{1}{n}\tilde F'\tilde U_{t, m}  )^2 \right)^2 \\
&= O_P\left(\frac{\log n}{npT}+\frac{\Delta_F^2\log n}{n}+\frac{ \Delta_{\max}^2}{n}+\frac{1}{n^2}+\frac{1}{n^2}\sqrt{\frac{\log n}{T}}\right) \\
&= O_P\left(\frac{\log n}{npT}+\frac{\log n}{n^2T^2}+ \frac{1}{n^2}+\frac{1}{n^2}\sqrt{\frac{\log n}{T}}\right)
\end{align*}
where we use $\max_i\|\widehat f_i-H'\tilde f_i\|_2=O_P (  \Delta_{\max})=O_P \left(  \frac{1}{\sqrt{n}}  +\sqrt{\frac{\log n}{Tp}}\right)$ from Lemma \ref{lc.8}.
 

(iii) By (\ref{ee.14}), $\frac{1}{n}\sum_{i=1}^n(\frac{1}{pT}\sum_{t=1}^T \widehat \Lambda_t'\tilde U_{it}^*)^2=O_{P^*}(\Delta_F^2)$ and $\frac{1}{n}\sum_{i=1}^n(\frac{1}{pT}\sum_{t=1}^T \widehat \Lambda_t'\widehat U_{it})^2=O_{P}(\Delta_F^2)$. Therefore,
\begin{align*} 
E^*\left|\frac{1}{nTp}\sum_{j=1}^n\sum_{t=1}^T\widehat f_j\widehat \Lambda_t'\tilde U_{jt}^*\right|^2
&= \frac{1}{n^2}\sum_{j=1}^n\left(\frac{1}{Tp}\sum_{t=1}^T\widehat f_j\widehat \Lambda_t'\widehat  U_{jt} \right)^2\Var^*(w_j^U) \\
&= \frac{1}{n^2}\sum_{j=1}^n\left(\frac{1}{Tp}\sum_{t=1}^T\widehat \Lambda_t'\widehat  U_{jt} \right)^2\widehat f_j^2 \\
&\leq \frac{1}{n^2}\sum_{j=1}^n \left(\frac{1}{Tp}\sum_{t=1}^T\widehat \Lambda_t'\widehat  U_{jt} \right)^2\max_j\widehat f_j^2 \\
&=O_P\left(\frac{\Delta_F^2\log n}{n}\right).
\end{align*}
Hence,
\begin{align*} 
\left\|\frac{1}{nTp}\sum_{j=1}^n\sum_{t=1}^T\widehat f_j^*\widehat \Lambda_t'\tilde U_{jt}^*\right\|_2^2&\leq O_{P^*}(\Delta_F^2)\frac{1}{n}\sum_{j=1}^n\left(\frac{1}{pT}\sum_{t=1}^T\widehat\Lambda_t'\tilde U_{jt}^*\right)^2 + \left|\frac{1}{nTp}\sum_{j=1}^n\sum_{t=1}^T\widehat f_j\widehat \Lambda_t'\tilde U_{jt}^*\right|^2O_{P^*}(1) \\
&\leq O_{P^*}\left(\Delta_F^4+\frac{\Delta_F^2\log n}{n}\right).
\end{align*}


(iv) For notational simplicity, we assume $\widehat h_{sk}$ to be a scalar.  Then
$$
\max_{mklt}|\frac{1}{nT} \sum_{i=1}^n\sum_{s=1}^T(\tilde U_{it,l}^* \tilde U_{is,m}^*-E^*\tilde U_{it,l}^* \tilde U_{is,m}^*)  \widehat h_{sk}|  =\max_{mklt}|\frac{1}{n}\sum_{i=1}^n\Gamma_{i,mklt}|
$$
where $\Gamma_{i,mklt}=\frac{1}{T}\sum_{s=1}^T\widehat U_{it,l} \widehat U_{is,m} \widehat h_{sk}\left[(w_i^U)^2-  E^*(w_i^U)^2\right]$. We  still use the Bernstein inequality for independent data to achieve 
$$
\max_{mklt}|\frac{1}{n}\sum_{i=1}^n\Gamma_{i,mklt}|=O_{P^*}(\sqrt{\max_{mklt}\frac{1}{n}\sum_{i=1}^n\Var^*(\Gamma_{i,mklt})\frac{\log(pT)}{n}}).
$$
So we now bound $\max_{mklt}\frac{1}{n}\sum_{i=1}^n\Var^*(\Gamma_{i,mklt})$ below.  Recall the equality:
 $$
  \tilde U_{it, m}-\widehat U_{it, m}=
   \widehat  f_i' (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tm}
    +   \widehat  f_i' 		 (\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, m}
    +\tilde\lambda_{tm}'H^{'-1}(\widehat  f_i -H'\tilde f_i),$$   
\begin{align}
\nonumber
\max_{mklt}\frac{1}{n}\sum_{i=1}^n\Var^*(\Gamma_{i,mklt}) &= \max_{mklt}\frac{1}{n}\sum_{i=1}^n\widehat U_{it,l}^2\left(\frac{1}{T}\sum_{s=1}^T \widehat U_{is,m} \widehat h_{sk}\right)^2\Var^*\left( (w_i^U)^2\right) \\
\nonumber
&\leq O_P(1)\max_{imk} \left(\frac{1}{T}\sum_{s=1}^T \widehat U_{is,m} \widehat h_{sk}\right)^2 \\
\nonumber
&\leq O_P(1)\max_{imk} \left|\frac{1}{T}\sum_{s=1}^T \tilde U_{is,m} \widehat h_{sk}\right|^2 \\
\nonumber
&\qquad + \max_{mk} \frac{1}{T}\sum_{t=1}^T \left|\tilde \lambda_{tm} \widehat h_{tk}\right|^2 O_P\left(\Delta_F^2\log n+\Delta_{\max}^2\right) \\
\nonumber
&\qquad + O_P(\log n)\max_{m}  \frac{1}{T}\sum_{t=1}^T \left| \frac{1}{n}\widehat F'\tilde U_{t, m}\right|^2 \\
\label{ee.15a}
\begin{split}
&\leq O_P\left(\Delta_F^2\log n+\Delta_{\max}^2+{\frac{\log n\log (pT)}{n}}+\frac{\log(np)}{T}\right) \\
&\qquad +\max_{imk} \left|\frac{1}{T}\sum_{t=1}^T \tilde U_{it,m} (\widehat h_{tk}-H^{-1}\tilde h_{tk})\right|^2.
\end{split}
\end{align}
In the derivation above, we used  
$$
\max_{m\leq p}\frac{1}{n^2T}\sum_t \|\tilde U_{t,m}'\widehat F\|_2^2=  O_P\left({\frac{\log (pT)}{n}}+\Delta^2_{fum}\right) =O_P \left({\frac{\log (pT)}{n}}+\Delta^2_F\right) 
$$ 
from Lemma \ref{la.1} and $\max_{imk} \frac{1}{T}\sum_{t=1}^T |\tilde \lambda_{tm} \widehat h_{tk}|=O_P(1)$ which follows from  
$\max_{tm}|\widehat\lambda_{tm}|=O_P(1)$ and  {$\max_t|\widehat h_{tk}|=O_P(1)$} for $\widehat h_{tk}\in\{\widehat\lambda_{tm},\widehat \delta_{yt} \}$. We now bound the second term on the right in  (\ref{ee.15a}) in two cases. 

