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\newtheorem*{assumext_4.2}{Assumption 4.2}
\newtheorem*{assumext_4.3}{Assumption 4.3}
\newtheorem*{assumext_4.4}{Assumption 4.4}
\newtheorem*{assumext_4.5}{Assumption 4.5}

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\begin{document}
%\large
\title{Supplement to ``Nonlinear Panel Data Models with Distribution-Free
Correlated Random Effects"
%\thanks{Helpful comments by Arthur Lewbel,
%and Matthew Shum are acknowledged. We also thank the editors, and two anonymous referees for useful
%comments. The authors are solely
%responsible for any remaining errors.   Yu-Chin Hsu gratefully acknowledges research support from the Ministry of Science and Technology of Taiwan (MOST107-2410-H-001-034-MY3) and Career Development Award of Academia Sinica, Taiwan.}
}
\author{Yu-Chin Hsu\thanks{Institute of Economics, Academia Sinica; Department of Finance, National Central
University; Department of Economics, National Chengchi University.
E-mail:
ychsu@econ.sinica.edu.tw. } \and Ji-Liang Shiu\thanks{Institute for
Economic and Social Research, Jinan University. E-mail: jishiu.econ@gmail.com.} \\
}
\date{\today}
\maketitle

%\begin{abstract}
%\noindent
%This supplement includes proofs of some results that are not proved in the main text, additional results, and Monte-Carlo simulations.
%\end{abstract}

\appendix
\setcounter{section}{1}
This supplement complements Appendix A  of the main paper, and we use a sequential numbering that follows Appendix A.
In Appendix B, we provide primitive conditions for Assumption 2.6 and work through the binary choice model in Eq.\ (6) of the main text. In Appendix C, we provide results for the sieve maximum likelihood estimators proposed in the main paper.
In Appendix D, we provide a consistent estimator for $\Omega_{di}$ that is needed for the Hausman-type test proposed in Section 4.1.  We present the Monte Carlo simulation results in Appendix E.


\section{Discussion on Assumption 2.6}

\subsection{Primitive Conditions for Assumption 2.6}
The internally consistent semiparametric density of observable variables has the following form:
\begin{align}\label{parametric111}
f(y|x,\overline{w};\alpha )
&=\int_{\mathcal{C}}f_{Y|X,C}(y|x,c;\theta)f_{C|\overline{W}}(c|\overline{w};\alpha)dc,
\end{align}
where  $f_{C|\overline{W}}(c|\overline{w};\alpha)=\frac{1}{2\pi c_{\alpha}(\overline{w})}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)d\xi$ with
\begin{align}\label{illustration000}
\phi _{v;\alpha }(\xi)=\frac{
-\lambda_{1}\int_{\mathcal{Y}\times\mathcal{X}\times\overline{\mathcal{W}}}
e^{-i\xi\sum_{k=1}\lambda_{k}\overline{w}_{k}}
f_{Y|X,\overline{W}}(y|x,\overline{w})\Omega(y,x,\overline{w}_{-1})dydxd\overline{w}
}{\int_{\mathcal{Y}\times\mathcal{X}\times\mathcal{C}}e^{-i\xi
c}f_{Y|X,C}(y|x,c;\theta)\Omega(y,x)dydxdc}.
\end{align}
To illustrate, we consider $\alpha=(\theta_{1},\lambda_{1},\lambda_{2})$, i.e., $\theta$ is a scalar and  $\lambda$ is a $2\times 1$ vector. Similar lines of argument show that the same conclusion holds for
general cases. The scale factor can be written as follows:
\begin{align}\label{scale111}
c_{\alpha}(\overline{w})\equiv\int_{\mathcal{C}}
f_{v;\alpha}(c-\overline{w}\lambda)dc&=\frac{1}{2\pi }\int_{\mathcal{C}}\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)d\xi dc\\
&=\int_{-\infty}^{\infty}\left(\frac{1}{2\pi }\int_{\mathcal{C}}e^{-i\xi c} dc\right)e^{i\xi \overline{w}\lambda}\phi_{v;\alpha }(\xi)d\xi.\notag
\end{align}
The gradient of the log likelihood at the true value $\alpha_{0}$ in this simple case is
\begin{align}\label{gradient111}
&\frac{\partial}{\partial\alpha}\log f(Y|X,\overline{W};\alpha_{0} )\\
&=\frac{1}{f(Y|X,\overline{W};\alpha )}\Big(\frac{\partial}{\partial\theta_{1}}f(Y|X,\overline{W};\alpha_{0} ),\frac{\partial}{\partial\lambda_{1}}f(Y|X,\overline{W};\alpha_{0} ),\frac{\partial}{\partial\lambda_{2}}f(Y|X,\overline{W};\alpha_{0} )\Big)',\notag
\end{align}
with
\begingroup
\allowdisplaybreaks
\begin{align}
\frac{\partial}{\partial\theta_{1}}f(Y|X,\overline{W};\alpha_{0})
&=\int_{\mathcal{C}}\frac{\partial}{\partial\theta_{1}}f_{Y|X,C}(y|x,c;\theta_{0})
f_{C|\overline{W}}(c|\overline{w};\alpha_{0})dc\label{gradient111_partial}\\
&\qquad+\int_{\mathcal{C}}f_{Y|X,C}(y|x,c;\theta_{0})\underbrace{\frac{\partial}{
\partial\theta_{1}}f_{C|\overline{W}}(c|\overline{w};\alpha_{0})}_{
\parbox[t]{2.7cm}{\footnotesize\addtolength{\baselineskip}{-8pt}related to $f(y|x,\overline{w})$
and $f_{Y|X,C}(y|x,c;\theta)$}}dc,\notag\\
\frac{\partial}{\partial\lambda_{1}}f(Y|X,\overline{W};\alpha_{0})
&=\int_{\mathcal{C}}f_{Y|X,C}(y|x,c;\theta_{0})\frac{\partial}{
\partial\lambda_{1}}f_{C|\overline{W}}(c|\overline{w};\alpha_{0})dc,\\
\frac{\partial}{\partial\lambda_{2}}f(Y|X,\overline{W};\alpha)
&=\int_{\mathcal{C}}f_{Y|X,C}(y|x,c;\theta_{0})\frac{\partial}{
\partial\lambda_{2}}f_{C|\overline{W}}(c|\overline{w};\alpha_{0})dc.
\end{align}
\endgroup
Compared  with the conventional CRE approach that
$V\sim N(0,1)$, and
$f_{C|\overline{W}}(c|\overline{w};\lambda)=\phi(c-\overline{w}\lambda)$
where $\phi(\cdot)$ denotes the density function of the standard normal, the derivative term
$\frac{\partial}{
\partial\theta_{1}}f_{C|\overline{W}}(c|\overline{w};\alpha)$ in Eq.\ (\ref{gradient111_partial}) in our model is generally nonzero and related to the sample density and the proposed model. This implies that the term $\int_{\mathcal{C}}f_{Y|X,C}(y|x,c;\theta_{0})\frac{\partial}{
\partial\theta_{1}}f_{C|\overline{W}}(c|\overline{w};\alpha_{0})$ in our model is used to accommodate the modeling of the panel data structure
$f_{Y|X,C}(y|x,c;\theta)$ and the distribution of the unobserved heterogeneity.

To provide a more primitive condition on the densities, we need to write out the derivatives of the composite distribution of the unobserved heterogeneity $\frac{\partial}{
\partial\theta_{1}}f_{C|\overline{W};\alpha}$, $\frac{\partial}{\partial\lambda_{1}}f_{C|\overline{W};\alpha}$, and $\frac{\partial}{\partial\lambda_{2}}f_{C|\overline{W};\alpha}$ at the true value $\alpha_{0}$, where
$f_{C|\overline{W};\alpha}\equiv f_{C|\overline{W}}(c|\overline{w};\alpha)$.
We have
\begin{align*}%\label{ill_D_theta0}
\frac{\partial \phi_{v;\alpha }(\xi)}{
\partial\theta_{1}}&=\phi_{v;\alpha }(\xi)\times\frac{-\int_{\mathcal{Y}\times\mathcal{X}\times\mathcal{C}}e^{-i\xi
c}\frac{\partial f_{Y|X,C}(y|x,c;\theta)}{
\partial\theta_{1}}\Omega(y,x)dydxdc
}{\int_{\mathcal{Y}\times\mathcal{X}\times\mathcal{C}}e^{-i\xi
c}f_{Y|X,C}(y|x,c;\theta)\Omega(y,x)dydxdc}\equiv \phi_{v;\alpha }(\xi)\gamma_{\theta_{1}}(\xi;\theta),\\
\frac{\partial \phi_{v;\alpha }(\xi)}{
\partial\lambda_{1}}
&=\frac{1}{\lambda_{1}}\phi_{v;\alpha }(\xi)+\frac{
i\lambda_{1}\xi\int_{\mathcal{Y}\times\mathcal{X}\times\overline{\mathcal{W}}}
e^{-i\xi\sum_{k=1}\lambda_{k}\overline{w}_{k}}\overline{w}_{1}
f_{Y|X,\overline{W}}(y|x,\overline{w})\Omega(y,x,\overline{w}_{-1})dydxd\overline{w}
}{\int_{\mathcal{Y}\times\mathcal{X}\times\mathcal{C}}e^{-i\xi
c}f_{Y|X,C}(y|x,c;\theta)\Omega(y,x)dydxdc}\\
&\equiv\frac{1}{\lambda_{1}}\phi_{v;\alpha }(\xi)+i\lambda_{1}\xi \gamma_{\lambda_{1}}(\xi;\alpha),\\
\frac{\partial \phi_{v;\alpha }(\xi)}{
\partial\lambda_{2}}
&=\frac{
i\lambda_{1}\xi\int_{\mathcal{Y}\times\mathcal{X}\times\overline{\mathcal{W}}}
e^{-i\xi\sum_{k=1}\lambda_{k}\overline{w}_{k}}\overline{w}_{2}
f_{Y|X,\overline{W}}(y|x,\overline{w})\Omega(y,x,\overline{w}_{-1})dydxd\overline{w}
}{\int_{\mathcal{Y}\times\mathcal{X}\times\mathcal{C}}e^{-i\xi
c}f_{Y|X,C}(y|x,c;\theta)\Omega(y,x)dydxdc}\\
&\equiv i\lambda_{1}\xi \gamma_{\lambda_{2}}(\xi;\alpha).
\end{align*}
We note that these derivatives are mainly stated in terms of the proposed parametric nonlinear panel
data model $f_{Y|X,C}(y|x,c;\theta)$ and the density of observables
$f_{Y|X,\overline{W}}(y|x,\overline{w})$. These derivatives evaluated at the true parameter $\alpha_{0}$ are
\begin{align*}%\label{ill_D_theta0}
\frac{\partial \phi_{v;\alpha_{0} }(\xi)}{
\partial\theta_{1}}&= \phi_{v}(\xi)\gamma_{\theta_{1}}(\xi;\theta_{0}),\\
\frac{\partial \phi_{v;\alpha_{0} }(\xi)}{
\partial\lambda_{1}}
&=\frac{1}{\lambda_{1}}\phi_{v }(\xi)+i\lambda_{01}\xi \gamma_{\lambda_{1}}(\xi;\alpha_{0}),\\
\frac{\partial \phi_{v;\alpha_{0} }(\xi)}{
\partial\lambda_{2}}
&= i\lambda_{01}\xi \gamma_{\lambda_{2}}(\xi;\alpha).
\end{align*}
It follows that
\begingroup
\allowdisplaybreaks
\begin{align}\label{ill_D000}
&\frac{\partial}{
\partial\theta_{1}}f_{C|\overline{W};\alpha}\\
&=\frac{-1}{2\pi c_{\alpha}(\overline{w})^{2}}
\frac{\partial c_{\alpha}(\overline{w})}{
\partial\theta_{1}}\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)d\xi+
\frac{1}{2\pi c_{\alpha}(\overline{w})}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \frac{\partial \phi_{v;\alpha }(\xi)}{
\partial\theta_{1}}d\xi \notag\\
&=\frac{-1}{2\pi c_{\alpha}(\overline{w})^{2}}
\left(
\int_{-\infty}^{\infty}(\frac{1}{2\pi }\int_{\mathcal{C}}e^{-i\xi c} dc)e^{i\xi \overline{w}\lambda}\frac{\partial \phi_{v;\alpha }(\xi)}{
\partial\theta_{1}}d\xi
\right)
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)d\xi\notag\\
&\quad+
\frac{1}{2\pi c_{\alpha}(\overline{w})}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \frac{\partial \phi_{v;\alpha }(\xi)}{
\partial\theta_{1}}d\xi\notag\\
&=\frac{-1}{2\pi c_{\alpha}(\overline{w})^{2}}
\left(
\int_{-\infty}^{\infty}(\frac{1}{2\pi }\int_{\mathcal{C}}e^{-i\xi c} dc)e^{i\xi \overline{w}\lambda}\phi_{v;\alpha }(\xi)\gamma_{\theta_{1}}(\xi;\theta)d\xi
\right)
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)d\xi\notag\\
&\quad+
\frac{1}{2\pi c_{\alpha}(\overline{w})}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)\gamma_{\theta_{1}}(\xi;\theta)d\xi.\notag\\%%11111111111111
&\frac{\partial}{
\partial\lambda_{1}}f_{C|\overline{W};\alpha}\\
&=\frac{-1}{2\pi c_{\alpha}(\overline{w})^{2}}
\frac{\partial c_{\alpha}(\overline{w})}{
\partial\lambda_{1}}\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)d\xi+
\frac{1}{2\pi c_{\alpha}(\overline{w})}
\int_{-\infty}^{\infty}i\xi \overline{w}_{1}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)d\xi\notag\\
&\quad+
\frac{1}{2\pi c_{\alpha}(\overline{w})}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \frac{\partial \phi_{v;\alpha }(\xi)}{
\partial\lambda_{1}}d\xi\notag\\
&=\frac{-1}{2\pi c_{\alpha}(\overline{w})^{2}}
\left(
\int_{-\infty}^{\infty}(\frac{1}{2\pi }\int_{\mathcal{C}}e^{-i\xi c} dc)e^{i\xi \overline{w}\lambda}
(i\xi \overline{w}_{1} \phi_{v;\alpha }(\xi)+
\frac{\partial \phi_{v;\alpha }(\xi)}{
\partial\lambda_{1}}
)
d\xi
\right)
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)d\xi.\notag\\
&\quad+
\frac{1}{2\pi c_{\alpha}(\overline{w})}
\int_{-\infty}^{\infty}i\xi \overline{w}_{1}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)d\xi+
\frac{1}{2\pi c_{\alpha}(\overline{w})}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \frac{\partial \phi_{v;\alpha }(\xi)}{
\partial\lambda_{1}}d\xi. \notag\\%%22222
&\frac{\partial}{
\partial\lambda_{2}}f_{C|\overline{W};\alpha}\\
&=\frac{-1}{2\pi c_{\alpha}(\overline{w})^{2}}
\left(
\int_{-\infty}^{\infty}(\frac{1}{2\pi }\int_{\mathcal{C}}e^{-i\xi c} dc)e^{i\xi \overline{w}\lambda}
(i\xi \overline{w}_{2} \phi_{v;\alpha }(\xi)+
\frac{\partial \phi_{v;\alpha }(\xi)}{
\partial\lambda_{2}}
)
d\xi
\right)
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)d\xi\notag\\
&\quad+
\frac{1}{2\pi c_{\alpha}(\overline{w})}
\int_{-\infty}^{\infty}i\xi \overline{w}_{2}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)d\xi+
\frac{1}{2\pi c_{\alpha}(\overline{w})}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \frac{\partial \phi_{v;\alpha }(\xi)}{
\partial\lambda_{2}}d\xi.\notag
\end{align}
\endgroup
%% consider these derivatives at the true value \alpha_{0}
The derivatives at the true value $\alpha_{0}$ can be stated as follows
\begingroup
\allowdisplaybreaks
\begin{align}
&\frac{\partial}{
\partial\theta_{1}}f_{C|\overline{W};\alpha_{0}}=\frac{-1}{2\pi }
\left(
\int_{-\infty}^{\infty}(\frac{1}{2\pi }\int_{\mathcal{C}}e^{-i\xi c} dc)e^{i\xi \overline{w}\lambda_{0}}\phi_{v}(\xi)\gamma_{\theta_{01}}(\xi;\theta_{0})d\xi
\right)
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})} \phi_{v}(\xi)d\xi\label{ill_D111}\\
&\qquad\qquad\qquad+
\frac{1}{2\pi }
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})} \phi_{v }(\xi)\gamma_{\theta_{1}}(\xi;\theta_{0})d\xi,\notag\\%%11111111111111
&\frac{\partial}{
\partial\lambda_{1}}f_{C|\overline{W};\alpha_{0}}\\
&=\frac{-1}{2\pi }
\left(
\int_{-\infty}^{\infty}(\frac{1}{2\pi }\int_{\mathcal{C}}e^{-i\xi c} dc)e^{i\xi \overline{w}\lambda_{0}}
(i\xi \overline{w}_{01} \phi_{v }(\xi)+
\frac{\partial \phi_{v;\alpha_{0} }(\xi)}{
\partial\lambda_{1}}
)
d\xi
\right)
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})} \phi_{v }(\xi)d\xi\notag\\
&\quad+
\frac{1}{2\pi }
\int_{-\infty}^{\infty}i\xi \overline{w}_{01}e^{-i\xi (c-\overline{w}\lambda_{0})} \phi_{v }(\xi)d\xi+
\frac{1}{2\pi }
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})} \frac{\partial \phi_{v;\alpha_{0} }(\xi)}{
\partial\lambda_{1}}d\xi,\notag\\%%22222
&\frac{\partial}{
\partial\lambda_{2}}f_{C|\overline{W};\alpha_{0}}\label{ill_D333}\\
&=\frac{-1}{2\pi }
\left(
\int_{-\infty}^{\infty}(\frac{1}{2\pi }\int_{\mathcal{C}}e^{-i\xi c} dc)e^{i\xi \overline{w}\lambda_{0}}
(i\xi \overline{w}_{02} \phi_{v}(\xi)+
\frac{\partial \phi_{v;\alpha_{0} }(\xi)}{
\partial\lambda_{2}}
)
d\xi
\right)
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})} \phi_{v }(\xi)d\xi\notag\\
&\quad+
\frac{1}{2\pi }
\int_{-\infty}^{\infty}i\xi \overline{w}_{02}e^{-i\xi (c-\overline{w}\lambda_{0})} \phi_{v }(\xi)d\xi+
\frac{1}{2\pi }
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})} \frac{\partial \phi_{v;\alpha_{0} }(\xi)}{
\partial\lambda_{2}}d\xi,\notag
\end{align}
\endgroup
where $c_{\alpha_{0}}(\overline{w})=1$, and $\phi_{v;\alpha_{0} }(\xi)=\phi_{v }(\xi)$.
%%the gradient evaluated at the true value \alpha_{0}

