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\newtheorem*{lemmaA4}{Lemma A.4}
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\begin{document}

\title{{\huge Online Supplemental Material for \textquotedblleft A
Max-Correlation White Noise Test for Weakly Dependent Time Series"}}
\author{ \ \ Jonathan B. Hill \thanks{%
Department of Economics, University of North Carolina at Chapel Hill;
\texttt{jbhill@email.unc.edu}; \texttt{https://jbhill.web.unc.edu}.}\qquad\
\ \ \ \ and\qquad\ \ Kaiji Motegi\thanks{%
Graduate School of Economics, Kobe University. E-mail: \texttt{%
motegi@econ.kobe-u.ac.jp}} \\
%EndAName
University of North Carolina\qquad\ \qquad Kobe University}
\date{\bigskip{\normalsize This Draft: \today}}
\maketitle
\tableofcontents
\listoftables
\listoffigures

\newpage

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\section{Outline and Assumptions}

In this supplemental material, we present theoretical and numerical details
that are omitted in the main paper \cite{HillMotegi_maxcorr_main_paper}. In
the present appendix, we state our assumptions on the data generating
process and plug-in estimator.

Appendix \ref{app:nedmc} contains a comparison between our imposed Near
Epoch Dependence [NED] property with the moment contraction property invoked
in \cite{Shao2011_JoE}, \cite{GuayGuerreLazarova2013} and \cite{XiaoWu2014}.

Appendix \ref{app:examples} contains several examples of data generating
processes, filters, and the resulting correlation expansion form.

Appendix \ref{app:local_power} details local asymptotic power for the
max-correlation test.

Appendix \ref{app:kern_var} presents a kernel variance estimator for
self-standardization of the sample correlation as an option for weighting.

Appendix \ref{app:proofs} presents omitted proofs for Lemma A.4 (expansion),
Lemma A.5 (convergence in finite dimensional distributions) and Lemma A.6
(ULLN's).

Appendix \ref{app:max_L} presents an environment in which an upper bound on
the maximum lag rate of increase $\mathcal{L}_{n}$ $\rightarrow $ $\infty $
is available.

Appendix \ref{app:sim_all_supp_mat} contains the complete Monte Carlo
study.\medskip

Throughout $|\cdot |$ is the $l_{1}$-matrix norm; $||\cdot ||$ is the $l_{2}$%
-matrix norm; $||\cdot ||_{p}$ is the $L_{p}$-norm. $I(\cdot )$ is the
indicator function: $I(A)$ $=$ $1$ if $A$ is true, else $I(A)$ $=$ $0$. $%
\mathcal{F}_{t}$ $\equiv $ $\sigma (y_{\tau },x_{\tau }$ $:$ $\tau $ $\leq $
$t)$. All random variables lie in a complete probability measure space $%
(\Omega ,\mathcal{P},\mathcal{F})$, hence $\sigma (\cup _{t\in \mathbb{Z}}%
\mathcal{F}_{t})$ $\subseteq $ $\mathcal{F}$. We drop the (pseudo) true
value $\theta _{0}$ from function arguments when there is no confusion. $K$
is a positive constant the value of which may be different in different
places.

\paragraph{Assumptions}

The class of time series models considered is:%
\begin{equation}
y_{t}=f(x_{t-1},\phi _{0})+u_{t}\text{ \ and \ }u_{t}=\epsilon _{t}\sigma
_{t}(\theta _{0})  \label{reg_model}
\end{equation}%
where $\phi $ $\in $ $\mathbb{R}^{k_{\phi }}$, $k_{\phi }$ $\geq $ $0$, and $%
f(x,\phi )$ is a level response function. $\epsilon _{t}$ satisfies $%
E[\epsilon _{t}]$ $=$ $0$, $E[\epsilon _{t}^{2}]$ $<$ $\infty $, and the
regressors are $x_{t}$ $\in $ $\mathbb{R}^{k_{x}}$, $k_{x}$ $\geq $ $0$. $%
\{x_{t},y_{t}\}$ are strictly stationary in order to focus ideas. $\sigma
_{t}^{2}(\theta _{0})$ is a process measurable with respect to $\mathcal{F}%
_{t-1}$ $\equiv $ $\sigma (y_{\tau },x_{\tau }$ $:$ $\tau $ $\leq $ $t$ $-$ $%
1)$, where $\theta _{0}$ is decomposed as $[\phi _{0}^{\prime },\delta
_{0}^{\prime }]$ and $\delta _{0}\in $ $\mathbb{R}^{k_{\delta }}$ are
volatility-specific parameters, $k_{\delta }$ $\geq $ $0$.