%\textcolor{blue}{DOUBLE CHECK change from $\max_t|\widehat \delta_{yt}|$ to $\max_t|\widehat h_{tk}|$ above.}

 
Case 1: $\widehat h_{tk}=\widehat\delta_{yt}$ for all $k\leq p$.  Using the equality
$$
\widehat\delta_{yt}-H^{-1}\tilde\delta_{yt}=
(\widehat F'\widehat F)^{-1}\widehat F' (\tilde FH-\widehat F)H^{-1}\tilde\delta_{yt}
+(\widehat F'\widehat F)^{-1}\widehat F' \tilde U_t\gamma_y
+(\widehat F'\widehat F)^{-1}\widehat F' \tilde e_t,$$
we have       
\begin{align*}
\max_{imk} \left|\frac{1}{T}\sum_{t=1}^T \tilde U_{it,m} (\widehat h_{tk}-H^{-1}\tilde h_{tk}) \right|^2
&\leq \max_{im} \left|\frac{1}{T}\sum_{t=1}^T \tilde U_{it,m}  \tilde\delta_{yt} \right|^2 O_P(\Delta_F^2)  +\max_{im} \left|\frac{1}{T}\sum_{t=1}^T \tilde U_{it,m} \frac{1}{n}\widehat F' \tilde U_t\gamma_y \right|^2 \\
&\qquad +\max_{im} \left|\frac{1}{T}\sum_{t=1}^T \tilde U_{it,m}\frac{1}{n}\widehat F' \tilde e_t\right|^2 \\
&\leq \frac{\log(np)}{T} O_P(\Delta_F^2) + \max_{im}\frac{1}{T}\sum_t\tilde U_{it,m}^2 O_P\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^2\right) \\
&= \frac{\log(np)}{T}O_P(\Delta_F^2) + \left(1+\sqrt{\frac{\log (np)}{T}}\right)O_P\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^2\right),
\end{align*}
where we used Lemma \ref{la.1} (iv) that $\frac{1}{T}\sum_t\|\frac{1}{n}\widehat F'\tilde e_t\|_2^2+\frac{1}{T}\sum_t\|\frac{1}{n}\widehat F'\tilde U_t\gamma\|_2^2=O_P\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^2\right).$


Case 2: $\widehat h_{tk}=\widehat\lambda_{tk}$.  Using the equality
$$
\widehat\lambda_{tk}-H^{-1}\tilde \lambda_{tk}=(\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{tk}+(\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{t, k},
$$
we have
\begin{align*}
\max_{imk} \left|\frac{1}{T}\sum_{t=1}^T \tilde U_{it,m} (\widehat h_{tk}-H^{-1}\tilde h_{tk})  \right|^2
&\leq  \max_{imk} \left|\frac{1}{T}\sum_{t=1}^T \tilde U_{it,m}  \tilde \lambda_{tk} \right|^2O_P(\Delta_F^2) \\
&\qquad +\max_{imk} \left|\frac{1}{T}\sum_{t=1}^T \tilde U_{it,m} \frac{1}{n}\widehat F'\tilde U_{t, k}  \right|^2 \\
&\leq O_P\left(\frac{\Delta_F^2\log(np)}{T}\right) + \max_{imk} \frac{1}{T}\sum_{t=1}^T \tilde U_{it,m}^2\frac{1}{T}\sum_t\left( \frac{1}{n}\widehat F'\tilde U_{t, k}\right)^2 \\
&= O_P\left(\frac{\Delta_F^2\log(np)}{T}\right) + \left(1+\sqrt{\frac{\log (pn)}{T}}\right)\left(\Delta_F^2+\frac{\log(pT)}{n}\right).
\end{align*}
Hence $\max_{imk} |\frac{1}{T}\sum_{t=1}^T \tilde U_{it,m} (\widehat h_{tk}-H^{-1}\tilde h_{tk})|^2=O_P(a_n)$, where 
$$
a_n:=\frac{\log(np)}{T}(\Delta_F^2) + \left(1+\sqrt{\frac{\log (np)}{T}}\right)\left(\frac{|J|_0}{n}+|J|_0^2\Delta_F^2+\frac{\log(pT)}{n}\right).
$$
Substituting this expression into (\ref{ee.15a}), we obtain
\begin{align}\label{ee.16}
\begin{split}
\max_{mklt}\frac{1}{n}\sum_{i=1}^n\Var^*(\Gamma_{i,mklt}) &= O_P(1)\max_{imk} \left(\frac{1}{T}\sum_{s=1}^T \widehat U_{is,m} \widehat h_{sk}\right)^2 \\
&= O_P\left(\frac{\log n\log (pT)}{n}+\frac{\log(np)}{T}\right).
\end{split}
\end{align}
Hence, $\max_{mklt}\left|\frac{1}{n}\sum_{i=1}^n\Gamma_{i,mklt}\right|=O_{P^*}\left(\sqrt{\frac{\log (pT)}{n}\left(\frac{\log n\log (pT)}{n}+\frac{\log(np)}{T}\right)}\right)$.
 