When $\mathcal{C}=\mathbb{R}$, or the domain of $C$ is a real line, using the relationship in Eq.\ (A.11) we obtain
\begin{align}\label{illustration111}
f_{C|\overline{W}}(c|\overline{w};\alpha)
&=\frac{1}{2\pi \phi_{v;\alpha }(0)}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)d\xi.
\end{align}
Applying the relations $\frac{1}{2\pi}\int_{\mathcal{C}}e^{-i\xi
c}dc=\frac{1}{2\pi}\int_{-\infty}^{\infty} e^{-i\xi
c}dc=\delta(\xi)$ and $\int
f(\xi)\delta(\xi)dt=f(0)$ into Eqs.\ (\ref{ill_D111})-(\ref{ill_D333}) yields
\begingroup
\allowdisplaybreaks
\begin{align}
&\frac{\partial}{
\partial\theta_{1}}f_{C|\overline{W};\alpha_{0}}=\frac{-1}{2\pi }
\phi_{v}(0)\gamma_{\theta_{01}}(0;\theta_{0})
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})} \phi_{v}(\xi)d\xi \\
&\qquad\qquad\quad~~+
\frac{1}{2\pi }
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})} \phi_{v }(\xi)\gamma_{\theta_{1}}(\xi;\theta_{0})d\xi,\notag\\
&\qquad\qquad\quad=\frac{1}{2\pi }
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})} \phi_{v }(\xi)\left(\gamma_{\theta_{1}}(\xi;\theta_{0})-\gamma_{\theta_{01}}(0;\theta_{0})\right)d\xi
,\notag\\%%11111111111111
&\frac{\partial}{
\partial\lambda_{1}}f_{C|\overline{W};\alpha_{0}}=\frac{-1}{2\pi }
\frac{1}{\lambda_{1}}\phi_{v }(0)
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})} \phi_{v }(\xi)d\xi
+
\frac{1}{2\pi }
\int_{-\infty}^{\infty}i\xi \overline{w}_{01}e^{-i\xi (c-\overline{w}\lambda_{0})} \phi_{v }(\xi)d\xi\notag\\&
\qquad\qquad\qquad+
\frac{1}{2\pi }
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})} \left(\frac{1}{\lambda_{1}}\phi_{v }(\xi)+i\lambda_{01}\xi \gamma_{\lambda_{1}}(\xi;\alpha_{0})\right)d\xi\notag\\&
\qquad\qquad\quad=
\frac{1}{2\pi }
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})} \left(i\xi \overline{w}_{01}+i\lambda_{01}\xi \gamma_{\lambda_{1}}(\xi;\alpha_{0})\right)d\xi
,\notag\\%%22222
&\frac{\partial}{
\partial\lambda_{2}}f_{C|\overline{W};\alpha_{0}}=
\frac{1}{2\pi }
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})} \left(i\xi \overline{w}_{02}\phi_{v }(\xi)+i\lambda_{01}\xi \gamma_{\lambda_{2}}(\xi;\alpha_{0})\right)d\xi.\notag
\end{align}
\endgroup
%%the gradient is expressed in terms of the proposed model and sample density
The gradient of the log likelihood in Eq.\ (\ref{gradient111}) is therefore
expressed in terms of the population panel
data model $f_{Y|X,C}(y|x,c;\theta_{0})$ and the density of observables
$f_{Y|X,\overline{W}}(y|x,\overline{w})$.
The negative definiteness of the information matrix
in Assumption 2.6 is equivalent to the positive definiteness of the outer product of the gradient of the log likelihood.
%%sufficient condition for the positiveness of the outer product of the gradient
%% by using Theorem 6.1.10
The property of the positive definiteness
can be imposed by choosing the outer product  $\text{E}\Big[\frac{\partial}{\partial\alpha}\log
f(Y|X,\overline{W};\alpha_{0})
\cdot\frac{\partial}{\partial\alpha}\log
f(Y|X,\overline{W};\alpha_{0})'\Big|X=x,\overline{W}=\overline{w}\Big]$ as a strictly diagonally dominant matrix. To provide a more transparent condition,
we define the vector of the derivatives of the average likelihood as
\begin{align}\label{deri_average111}
Df(y|x,\overline{w};\alpha_{0} )=\left(
\begin{tabular}{c}
  % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
$\int_{\mathcal{C}}\frac{\partial}{\partial\theta_{1}}f_{Y|X,C}(y|x,c;\theta_{0})
f_{C|\overline{W}}(c|\overline{w};\alpha_{0})dc$ \\
$\int_{\mathcal{C}}f_{Y|X,C}(y|x,c;\theta_{0})
\frac{\partial}{
\partial\theta_{1}}f_{C|\overline{W}}(c|\overline{w};\alpha_{0})dc$\\
$\int_{\mathcal{C}}f_{Y|X,C}(y|x,c;\theta_{0})\frac{\partial}{
\partial\lambda_{1}}f_{C|\overline{W}}(c|\overline{w};\alpha_{0})dc$ \\
$\int_{\mathcal{C}}f_{Y|X,C}(y|x,c;\theta_{0})\frac{\partial}{
\partial\lambda_{2}}f_{C|\overline{W}}(c|\overline{w};\alpha_{0})dc$ \\
\end{tabular}
\right).
\end{align}
The detailed formulae of $\frac{\partial}{
\partial\theta_{1}}f_{C|\overline{W}}(c|\overline{w};\alpha_{0})$, $\frac{\partial}{
\partial\lambda_{1}}f_{C|\overline{W}}(c|\overline{w};\alpha_{0})$, and $\frac{\partial}{
\partial\lambda_{2}}f_{C|\overline{W}}(c|\overline{w};\alpha_{0})$ are in
Eqs. (\ref{ill_D111})-(\ref{ill_D333}) respectively.