Unless $y_{t}$ $=$ $\epsilon _{t}$ such that $y_{t}$ is known to have a zero
mean, let $\hat{\theta}_{n}$ $=$ $[\hat{\phi}_{n}^{\prime },\hat{\delta}%
_{n}^{\prime }]$ estimate $\theta _{0}$ where $n$ is the sample size, and
define the residual, and its sample serial covariance and correlation at lag
$h$ $\geq $ $1$:
\begin{equation*}
\epsilon _{t}(\hat{\theta}_{n})\equiv \frac{u_{t}(\hat{\phi}_{n})}{\sigma
_{t}(\hat{\theta}_{n})}\equiv \frac{y_{t}-f(x_{t-1},\hat{\phi}_{n})}{\sigma
_{t}(\hat{\theta}_{n})}\text{ \ and \ }\hat{\gamma}_{n}(h)\equiv \frac{1}{n}%
\sum_{t=1+h}^{n}\epsilon _{t}(\hat{\theta}_{n})\epsilon _{t-h}(\hat{\theta}%
_{n})\text{ \ and }\hat{\rho}_{n}(h)\equiv \frac{\hat{\gamma}_{n}(h)}{\hat{%
\gamma}_{n}(0)}.
\end{equation*}%
In the pure volatility model set $f(x_{t-1},\hat{\phi}_{n})$ $=$ $0$, and in
the level model set $\sigma _{t}(\hat{\theta}_{n})$ $=$ $1$.

The general test statistic is based on the measurable function:
\begin{equation}
\vartheta :\mathbb{R}^{\mathcal{L}}\rightarrow \lbrack 0,\infty )
\label{phi_map}
\end{equation}%
that satisfies the following: $\vartheta (x)$ is continuous; lower bound $%
\vartheta (a)$ $=$ $0$ \emph{if and only if} $a$ $=$ $0$; upper bound $%
\vartheta (a)$ $\leq $ $K\mathcal{LM}$ for some $K$ $>$ $0$ and any $a$ $=$ $%
[a_{h}]_{h=1}^{\mathcal{L}}$ such that $|a_{h}|$ $\leq $ $\mathcal{M}$ for
each $h$; divergence $\vartheta (a)$ $\rightarrow $ $\infty $ as $||a||$ $%
\rightarrow $ $\infty $; monotonicity $\vartheta (a_{\mathcal{L}_{1}})$ $%
\leq $ $\vartheta ([a_{\mathcal{L}_{1}}^{\prime },c_{\mathcal{L}_{2}-%
\mathcal{L}_{1}}^{\prime }]^{\prime })$ where $(a_{\mathcal{L}},c_{\mathcal{L%
}})$ $\in $ $\mathbb{R}^{\mathcal{L}}$, $\forall \mathcal{L}_{2}$ $\geq $ $%
\mathcal{L}_{1}$ and any $c_{\mathcal{L}_{2}-\mathcal{L}_{1}}$ $\in $ $%
\mathbb{R}^{\mathcal{L}_{2}-\mathcal{L}_{1}}$; and the triangle inequality $%
\vartheta (a$ $+$ $b)$ $\leq $ $\vartheta (a)$ $+$ $\vartheta (b)$ $\forall
a,b$ $\in $ $\mathbb{R}^{\mathcal{L}_{n}}$. Examples include the maximum $%
\vartheta (a)$ $=$ $\max_{1\leq h\leq \mathcal{L}}|a_{h}|$ and sum $%
\vartheta (a)$ $=$ $\sum_{h=1}^{\mathcal{L}}|a_{h}|$, where $a$ $=$ $%
[a_{h}]_{h=1}^{\mathcal{L}}$. The general test statistic is:%
\begin{equation*}
\mathcal{\hat{T}}_{n}\equiv \vartheta \left( \left[ \sqrt{n}\hat{\omega}%
_{n}(h)\hat{\rho}_{n}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) \text{ with
weights }\hat{\omega}_{n}(h)>0\text{ and }\hat{\omega}_{n}(h)\overset{p}{%
\rightarrow }\omega (h)\in \left( 0,\infty \right) .
\end{equation*}

Let $\{\upsilon _{t}\}$ be a stationary $\alpha $-mixing process with $%
\sigma $-fields $\mathfrak{V}_{s}^{t}$ $\equiv $ $\sigma (\upsilon _{\tau }$
$:$ $s$ $\leq $ $\tau $ $\leq $ $t)$ and $\mathfrak{V}_{t}\equiv \mathfrak{V}%
_{-\infty }^{t}$ and coefficients
\begin{equation*}
\alpha _{h}^{(\upsilon )}=\sup_{t\in \mathbb{Z}}\sup_{\mathcal{A}\subset
\mathfrak{V}_{t}^{\infty },\mathcal{B}\subset \mathfrak{V}_{-\infty
}^{t-h}}\left\vert P\left( \mathcal{A}\cap \mathcal{B}\right) -P\left(
\mathcal{A}\right) P\left( \mathcal{B}\right) \right\vert .
\end{equation*}%
We say $\{\epsilon _{t}\}$ is stationary $L_{q}$-NED with size $\lambda $ on
a mixing \textit{base} $\{\upsilon _{t}\}$ when $\epsilon _{t}$ is $L_{q}$%
-bounded and%
\begin{equation*}
\left\Vert \epsilon _{t}-E[\epsilon _{t}|\mathfrak{V}_{t-h}^{t+h}]\right%
\Vert _{q}\leq K\psi _{h}\text{ where }\psi _{h}=O\left( h^{-\lambda -\iota
}\right) \text{ for tiny }\iota >0.
\end{equation*}%
If $\epsilon _{t}$ $=$ $\upsilon _{t}$ then $||\epsilon _{t}$ $-$ $%
E[\epsilon _{t}|\mathfrak{V}_{t-h}^{t+h}]||_{q}$ $=$ $0$, hence NED includes
mixing sequences, but it also includes non-mixing sequences since it covers
infinite lag functions of mixing sequences that need not be mixing %
\citep[see, e.g.,][]{Doukhan94}. See \cite{Davidson1994} for historical
references and deep results.