(v) We assume $\widehat h_{sk}$ to be a scalar for notational simplicity. 
\begin{align*}
\max_{m, k\leq p} & \frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T E^*(\tilde U_{it,l}^* \tilde U_{is,m}^*)  \widehat h_{sk} \right| \\
&= \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\widehat U_{it,l}\widehat U_{is,m} \widehat h_{sk} \right| \\
%&\leq& \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T |\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T(\widehat U_{it,l}\widehat U_{is,m} -\tilde U_{it,l}\tilde U_{is,m})\widehat h_{sk} |+ \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T |\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T(\tilde U_{it,l}\tilde U_{is,m} )\widehat h_{sk} |\cr
&\leq \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T(\tilde U_{it,l}\tilde U_{is,m} )\widehat h_{sk} \right| \\
&\qquad + \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\tilde U_{is,m}(\widehat U_{it,l} -\tilde U_{it,l})\widehat h_{sk} \right| \\
&\qquad + \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{n} \sum_{i=1}^n  \widehat U_{it,l} \widehat  f_i'\right| O_P(\Delta_F) \\
&\qquad + \frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{n} \sum_{i=1}^n \widehat U_{it,l}
   \widehat  f_i' \right|	\max_{m, k\leq p}\left|\frac{1}{Tn} \sum_{s=1}^T\tilde F'\tilde U_{s, m}  \widehat h_{sk} \right| \\
&\qquad + \frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{n} \sum_{i=1}^n  \tilde  U_{it,l}
 (\widehat  f_i -H'\tilde f_i)    \right| O_P(1) \\
&\leq O_P(\Delta_F) + \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T(\tilde U_{it,l}\tilde U_{is,m} )\widehat h_{sk} \right| \\
&\qquad + \frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{n} \sum_{i=1}^n \widehat U_{it,l}
\tilde   f_i' \right|	\max_{m, k\leq p}\left| 	\frac{1}{Tn} \sum_{s=1}^T\tilde F'\tilde U_{s, m}  \widehat h_{sk} \right| \\
&\leq O_P(\Delta_F)+\max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T(\tilde U_{it,l}\tilde U_{is,m} )\widehat h_{sk} \right| \\
&\qquad + \max_{m, k\leq p} \left|\frac{1}{Tn} \sum_{s=1}^T\tilde F'\tilde U_{s, m}  \widehat h_{sk} \right|O_P\left(\frac{1}{\sqrt{n}}\right) \\
&\leq O_P\left(\Delta_F+\frac{\sqrt{\log p}}{n}\right) + \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\tilde U_{it,l}\tilde U_{is,m} \widehat h_{sk} \right| \\
&\leq O_P\left(\Delta_F+\frac{\sqrt{\log p}}{n}\right) +O_P(1) \max_{m\leq p}\frac{1}{p}\sum_{l=1}^p  \left|\frac{1}{n} \sum_{i=1}^n  \bar U_{i\cdot,l}\bar U_{i\cdot,m} \right| \\ 
&\qquad +O_P(1) \max_{m\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{n} \sum_{i=1}^n   U_{it,l}\bar U_{i\cdot,m} \right| + \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T  U_{it,l} U_{is,m}  \widehat h_{sk} \right| \\
&\qquad + \max_{m, k\leq p}\frac{1}{p}\sum_{l=1}^p  \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\bar U_{i\cdot,l} U_{is,m} \widehat h_{sk} \right| \\
&:= O_P\left(\Delta_F+\frac{\sqrt{\log p}}{n}\right)+\sum_{l=1}^4 D_l O_P(1).
%&&     + \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T |\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\bar U_{\cdot t,l} U_{is,m} \widehat h_{sk} |\cr+ 
   %  &&+ \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T |\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\bar{ \bar U}_l U_{is,m} \widehat h_{sk} | 
\end{align*}
 
         %   $$
  %\tilde U_{is, m}-\widehat U_{is, m}=
  % \widehat  f_i' (\widehat F'\widehat F)^{-1}\widehat F'(\tilde FH-\widehat F)H^{-1}\tilde \lambda_{sm}
    %+   \widehat  f_i' 		 (\widehat F'\widehat F)^{-1}\widehat F'\tilde U_{s, m}
    %+  \tilde\lambda_{sm}'H^{'-1}(\widehat  f_i -H'\tilde f_i)   .$$
We now bound each of $D_1-D_4$. 
\begin{align*}
D_1 &= \max_{m\leq p}\frac{1}{p}\sum_{l=1}^p  \left|\frac{1}{n} \sum_{i=1}^n  \bar U_{i\cdot,l}\bar U_{i\cdot,m}   \right| \\
&\leq  \frac{1}{pT}    \max_{ms}\sum_t \sum_{l=1}^p |EU_{is,l} U_{it,m}   | \\
&\qquad + \max_{m,l\leq p}   \left|\frac{1}{n} \sum_{i=1}^n  \frac{1}{T^2}\sum_{s,t} (U_{is,l} U_{it,m} -EU_{is,l} U_{it,m})  \right| \\
&=O\left(\frac{1}{pT}\right)+ \max_{m,l\leq p}   \left|\frac{1}{n} \sum_{i=1}^nX_{i,ml}  \right|  
\end{align*}
where $X_{i,ml}=  \frac{1}{T^2}\sum_{s,t} (U_{is,l} U_{it,m} -EU_{is,l} U_{it,m})$ with
$
\max_{ml}\frac{1}{n}\sum_i\Var(X_{i,ml})=O(\frac{1}{T^2})$.  Thus, $D_1=O_P\left(\frac{1}{T}\sqrt{\frac{\log p}{n}}+\frac{1}{pT}\right)$.