%\begin{defT}\label{definition222}
%A square matrix $A=[a_{i j}]$ is strictly diagonally dominant if
%$$
%\left|a_{i i}\right| > \sum_{j \neq i}\left|a_{i j}\right| \quad \text { for all } i.
%$$
%\end{defT}
%statement of Assumption
\begin{assum}\label{assu:strictly_diagonal} (Strictly Diagonally Dominant)\\
Assume that every element of $Df(y|x,\overline{w};\alpha_{0} )$ is nonzero and the outer product of the derivatives of the average likelihood
\begin{align}\label{strictly_diagonal}
\text{E}\Big[Df(y|x,\overline{w};\alpha_{0} )
\cdot Df(y|x,\overline{w};\alpha_{0} )'\Big|X=x,\overline{W}=\overline{w}\Big]
\end{align}
is strictly diagonally dominant.\footnote{A square matrix $A=[a_{i j}]$ is strictly diagonally dominant if
$\left|a_{i i}\right| > \sum_{j \neq i}\left|a_{i j}\right|$  for all  $i$. }
\end{assum}
%\begin{assumext_2.6_prime}(Strictly Diagonally Dominant)\\
%Assume that every element of $Df(y|x,\overline{w};\alpha_{0} )$ is non-zero and the outer product of the derivatives of the average likelihood
%\begin{align}\label{strictly_diagonal}
%\text{E}\Big[Df(y|x,\overline{w};\alpha_{0} )
%\cdot Df(y|x,\overline{w};\alpha_{0} )'\Big|X=x,\overline{W}=\overline{w}\Big]
%\end{align}
%is strictly diagonally dominant.
%\end{assumext_2.6_prime}
%%the discussion the Assumption 2.6'
Assumption \ref{assu:strictly_diagonal} implies that the expectation of the squared term of each derivative of the average likelihood
is larger than the sum of the expectation of the
magnitudes of the products of the derivative and other derivatives of the average likelihood. It is straightforward to verify that Assumption \ref{assu:strictly_diagonal} implies that every element of the gradient of the log
likelihood $\frac{\partial}{\partial\alpha}\log
f(Y|X,\overline{W};\alpha_{0})$ is nonzero and its
outer product is strictly diagonally dominant.
%%sufficient condition for the positiveness
Because every diagonal element of the matrix $\text{E}\Big[\frac{\partial}{\partial\alpha}\log
f(Y|X,\overline{W};\alpha_{0})
\cdot\frac{\partial}{\partial\alpha}\log
f(Y|X,\overline{W};\alpha_{0})'\Big|X=x,\overline{W}=\overline{w}\Big]$ is positive,
symmetric, and strictly diagonally dominant, the matrix is positive definite.\footnote{%
The result
can be found as Corollary 6.1.10. in \cite{horn1985matrix}.}
\par

\subsection{Primitive Conditions for the Binary Choice Model}
To better elucidate
these sufficient conditions, we work through the binary choice model in Eq.\ (6) by
assuming that $\varepsilon_{t}$ is symmetrically distributed, and $m(X_{t},C;\theta)=\theta X_{t}+C$.
It follows that
\begin{align*}
f_{Y_{t}|X_{t},C}(y_{t}|x_{t},c;\theta)=\left(
1-F_{\varepsilon _{t}}\left( \theta X_{t}+C
\right) \right) ^{1-y_{t}}F_{\varepsilon _{t}}\left(
\theta X_{t}+C \right) ^{y_{t}}.
\end{align*}
Denote $\vec{1}=(1_1,\ldots, 1_T )$, and suppose that the conditional density function
$f(x,\overline{w}_{-1}|y=\vec{1})\neq0$.
We choose a weighting function concentrating at $y=\vec{1}$, i.e.,
$\Omega(y=\vec{1},x,\overline{w}_{-1})=f(x,\overline{w}_{-1}|y=\vec{1})$ for all $(x,\overline{w}_{-1})$ and zero elsewhere. It follows that $\Omega(y=\vec{1},x)=f(x|y=\vec{1})$ for all $\overline{w}_{-1}$ and zero elsewhere.

The denominator term of $\phi _{v;\alpha }(\xi)$ in Eq.\ (\ref{illustration000}) becomes
\begin{align*}
\int_{\mathcal{Y}\times\mathcal{X}\times\mathcal{C}}e^{-i\xi
c}f_{Y|X,C}(y|x,c;\theta)\Omega(y,x)dydxdc
&=\int_{\mathcal{X}\times\mathcal{C}}e^{-i\xi
c}\prod_{t=1}^{T}F_{\varepsilon _{t}}\left(
\theta x_{t}+c \right)f(x|y=\vec{1})dxdc\\
&=\text{E}\Big[\int_{\mathcal{C}}e^{-i\xi
c}\prod_{t=1}^{T}F_{\varepsilon _{t}}\left(
\theta X_{t}+c \right)dc\Big|Y=\vec{1}\Big].
\end{align*}
Similarly, the numerator term of $\phi _{v;\alpha }(\xi)$ becomes
\begin{align*}
&-\lambda_{1}\int_{\mathcal{Y}\times\mathcal{X}\times\overline{\mathcal{W}}}
e^{-i\xi\sum_{k=1}\lambda_{k}\overline{w}_{k}}
f_{Y|X,\overline{W}}(y|x,\overline{w})\Omega(y,x,\overline{w}_{-1})dydxd\overline{w}\\
=&-
\lambda_{1}\int_{\mathcal{X}\times\overline{\mathcal{W}}_{-1}}\left(\int_{\overline{\mathcal{W}}_{1}}
e^{-i\xi\sum_{k=1}\lambda_{k}\overline{w}_{k}}
f_{Y|X,\overline{W}}(y=\vec{1}|x,\overline{w}_{1},\overline{w}_{-1})d\overline{w}_{1}\right)
f(x,\overline{w}_{-1}|y=\vec{1})dxd\overline{w}_{-1}\\
=&-\lambda_{1}\text{E}\Big[\int_{\overline{\mathcal{W}}_{1}}
e^{-i\xi\left(\lambda_{1}\overline{w}_{1}+\sum_{k=2}\lambda_{k}\overline{W}_{k}\right)}
f_{Y|X,\overline{W}}(Y|X,\overline{w}_{1},\overline{W}_{-1})d\overline{w}_{1}\Big|Y=\vec{1}\Big].
\end{align*}
Combining the results yields
\begin{align*}
\phi _{v;\alpha }(\xi)=\frac{-\lambda_{1}\text{E}\Big[\int_{\overline{\mathcal{W}}_{1}}
e^{-i\xi\left(\lambda_{1}\overline{w}_{1}+\sum_{k=2}\lambda_{k}\overline{W}_{k}\right)}
f_{Y|X,\overline{W}}(Y|X,\overline{w}_{1},\overline{W}_{-1})d\overline{w}_{1}\Big|Y=\vec{1}\Big]
}{\text{E}\Big[\int_{\mathcal{C}}e^{-i\xi
c}\prod_{t=1}^{T}F_{\varepsilon _{t}}\left(
\theta X_{t}+c \right)dc\Big|Y=\vec{1}\Big]}\equiv\frac{-\lambda_{1}D(\xi;\lambda)}{M(\xi;\theta )},
\end{align*}%
where $D(\xi;\lambda)$ is related to the population density and $M(\xi;\theta )$ is related to the panel data model.
Assume that the support $\mathcal{C}$ is compact and the domain of $C$ is a real line. Thus,
we have
\begin{align*}
c_{\alpha}(\overline{w})&=\phi_{v;\alpha }(0)=\frac{-\lambda_{1}\text{E}\Big[
\int_{\overline{\mathcal{W}}_{1}}
f_{Y|X,\overline{W}}(Y|X,\overline{w}_{1},\overline{W}_{-1})d\overline{w}_{1}\Big|Y=\vec{1}\Big]
}{\text{E}\Big[\int_{\mathcal{C}}\prod_{t=1}^{T}F_{\varepsilon _{t}}\left(
\theta X_{t}+c \right)dc\Big|Y=\vec{1}\Big]}\equiv\frac{-\lambda_{1}D(0)}{M(0;\theta )}.
\end{align*}
Therefore,
\begin{align*}
f_{C|\overline{W}}(c|\overline{w};\alpha)
=\frac{1}{2\pi c_{\alpha}(\overline{w})}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)} \phi_{v;\alpha }(\xi)d\xi=\frac{M(0;\theta )}{2\pi D(0)}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda)}\frac{D(\xi;\lambda )
}{M(\xi;\theta )}d\xi.
\end{align*}
We consider $\alpha=(\theta_{1},\lambda_{1},\lambda_{2})$ and denote the derivatives of $D(\xi;\lambda )$ and $M(\xi;\theta )$ with respect to $\lambda$ and $\theta$, respectively, as
\begin{align*}
&\frac{\partial M(\xi;\theta )}{
\partial\theta_{1}}=\text{E}\Big[\int_{\mathcal{C}}e^{-i\xi
c}\sum_{s=1}^{T}\prod_{t=1,t\neq s}^{T}F_{\varepsilon _{t}}\left(
\theta X_{t}+c \right)f_{\varepsilon _{s}}\left(
\theta X_{s}+c \right)X_{s}dc\Big|Y=\vec{1}\Big],\\
&\frac{\partial D(\xi;\lambda )}{
\partial\lambda_{1}}=-\text{E}\Big[\int_{\overline{\mathcal{W}}_{1}}i\xi\overline{w}_{1}
e^{-i\xi\left(\lambda_{1}\overline{w}_{1}+\sum_{k=2}\lambda_{k}\overline{W}_{k}\right)}
f_{Y|X,\overline{W}}(Y|X,\overline{w}_{1},\overline{W}_{-1})d\overline{w}_{1}\Big|Y=\vec{1}\Big],\\
&\frac{\partial D(\xi;\lambda )}{
\partial\lambda_{2}}=-\text{E}\Big[\int_{\overline{\mathcal{W}}_{1}}i\xi\overline{W}_{2}
e^{-i\xi\left(\lambda_{1}\overline{w}_{1}+\sum_{k=2}\lambda_{k}\overline{W}_{k}\right)}
f_{Y|X,\overline{W}}(Y|X,\overline{w}_{1},\overline{W}_{-1})d\overline{w}_{1}\Big|Y=\vec{1}\Big].
\end{align*}
The derivatives of $f_{C|\overline{W}}(c|\overline{w};\alpha)$ with respect to $\theta_{1}$, $\lambda_{1}$, and $\lambda_{2}$ at the true value $\alpha_{0}$ can then be stated as
\begingroup
\allowdisplaybreaks
\begin{align*}
&\frac{\partial}{
\partial\theta_{1}}f_{C|\overline{W};\alpha_{0}}=\frac{\frac{\partial M(0;\theta_{0} )}{
\partial\theta_{1}}}{2\pi D(0)}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})}\frac{D(\xi;\lambda_{0} )
}{M(\xi;\theta_{0} )}d\xi-\frac{M(0;\theta_{0} )}{2\pi D(0)}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})}\frac{D(\xi;\lambda_{0} )
}{M(\xi;\theta_{0} )^{2}}\frac{\partial M(\xi;\theta_{0} )}{
\partial\theta_{1}}d\xi,\\
&\frac{\partial}{
\partial\lambda_{1}}f_{C|\overline{W};\alpha_{0}}=\frac{M(0;\theta_{0} )}{2\pi D(0)}
\int_{-\infty}^{\infty}i\xi\overline{w}_{1}e^{-i\xi (c-\overline{w}\lambda_{0})}\frac{D(\xi;\lambda_{0} )
}{M(\xi;\theta_{0} )}d\xi+\frac{M(0;\theta_{0} )}{2\pi D(0)}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})}\frac{\frac{\partial D(\xi;\lambda _{0} )}{
\partial\lambda_{1}}
}{M(\xi;\theta_{0} )}d\xi,\\
&\frac{\partial}{
\partial\lambda_{2}}f_{C|\overline{W};\alpha_{0}}=\frac{M(0;\theta_{0} )}{2\pi D(0)}
\int_{-\infty}^{\infty}i\xi\overline{w}_{2}e^{-i\xi (c-\overline{w}\lambda_{0})}\frac{D(\xi;\lambda_{0} )
}{M(\xi;\theta_{0} )}d\xi+\frac{M(0;\theta_{0} )}{2\pi D(0)}
\int_{-\infty}^{\infty}e^{-i\xi (c-\overline{w}\lambda_{0})}\frac{\frac{\partial D(\xi;\lambda_{0} )}{
\partial\lambda_{2}}
}{M(\xi;\theta_{0} )}d\xi.
\end{align*}
\endgroup
Let $f_{\varepsilon _{t}}$ be the PDF of $\varepsilon _{t}$. In the binary choice model, we have
\begingroup
\allowdisplaybreaks
\begin{align*}
f_{Y|X,C}(y|x,c;\theta_{0})
&=\prod_{t=1}^{T}\left(
1-F_{\varepsilon _{t}}\left( \theta_{0} X_{t}+C
\right) \right) ^{1-y_{t}}F_{\varepsilon _{t}}\left(
\theta_{0} X_{t}+C \right) ^{y_{t}},\\
\frac{\partial}{\partial\theta_{1}}f_{Y|X,C}(y|x,c;\theta_{0})
&=\sum_{s=1}^{T}\prod_{t=1,t\neq s}^{T}\frac{\partial}{\partial\theta_{1}}f_{Y_{t}|X_{t},C}(y_{t}|x_{t},c;\theta_{0})\\
&=\sum_{s=1}^{T}\prod_{t=1,t\neq s}^{T}
\frac{f_{\varepsilon _{t}}\left(
\theta_{0} X_{t}+C \right)X_{t}\left(
y_{t}-F_{\varepsilon _{t}}\left( \theta_{0} X_{t}+C
\right) \right)}{\left(
1-F_{\varepsilon _{t}}\left( \theta_{0} X_{t}+C
\right) \right) ^{y_{t}}F_{\varepsilon _{t}}\left(
\theta_{0} X_{t}+C \right) ^{1-y_{t}}}.
\end{align*}
\endgroup
With the detailed formulae, we can construct the vector of the derivatives of the average likelihood $Df(y|x,\overline{w};\alpha_{0} )$ defined in Eq.\ (\ref{deri_average111}) for the binary choice model.
Assumption B.1 requires that the outer product of $Df(y|x,\overline{w};\alpha_{0} )$
is strictly diagonally dominant.
Thus, Assumption \ref{assu:strictly_diagonal} is intuitive and provides a sufficient condition for Assumption 2.6.