\begin{assumption}[data generating process]
\label{assum:dgp}$\medskip $\newline
$a$. $\{x_{t},y_{t}\}$ are stationary, ergodic, and $L_{2+\delta }$-bounded
for tiny $\delta $ $>$ $0.\medskip $\newline
$b$. $\epsilon _{t}$ is stationary, ergodic, $E[\epsilon _{t}]$ $=$ $0$, $%
L_{r}$-bounded, $r$ $>$ $4$, and $L_{4}$-NED with size $1/2$ on stationary $%
\alpha $-mixing $\{\upsilon _{t}\}$ with coefficients $\alpha
_{h}^{(\upsilon )}$ $=$ $O(h^{-r/(r-2)-\iota })$ for tiny $\iota $ $>$ $%
0.\medskip $\newline
$c$. The weights satisfy $\hat{\omega}_{n}(h)$ $>$ $0$ $a.s.$ and $\hat{%
\omega}_{n}(h)$ $=$ $\omega (h)+O_{p}(1/n^{\kappa })$ for some $\kappa $ $>$
$0$ and non-random $\omega (h)$ $\in $ $(0,\infty )$, for each $h$.
\end{assumption}

We require notation that makes use of estimating equations $m_{t}$ $\in $ $%
\mathbb{R}^{k_{m}}$ and a matrix $\mathcal{A}$ $\in $ $\mathbb{R}^{k_{\theta
}\times k_{m}}$ defined under Assumption \ref{assum:plug}.c. Define%
\begin{equation}
\mathcal{D}(h)\equiv E\left[ \left( \epsilon _{t}s_{t}+\frac{G_{t}}{\sigma
_{t}}\right) \epsilon _{t-h}\right] +E\left[ \epsilon _{t}\left( \epsilon
_{t-h}s_{t-h}+\frac{G_{t-h}}{\sigma _{t-h}}\right) \right] \in \mathbb{R}%
^{k_{\theta }},  \label{Dh}
\end{equation}%
and
\begin{equation*}
z_{t}(h)\equiv r_{t}(h)-\rho (h)r_{t}(0)\text{ where }r_{t}(h)\equiv \frac{%
\epsilon _{t}\epsilon _{t-h}-E\left[ \epsilon _{t}\epsilon _{t-h}\right] -%
\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}}{E\left[ \epsilon _{t}^{2}\right] }%
\text{ and }\rho (h)\equiv \frac{E[\epsilon _{t}\epsilon _{t-h}]}{E[\epsilon
_{t}^{2}]}.
\end{equation*}%
The process that arises in the key approximation is:
\begin{equation}
\mathcal{Z}_{n}(h)\equiv \frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}z_{t}(h).
\label{Zn}
\end{equation}