\begin{align*}
D_2 &= \max_{m\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{n} \sum_{i=1}^n   U_{it,l}\bar U_{i\cdot,m}\right| \\
&\leq  \max_{m\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{n} \sum_{i=1}^n\frac{1}{T}\sum_sE   U_{it,l} U_{is,m} \right| \\
&\qquad + \max_{mlt}\left|\frac{1}{n} \sum_{i=1}^n\frac{1}{T}\sum_s  ( U_{it,l} U_{is,m}  -EU_{it,l} U_{is,m} ) \right| \\
&= O\left(\frac{1}{pT}\right)+ \max_{mlt}\left|\frac{1}{n} \sum_{i=1}^nX_{i,mtl}\right|  
\end{align*}
where $X_{i,mtl} =\frac{1}{T}\sum_s  ( U_{it,l} U_{is,m}  -EU_{it,l} U_{is,m} )$ with $\max_{mtl}\frac{1}{n}\sum_i\Var(X_{i,mtl})=O\left(\frac{1}{T}\right)$.
Hence, $D_2=O_P\left( \sqrt{\frac{\log (pT)}{nT}}+\frac{1}{pT}\right)$.

\begin{align*}
D_3+D_4 &= \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T  U_{it,l} U_{is,m}  \widehat h_{sk} \right| \\
&\qquad + \max_{m, k\leq p}\frac{1}{p}\sum_{l=1}^p  \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\bar U_{i\cdot,l} U_{is,m} \widehat h_{sk} \right| \\
&\leq % \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T |\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T  EU_{it,l} U_{is,m}    \tilde h_{sk} |+ \max_{m, k\leq p}\frac{1}{p}\sum_{l=1}^p  |\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\frac{1}{T}\sum_{t=1}^TE U_{it,l} U_{is,m} \tilde  h_{sk} |\cr
O\left(\frac{1}{pT}+\sqrt{\frac{\log (pT)}{nT}}\right)
%+  \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T |\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T ( U_{it,l} U_{is,m}-EU_{it,l} U_{is,m} )   \tilde h_{sk} |\cr
%&&+ \max_{m, k\leq p}\frac{1}{p}\sum_{l=1}^p  |\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\frac{1}{T}\sum_{t=1}^T (U_{it,l} U_{is,m} -EU_{it,l} U_{is,m})\tilde  h_{sk} |\cr
+ \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T  U_{it,l} U_{is,m} ( \widehat h_{sk} -H^{-1}\tilde h_{sk})\right| \\
&\qquad + \max_{m, k\leq p}\frac{1}{p}\sum_{l=1}^p  \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\bar U_{i\cdot,l} U_{is,m} (\widehat h_{sk}-H^{-1}\tilde h_{sk}) \right|.
\end{align*}
To bound the last two terms on the right-hand-side of the last inequality, we discuss two cases.
            
Case 1:   $\widehat h_{tk}=\widehat\delta_{yt}$ for all $k\leq p$.  We have
\begin{align*}
\max_{m, k\leq p}&\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T  U_{it,l} U_{is,m} ( \widehat h_{sk} -H^{-1}\tilde h_{sk})\right| \\
&+\frac{1}{p}\sum_{l=1}^p  \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\bar U_{i\cdot,l} U_{is,m} (\widehat h_{sk}-H^{-1}\tilde h_{sk}) \right| \leq O_P\left(\Delta_F+\frac{\sqrt{|J|_0\log(pT)}}{n}\right).  
%\max_{mtls}   | \frac{1}{n}  \sum_{i=1}^n ( U_{it,l} U_{is,m} -EU_{it,l} U_{is,m}) | O_P(\sqrt{\frac{|J|_0}{n}})\cr
%&& \max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T |\frac{1}{T}  \sum_{s=1}^T\frac{1}{n}   \sum_{i=1}^n (U_{it,l} U_{is,m}-EU_{it,l} U_{is,m})\frac{1}{n}\tilde F' \tilde e_s  |\cr
%&&+ \max_{m, k\leq p}\frac{1}{p}\sum_{l=1}^p  |\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\frac{1}{T}\sum_{t=1}^T ( U_{it,l} U_{is,m}  -EU_{it,l} U_{is,m}  )|O_P(\sqrt{\frac{|J|_0}{n}})\cr
%&&+ \max_{m, k\leq p}\frac{1}{p}\sum_{l=1}^p  |\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\bar U_{i\cdot,l} U_{is,m}\frac{1}{n}\tilde  F' \tilde e_s |
\end{align*}

Case 2: $\widehat h_{tk}=\widehat\lambda_{tk}$.  We have    
\begin{align*}
\max_{m, k\leq p} & \frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T  U_{it,l} U_{is,m} ( \widehat h_{sk} -H^{-1}\tilde h_{sk})\right| \\
& + \frac{1}{p}\sum_{l=1}^p  \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T\bar U_{i\cdot,l} U_{is,m} (\widehat h_{sk}-H^{-1}\tilde h_{sk}) \right| 
\leq  O_P\left(\Delta_F+\frac{\sqrt{\log (pT)\log p}}{n} \right). 
\end{align*}

Combining all of the above bounds yields
$$
\max_{m, k\leq p}\frac{1}{pT}\sum_{l=1}^p\sum_{t=1}^T \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^TE^*(\tilde U_{it,l}^* \tilde U_{is,m}^*)  \widehat h_{sk} \right|=O_P\left(\Delta_F+\frac{\sqrt{\log(pT)\log p}}{n}+ \sqrt{\frac{\log (pT)}{nT}}\right).
$$
 