\section{Sieve Maximum Likelihood Estimators}\label{app_a2}
\subsection{Sufficient Conditions and Proof for Theorem 3.1}
We provide sufficient conditions and proof for Theorem 3.1.
%Let $g=(\theta,\lambda,f_1)$ and $g_0=(\theta_0,\lambda_0,f_V)$. Let $\mathcal{G}$ denote the space of $g$ and $(\mathcal{G},\|\cdot\|_{\mathcal{G}})$ be an infinite dimensional separable complete metric space.  Let
%$Z=(Y,X,\overline{W})$ and
%$\psi(Z,g)=\log f_{Y|X,\overline{W}}(y|x,\overline{w};\theta,\lambda,f_1)$.  We have for all $g$,
%$\psi(Z,g)-\psi(Z,g_0)$ are approximated by $\Delta_\psi (Z,g_0)[g-g_0]$ such that
%$\Delta_\psi (Z,g_0)[g-g_0]$ is linear in $g-g_0$.  Specifically, in our case, we have
%\begin{align*}
%\Delta_\psi (Z,g_0)[g-g_0]
%=&
%\frac{1}{\int_{\mathcal{C}}f_{Y|X,C}(y|x,c;\theta_0)f_{V}(c-\overline{w}\lambda_0)dc}
%\Big\{\int_{\mathcal{C}}\bigtriangledown_\theta f_{Y|X,C}(y|x,c;\theta_0) [\theta-\theta_0]
%f_{V}(c-\overline{w}\lambda_0)dc\\
%&-\int_{\mathcal{C}}f_{Y|X,C}(y|x,c;\theta_0)f'_{V}(c-\overline{w}\lambda_0)
%\overline{w}[\lambda-\lambda_0]dc\\
%&+
%\int_{\mathcal{C}} f_{Y|X,C}(y|x,c;\theta_0) \big(f_{1}(c-\overline{w}\lambda_0)-f_V(c-\overline{w}\lambda_0)\big)dc\Big\}.
%\end{align*}
%For any $v_1, v_2 \in \mathcal{G}-g_0$ , we define their inner product as
%\begin{align*}
%\langle v_1, v_2 \rangle_\psi=\frac{-d E[\Delta_\psi (Z,g_0+\tau v_1)[v_2]]}{d\tau}\Big| _{\tau=0}.
%\end{align*}
%The expression is tedious so we omit it.
%Note that we can use this inner product to define a norm on $\mathcal{G}$ which is
%$\|g_1-g_2\|^2_\psi=\langle (g_1-g_2),(g_1-g_2)\rangle_\psi$.
%
%For any $g\in\mathcal{G}$ and
%for any $\gamma\in R^{d_\alpha}$, let $\rho_{\gamma}(g)=\gamma'\alpha$ so that
%$d\rho_{\gamma}(g_0)[v]=d\rho(g_0+\tau v)/d\tau\big|_{\tau=0}=\gamma' v$, which is a linear functional of $g$. By the Riesz representation theorem,  there is a sieve Riesz representer $v^*=(v_1^*,\ldots, v^*_{d_\alpha})\in
%\mathcal{V}^{d_\alpha}$ where  $\mathcal{V}$ is the closed linear span of $\mathcal{G}-g_o$ such that
%for all $v\in \mathcal{V}$
%\begin{align*}
%d\rho_{\gamma}(g_0)[v]=\langle \gamma' v^*,v\rangle_\psi,
%\end{align*}
%and $v_j^*$ satisfies that
%\begin{align*}
%d\rho_j(g_0)[v]=\langle v^*_j,v\rangle_\psi,
%\end{align*}
%and $\rho_j(g)=\alpha_j$ for $j=1,\ldots,d_\alpha$.
%
%Define
%\begin{align*}
%\sigma^2_{\gamma}=Var\Big(\gamma' \big(\Delta_\psi (Z,g_0)[v_1^*],\ldots, \Delta_\psi (Z,g_0)[v_{d_\alpha}^*]\big)\Big)=\gamma' \Omega \gamma,
%\end{align*}
%where $\Omega=Var\big( (\Delta_\psi (Z,g_0)[v_1^*],\ldots, \Delta_\psi (Z,g_0)[v_{d_\alpha}^*])\big)$.
%We have
%\begin{align}
%\frac{\gamma \sqrt{N}(\widehat{\alpha}-\alpha_0)}{\sigma^2_{\gamma}}\stackrel{d}{\rightarrow} \mathcal{N}(0,1).\label{eq: wold limit}
%\end{align}
%Therefore, (\ref{eq: wold limit}) implies that
%\begin{align}
%\sqrt{N}(\widehat{\alpha}-\alpha_0)\stackrel{d}{\rightarrow} \mathcal{N}(0,\Omega),
%\label{eq: limiting dis}
%\end{align}
%where $\Omega$ is the asymptotic covariance matrix and
%$\Omega=Var\big( (\Delta_\psi (Z,g_0)[v_1^*],\ldots, \Delta_\psi (Z,g_0)[v_{d_\alpha}^*])\big)$.
%(\ref{eq: limiting dis}) gives the result we need.
%
%
%We derive the limiting distribution of $\sqrt{N}(\widehat{\alpha}-\alpha_0)$.
We first present conditions such that the estimator is consistent.  Let $\widehat{g}\equiv(\widehat{\alpha},\widehat{f}_1)$.

\begin{assum}\label{assu: identi}
For any $\epsilon>0$, there exists a nonincreasing positive sequence $c_N(\epsilon)$
such that for all $N\geq 1$ we have
\begin{align}
E[\psi(Z,g_o)]-\sup_{g\in\mathcal{G}_N:\|g-g_o\|_{\mathcal{G}}\geq \epsilon}E[\psi(Z,g)] \geq c_N(\epsilon),
\end{align}
and $\lim\inf c_N(\epsilon)>0$.
\end{assum}

Define $\mu_N(f)=N^{-1}\sum_{i=1}^N [ f_(Z_i)-E[f(Z_i)]]$, which denotes the empirical process indexed by $f$.
\begin{assum} \label{assu: ULLN}
Assume that (i)
for all $N$, there exists some $g_N\in\mathcal{G}_N$ such that
\begin{align}
| E[\psi(Z,g_N)]-E[\psi(Z,g_o)]|=o(1);
\end{align}
(ii) $\sup_{g\in\mathcal{G}_N}| \mu_N( \psi(Z,g))|=o_P(1)$.
\end{assum}

\begin{thm}\label{thm: consistency}
If Assumptions \ref{assu: identi} and \ref{assu: ULLN} hold, then $\|\widehat{g}-g_o\|=o_p(1).$
\end{thm}


Let $C$ denote a positive finite number. Define $\mathcal{N}_{g,C}=\{g\in\mathcal{G}: \|g-g_o\|\leq C\}$ and
$\mathcal{N}_{g,N,C}=\{g\in\mathcal{G}_N: \|g-g_o\|\leq C\}$. By Theorem \ref{thm: consistency}, we have $\widehat{g}\in \mathcal{N}_{g,N,C}$ with probability approaching 1.  Next, we provide conditions to derive the convergence rate of $\widehat{g}$.


%\begin{assum}\label{assu: convergence rate}
%There exist some finite, positive and non-increasing sequences $\delta_{1N}$, $\delta_{2N}$,  $\delta_{3N}$ and $\delta_{4N}$ that are all $o(1)$ such that
%(i)
%\begin{align}
%|E[\psi(Z,g_N)]-E[\psi(Z,g_o)]|=O(\delta_{2N}^2);
%\end{align}
%(ii)
%for any small constant $\delta,{\delta}>0$ and for any $g\in\mathcal{N}_{g,N,C}$ with $0<\tilde{\delta}<\|g-g_o\|_{\mathcal{G}}<\delta$,
%\begin{align}
%E[\psi(Z,g)-\psi(Z,g_o)]\leq c_1\delta_{1N}\delta-c_2\delta^2
%\end{align}
%where $c_1,c_2$ are finite positive constants;
%(iii)
%for all $N$ large enough and for any $\delta>0$ small enough,
%\begin{align}
%E\Big[  \sup_{g\in\mathcal{N}_{g,N,C}: \|g-g_o\|_{\mathcal{G}}\leq \delta}
%\big| \mu_N( \psi(Z,g)-\psi(Z,g_o))\big|\Big]\leq \frac{c_3\phi_N(\delta)}{\sqrt{N}}
%\end{align}
%where $c_3$ is some positive number and $\phi_N(\cdot)$ is some function such that $\delta^{\nu} \psi_N(\delta)$ is a decreasing function for some $\nu\in(0,2)$;
%(iv)
%$\|g_n-g_o\|_{\mathcal{G}}=O(\delta_{3,N})$;
%(v)
%$\delta_{4N}^{-2}\phi(\delta_{4N}) \leq c_4 \sqrt{N}$
%for some finite positive $c_4>0$
%\end{assum}

\begin{assum}\label{assu: convergence rate}
There exist some finite, positive, and nonincreasing sequences $\delta_{1N}$, $\delta_{2N}$,  and $\delta_{3N}$  that are all $o(1)$ such that
(i)
\begin{align}
\sup_{g\in \mathcal{N}_{g,N,C}}\Big|\mu_N(\psi(Z,g)-\psi(Z,g_o))\Big|=O_p(\delta_{1N}^2);
\end{align}
(ii)
for all $N$ large enough and for any $\delta>0$ small enough,
\begin{align}
E\Big[  \sup_{g\in\mathcal{N}_{g,N,C}: \|g-g_o\|_{\mathcal{G}}\leq \delta}
\big| \mu_N( \psi(Z,g)-\psi(Z,g_o))\big|\Big]\leq \frac{c_1\phi_N(\delta)}{\sqrt{N}},
\end{align}
where $c_1$ is some positive number and $\phi_N(\cdot)$ is some function such that $\delta^{\nu} \psi_N(\delta)$ is a decreasing function for some $\nu\in(0,2)$;


(iii)
$\delta_{2N}^{-2}\phi(\delta_{2N}) \leq c_2 \sqrt{N}$
for some finite positive $c_2>0$;
(iv)
$\|g_N-g_o\|_{\mathcal{G}}=O(\delta_{3N})$.
\end{assum}

\begin{thm}\label{thm: convergence rate}
Suppose that Assumptions \ref{assu: identi}, \ref{assu: ULLN} and \ref{assu: convergence rate} hold.  Then,
we have $\|\widehat{g}-g_o\|=O_p(\epsilon^*_N)$, where $\epsilon^*_N=\max\{\delta_{1N},\delta_{2N},\delta_{3N}\}=o(1)$.
\end{thm}

\begin{assum}\label{assu: metrics}
Assume that there exists $c_\psi<\infty$ such that $\|v\|_\psi\leq c_\psi \|v\|_{\mathcal{G}}$.
\end{assum}

Let $\epsilon_N=\epsilon^*_N\cdot \delta_N$ with $\delta_N\rightarrow \infty$ such that $\epsilon_N$ remains $o_p(1)$.  Let $\mathcal{G}_N\equiv\mathcal{A}\times\mathcal{F}_{1N}$.  Let $\mathcal{N}_g=\{(g\in\mathcal{G}: \|g-g_o\|_{\mathcal{G}}\leq \epsilon_N)\}$ and $\mathcal{N}_{g,N}=\mathcal{N}_g \cap \mathcal{G}_N$.  It is true that $\widehat{g}\in \mathcal{N}_{g,N}$ with probability approaching one.  Let $\Pi_N$ denote the projection of $g$ on $\mathcal{G}_N$ under the norm $\|\cdot\|_{\psi}$ and let $g_{o,N}=\Pi_N g_o$.  Assumption \ref{assu: metrics} implies that $g_{o,N}\in \mathcal{N}_{g,N}$.

$\Pi_N$ is also the projection of $v$ on $\mathcal{V}_N$, which is the closed linear span of $\mathcal{G}_N-g_{o,N}$.
Let $v^*_N\equiv(\Pi_N\cdot v_1^*,\ldots, \Pi_N\cdot v^*_{d_\alpha})\in\mathcal{V}_N^{d_\alpha}$. By construction, we have for all $v\in \mathcal{V}_N$
\begin{align*}
d\rho_{\gamma}(g_0)[v]=\langle \gamma' v^*_N,v\rangle_\psi.
\end{align*}
Let $\kappa_N$ denote a sequence of positive numbers and $\kappa_N=o(N^{-1/2})$.  For any $g$, let $g^*_\lambda=g\pm\kappa_N\cdot \gamma' v^*_N$.
\begin{assum}\label{assu: regularity}
Assume that (i)
\begin{align}
&\sup_{g\in\mathcal{N}_{g,N}}\Big| \mu_N\Big(\psi(Z,g^*)-\psi(Z,g)- \Delta_\psi(Z,g)[\pm \kappa_N \cdot \gamma' v^*_N] \Big)\Big|=O_p(\kappa_N^2),\\
&
\sup_{g\in\mathcal{N}_{g,N}}\Big| \mu_N\Big(\Delta_\psi(Z,g)[\gamma' v^*_N]
-\Delta_\psi(Z,g_o)[\gamma' v^*_N]\Big)\Big|=O_p(\kappa_N);
\end{align}
(ii)
let $K_{\psi}(g)\equiv E[ \psi(Z,g)-\psi(Z,g_o)]$; then,
\begin{align}
\sup_{g\in\mathcal{N}_{g,N}}\Big|
K_{\psi}(g)-K_{\psi}(g_o)- \frac{\|g^*-g_o\|^2_\psi-\|g-g_o\|^2_\psi  }{2}\Big|=O_p(\kappa_N^2);
\end{align}
(iii) $\kappa_N/ \epsilon^*_N=o(1)$ and (iv) $\|v^*_j\|_\psi < \infty$ for all $j=1,\ldots,d_\alpha$.
\end{assum}

\begin{thm}\label{thm: asymptotic normality-app}
Suppose that Assumptions  \ref{assu: identi}, \ref{assu: ULLN}, \ref{assu: convergence rate},
\ref{assu: metrics}, and \ref{assu: regularity} hold.  Then
\begin{align*}
\lambda\sqrt{N}(\widehat{\alpha}-\alpha_0)\stackrel{d}{\rightarrow}\mathcal{N}(0, \sigma^2_{\gamma})=
\mathcal{N}(0, \gamma' \Omega \gamma).
\end{align*}
\end{thm}
The proofs of Theorem \ref{thm: consistency}, \ref{thm: convergence rate}, and \ref{thm: asymptotic normality-app} follow the same arguments as \cite{CLS2014} and \cite{HLR2018}, so we omit the details.
By the Cram{\'e}r-Wold theorem, Theorem \ref{thm: asymptotic normality-app} implies Theorem 3.1.