\begin{assumption}[plug-in: response and identification]
\label{assum:plug} $\ \ \ \medskip $\newline
$a$. \emph{Level response.} $f$ $:$ $\mathbb{R}^{k_{x}}\times \Phi $ $%
\rightarrow $ $\mathbb{R}$, where $\Phi $ is a compact subset of $\mathbb{R}%
^{k_{\phi }}$, $k_{\phi }$ $\geq $ $0$; $f(x,\phi )$ is Borel measurable for
each $\phi $, and for each $x$ three times continuously differentiable,
where $(\partial /\partial \phi )^{j}f(x,\phi )$ is Borel measurable for
each $\phi $ and $j$ $=$ $1,2,3$; $E[\sup_{\phi \in \mathcal{N}_{\phi
_{0}}}|(\partial /\partial \phi )^{j}f(x_{t},\phi )|^{6}]$ $<$ $\infty $ for
$j$ $=$ $0,1,2,3$ and some compact set with positive measure $\mathcal{N}%
_{\phi _{0}}\subseteq \Phi $ containing $\phi _{0}$.\medskip \newline
$b$. \emph{Volatility.} $\sigma _{t}^{2}$ $:\Theta $ $\rightarrow $ $%
[0,\infty )$ where $\Theta $ $=$ $\Phi $ $\times $ $\Delta $ $\in $ $\mathbb{%
R}^{k_{\theta }}$, and $\Delta $\ is a compact subset of $\mathbb{R}%
^{k_{\delta }}$, $k_{\delta }$ $\geq $ $0$; $\sigma _{t}^{2}(\theta )$ is $%
\mathcal{F}_{t-1}$-measurable, continuous, and three times continuously
differentiable, where $(\partial /\partial \theta )^{j}\ln \sigma
_{t}^{2}(\theta )$ is Borel measurable for each $\theta $ and $j$ $=$ $%
0,1,2,3$; $\inf_{\theta \in \Theta }|\sigma _{t}^{2}(\theta )|$ $\geq $ $%
\iota $ $>$ $0$ $a.s.$ and $E[\sup_{\theta \in \mathcal{N}_{\theta
_{0}}}|(\partial /\partial \theta )^{j}\ln \sigma _{t}^{2}(\theta )|^{4}]$ $%
< $ $\infty $ for $j$ $=$ $0,1,2,3$ and some compact subset $\mathcal{N}%
_{\theta _{0}}\subseteq \Theta $ containing $\theta _{0}$.\medskip \newline
$c$. \emph{Estimator}$.$ $\hat{\theta}_{n}$ $\in $ $\Theta $ for each $n$,
and for a unique interior point $\theta _{0}$ $\in $ $\Theta $ we have $%
\sqrt{n}(\hat{\theta}_{n}$ $-$ $\theta _{0})$ $=$ $\mathcal{A}%
n^{-1/2}\sum_{t=1}^{n}m_{t}(\theta _{0})$ $+$ $\mathcal{R}_{m}(n)$, where
the $k_{m}$ $\times $ $1$ stochastic remainder $\mathcal{R}_{m}(n)$ $=$ $%
O_{p}(n^{-\zeta })$\ for some $\zeta $ $>$ $0$, with $\mathcal{F}_{t}$%
-measurable estimating equations $m_{t}$ $=$ $[m_{i,t}]_{i=1}^{k_{m}}$ $:$ $%
\Theta $ $\rightarrow $ $\mathbb{R}^{k_{m}}$ for $k_{m}$ $\geq $ $k_{\theta
} $, and non-stochastic $\mathcal{A}$ $\in $ $\mathbb{R}^{k_{\theta }\times
k_{m}}$. Moreover, zero mean $m_{t}(\theta _{0})$ is stationary, ergodic, $%
L_{r/2}$-bounded and $L_{2}$-NED with size $1/2$ on $\{\upsilon _{t}\}$,
where $r$ $>$ $4$ and $\{\upsilon _{t}\}$\ appear in Assumption \ref%
{assum:dgp}.b.\medskip \newline
$d$. \emph{Nondegenerate Finite Dimensional Variance}$.$ Let $\mathcal{L}$ $%
\in $ $\mathbb{N}$ be arbitrary, and let $\lambda $ $\equiv $ $[\lambda
_{h}]_{h=1}^{\mathcal{L}}$ $\in $ $\mathbb{R}^{\mathcal{L}}$. Then $\lim
\inf_{n\rightarrow \infty }\inf_{\lambda ^{\prime }\lambda =1}E[(\sum_{h=1}^{%
\mathcal{L}}\lambda _{h}\mathcal{Z}_{n}(h))^{2}]$ $>$ $0$.
\end{assumption}

\begin{assumption2c'}
$\hat{\theta}_{n}$ $\in $ $\Theta $ for each $n$, and for a unique interior
point $\theta _{0}$ $\in $ $\Theta $ we have $\sqrt{n}(\hat{\theta}_{n}$ $-$
$\theta _{0})$ $=$ $\mathcal{A}n^{-1/2}\sum_{t=1}^{n}m_{t}(\theta _{0})$ $+$
$\mathcal{R}_{m}(n)$ where the $k_{m}$ $\times $ $1$ stochastic remainder $%
\mathcal{R}_{m}(n)$ $=$ $O_{p}(n^{-\zeta })$\ for some $\zeta $ $>$ $0$,
with $\mathcal{F}_{t}$-measurable estimating equations $m_{t}$ $=$ $%
[m_{i,t}]_{i=1}^{k_{m}}$ $:$ $\Theta $ $\rightarrow $ $\mathbb{R}^{k_{m}}$
for $k_{m}$ $\geq $ $k_{\theta }$; and non-stochastic $\mathcal{A}$ $\in $ $%
\mathbb{R}^{k_{\theta }\times k_{m}}$. $m_{t}(\theta )$ is twice
continuously differentiable, $(\partial /\partial \theta )^{j}m_{t}(\theta )$
is Borel measurable for each $\theta $ and $j$ $=$ $1,2$, and $%
E[\sup_{\theta \in \Theta }|(\partial /\partial \theta )^{i}m_{j,t}(\theta
)|]$ $<$ $\infty $ for each $i$ $=$ $0,1,2$ and $j$ $=$ $1,...,k_{m}$.
Moreover, zero mean $m_{t}$ is stationary, ergodic, $L_{r/2}$-bounded and $%
L_{2}$-NED with size $1/2$ on $\{\upsilon _{t}\}$, where $r$ $>$ $4$ and $%
\{\upsilon _{t}\}$\ appear in Assumption \ref{assum:dgp}.b.
\end{assumption2c'}