(vi) We assume $\widehat\lambda_t'$ is a row vector and $\widehat h_{sk}$ is a scalar.  Now let
$$ 
\max_{m, k\leq p} \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}\sum_{t=1}^T\widehat \Lambda_t'(\tilde U_{it}^* \tilde U_{is,m}^* -E^*\tilde U_{it}^* \tilde U_{is,m}^*) \widehat h_{sk}' \right| = \max_{m, k\leq p}\left|\frac{1}{n}\sum_i X_{i,mk} \right|
$$
where $X_{i,mk}= \frac{1}{T} \sum_{s=1}^T \frac{1}{pT} \sum_{t=1}^T  \widehat \Lambda_t'\widehat U_{it}\widehat U_{is,m} \widehat h_{sk}(w_i^{U2}-E^*w_i^{U2})$. Note that $X_{i,mk}$ is mean zero and satisfies
\begin{align*}
\max_{mk}\frac{1}{n}\sum_{i=1}^n\Var^*(X_{i,mk}) &= \max_{mk}\frac{1}{n}\sum_{i=1}^n  \left(\frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'\widehat U_{it}\right)^2\left( \frac{1}{T}  \sum_{s=1}^T\widehat U_{is,m} \widehat h_{sk}\right)^2\Var^*(w_i^{U2}  ) \\
&\leq^{(\ref{ee.16})} \frac{1}{n}\sum_{i=1}^n \left(  \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'\widehat U_{it}\right)^2 O_P\left(\frac{\log n\log (pT)}{n}+\frac{\log(np)}{T}\right) \\
&=^{(\ref{ee.14})} O_P\left(\frac{\log n\log (pT)}{n}+\frac{\log(np)}{T}\right)\Delta_F^2.
\end{align*}
Hence, $\max_{m, k\leq p}|\frac{1}{n}\sum_i X_{i,mk}| = O_{P^*}\left(\sqrt{\frac{\log p}{n}\left(\frac{\log n\log (pT)}{n}+\frac{\log(np)}{T}\right)\Delta_F^2}\right)$.
  $\blacksquare$

\begin{lemma}\label{le.12} For $\widehat h_{sk} \in \{\widehat\delta_{ys}, \widehat\delta_{ds},\widehat\lambda_{sk}\}$, $s\leq T, k\leq p$:\\
(i) 
\begin{align*}
\max_{m, k\leq p} & \left|\frac{1}{nT} \frac{1}{pT}    \sum_{ist} \tilde  \Lambda_t'\tilde   U_{it} \tilde U_{is,m} \widehat h_{sk} \right| \\
&= O_P\left(\frac{1}{pT}+\sqrt{\frac{\log p}{npT^2}}+ \left(\sqrt{\frac{\log (pT)}{npT}}\right)\left( \sqrt{\frac{|J|_0}{n}}+\sqrt{\frac{\log(pT)}{n}}\right) \right)
\end{align*}
(ii)  $ \max_{m, k\leq p}\left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n    \frac{1}{pnT}    \sum_{st}
\widehat F'\tilde U_{t, v}  \tilde   U_{it,v} \tilde U_{is,m} \widehat h_{sk} \right| = O_P\left(\sqrt{\frac{\log p }{npT}}+\frac{\sqrt{\log p}}{n}+\frac{1}{pT}\right)$\\
(iii) 
\begin{align*} 
\max_{m, k\leq p} & \left\|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'E^*\tilde U_{it}^* \tilde U_{is,m}^*  \widehat h_{sk}'  \right\|_F \\
&= O_P \left(\frac{1}{pT}+\sqrt{\frac{\log p}{npT}}+\frac{\sqrt{\log p}}{n}+\frac{\sqrt{|J|_0}}{nT}+\sqrt{\frac{|J|_0}{npT}}\right)+ O_P(\Delta_{ud})
\end{align*}
\end{lemma}


\proof 
(i)   $\max_{m, k\leq p}|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \tilde  \Lambda_t'\tilde   U_{it} \tilde U_{is,m} \widehat h_{sk} | $ is bounded by 
\begin{eqnarray}\label{LJ9.i}
&& \max_{m, k\leq p}\left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \tilde  \Lambda_t'   (E U_{it}   U_{is,m}) \widehat h_{sk} \right|  +\max_{m, k\leq p} \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \tilde  \Lambda_t'    U_{it} \bar U_{i\cdot,m} \widehat h_{sk} \right| \cr
&& +\max_{m, k\leq p} \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \tilde  \Lambda_t'    (U_{it}   U_{is,m} -EU_{it}   U_{is,m})\widehat h_{sk} \right|+ O_P\left(\frac{1}{\sqrt{npT}}\sqrt{\frac{\log p}{n}}\right).
\end{eqnarray}
The first term in (\ref{LJ9.i}) is $O_P\left(\frac{1}{pT}\right)$. The second term in (\ref{LJ9.i}) is stochastically bounded by $O_P\left(\frac{1}{pT}\right)$ plus the third term.
Note that 
$$ 
\max_{m, k\leq p} \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \tilde  \Lambda_t'    (U_{it}   U_{is,m} -EU_{it}   U_{is,m})\tilde h_{sk} \right| = O_P\left(\sqrt{\frac{\log p}{npT^2}}\right) 
$$ 
by (\ref{ee.9a5}). Hence, the third term in (\ref{LJ9.i}) is bounded by 
\begin{align*}
O_P&\left(\sqrt{\frac{\log p}{npT^2}}\right) +
\max_{m, k\leq p} \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT} \sum_{t=1}^T \tilde  \Lambda_t'(U_{it}   U_{is,m} -EU_{it}   U_{is,m})(\widehat h_{sk}- H^{-1}\tilde h_{sk}) \right| \\
&\leq O_P\left(\sqrt{\frac{\log p}{npT^2}}\right) + \left[\max_{m,  s }\left|\frac{1}{n } \sum_{i=1}^n\frac{1}{pT}\sum_{t=1}^T \tilde  \Lambda_t'(U_{it}   U_{is,m} -EU_{it}   U_{is,m}) \right| \times \right. \\
&\qquad \left. \max_{ k\leq p}\frac{1}{T}\sum_s\left(\left| \frac{1}{n}\widehat F'\tilde U_{s, k}\right| + \left|  \frac{1}{n}\widehat F'\tilde U_{t}\gamma \right| + \left| \frac{1}{n}\widehat F' \tilde e_s\right|+\Delta_F\right)\right] \\
&\leq O_P\left(\sqrt{\frac{\log p}{npT^2}}\right)+O_P\left(\sqrt{\frac{\log (pT)}{npT}}\right)\left( \sqrt{\frac{|J|_0}{n}}+\sqrt{\frac{\log(pT)}{n}}\right) 
\end{align*}
regardless of whether $h_{sk}=\delta_t$ or $\lambda_{tk}$.
Hence, 
\begin{align*}
\max_{m, k\leq p} & \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \tilde  \Lambda_t'\tilde   U_{it} \tilde U_{is,m} \widehat h_{sk} \right| \\
&= O_P\left(\frac{1}{pT}+\sqrt{\frac{\log p}{npT^2}}+ \left(\sqrt{\frac{\log (pT)}{npT}}\right)\left( \sqrt{\frac{|J|_0}{n}}+\sqrt{\frac{\log(pT)}{n}}\right) \right).
\end{align*}
 