\subsection{A Consistent Estimator for $\Omega$}\label{app_Omega}
To make inference, one would need a consistent estimator for $\Omega$ when the sieve MLE is implemented based on a Hermite polynomial series.  The estimator is a sample analog.
To be specific, for $j=1,\ldots,d_\alpha$, define the empirical Riesz representer $\widehat{v}_j^*$ as
\begin{align*}
&d\rho_j(\widehat{g})[v]=\langle \widehat{v}_j^*, v \rangle_{N,\psi},~\text{where}\\
&\langle v_1, v_2 \rangle_{N,\psi}=\frac{1}{N}\sum_{i=1}^N
-\frac{d \Delta_\psi (Z_i,\widehat{g}+\tau v_1)[v_2]}{d\tau}\Big| _{\tau=0}.
\end{align*}
A consistent estimator for $\widehat{\Omega}$ is given as
\begin{align}\label{eq: var esti}
\widehat{\Omega}=\frac{1}{N}\sum_{i=1}^N
\big(\Delta_\psi (Z_i,\widehat{g})[\widehat{v}_1^*],\ldots,\Delta_\psi (Z_i,\widehat{g})[\widehat{v}_{d_\alpha}^*]\big)'
\big(\Delta_\psi (Z_i,\widehat{g})[\widehat{v}_1^*],\ldots,\Delta_\psi (Z_i,\widehat{g})[\widehat{v}_{d_\alpha}^*]\big).
\end{align}
The consistency of $\widehat{\Omega}$ can be proven by Theorem 4.1 of \cite{HLR2018}.  More important, even if (\ref{eq: var esti}) appears complicated, one can apply Theorem 6.1 and Remark 6.1 of \cite{HLR2018} to use the variance estimator of the parametric MLE model when the sieve space is generated by a finite number of basis functions such as a Hermite polynomial series.
Specifically, suppose that $f_1$ is approximated by $f_1(v; \beta)$ where $\beta$ has $K(N)$ dimensions.
We then write
\begin{align*}
f_{Y|X,\overline{W}}(y|x,\overline{w};\alpha,\beta)
&=\int_{\mathcal{C}}f_{Y|X,C}(y|x,c;\theta)f_1(c-\overline{w}\lambda;\beta)) dc,
\end{align*}
and the sieve estimator is given by
\begin{align*}
&(\widehat{\alpha},\widehat{\beta})\equiv
\arg\max_{\alpha,\beta}
\frac{1}{N}\sum\limits_{i=1}^{N} \log \big(f_{Y|X,\overline{W}}(Y_i|X_i,\overline{W}_i;\theta,\lambda,\beta)\big),~\text{and}\\
&\widehat{f}_1(v)=f_1(v; \widehat{\beta}).
\end{align*}

Let
%$\nabla f_{Y|X,\overline{W}}(y|x,\overline{w};\theta,\lambda,\beta)$ be the gradient of  $f_{Y|X,\overline{W}}(y|x,\overline{w};\theta,\lambda,\beta)$ with respect to $(\theta,\alpha, \beta)$ and
$\widehat{s}_i=\nabla \log \big(f_{Y|X,\overline{W}}(Y_i|X_i,\overline{W}_i;\alpha,\beta)\big)$ and
$\widehat{H}_i=\nabla^2 \log \big(f_{Y|X,\overline{W}}(Y_i|X_i,\overline{W}_i;\alpha,\beta)\big)$ be the gradient and the Hessian of  $\log \big(f_{Y|X,\overline{W}}(Y_i|X_i,\overline{W}_i;\alpha,\beta)\big)$, respectively, evaluated at $(\widehat{\alpha},\widehat{\beta})$.  Let $\widehat{V}$ be the variance matrix estimator of the MLE estimator that is given by
\begin{align*}
\widehat{V}=\Big(\frac{1}{N} \sum_{i=1}^N - \widehat{H}_i\Big)^{-1} \Big(\frac{1}{N}\sum_{i=1}^N \widehat{s}_i\widehat{s}^\prime_i \Big)\Big(\frac{1}{N} \sum_{i=1}^N - \widehat{H}_i\Big)^{-1}.
\end{align*}
The expression for $\widehat{\Omega}$ will then be the upper-left $d_\alpha\times d_\alpha$ matrix of
$\widehat{V}$.
Equivalently, let $\widehat{s}_{\alpha,i}$ be the first $d_\alpha$ row of the vector of $\widehat{\mathcal{H}}^{-1}\cdot \widehat{s}_i$ with $\widehat{\mathcal{H}}=N^{-1} \sum_{i=1}^N - \widehat{H}_i$,
and $\widehat{\Omega}$ is given by $N^{-1}\sum_{i=1}^N \widehat{s}_{\alpha,i}\widehat{s}^\prime_{\alpha,i}$.


\section{A Consistent Estimator for $\Omega_{di}$}\label{app_hausman}
In this section, we provide a consistent estimator for $\Omega_{di}$.  Recall that the parameter MLE estimator given in Section 4.1 is
\begin{align*}
&(\widehat{\alpha}_{pa}, \widehat{\tau}_{pa})\equiv{\arg\max}_{\alpha\in\mathcal{A},\tau\in \mathcal{T}}
\frac{1}{N}\sum_{i=1}^Nf_{pa}(Y_{i}|X_{i},\overline{W}_{i};\alpha,\tau ),\\
&f_{pa}(y|x,\overline{w};\alpha,\tau )=
\int_{\mathcal{C}}f_{Y|X,C}
(y|x,c;\theta) f_V(c-\overline{W}\lambda;\tau)dc.
\end{align*}
Let
$\widehat{s}_{pa,i}=\nabla \log \big(f_{pa}(Y_i|X_i,\overline{W}_i;\alpha,\beta)\big)$ and
$\widehat{H}_{pa,i}=\nabla^2 \log \big(f_{pa}(Y_i|X_i,\overline{W}_i;\alpha,\beta)\big)$ be the gradient and the Hessian of  $\log \big(f_{pa}(Y_i|X_i,\overline{W}_i;\alpha,\beta)\big)$, respectively, evaluated at $(\widehat{\alpha}_{pa},\widehat{\tau}_{pa})$.  Let $\widehat{V}_{pa}$ be the variance matrix estimator of the MLE estimator that is given by
\begin{align*}
\widehat{V}_{pa}=\Big(\frac{1}{N} \sum_{i=1}^N - \widehat{H}_{pa,i}\Big)^{-1} \Big(\frac{1}{N}\sum_{i=1}^N \widehat{s}_{pa,i}\widehat{s}_{pa,i}' \Big)\Big(\frac{1}{N} \sum_{i=1}^N - \widehat{H}_{pa,i}\Big)^{-1}.
\end{align*}
The expression for $\widehat{\Omega}_{pa}$ will then be the upper-left $d_\alpha\times d_\alpha$ matrix of
$\widehat{V}_{pa}$.
Equivalently, let $\widehat{s}_{pa,\alpha,i}$ be the first $d_\alpha$ row of the vector of $\widehat{\mathcal{H}}_{pa}^{-1}\cdot \widehat{s}_{pa,i}$ with
$\widehat{\mathcal{H}}_{pa}=N^{-1} \sum_{i=1}^N - \widehat{H}_{pa,i}$, and $\widehat{\Omega}_{pa}$ is given by $N^{-1}\sum_{i=1}^N \widehat{s}_{pa,\alpha,i}\widehat{s}_{pa,\alpha,i}'$.

Given these results, a consistent estimator for $\Omega_{di}$ is given by
\begin{align*}
\widehat{\Omega}_{di}=\frac{1}{N}\sum_{i=1}^N \big(\widehat{s}_{pa,\alpha,i}-\widehat{s}_{\alpha,i}\big)\cdot\big(\widehat{s}_{pa,\alpha,i}-\widehat{s}_{\alpha,i}\big)'.
\end{align*}


\section{Monte Carlo Simulation}
\label{section5}

In this section, we present simulation results to illustrate the
finite-sample performance of the proposed sieve ML estimation
procedure of a panel data model in Section 3. We
consider both panel data probit models and panel data Poisson models.


\subsection{Panel Data Probit Models}\label{section51}
We demonstrate our simulation results through static and dynamic settings.
The \emph{static} data generating process (DGP) is defined as
follows:
\begin{align*}
&Y_{t} =\textbf{1} \left(\theta X_{t} +C+\varepsilon _{t}\geq
0\right),~~ \text{for $t=1,2$,}\\
&C=\lambda \overline{W}+V,~~\overline{W}=\frac{1}{2}\sum_{t=1}^{2}X_{t},\\
&X_2=0.5X_1+\xi,~~ X_1\sim U(0,2),~~ \xi\sim N(0,1),\\
&(\varepsilon_1,\varepsilon_2)\sim N(0,I_2),
\end{align*}
where $I_2$ is the $2\times 2$ identity matrix and
$(\theta,\lambda)=(-0.5,0.5)$. For a random variable $Q$, we denote
the corresponding truncated random variable over interval $[a,b]$ as
$Trun(Q,[a,b])$.\footnote{$Trun(Q,[a,b])$ is a random variable
generated by $F_Q ^{-1}(u\cdot(F_Q (b)-F_Q(a))+F_Q(a))$, where $F_Q$
is the CDF of the $Q$ random variable, $F_Q^{-1}$ is the inverse of
$F_Q$, and $u$ is a uniform random variable on $[0,1]$.} Let
$\mu_{\omega}$ be the mean of $\omega$. Three specifications of $V$
are considered:
\begingroup
\allowdisplaybreaks
\begin{align}
&\text{DGP I}\text{: } V \sim Trun(N(0,1), [-1,1]), \label{DGP I}\\
&\text{DGP II}\text{: } V=\omega-\mu_{\omega} \text{ with }\omega\sim Trun\big(H, [-2,2])\big) ~\text{and}~\ln H=N(0,5),\label{DGP II}\\
&\text{DGP III}\text{: } V=\omega-\mu_{\omega} \text{ with
}\sqrt{\omega}\sim  Trun(Rayleigh(1), [-2,15]).\label{DGP III}
\end{align}
\endgroup
The unobserved heterogeneities in all the simulation designs have bounded supports, so Assumption 2.4(ii) is
satisfied in all cases. We consider sample sizes of 500 and 1,000, and
for each case, we consider 1000 simulation replications. For
comparison, we also consider two other estimators. The first
is an infeasible estimator that treats $V$ as known. The second
is the conventional random effects estimator, which specifies the
unobserved heterogeneity to be normally distributed. The simulation
results for parameters and APE are presented in Tables
\ref{Monte_carlo1}--\ref{Monte_carlo2} and
\ref{Monte_carlo3}--\ref{Monte_carlo4}, respectively.

The estimation results of the parameters in DGP I show a small bias
in all three estimators. In this case, the normal specification
in the conventional random effects estimator is close to the true
distribution of the data, and the estimation does not suffer from the
misspecification of the estimator. The proposed sieve ML estimator
exhibits small degrees of bias in DGPs II \& III, but the conventional
random effects estimator exhibits conspicuous bias in
$\theta$, $\lambda$, and $\sigma$ for all sample sizes.

The simulation results overall show that the proposed sieve ML
estimator works well in simulation designs. As expected, the
infeasible estimator outperforms the proposed estimator in RMSE. The
conventional estimator performs well at estimating $\theta$ and
$\lambda$ in DGP I, but causes bias in DGPs II \& III. The estimation
results for APEs in Tables \ref{Monte_carlo3}--\ref{Monte_carlo4}
present a similar pattern. While the infeasible estimator and the
proposed sieve ML estimator perform well in all simulations, the
conventional estimator performs well only in DGP I.

We also consider the Hausman-type test proposed in Section
4.1 for the normality assumption of $V$.
The results are summarized in Table \ref{Monte_carlo9}. For DGP
I, the rejection rates are 0.051 and 0.033, which are close to the
nominal size of 5\% given that the normality assumption
holds.\footnote{We use an
empirical covariance matrix, which is the average of the 1000
simulated estimators, to conduct the Hausman test in the
simulations. } For DGPs II and III, the rejection rates are much
higher than the nominal size of 5\% and increase with sample size,
indicating that our test is consistent when the normality
assumption is violated.