We use the following variance bound for NED sequences repeatedly and without
further citation. If $w_{t}$\ is zero mean, stationary, $L_{p}$-bounded for
some $p$ $>$ $2$, and $L_{2}$-NED with size $1/2$, on an $\alpha $-mixing
base with decay $O(h^{-p/(p-2)-\iota })$, then by Theorem 17.5 in \cite%
{Davidson1994} and Theorem 1.6 in \cite{McLeish1975}:%
\begin{equation}
E\left[ \left( \frac{1}{\sqrt{n}}\sum_{t=1}^{n}w_{t}\right) ^{2}\right]
=O(1).  \label{NED_var}
\end{equation}

\setcounter{equation}{0}

\section{Moment Contraction and NED\label{app:nedmc}}

Consider Wu's (\citeyear{Wu2005}) physical dependence measure
\citep[see
also][]{WuShao2004,WuMin2005} exploited in \cite{Shao2011_JoE}, \cite%
{XiaoWu2014} and \cite{GuayGuerreLazarova2013} for white noise tests. Assume
there exists a measurable function $g$ and random variables $\{v_{t}\}_{t\in
\mathbb{Z}}$ such that
\begin{equation*}
\epsilon _{t}=g(\nu _{t},\nu _{t-1},...)
\end{equation*}%
is a well-defined random variable. If $v_{t}$ is iid then $\{\epsilon _{t}\}$
is a stationary and ergodic process. Let $\epsilon _{t}^{\prime }$ $=$ $%
g(\nu _{t},\nu _{t-1},...,v_{0}^{\prime },v_{-1}^{\prime },...)$ where $%
\{v_{t}^{\prime }\}_{t\in \mathbb{Z}}$ is an iid copy of $\{v_{t}\}_{t\in
\mathbb{Z}}$, thus $\epsilon _{t}^{\prime }$ is a coupled version of $%
\epsilon _{t}$. The dependence quantity
\begin{equation*}
d_{t}^{(\epsilon )}(q)\equiv \left\Vert \epsilon _{t}-\epsilon _{t}^{\prime
}\right\Vert _{q}
\end{equation*}%
measures the impact of a distant past on $\epsilon _{t}$ as $t$ $\rightarrow
$ $\infty $. We say that a random variable $z_{t}$ is $L_{q}$-\textit{moment
contracting}, or MC$_{q}$, when $z_{t}$ is $L_{q}$-bounded and
\begin{equation*}
\sum_{k=0}^{\infty }d_{k}^{(z)}(q)<\infty .
\end{equation*}%
Under stationary MC$_{2}$, $\{z_{t}\}$ is a short memory process since
\citep[see, e.g.][Remark
2.1]{Shao2011_JoE}:
\begin{equation*}
\sum_{h=0}^{\infty }\left\vert E[z_{t}z_{t-h}]\right\vert \leq \left(
\sum_{k=0}^{\infty }d_{k}^{(z)}(2)\right) ^{2}<\infty .
\end{equation*}%
We say $z_{t}$ is geometrically MC$_{q}$, or GMC$_{q}$, if $E|\epsilon _{t}$
$-$ $\epsilon _{t}^{\prime }|^{q}$ $\leq $ $K\rho ^{t}$ for some $\rho $ $%
\in $ $(0,1)$.

It is easy to show
\begin{equation*}
\left\Vert \epsilon _{t}\epsilon _{t-h}-\epsilon _{t}^{\prime }\epsilon
_{t-h}^{\prime }\right\Vert _{q/2}\leq K\left\{ d_{t}(q)+d_{t-h}(q)\right\} ,
\end{equation*}%
hence $\epsilon _{t}\epsilon _{t-h}$ is MC$_{2}$\ when $\epsilon _{t}$ is MC$%
_{4}$, and therefore $\{\epsilon _{t}\epsilon _{t-h}\}$ is a short memory
process when $\epsilon _{t}$ is MC$_{4}$.