(ii) Note that $\sum_s\tilde h_{sk}=0$ for $h \in \{\delta, \lambda\}$.  We then have that 
\begin{align*}
\max_{m, k\leq p} & \left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v}  \tilde   U_{it,v} \tilde U_{is,m} \widehat h_{sk} \right| \\ 
&\leq  \max_{m, k\leq p}\left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT} \sum_{t=1}^T \frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v}  U_{it,v} \tilde U_{is,m} \widehat h_{sk} \right| \\
&\qquad + \max_{m, k\leq p} \left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v}  \bar  U_{t,v} \tilde U_{is,m} \widehat h_{sk} \right| \\
&\leq O_P\left(\Delta_F\frac{\sqrt{\log p}}{\sqrt{n}}+ {\frac{\sqrt{\log p}}{n}}\right) \\
&\qquad + \max_{m, k\leq p}\left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT} \sum_{t=1}^T \frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v} (E U_{it,v}  U_{is,m}) \widehat h_{sk} \right| \\
%&&+\max_{m, k\leq p}|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T  \frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v}  U_{it,v} \bar U_{s,m} \widehat h_{sk} |\cr
&\qquad +\max_{m \leq p} \left|\frac{1}{n } \sum_{v=1}^p\sum_{i=1}^n   \frac{1}{pT}    \sum_{t=1}^T 
\frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v}  U_{it,v} \bar U_{i,m} \right| \\
&\qquad +\max_{m, k\leq p} \left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v} ( U_{it,v}  U_{is,m} -EU_{it,v}  U_{is,m})\widehat h_{sk} \right| \\
%&&+\max_{k\leq p}|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v}  U_{it,v}   \widehat h_{sk} | \max_m|\bar{\bar U}_m|\cr
&\leq^{(a)} O_P\left(\Delta_F\frac{\sqrt{\log p}}{\sqrt{n}}+ {\frac{\sqrt{\log p}}{n}}+\frac{1}{pT}\right) + O_P\left( \frac{1}{T\sqrt{p}}  + \frac{1}{\sqrt{n}} \right) \left(\sqrt{\frac{\log(pT)}{nT}}\right) \\
&\qquad + \max_{m, k\leq p}\left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v} ( U_{it,v}  U_{is,m} -EU_{it,v}  U_{is,m})(\widehat h_{sk}-H^{-1}\tilde h_{sk}) \right| \\
&\leq^{(b)} O_P\left(\Delta_F\frac{\sqrt{\log p}}{\sqrt{n}}+ {\frac{\sqrt{\log p}}{n}}+\frac{1}{pT}\right)+ O_P\left( \frac{1}{T\sqrt{p}}  + \frac{1}{\sqrt{n}} \right) \left(\sqrt{\frac{\log(pT)}{nT}}\right) \\
&\qquad +\max_{m, k\leq p}\left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v} ( U_{it,v}  U_{is,m} -EU_{it,v}  U_{is,m})\tilde h_{sk} \right|O_P(\Delta_F) \\
&\qquad +\max_{m, k\leq p}\left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v} ( U_{it,v}  U_{is,m} -EU_{it,v}  U_{is,m})\frac{1}{n}\widehat F'\tilde U_{s, k} \right| \\
&\qquad +\max_{m, k\leq p}\left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v} ( U_{it,v}  U_{is,m} -EU_{it,v}  U_{is,m}) \frac{1}{n}\widehat F' \tilde U_s\gamma \right| \\
&\qquad +\max_{m, k\leq p}\left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v} ( U_{it,v}  U_{is,m} -EU_{it,v}  U_{is,m})\frac{1}{n}\widehat F' \tilde e_s \right| \\
&= O_P\left(\Delta_F\frac{\sqrt{\log p}}{\sqrt{n}}+ {\frac{\sqrt{\log p}}{n}}+\frac{1}{pT}\right)+O_P\left( \frac{1}{T\sqrt{p}}  + \frac{1}{\sqrt{n}} \right) \left(\sqrt{\frac{\log(pT)}{nT}}\right) \\
&+O_P\left(\Delta_F+\frac{1}{\sqrt{n}}\right)\left(\Delta_F+\sqrt{\frac{\log (pT)}{n}}+\sqrt{\frac{|J|_0}{n}}\right)\sqrt{\frac{\log (pT)}{n}} \\
&=O_P\left(\sqrt{\frac{\log p }{npT}}+\frac{\sqrt{\log p}}{n}+\frac{1}{pT}\right).
\end{align*}
In (a) above, we used the bound in (\ref{ee.9a2}) for $C_2$. Also, we did not formally treat the term 
$$
\max_{m \leq p}\left|\frac{1}{n } \sum_{v=1}^p\sum_{i=1}^n   \frac{1}{pT}    \sum_{t=1}^T 
\frac{1}{n}\sum_{j=1}^n\widehat f_j\tilde U_{jt, v}  U_{it,v} \bar U_{i,m} \right|
$$
as can be bounded similarly to the term $C_2$ in (\ref{ee.9a2}). The inequality (b) holds for $h \in \{\delta, \lambda\}$.
 