The plots of the simulation
results for the remaining error in the CRE specification $f_{V}$ of DGPs I, II, and III in the static probit models appear in Figures
1--3. While the distribution of $V$ in DGP I is symmetric, the distributions of $V$ in DGPs II and III are nonsymmetric.
These plots show that the sieve ML estimators
perform poorly in DGPs II and III simulation designs because not only do the average results miss the shape of the function, but the confidence bands also do not offer a view of accuracy.
This implies that to obtain a better nonparametric estimate of $f_{V}$, a sample size larger than 1000 may be needed.
However, the visual displays of the average estimator capture the basic features of the true $f_{V}$ such as the location of the mode and symmetry or nonsymmetry.
The black dashed lines are
constructed by 1000 sieve ML estimated curves from 1000 replications in the 10th and 90th
percentiles.


%The shapes of the mean estimates (blue hexagram lines) are
%similar to the true functions (red dashed lines).
\par
The simulation design for dynamic panel data probit models is close
to the static panel data probit models.
We define the DGP for \emph{dynamic}
models as
\begin{align*}
&Y_{t} =\textbf{1} \left(\gamma Y_{t-1}+\theta X_{t} +C+\varepsilon _{t}\geq
0\right),~~ \text{for $t=1,...,7$,}
\end{align*}
where $(\gamma,\theta,\lambda)=(-0.5,0.5,0.5)$ and DGPs for $X_{t}$
and $C$ are the same as those in the static models.

Tables
\ref{Monte_carlo6}--\ref{Monte_carlo7} and
\ref{Monte_carlo8}--\ref{Monte_carlo9} present the estimation
results for parameters and the magnitudes of state dependence. We
reach the same conclusion as the estimation results of the static
models. While the proposed sieve ML estimator performs well in all
DGPs, the conventional random effects estimator cannot deliver a
consistent estimation for the parameters $\gamma$, $\lambda$, and
$\sigma$ in DGPs II \& III. The simulation results for the Hausman-type test
are also similar to the static case.

\subsection{Dynamic Panel Data Poisson Models}
\label{section52}
We consider a data generating process for dynamic panel data Poisson models as follows:
\begin{align}
m(y_{t-1},x_{t},c;\theta)=\gamma Y_{t-1}+\theta X_{t} +C
\text{ with}\hspace{0.5cm} y_{t}=0,1,...,
\end{align}%
where $m(y_{t-1},x_{t},c;\theta)=\text{E}(Y_{t}|y_{t-1},x_{t},c)$ and $(\gamma,\theta,\lambda)=(-0.5,0.5)$. While the DGP for $X_{t}$
is $X_1\sim U(0,2)$, $X_t\sim N(0,1)$ for $t>1$ , the DGP for $C$ is
$C=0.5 \overline{W}+V,~~\overline{W}=\frac{1}{T}\sum_{t=1}^{T}X_{t}+z$ with $z\sim N(0,1)$.
In this case, we consider unbounded support for $V$ and three specifications:
\begingroup
\allowdisplaybreaks
\begin{align}
&\text{DGP IV}\text{: } V \sim N(0,1), \label{DGP IV}\\
&\text{DGP V}\text{: } V=\omega-\mu_{\omega} \text{ with }\omega\sim \text{ Student's $t$ with 100 degrees of freedom},\label{DGP V}\\
&\text{DGP VI}\text{: } V=\omega-\mu_{\omega} \text{ with
}\omega=\omega_{1}+\omega_{2},\omega_{1}\sim N(0,1),\omega_{2}\sim  Trun(Rayleigh(1), [-2,12]).\label{DGP VI}
\end{align}
\endgroup
The finite-sample performance is evaluated over two different time dimensions: $T=2$ and $T=4$.


While Tables
\ref{Monte_carlo10}--\ref{Monte_carlo13} report the estimated results for panel data for two periods, Tables
\ref{Monte_carlo14}--\ref{Monte_carlo17} report the estimated results for panel data for four periods. From the estimation results, we find that in the
designs of panel data for two periods, the proposed estimator outperforms the conventional estimator in the parameter estimation. However, for longer periods, such as four periods, the conventional estimator performs well and is close to the proposed estimator.
Table
\ref{Monte_carlo18} reports the Hausman-type test proposed for the normality assumption of $V$.
In the simulated data for two periods, the rejection rates of the proposed Hausman-type test are much higher than the nominal size of 5\% and increase with sample size in all designs.
This implies that the conventional estimator performs poorly even for the case in which $V$ is normally distributed. However, in the simulated data for four periods with a sample size of 1000,
the estimation results of the conventional estimator are close to those of the proposed estimator in DGP I, and the rejection rate is 0.080. This suggests under an unbounded assumption of $V$ that the conventional estimator may perform better for panel data with a longer period and with a larger sample size.


\bibliographystyle{econometrica}
%\bibliographystyle{ecta}
%\bibliography{bibfile_hsu_Shiu}
\bibliography{bibfile111}




\clearpage
\extrarowheight=-10pt
\begin{longtable}{lccccccc}\label{Monte_carlo1}\\
\caption{Simulations of Static Probit Models}\\
\toprule & \multicolumn{2}{c}{Infeasible}&\multicolumn{3}{c}{Conventional}&\multicolumn{2}{c}{Sieve ML }\\[-3pt]
\cmidrule(l){2-3}\cmidrule(l){4-6}\cmidrule(l){7-8} N=500&$\theta$&$\lambda$&$\theta$&$\lambda$&$\sigma$&$\theta$&$\lambda$\\[-3pt]
\midrule\multicolumn{1}{c}{True}&$-0.5$&$0.5$&$-0.5$&$0.5$&$1$&$-0.5$&$0.5$\\[-3pt]
%Probit Model:&&&&&\\[-1pt]
DGP I: &&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.500 &0.500 &-0.502 &0.505 &0.920 &-0.423 &0.404 \\
\multicolumn{1}{l}{Std.dev.}&-0.499 &0.500 &-0.500 &0.503 &0.919 & -0.420 &0.408 \\
\multicolumn{1}{l}{RMSE}&0.049 &0.057 &0.065 &0.079 &0.141 &0.165 &0.143 \\
DGP II:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.499 &0.496 &-0.607 &0.197 &1.879 &-0.419 &0.403 \\
\multicolumn{1}{l}{Std.dev.}&-0.497 &0.496 &-0.600 &0.200 &1.868 & -0.415 &0.416 \\
\multicolumn{1}{l}{RMSE}&0.060 &0.062 &0.141 &0.321 &0.911 &0.176 &0.158 \\
DGP III:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.499 &0.496 &-0.594 &0.228 &1.768 &-0.417 &0.405 \\
\multicolumn{1}{l}{Std.dev.}&-0.496 &0.496 &-0.590 &0.229 &1.755 & -0.410 &0.421 \\
\multicolumn{1}{l}{RMSE}&0.057 &0.061 &0.129 &0.292 &0.797 &0.184 &0.163 \\				
\bottomrule \multicolumn{8}{l}{
\parbox[t]{11cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP I, DGP II, and DGP III represent the data generating processes in Eqs.\ (\ref{DGP I}), (\ref{DGP II}), and (\ref{DGP III}), respectively. }}
\end{longtable}

\begin{longtable}{lccccccc}\label{Monte_carlo2}\\
\caption{Simulations of Static Probit Models}\\
\toprule & \multicolumn{2}{c}{Infeasible}&\multicolumn{3}{c}{Conventional}&\multicolumn{2}{c}{Sieve ML }\\[-3pt]
\cmidrule(l){2-3}\cmidrule(l){4-6}\cmidrule(l){7-8} N=1000&$\theta$&$\lambda$&$\theta$&$\lambda$&$\sigma$&$\theta$&$\lambda$\\[-3pt]
\midrule\multicolumn{1}{c}{True}&$-0.5$&$0.5$&$-0.5$&$0.5$&$1$&$-0.5$&$0.5$\\[-3pt]
%Probit Model:&&&&&\\[-1pt]
DGP I: &&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.501 &0.500 &-0.504 &0.506 &0.923 &-0.489 &0.558 \\
\multicolumn{1}{l}{Std.dev.}&-0.502 &0.500 &-0.503 &0.504 &0.926 & -0.491 &0.552 \\
\multicolumn{1}{l}{RMSE}&0.034 &0.038 &0.045 &0.055 &0.114 &0.116 &0.161 \\
DGP II:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.502 &0.501 &-0.608 &0.206 &1.866 &-0.474 &0.543 \\
\multicolumn{1}{l}{Std.dev.}&-0.500 &0.500 &-0.606 &0.207 &1.853 & -0.465 &0.546 \\
\multicolumn{1}{l}{RMSE}&0.041 &0.044 &0.128 &0.304 &0.883 &0.114 &0.111 \\
DGP III:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.501 &0.501 &-0.594 &0.235 &1.758 &-0.469 &0.545 \\
\multicolumn{1}{l}{Std.dev.}&-0.500 &0.499 &-0.592 &0.234 &1.749 & -0.462 &0.551 \\
\multicolumn{1}{l}{RMSE}&0.039 &0.043 &0.114 &0.275 &0.773 &0.114 &0.112 \\																			\bottomrule \multicolumn{8}{l}{
\parbox[t]{11cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP I, DGP II, and DGP III represent the data generating processes in Eqs.\ (\ref{DGP I}), (\ref{DGP II}), and (\ref{DGP III}), respectively.}}
\end{longtable}

\clearpage


%
%
%

\begin{longtable}{lccc}\label{Monte_carlo3}\\
\caption{Simulation of $\text{APE}(\bar{x})$ in Static Probit Models}\\
\toprule & \multicolumn{1}{c}{Infeasible}&\multicolumn{1}{c}{Conventional}&\multicolumn{1}{c}{Sieve ML}\\[-3pt]
N=500& \multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}\\[-3pt]
\midrule
%Probit Model:&&&\\[-1pt]
DGP I: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.168&-0.143&-0.126\\
\multicolumn{1}{l}{Std. dev.}&0.001&0.017&0.042\\
\multicolumn{1}{l}{RMSE}&--&0.017&0.059\\
DGP II: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.188&-0.113&-0.125\\
\multicolumn{1}{l}{Std. dev.}&0.001&0.017&0.045\\
\multicolumn{1}{l}{RMSE}&--&0.026&0.072\\
DGP III:  &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.188&-0.116&-0.124\\
\multicolumn{1}{l}{Std. dev.}&0.001&0.016&0.046\\
\multicolumn{1}{l}{RMSE}&--&0.025&0.079\\				
\bottomrule\multicolumn{4}{l}{
\parbox[t]{8cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP I, DGP II, and DGP III represent the data generating processes in Eqs.\ (\ref{DGP I}), (\ref{DGP II}), and (\ref{DGP III}), respectively.}}
\end{longtable}

\begin{longtable}{lccc}\label{Monte_carlo4}\\
\caption{Simulation of $\text{APE}(\bar{x})$ in Static Probit Models}\\
\toprule & \multicolumn{1}{c}{Infeasible}&\multicolumn{1}{c}{Conventional}&\multicolumn{1}{c}{Sieve ML}\\[-3pt]
N=1000& \multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}\\[-3pt]
\midrule
%Probit Model:&&&\\[-1pt]
DGP I: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.168&-0.144&-0.141\\
\multicolumn{1}{l}{Std. dev.}&0.001&0.011&0.034\\
\multicolumn{1}{l}{RMSE}&--&0.011&0.043\\
DGP II: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.188&-0.113&-0.136\\
\multicolumn{1}{l}{Std. dev.}&0.001&0.011&0.032\\
\multicolumn{1}{l}{RMSE}&--&0.024&0.055\\
DGP III:  &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.188&-0.116&-0.135\\
\multicolumn{1}{l}{Std. dev.}&0.001&0.011&0.031\\
\multicolumn{1}{l}{RMSE}&--&0.022&0.062\\							
\bottomrule\multicolumn{4}{l}{
\parbox[t]{8cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviation of the estimates across 1000 simulations. The notation DGP I, DGP II, and DGP III represent the data generating processes in Eqs.\ (\ref{DGP I}), (\ref{DGP II}), and (\ref{DGP III}), respectively.}}
\end{longtable}


\begin{figure}[ptb]
\begin{center}
\includegraphics[width=0.45\textwidth]{figure111.pdf}
\end{center}
\caption{Average Estimates of $f_{V}$ in the CRE Specification of DGP I in Eq.\ (\ref{DGP I}) in Section \ref{section51}}%
\label{density1}%
\end{figure}

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=0.45\textwidth]{figure222.pdf}
\end{center}
\caption{Average Estimates of $f_{V}$ in the CRE Specification of DGP II in Eq.\ (\ref{DGP II}) in Section \ref{section51}}%
\label{density2}%
\end{figure}

\begin{figure}[ptb]
\begin{center}
\includegraphics[width=0.45\textwidth]{figure333.pdf}
\end{center}
\caption{Average Estimates of $f_{V}$ in the CRE Specification of DGP III in Eq.\ (\ref{DGP III}) in Section \ref{section51}}%
\label{density3}%
\end{figure}