If $\epsilon _{t}\epsilon _{t-h}$ is $L_{2}$-NED with size $1/2$\ then by
Theorem 1.6 in \cite{McLeish1975}, cf. \citet[Theorem
17.2]{Davidson1994}:
\begin{equation*}
\mathcal{S}\equiv \lim_{n\rightarrow \infty }E\left[ \left( \frac{1}{\sqrt{n}%
}\sum_{t=1}^{n}\left\{ \epsilon _{t}\epsilon _{t-h}-E[\epsilon _{t}\epsilon
_{t-h}]\right\} \right) ^{2}\right] <\infty .
\end{equation*}%
Conversely, MC$_{4}$ does not generally imply $\mathcal{S}$ $<$ $\infty $,
hence $\mathcal{S}$ $<$ $\infty $ would have to be assumed under an MC
assumption. In the present context, $\mathcal{S}$ $<$ $\infty $ holds if:
(i) a standard cumulant series bound holds which essentially reduces to $%
\mathcal{S}$ $<$ $\infty $ \citep{WuShao2004,Shao2011_JoE}; or (ii) $%
\epsilon _{t}$ is GMC$_{4}$ (Wu and Shao, 2004, Proposition 2, cf. Wu, 2005)%
\nocite{WuShao2004}\nocite{Wu2005}; or (iii) $\epsilon _{t}$ $=$ $%
\sum_{i=0}^{\infty }\psi _{i}\upsilon _{t-i}$ with zero mean $\upsilon _{t}$
$=$ $g(\omega _{t},\omega _{t-1},...)$, iid $\omega _{t}$, $%
\sum_{i,j=0}^{\infty }|\psi _{i}\psi _{j}E[\upsilon _{0}\upsilon _{i-j}]|$ $%
< $ $\infty $ and $\sum_{i=0}^{\infty }i^{1/2}|\psi _{i}|$ $<$ $\infty $,
provided $\upsilon _{t}$ and $\epsilon _{t}\epsilon _{t-h}$ are respectively
$L_{2}$- and $L_{4}$-\textit{weakly dependent}
\citep[Theorem 3, Proposition
1]{WuMin2005}. See \cite{Hannan1973} and \cite{WuMin2005} for definitions of
$L_{p}$-\textit{weak dependence}. If $\epsilon _{t}\epsilon _{t-h}$ is MC$%
_{2}$ then it is $L_{2}$-weakly dependent \citep[Theorem 1]{Wu2005}, hence
if $\mathcal{S}$ $<$ $\infty $ then central limit theorems in \cite%
{Hannan1973} and \cite{WuMin2005} apply.

Now define
\begin{equation*}
\mathcal{P}_{0}\epsilon _{t}\equiv E[\epsilon _{t}|\mathfrak{V}%
_{0}]-E[\epsilon _{t}|\mathfrak{V}_{-1}]\text{ where }\mathfrak{V}_{t}\equiv
\sigma (\upsilon _{\tau }:\tau \leq t).
\end{equation*}%
By Theorem 1 in \cite{Wu2005}, $||\mathcal{P}_{0}\epsilon _{t}||_{q}$ $\leq $
$d_{t}^{(\epsilon )}(q)$, hence MC$_{q}$ implies $\sum_{k=0}^{\infty }||%
\mathcal{P}_{0}\epsilon _{t}||_{q}$ $<$ $\infty $. Similarly, if $\epsilon
_{t}$ is $L_{q}$-NED with size $1$ on $\{\nu _{t}\}$\ then $\epsilon _{t}$
is an $\mathfrak{V}_{t}$-adapted $L_{q}$-mixingale with size $1$ and
therefore $\sum_{k=0}^{\infty }||E[\epsilon _{k}|\mathfrak{F}_{0}]||_{q}$ $<$
$0$ hence $\sum_{k=0}^{\infty }||\mathcal{P}_{0}\epsilon _{t}||_{q}$ $<$ $%
\infty $ \citep[see, e.g.,][]{MengLin2009}. Thus, MC$_{q}$ and $L_{q}$-NED
with size $1$ equally imply a sufficient condition for linear processes of $%
\epsilon _{t}$ to satisfy a central limit theorem %
\citep{Hannan1973,WuMin2005}. However, in Assumption \ref{assum:dgp}\ we
only need $\epsilon _{t}$ to be $L_{4}$-NED with size $1/2$, which allows
for slower memory decay than MC$_{4}$.

In addition to MC$_{8}$ and therefore a finite $8^{th}$ moment, \cite%
{Shao2011_JoE} imposes an eighth order cumulant summability condition which
ultimately ensures a cross product partial sum has a finite second moment.
As discussed above, $L_{4}$-NED \ under Assumption 1.b implies $\mathcal{S}$
$<$ $\infty $, hence no additional cumulant condition is required.

\cite{GuayGuerreLazarova2013} impose an MC$_{12}$ condition and therefore
require at least a $12^{th}$ moment. Further, rather than impose sufficient
conditions for weak convergence as in \cite{Lobato2001}, \cite{Shao2011_JoE}%
, and in the present paper under Assumption \ref{assum:dgp}, they assume it
under their Assumption M in the form of joint weak convergence of a
covariance process and the plug-in estimator. They claim their Assumption M
holds for linear processes and a least squares estimator, but no further
processes or estimators are considered.