 
(iii)
\begin{align*}
\max_{m, k\leq p} & \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'E^*\tilde U_{it}^* \tilde U_{is,m}^*  \widehat h_{sk} \right| \\
&= \max_{m, k\leq p}\left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'\widehat   U_{it} \widehat U_{is,m} \widehat h_{sk} \right| \\
&\leq \max_{m, k\leq p}\left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'\widehat   U_{it} \tilde U_{is,m} \widehat h_{sk} \right| + \left|\frac{1}{n} \sum_{i=1}^n \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'\widehat   U_{it}   \widehat  f_i\right| O_P(\Delta_F) \\
&\qquad + \left|\frac{1}{n } \sum_{i=1}^n  \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'\widehat   U_{it}  \widehat  f_i\right|	\max_{m, k\leq p}\left|	\frac{1}{T} \sum_{s=1}^T\frac{1}{n}\tilde F'\tilde U_{s, m} \widehat h_{sk} \right| \\
&\qquad + \left(\frac{1}{n } \sum_{i=1}^n  \left( \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'\widehat   U_{it}  \right)^2\right)^{1/2}O_P(\Delta_F).
\end{align*}
Note that  $\frac{1}{n}\sum_{i=1}^n \left(\frac{1}{pT}\sum_{t=1}^T \widehat \Lambda_t'\widehat  U_{it}\right)^2=O_P(\Delta_F)$ by (\ref{ee.14}) and $	\max_{m, k\leq p} \left|	\frac{1}{T} \sum_{s=1}^T\frac{1}{n}\tilde F'\tilde U_{s, m} \widehat h_{sk} \right|=O_P\left(\sqrt{\frac{\log p}{n}}\right)$, so $\left|\frac{1}{n } \sum_{i=1}^n  \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'\widehat   U_{it}  \widehat  f_i\right|=O_P(\Delta_F)$.  We may then complete the bound as
\begin{align*}
\max_{m, k\leq p} & \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'E^*\tilde U_{it}^* \tilde U_{is,m}^*  \widehat h_{sk} \right| \\
&\leq \max_{m, k\leq p}\left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'\widehat   U_{it} \tilde U_{is,m} \widehat h_{sk} \right|  +O_P\left(\Delta_F+\sqrt{\frac{\log p}{n}}\right)\Delta_F \\
&\leq O_P\left(\Delta_F+\sqrt{\frac{\log p}{n}}\right)\Delta_F + \max_{m, k\leq p} \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T  \widehat \Lambda_t'\tilde   U_{it} \tilde U_{is,m} \widehat h_{sk} \right|  \\
&\qquad + \max_{m, k\leq p} \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T  (\widehat  f_i -H'\tilde f_i) \tilde U_{is,m} \tilde h_{sk} \right|  O_P(1) \\
&\qquad + \max_{m, k\leq p} \left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T  (\widehat  f_i -H'\tilde f_i) \tilde U_{is,m} (\widehat h_{sk}-H^{-1}\tilde h_{sk}) \right| \\
&\qquad +\left(\frac{1}{p }    \sum_{v=1}^p\frac{1}{T}\sum_{t=1}^T  \left|\frac{1}{n}\tilde F'\tilde U_{t, v}   \right|^2+  \left|\frac{1}{p }    \sum_{v=1}^p\frac{1}{T}\sum_{t=1}^T  \frac{1}{n}\tilde F'\tilde U_{t, v}\tilde \lambda_{tv}  \right|\right) \max_{m, k\leq p}\left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \tilde  f_i' \tilde U_{is,m} \widehat h_{sk} \right| \\
&\leq O_P\left(\Delta_F+\sqrt{\frac{\log p}{n}}+\sqrt{\frac{|J|_0}{n}}\right)\Delta_F + O_P\left(\frac{1}{n}+\frac{1}{\sqrt{npT}}\right)\sqrt{\frac{\log p}{n}}+ O_P(\Delta_{ud}) \\
&\qquad +\max_{m, k\leq p}\left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \tilde  \Lambda_t'\tilde   U_{it} \tilde U_{is,m} \widehat h_{sk} \right| \\
&\qquad +\max_{m, k\leq p}\left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n   \sum_{t=1}^T 
 \tilde \lambda_{tv}  \tilde   U_{it,v}  \sum_{s=1}^T \frac{1}{pT}   \tilde U_{is,m} \widehat h_{sk} \right| O_P(\Delta_F) \\
&\qquad + \max_{m, k\leq p}\left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \frac{1}{n}\widehat F'\tilde U_{t, v}  \tilde   U_{it,v} \tilde U_{is,m} \widehat h_{sk} \right| \\
&\leq O_P\left(\Delta_F+\sqrt{\frac{\log p}{n}}+\sqrt{\frac{|J|_0}{n}}+\frac{1}{T\sqrt{p}}\right)\Delta_F + O_P\left(\frac{1}{n}+\frac{1}{\sqrt{npT}}\right)\sqrt{\frac{\log p}{n}}+ O_P(\Delta_{ud}) \\
&\qquad +\max_{m, k\leq p}\left|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \tilde  \Lambda_t'\tilde   U_{it} \tilde U_{is,m} \widehat h_{sk} \right| \\ 
&\qquad + \max_{m, k\leq p}\left|\frac{1}{nT} \sum_{v=1}^p\sum_{i=1}^n  \sum_{s=1}^T \frac{1}{pT}    \sum_{t=1}^T \frac{1}{n}\widehat F'\tilde U_{t, v}  \tilde   U_{it,v} \tilde U_{is,m} \widehat h_{sk} \right| \\
&\leq^{(i) (ii)} O_P\left(\Delta_F+\sqrt{\frac{\log p}{n}}+\sqrt{\frac{|J|_0}{n}}+\frac{1}{T\sqrt{p}}\right)\Delta_F + O_P\left(\frac{1}{n}+\frac{1}{\sqrt{npT}}\right)\sqrt{\frac{\log p}{n}}+ O_P(\Delta_{ud}) \\
&\qquad +O_P\left(\frac{1}{pT}+\sqrt{\frac{\log p}{npT^2}}+ \left(\sqrt{\frac{\log (pT)}{npT}}\right)\left( \sqrt{\frac{|J|_0}{n}}+\sqrt{\frac{\log(pT)}{n}}\right) +\sqrt{\frac{\log p }{npT}}+\frac{\sqrt{\log p}}{n}\right) \\
&= O_P\left(\frac{1}{pT}+\sqrt{\frac{\log p}{npT}}+\frac{\sqrt{\log p}}{n}+\frac{\sqrt{|J|_0}}{nT}+\sqrt{\frac{|J|_0}{npT}}\right)+ O_P(\Delta_{ud})
\end{align*}
where $\max_{m, k\leq p}|\frac{1}{nT} \sum_{i=1}^n  \sum_{s=1}^T  (\widehat  f_i -H'\tilde f_i) \tilde U_{is,m} \tilde h_{sk} |=O_P(\Delta_{ud})$ is given in Lemma \ref{lc.7}.