\clearpage \extrarowheight=-10pt
\begin{longtable}{lcccccccccc}\label{Monte_carlo5}\\
\caption{Simulations of Dynamic Probit  Models}\\
\toprule & \multicolumn{3}{c}{Infeasible}&\multicolumn{4}{c}{Conventional}&\multicolumn{3}{c}{Sieve ML }\\[-3pt]
\cmidrule(l){2-4}\cmidrule(l){5-8}\cmidrule(l){9-11} N=500&$\gamma$&$\theta$&$\lambda$&$\gamma$&$\theta$&$\lambda$&$\sigma$&$\gamma$&$\theta$&$\lambda$\\[-3pt]
\midrule\multicolumn{1}{c}{True}&$-0.5$&$0.5$&$0.5$&$-0.5$&$0.5$&$0.5$&$1$&$-0.5$&$0.5$&$0.5$\\[-3pt]
%Probit Model:&&&&&\\[-1pt]
DGP I: &&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.502 &0.498 &0.498 &-0.382 &0.378 &0.198 &0.301 &-0.555 &0.444 &0.490 \\ \multicolumn{1}{l}{Std.dev.}&-0.501 &0.500 &0.496 &-0.379 &0.376 &0.198 &0.334 &-0.551 &0.453 & 0.464 \\
\multicolumn{1}{l}{RMSE}&0.034 &0.066 &0.057 &0.125 &0.138 &0.307 &0.722 &0.158 &0.156 &0.191 \\
DGP II:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.503 &0.497 &0.500 &-0.383 &0.376 &0.198 &0.306 &-0.553 &0.449 &0.501 \\ \multicolumn{1}{l}{Std.dev.}&-0.501 &0.498 &0.500 &-0.381 &0.378 &0.195 &0.338 &-0.545 &0.459 & 0.460 \\
\multicolumn{1}{l}{RMSE}&0.033 &0.067 &0.059 &0.124 &0.140 &0.306 &0.718 &0.153 &0.177 &0.220 \\
DGP III:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.505 &0.500 &0.500 &-0.372 &0.352 &0.155 &0.317 &-0.543 &0.454 &0.512 \\ \multicolumn{1}{l}{Std.dev.}&-0.502 &0.501 &0.500 &-0.370 &0.349 &0.154 &0.351 &-0.536 &0.463 & 0.467 \\
\multicolumn{1}{l}{RMSE}&0.034 &0.068 &0.062 &0.135 &0.163 &0.349 &0.710 &0.160 &0.172 &0.232 \\					\bottomrule \multicolumn{11}{l}{
\parbox[t]{15cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP I, DGP II, and DGP III represent the data generating processes in Eqs.\ (\ref{DGP I}), (\ref{DGP II}), and (\ref{DGP III}), respectively.}}
\end{longtable}

\begin{longtable}{lcccccccccc}\label{Monte_carlo6}\\
\caption{Simulations of Dynamic Probit  Models}\\
\toprule & \multicolumn{3}{c}{Infeasible}&\multicolumn{4}{c}{Conventional}&\multicolumn{3}{c}{Sieve ML }\\[-3pt]
\cmidrule(l){2-4}\cmidrule(l){5-8}\cmidrule(l){9-11} N=1000&$\gamma$&$\theta$&$\lambda$&$\gamma$&$\theta$&$\lambda$&$\sigma$&$\gamma$&$\theta$&$\lambda$\\[-3pt]
\midrule\multicolumn{1}{c}{True}&$-0.5$&$0.5$&$0.5$&$-0.5$&$0.5$&$0.5$&$1$&$-0.5$&$0.5$&$0.5$\\[-3pt]
%Probit Model:&&&&&\\[-1pt]
DGP I: &&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.501 &0.498 &0.498 &-0.372 &0.352 &0.155 &0.317 &-0.510 &0.500 &0.512 \\ \multicolumn{1}{l}{Std.dev.}&-0.500 &0.499 &0.500 &-0.370 &0.349 &0.154 &0.351 &-0.513 &0.495 & 0.513 \\
\multicolumn{1}{l}{RMSE}&0.024 &0.045 &0.042 &0.135 &0.163 &0.349 &0.710 &0.105 &0.102 &0.098 \\
DGP II:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.501 &0.499 &0.499 &-0.380 &0.381 &0.198 &0.328 &-0.506 &0.505 &0.507 \\ \multicolumn{1}{l}{Std.dev.}&-0.500 &0.497 &0.499 &-0.379 &0.382 &0.198 &0.354 &-0.505 &0.505 & 0.507 \\
\multicolumn{1}{l}{RMSE}&0.023 &0.045 &0.043 &0.123 &0.127 &0.304 &0.687 &0.104 &0.103 &0.103 \\
DGP III:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.501 &0.497 &0.497 &-0.371 &0.360 &0.159 &0.339 &-0.508 &0.503 &0.508 \\ \multicolumn{1}{l}{Std.dev.}&-0.500 &0.499 &0.495 &-0.368 &0.360 &0.158 &0.367 &-0.506 &0.504 & 0.508 \\
\multicolumn{1}{l}{RMSE}&0.023 &0.050 &0.044 &0.133 &0.147 &0.343 &0.679 &0.102 &0.099 &0.107 \\													\bottomrule \multicolumn{11}{l}{
\parbox[t]{15cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP I, DGP II, and DGP III represent the data generating processes in Eqs.\ (\ref{DGP I}), (\ref{DGP II}), and (\ref{DGP III}), respectively.}}
\end{longtable}

\clearpage

\begin{longtable}{lccc}\label{Monte_carlo7}\\
\caption{Simulations of State Dependence in the Dynamic Probit  Models}\\
\toprule & \multicolumn{1}{c}{Infeasible}&\multicolumn{1}{c}{Conventional}&\multicolumn{1}{c}{Sieve ML}\\[-3pt]
N=500& \multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}\\[-3pt]
\midrule
%Probit Model:&&&\\[-1pt]
DGP I: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.127&-0.138&-0.162\\
\multicolumn{1}{l}{Std. dev.}&0.003&0.013&0.042\\
\multicolumn{1}{l}{RMSE}&--&0.019&0.055\\
DGP II: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.126&-0.138&-0.161\\
\multicolumn{1}{l}{Std. dev.}&0.003&0.013&0.041\\
\multicolumn{1}{l}{RMSE}&--&0.019&0.054\\
DGP III:  &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.117&-0.135&-0.158\\
\multicolumn{1}{l}{Std. dev.}&0.003&0.013&0.044\\
\multicolumn{1}{l}{RMSE}&--&0.024&0.060\\
\bottomrule\multicolumn{4}{l}{
\parbox[t]{8cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP I, DGP II, and DGP III represent the data generating processes in Eqs.\ (\ref{DGP I}), (\ref{DGP II}), and (\ref{DGP III}), respectively.}}
\end{longtable}


\begin{longtable}{lccc}\label{Monte_carlo8}\\
\caption{Simulations of State Dependence in the Dynamic Probit  Models}\\
\toprule & \multicolumn{1}{c}{Infeasible}&\multicolumn{1}{c}{Conventional}&\multicolumn{1}{c}{Sieve ML}\\[-3pt]
N=1000& \multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}\\[-3pt]
\midrule
%Probit Model:&&&\\[-1pt]
DGP I: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.127&-0.137&-0.141\\
\multicolumn{1}{l}{Std. dev.}&0.002&0.009&0.029\\
\multicolumn{1}{l}{RMSE}&--&0.016&0.033\\
DGP II: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.126&-0.137&-0.140\\
\multicolumn{1}{l}{Std. dev.}&0.002&0.009&0.029\\
\multicolumn{1}{l}{RMSE}&--&0.016&0.032\\
DGP III:  &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.117&-0.135&-0.141\\
\multicolumn{1}{l}{Std. dev.}&0.002&0.010&0.029\\
\multicolumn{1}{l}{RMSE}&--&0.022&0.037\\
\bottomrule\multicolumn{4}{l}{
\parbox[t]{8cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP I, DGP II, and DGP III represent the data generating processes in Eqs.\ (\ref{DGP I}), (\ref{DGP II}), and (\ref{DGP III}), respectively.}}
\end{longtable}

\clearpage
\begin{longtable}{lcccc}\label{Monte_carlo9}\\
\caption{Hausman-type Test for Normality: Empirical Size}\\
\toprule & \multicolumn{2}{c}{Static Probit Models}&\multicolumn{2}{c}{Dynamic Probit  Models}\\[-1pt]
& \multicolumn{1}{c}{N=500}&\multicolumn{1}{c}{N=1000}&\multicolumn{1}{c}{N=500}&\multicolumn{1}{c}{N=1000}\\[-3pt]
\midrule
DGP I: &0.051 &0.033 &0.039 &0.062 \\
DGP II: &0.145 &0.977 &0.420 &0.598 \\
DGP III:  &0.139 &0.957 &0.368 &0.545 \\
\bottomrule\multicolumn{5}{l}{
\parbox[t]{9.6cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
The $p$-values of 0.05 of Chi-distributions for static models and
dynamic models are 5.991 and 7.815, respectively. Empirical size
refers to the fraction of rejections when using these values as the
critical values. The notations DGP I, DGP II, and DGP III represent the data generating processes in Eqs. (\ref{DGP I}), (\ref{DGP II}), and (\ref{DGP III}), respectively. }}
\end{longtable}

\clearpage
\extrarowheight=-10pt
\begin{longtable}{lcccccccccc}\label{Monte_carlo10}\\
\caption{Simulations of Two-Period Dynamic Poisson Models}\\
\toprule & \multicolumn{3}{c}{Infeasible}&\multicolumn{4}{c}{Conventional}&\multicolumn{3}{c}{Sieve ML }\\[-3pt]
\cmidrule(l){2-4}\cmidrule(l){5-8}\cmidrule(l){9-11} N=500&$\gamma$&$\theta$&$\lambda$&$\gamma$&$\theta$&$\lambda$&$\sigma$&$\gamma$&$\theta$&$\lambda$\\[-3pt]
\midrule\multicolumn{1}{c}{True}&$-0.5$&$0.5$&$0.5$&$-0.5$&$0.5$&$0.5$&$1$&$-0.5$&$0.5$&$0.5$\\[-3pt]
%Probit Model:&&&&&\\[-1pt]
DGP IV: &&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.502 &0.498 &0.498 &-0.382 &0.378 &0.198 &0.301 &-0.555 &0.444 &0.490 \\ \multicolumn{1}{l}{Std.dev.}&-0.501 &0.500 &0.496 &-0.379 &0.376 &0.198 &0.334 &-0.551 &0.453 & 0.464 \\
\multicolumn{1}{l}{RMSE}&0.034 &0.066 &0.057 &0.125 &0.138 &0.307 &0.722 &0.158 &0.156 &0.191 \\
DGP V:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.503 &0.497 &0.500 &-0.383 &0.376 &0.198 &0.306 &-0.553 &0.449 &0.501 \\ \multicolumn{1}{l}{Std.dev.}&-0.501 &0.498 &0.500 &-0.381 &0.378 &0.195 &0.338 &-0.545 &0.459 & 0.460 \\
\multicolumn{1}{l}{RMSE}&0.033 &0.067 &0.059 &0.124 &0.140 &0.306 &0.718 &0.153 &0.177 &0.220 \\
DGP VI:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.505 &0.500 &0.500 &-0.372 &0.352 &0.155 &0.317 &-0.543 &0.454 &0.512 \\ \multicolumn{1}{l}{Std.dev.}&-0.502 &0.501 &0.500 &-0.370 &0.349 &0.154 &0.351 &-0.536 &0.463 & 0.467 \\
\multicolumn{1}{l}{RMSE}&0.034 &0.068 &0.062 &0.135 &0.163 &0.349 &0.710 &0.160 &0.172 &0.232 \\					\bottomrule \multicolumn{11}{l}{
\parbox[t]{15cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP IV, DGP V, and DGP VI represent the data generating processes in Eqs.\ (\ref{DGP IV}), (\ref{DGP V}), and (\ref{DGP VI}), respectively.}}
\end{longtable}

\extrarowheight=-10pt
\begin{longtable}{lcccccccccc}\label{Monte_carlo11}\\
\caption{Simulations of Two-Period Dynamic Poisson Models}\\
\toprule & \multicolumn{3}{c}{Infeasible}&\multicolumn{4}{c}{Conventional}&\multicolumn{3}{c}{Sieve ML }\\[-3pt]
\cmidrule(l){2-4}\cmidrule(l){5-8}\cmidrule(l){9-11} N=1000&$\gamma$&$\theta$&$\lambda$&$\gamma$&$\theta$&$\lambda$&$\sigma$&$\gamma$&$\theta$&$\lambda$\\[-3pt]
\midrule\multicolumn{1}{c}{True}&$-0.5$&$0.5$&$0.5$&$-0.5$&$0.5$&$0.5$&$1$&$-0.5$&$0.5$&$0.5$\\[-3pt]
%Probit Model:&&&&&\\[-1pt]
DGP IV: &&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.501 &0.498 &0.498 &-0.372 &0.352 &0.155 &0.317 &-0.510 &0.500 &0.512 \\ \multicolumn{1}{l}{Std.dev.}&-0.500 &0.499 &0.500 &-0.370 &0.349 &0.154 &0.351 &-0.513 &0.495 & 0.513 \\
\multicolumn{1}{l}{RMSE}&0.024 &0.045 &0.042 &0.135 &0.163 &0.349 &0.710 &0.105 &0.102 &0.098 \\
DGP V:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.501 &0.499 &0.499 &-0.380 &0.381 &0.198 &0.328 &-0.506 &0.505 &0.507 \\ \multicolumn{1}{l}{Std.dev.}&-0.500 &0.497 &0.499 &-0.379 &0.382 &0.198 &0.354 &-0.505 &0.505 & 0.507 \\
\multicolumn{1}{l}{RMSE}&0.023 &0.045 &0.043 &0.123 &0.127 &0.304 &0.687 &0.104 &0.103 &0.103 \\
DGP VI:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.501 &0.497 &0.497 &-0.371 &0.360 &0.159 &0.339 &-0.508 &0.503 &0.508 \\ \multicolumn{1}{l}{Std.dev.}&-0.500 &0.499 &0.495 &-0.368 &0.360 &0.158 &0.367 &-0.506 &0.504 & 0.508 \\
\multicolumn{1}{l}{RMSE}&0.023 &0.050 &0.044 &0.133 &0.147 &0.343 &0.679 &0.102 &0.099 &0.107 \\												\bottomrule \multicolumn{11}{l}{
\parbox[t]{15cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP IV, DGP V, and DGP VI represent the data generating processes in Eqs.\ (\ref{DGP IV}), (\ref{DGP V}), and (\ref{DGP VI}), respectively.}}
\end{longtable}