\cite{XiaoWu2014} impose MC$_{4}$ and restrict the degree of dependence
based on the rate of increase of the maximum lag. We do not tie the degree
of dependence to the maximum lag, while $L_{4}$-NED with size $1/2$ allows
for slower memory decay.

\setcounter{equation}{0}

\section{Examples\label{app:examples}}

We give several examples of models under (\ref{reg_model}) in order to
verify the assumptions. Recall%
\begin{eqnarray*}
&&G_{t}(\phi )\equiv \left[ \frac{\partial }{\partial \phi ^{\prime }}%
f(x_{t-1},\phi ),\boldsymbol{0}_{k_{\delta }}^{\prime }\right] ^{\prime }\in
\mathbb{R}^{k_{\theta }}\text{ \ and \ }s_{t}(\theta )\equiv \frac{1}{2}%
\frac{\partial }{\partial \theta }\ln \sigma _{t}^{2}(\theta ) \\
&& \\
&&\mathcal{D}(h)\equiv E\left[ \left( \epsilon _{t}s_{t}+\frac{G_{t}}{\sigma
_{t}}\right) \epsilon _{t-h}\right] +E\left[ \epsilon _{t}\left( \epsilon
_{t-h}s_{t-h}+\frac{G_{t-h}}{\sigma _{t-h}}\right) \right] \in \mathbb{R}%
^{k_{\theta }}.
\end{eqnarray*}%
Under Assumption \ref{assum:plug}
\begin{equation*}
\sqrt{n}(\hat{\theta}_{n}-\theta _{0})=\mathcal{A}\frac{1}{n^{1/2}}%
\sum_{t=1}^{n}m_{t}(\theta _{0})+o_{p}(1),
\end{equation*}%
with $\mathcal{F}_{t}$-measurable estimating equations $m_{t}$ $=$ $%
[m_{i,t}]_{i=1}^{k_{m}}$ $:$ $\Theta $ $\rightarrow $ $\mathbb{R}^{k_{m}}$
for $k_{m}$ $\geq $ $k_{\theta }$, and non-stochastic $\mathcal{A}$ $\in $ $%
\mathbb{R}^{k_{\theta }\times k_{m}}$. Define%
\begin{equation*}
z_{t}(h)\equiv r_{t}(h)-\rho (h)r_{t}(0)\text{ where }r_{t}(h)\equiv \frac{%
\epsilon _{t}\epsilon _{t-h}-E\left[ \epsilon _{t}\epsilon _{t-h}\right] -%
\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}}{E\left[ \epsilon _{t}^{2}\right] }%
\text{ and }\rho (h)\equiv \frac{E[\epsilon _{t}\epsilon _{t-h}]}{E[\epsilon
_{t}^{2}]},
\end{equation*}%
and
\begin{equation*}
\mathcal{Z}_{n}(h)\equiv \frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}z_{t}(h).
\end{equation*}

Lemma 2.1 states that under Assumptions \ref{assum:dgp} and \ref{assum:plug}%
:
\begin{equation*}
\left\vert \vartheta \left( \sqrt{n}\left[ \hat{\rho}_{n}(h)-\rho (h)\right]
_{h=1}^{\mathcal{L}_{n}}\right) -\vartheta \left( \left[ \mathcal{Z}_{n}(h)%
\right] _{h=1}^{\mathcal{L}_{n}}\right) \right\vert \leq \vartheta \left( %
\left[ \sqrt{n}\{\hat{\rho}_{n}(h)-\rho (h)\}-\mathcal{Z}_{n}(h)\right]
_{h=1}^{\mathcal{L}_{n}}\right) \overset{p}{\rightarrow }0,
\end{equation*}%
for some non-unique sequence $\{\mathcal{L}_{n}\}$ of positive integers,
where $\mathcal{L}_{n}$ $\rightarrow $ $\infty $ and $\mathcal{L}_{n}$ $=$ $%
o(n)$.

We specifically characterize the form of $\mathcal{Z}_{n}(h)$ under the null
in order to compute the bootstrapped p-value:%
\begin{equation}
\mathcal{Z}_{n}(h)\equiv \frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}\frac{\left\{
\epsilon _{t}\epsilon _{t-h}-\mathcal{D}(h)^{\prime }\mathcal{A}%
m_{t}\right\} }{E\left[ \epsilon _{t}^{2}\right] }.  \label{Zn_H0}
\end{equation}%
The exact form of $\mathcal{Z}_{n}(h)$\ can be simplified depending on
whether $\epsilon _{t}$ is assumed independent under the null, the
regressors $\{x_{t}\}$ are independent of the sequence $\{\epsilon _{t}\}$,
the stochastic volatility component $\sigma _{t}$ is estimated, and the
level response $f$ is linear.

In the following we operate under the null.