   $\blacksquare$
   
   
   
 




}   
     
  
\footnotesize\linespread{.8}
\bibliographystyle{ims}
\bibliography{liaoBib_newest}
  

\clearpage
\begin{figure}
	\centering
		\includegraphics[width = \textwidth]{Fig_IV_RMSE.pdf}
	\caption{\footnotesize This figure shows the simulation RMSE of each of the estimators described in the text for estimating the coefficient of interest in an IV partial factor model. RMSE (truncated at 0.1) is shown in the vertical axis.  The horizontal axes give the fraction of the explanatory power in an infeasible regression of $Y$ on factors and factor residuals, ``\%Y,'' and the fraction of the explanatory power in an infeasible regression of $D$ on factors and factor residuals, ``\%D,'' where the infeasible regressions are described in the text.}
	\label{Fig: IV RMSE}
\end{figure}


\clearpage
\begin{figure}
	\centering
		\includegraphics[width = \textwidth]{Fig_IV_SIZE.pdf}
	\caption{\footnotesize This figure shows the simulation size of 5\% level tests based on each of the estimators described in the text for the IV partial factor model. Size (truncated at 0.3) is shown in the vertical axis.  The horizontal axes give the fraction of the explanatory power in an infeasible regression of $Y$ on factors and factor residuals, ``\%Y,'' and the fraction of the explanatory power in an infeasible regression of $D$ on factors and factor residuals, ``\%D,'' where the infeasible regressions are described in the text.}
	\label{Fig: IV SIZE}
\end{figure}

\clearpage


\begin{figure}
	\centering
		\includegraphics[width = \textwidth]{Fig_IV_BOOT.pdf}
	\caption{\footnotesize This figure shows the simulation size of 5\% level tests based on the factor-lasso estimator in the IV partial factor model and the asymptotic Gaussian approximation, the k-step bootstrap, and a score based bootstrap.  Size is shown in the vertical axis.  The horizontal axes give the fraction of the explanatory power in an infeasible regression of $Y$ on factors and factor residuals, ``\%Y,'' and the fraction of the explanatory power in an infeasible regression of $D$ on factors and factor residuals, ``\%D,'' where the infeasible regressions are described in the text.}
	\label{Fig: IV BOOT}
\end{figure}


\clearpage

\setlength{\tabcolsep}{10pt}
\begin{table}[h!]
\footnotesize{\caption{\footnotesize{Estimates of the First-Stage Relationship between Settler Mortality and Protection from Expropriation}}\label{AJRTableResults}
\begin{center}
\begin{tabular}{lccc}
\hline
  & $\widehat\pi$  & Estimated s.e. & Bootstrap C.I.  \\
\cline{2-4}
Latitude             & -0.549  &  (0.166)  & [-0.851,-0.246] \\
All Controls         & -0.218  &  (0.168)  & [-0.778,0.341]  \\
Double Selection     & -0.364  &  (0.178)  & [-0.885,0.158]  \\
Factor               & -0.475  &  (0.173)  & [-0.880,-0.070] \\
Factor-Lasso         & -0.353  &  (0.183)  & [-0.708.0.002]  \\
\hline
\end{tabular}
\end{center}}
\begin{flushleft}
\footnotesize{This table presents estimates of the coefficient on the instrument (Settler Mortality) in the first-stage regression of the endogenous variable from the \cite{acemoglu:colonial} example (Protection from Expropriation) on the instrument and geographic controls using different methods.  The row labeled ``Latitude'' uses the single variable distance from the equator to control for geography.  ``All Controls'' uses all 20 geographic controls without dimension reduction.  ``Double Selection'' uses the approach of \cite{belloni2014inference} to select important controls from among the 20 potential geography measures.  ``Factor'' reduces dimension through positing a conventional factor model.  ``Factor-Lasso'' makes use of the approach developed in this paper.  Point estimates from each method are provided in the column ``$\widehat\pi$'' and the associated estimated asymptotic standard errors are given in ``Estimated s.e.''.  The k-step bootstrap 95\% confidence interval is reported in ``Bootstrap C.I.''.}
\end{flushleft}
\end{table}




\end{document}