\clearpage


%
%
%

\begin{longtable}{lccc}\label{Monte_carlo12}\\
\caption{Simulations of $\text{APE}(\bar{x})$ in the Two-Period Dynamic Poisson Models}\\
\toprule & \multicolumn{1}{c}{Infeasible}&\multicolumn{1}{c}{Conventional}&\multicolumn{1}{c}{Sieve ML}\\[-3pt]
N=500& \multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}\\[-3pt]
\midrule
%Probit Model:&&&\\[-1pt]
DGP IV: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.127&-0.138&-0.162\\
\multicolumn{1}{l}{Std. dev.}&0.003&0.013&0.042\\
\multicolumn{1}{l}{RMSE}&--&0.019&0.055\\
DGP V: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.126&-0.138&-0.161\\
\multicolumn{1}{l}{Std. dev.}&0.003&0.013&0.041\\
\multicolumn{1}{l}{RMSE}&--&0.019&0.054\\
DGP VI: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.117&-0.135&-0.158\\
\multicolumn{1}{l}{Std. dev.}&0.003&0.013&0.044\\
\multicolumn{1}{l}{RMSE}&--&0.024&0.060\\
\bottomrule\multicolumn{4}{l}{
\parbox[t]{8cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP IV, DGP V, and DGP VI represent the data generating processes in Eqs.\ (\ref{DGP IV}), (\ref{DGP V}), and (\ref{DGP VI}), respectively.}}
\end{longtable}

\begin{longtable}{lccc}\label{Monte_carlo13}\\
\caption{Simulations of $\text{APE}(\bar{x})$ in the Two-Period Dynamic Poisson Models}\\
\toprule & \multicolumn{1}{c}{Infeasible}&\multicolumn{1}{c}{Conventional}&\multicolumn{1}{c}{Sieve ML}\\[-3pt]
N=1000& \multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}\\[-3pt]
\midrule
%Probit Model:&&&\\[-1pt]
DGP IV: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.127&-0.137&-0.141\\
\multicolumn{1}{l}{Std. dev.}&0.002&0.009&0.029\\
\multicolumn{1}{l}{RMSE}&--&0.016&0.033\\
DGP V: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.126&-0.137&-0.140\\
\multicolumn{1}{l}{Std. dev.}&0.002&0.009&0.029\\
\multicolumn{1}{l}{RMSE}&--&0.016&0.032\\
DGP VI: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.117&-0.135&-0.141\\
\multicolumn{1}{l}{Std. dev.}&0.002&0.010&0.029\\
\multicolumn{1}{l}{RMSE}&--&0.022&0.037\\	
\bottomrule\multicolumn{4}{l}{
\parbox[t]{8cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP IV, DGP V, and DGP VI represent the data generating processes in Eqs.\ (\ref{DGP IV}), (\ref{DGP V}), and (\ref{DGP VI}), respectively.}}
\end{longtable}





\clearpage \extrarowheight=-10pt
\begin{longtable}{lcccccccccc}\label{Monte_carlo14}\\
\caption{Simulations of Four-Period Dynamic Poisson Models}\\
\toprule & \multicolumn{3}{c}{Infeasible}&\multicolumn{4}{c}{Conventional}&\multicolumn{3}{c}{Sieve ML }\\[-3pt]
\cmidrule(l){2-4}\cmidrule(l){5-8}\cmidrule(l){9-11} N=500&$\gamma$&$\theta$&$\lambda$&$\gamma$&$\theta$&$\lambda$&$\sigma$&$\gamma$&$\theta$&$\lambda$\\[-3pt]
\midrule\multicolumn{1}{c}{True}&$-0.5$&$0.5$&$0.5$&$-0.5$&$0.5$&$0.5$&$1$&$-0.5$&$0.5$&$0.5$\\[-3pt]
%Probit Model:&&&&&\\[-1pt]
DGP IV: &&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.500&0.499&0.500&-0.524&0.494&0.500&1.004&-0.418&0.445&0.358\\
\multicolumn{1}{l}{Std. dev.}&-0.500&0.499&0.499&-0.522&0.494&0.499&1.004&-0.429&0.458&0.351\\
\multicolumn{1}{l}{RMSE}&0.020&0.023&0.018&0.035&0.036&0.039&0.046&0.133&0.157&0.196\\
DGP V:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.501&0.499&0.501&-0.525&0.493&0.499&1.013&-0.421&0.460&0.393\\
\multicolumn{1}{l}{Std. dev.}&-0.500&0.501&0.500&-0.525&0.492&0.500&1.011&-0.427&0.463&0.385\\
\multicolumn{1}{l}{RMSE}&0.020&0.023&0.018&0.036&0.040&0.040&0.047&0.121&0.130&0.152\\
DGP VI:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.502&0.500&0.500&-0.528&0.487&0.497&1.200&-0.463&0.454&0.366\\
\multicolumn{1}{l}{Std. dev.}&-0.502&0.500&0.500&-0.527&0.488&0.499&1.199&-0.475&0.461&0.359\\
\multicolumn{1}{l}{RMSE}&0.020&0.021&0.017&0.039&0.048&0.046&0.206&0.101&0.150&0.183\\		
\bottomrule \multicolumn{11}{l}{
\parbox[t]{15cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP IV, DGP V, and DGP VI represent the data generating processes in Eqs.\ (\ref{DGP IV}), (\ref{DGP V}), and (\ref{DGP VI}), respectively.}}
\end{longtable}

\begin{longtable}{lcccccccccc}\label{Monte_carlo15}\\
\caption{Simulations of Four-Period Dynamic Poisson Models}\\
\toprule & \multicolumn{3}{c}{Infeasible}&\multicolumn{4}{c}{Conventional}&\multicolumn{3}{c}{Sieve ML }\\[-3pt]
\cmidrule(l){2-4}\cmidrule(l){5-8}\cmidrule(l){9-11} N=1000&$\gamma$&$\theta$&$\lambda$&$\gamma$&$\theta$&$\lambda$&$\sigma$&$\gamma$&$\theta$&$\lambda$\\[-3pt]
\midrule\multicolumn{1}{c}{True}&$-0.5$&$0.5$&$0.5$&$-0.5$&$0.5$&$0.5$&$1$&$-0.5$&$0.5$&$0.5$\\[-3pt]
%Probit Model:&&&&&\\[-1pt]
DGP IV: &&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.500&0.500&0.499&-0.524&0.491&0.502&1.004&-0.511&0.470&0.462\\
\multicolumn{1}{l}{Std. dev.}&-0.500&0.500&0.500&-0.524&0.491&0.502&1.005&-0.516&0.472&0.460\\
\multicolumn{1}{l}{RMSE}&0.015&0.015&0.012&0.031&0.028&0.029&0.032&0.073&0.089&0.098\\
DGP V:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.500&0.499&0.500&-0.524&0.492&0.501&1.013&-0.421&0.472&0.389\\
\multicolumn{1}{l}{Std. dev.}&-0.501&0.499&0.500&-0.524&0.492&0.501&1.013&-0.426&0.484&0.382\\
\multicolumn{1}{l}{RMSE}&0.015&0.016&0.013&0.030&0.027&0.030&0.034&0.119&0.116&0.154\\
DGP VI:&&&&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.501&0.500&0.500&-0.527&0.484&0.494 &1.203&-0.471 &0.448 &0.365 \\
\multicolumn{1}{l}{Std. dev.}&-0.501&0.501&0.499&-0.526&0.486& 0.493&1.203&-0.484 &0.452 &0.357 \\
\multicolumn{1}{l}{RMSE}&0.015&0.015&0.011&0.033&0.038&0.032 &0.206&0.101 &0.149 &0.182 \\																			\bottomrule \multicolumn{11}{l}{
\parbox[t]{15cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP IV, DGP V, and DGP VI represent the data generating processes in Eqs.\ (\ref{DGP IV}), (\ref{DGP V}), and (\ref{DGP VI}), respectively.}}
\end{longtable}

\clearpage

\begin{longtable}{lccc}\label{Monte_carlo16}\\
\caption{Simulations of  State Dependence in the Four-Period Dynamic Poisson Models}\\
\toprule & \multicolumn{1}{c}{Infeasible}&\multicolumn{1}{c}{Conventional}&\multicolumn{1}{c}{Sieve ML}\\[-3pt]
N=500& \multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}\\[-3pt]
\midrule
%Probit Model:&&&\\[-1pt]
DGP IV: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.130&-0.133&-0.119\\
\multicolumn{1}{l}{Std. dev.}&0.003&0.007&0.030\\
\multicolumn{1}{l}{RMSE}&--&0.009&0.032\\
DGP V: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.129&-0.133&-0.120\\
\multicolumn{1}{l}{Std. dev.}&0.003&0.007&0.026\\
\multicolumn{1}{l}{RMSE}&--&0.009&0.027\\
DGP VI: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.119&-0.124&-0.132\\
\multicolumn{1}{l}{Std. dev.}&0.003&0.007&0.027\\
\multicolumn{1}{l}{RMSE}&--&0.010&0.030\\
\bottomrule\multicolumn{4}{l}{
\parbox[t]{8cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP IV, DGP V, and DGP VI represent the data generating processes in Eqs.\ (\ref{DGP IV}), (\ref{DGP V}), and (\ref{DGP VI}), respectively.}}
\end{longtable}


\begin{longtable}{lccc}\label{Monte_carlo17}\\
\caption{Simulations of State Dependence in the Four-Period Dynamic Poisson Models}\\
\toprule & \multicolumn{1}{c}{Infeasible}&\multicolumn{1}{c}{Conventional}&\multicolumn{1}{c}{Sieve ML}\\[-3pt]
N=1000& \multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}&\multicolumn{1}{c}{Estimator}\\[-3pt]
\midrule
%Probit Model:&&&\\[-1pt]
DGP IV: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.130&-0.133&-0.144\\
\multicolumn{1}{l}{Std. dev.}&0.002&0.005&0.020\\
\multicolumn{1}{l}{RMSE}&--&0.008&0.025\\
DGP V: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.129&-0.132&-0.120\\
\multicolumn{1}{l}{Std. dev.}&0.002&0.005&0.025\\
\multicolumn{1}{l}{RMSE}&--&0.007&0.027\\
DGP VI: &&&\\[-1pt]
\multicolumn{1}{l}{Mean}&-0.119&-0.123&-0.134\\
\multicolumn{1}{l}{Std. dev.}&0.002&0.005&0.028\\
\multicolumn{1}{l}{RMSE}&--&0.008&0.031\\	
\bottomrule\multicolumn{4}{l}{
\parbox[t]{8cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
Standard deviations of the parameters are computed by the standard
deviations of the estimates across 1000 simulations. The notations DGP IV, DGP V, and DGP VI represent the data generating processes in Eqs.\ (\ref{DGP IV}), (\ref{DGP V}), and (\ref{DGP VI}), respectively.}}
\end{longtable}

\clearpage
\begin{longtable}{lcccc}\label{Monte_carlo18}\\
\caption{Hausman-type Test for Normality: Empirical Size}\\
\toprule & \multicolumn{2}{c}{Two-Period Dynamic Poisson Models}&\multicolumn{2}{c}{ Four-Period Dynamic Poisson Models}\\[-1pt]
& \multicolumn{1}{c}{N=500}&\multicolumn{1}{c}{N=1000}&\multicolumn{1}{c}{N=500}&\multicolumn{1}{c}{N=1000}\\[-3pt]
\midrule
DGP I: &0.313 &0.833 &0.190 &0.080 \\
DGP II: &0.287 &0.807 &0.210 &0.200 \\
DGP II: &0.340 &0.867 &0.166 &0.139 \\
\bottomrule\multicolumn{5}{l}{
\parbox[t]{14cm}{\footnotesize\addtolength{\baselineskip}{-8pt} Note:
The $p$-values of 0.05 of Chi-distributions for static models and
dynamic models are 5.991 and 7.815, respectively. Empirical size
refers to the fraction of rejections when using these values as the
critical values. The notations DGP IV, DGP V, and DGP VI represent the data generating processes in Eqs.\ (\ref{DGP IV}), (\ref{DGP V}), and (\ref{DGP VI}), respectively.}}
\end{longtable}

\end{document}