\subsection{Level Response}

The level response model is
\begin{equation*}
y_{t}=f(x_{t-1},\phi _{0})+u_{t}.
\end{equation*}%
Assume $f(\cdot ,\phi )$ is three times continuously differentiable in $\phi
$ $\in $ $\mathbb{R}^{k_{\phi }}$, $E[u_{t}^{2}]$ $<$ $\infty $, and $%
E[G_{t}G_{t}^{\prime }]$ is finite and positive definite where $G_{t}$ $=$ $%
G_{t}(\phi _{0})$ $=$ $[G_{t,i}(\phi _{0})]_{i=1}^{k}$ $\equiv $ $(\partial
/\partial \phi )f(x_{t-1},\phi _{0})$. Define nonlinear least squares
estimating equations $m_{t}(\phi )$ $=$ $(y_{t}$ $-$ $f(x_{t-1},\phi ))$ $%
\times $ $(\partial /\partial \phi )f(x_{t-1},\phi )$. Assume $E[m_{t}(\phi
)]$ $=$ $0$ \emph{if and only if} $\phi =\phi _{0}$, a unique interior point
of compact $\Phi $.

Assume $\{u_{t},x_{t}\}$ are stationary $L_{r}$-bounded, $r$ $>$ $4$, and $%
\alpha $-mixing with coefficients $\alpha _{h}=$\linebreak $%
O(h^{-r/(r-4)}/\ln (h))$. Assume $E[\sup_{\phi \in \mathcal{N}_{\phi
_{0}}}|G_{t,i}(\phi )|^{r}]$ $<$ $\infty $ for each $i$, some $r$ $>$ $4$,
and some compact $\mathcal{N}_{\phi _{0}}\subseteq \Phi $ containing $\phi
_{0}$. Many nonlinear response functions satisfy this condition under $L_{r}$%
-boundedness of $\{u_{t},x_{t}\}$, including linear, logistic, and
trigonometric functions. Then $m_{t}$ is stationary, $L_{r}$-bounded, and $%
\alpha $-mixing. Sufficient conditions for stationary geometric ergodicity
of nonlinear AR-GARCH with iid innovations are in \cite{MeitzSaikkonen2008},
amongst others, cf. \citet[Chapt. 2.4.2]{Doukhan94}.

Define the nonlinear least squares estimator $\hat{\phi}_{n}$ $=$ $\arg
\min_{\phi \in \Phi }\{1/n\sum_{t=1}^{n}(y_{t}$ $-$ $f(x_{t-1},\phi ))^{2}\}$%
. By construction $s_{t}$ $=.5(\partial /\partial \theta )\ln \sigma
_{t}^{2} $ $=$ $0$ since $\sigma _{t}^{2}$ $=$ $1$, and $\mathcal{D}(h)$ $%
\mathcal{=}$ $E[u_{t-h}G_{t}]$ $+$ $E[u_{t}G_{t-h}]$ $\mathcal{=}$ $%
E[u_{t-h}G_{t}]$. Then Assumptions \ref{assum:dgp} and \ref{assum:plug}\
hold, with $m_{t}$ $=$ $u_{t}G_{t}$ and $\mathcal{A}$ $=(E[G_{t}G_{t}^{%
\prime }])^{-1}$, hence%
\begin{equation}
\sqrt{n}\hat{\rho}_{n}(h)=\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}\frac{%
u_{t}\left\{ u_{t-h}-E[u_{t-h}G_{t}]^{\prime }\left( E\left[
G_{t}G_{t}^{\prime }\right] \right) ^{-1}G_{t}\right\} }{E\left[ u_{t}^{2}%
\right] }+o_{p}(1).  \label{rho_ee_eGAm}
\end{equation}%
If additionally $u_{t}$ is independent of the sequence $\{x_{t}\}$, then $%
E[u_{t-h}G_{t}]$ $=$ $0$, hence%
\begin{equation*}
\sqrt{n}\hat{\rho}_{n}(h)=\linebreak \frac{1}{\sqrt{n}}%
\sum_{t=1+h}^{n}u_{t}u_{t-h}/E[u_{t}^{2}]+o_{p}(1),
\end{equation*}%
the well-known result that $\hat{\phi}_{n}$ does not impact the limit
distribution of $\sqrt{n}\hat{\rho}_{n}(h)$ \citep[cf.][]{Wooldridge1990}.

\subsection{Linear Response with Least Squares}

The model is
\begin{equation*}
y_{t}=\phi _{0}^{\prime }x_{t-1}+u_{t},\quad E[u_{t}]=0.
\end{equation*}%
Let $E[(y_{t}$ $-$ $\phi _{0}^{\prime }x_{t-1})u_{t}]$ $=$ $0$ for a unique
interior $\phi _{0}$ $\in $ $\Phi $, and assume $E[x_{t}x_{t}^{\prime }]$ is
positive definite. Assume $\{x_{t},u_{t}\}$ are stationary and ergodic, $%
L_{r}$-bounded, $p$ $>$ $4$, and $L_{4}$-NED on an $\alpha $-mixing base
with coefficient decay $O(h^{-r/(r-4)-\iota })$. An AR process with an iid
error that has a continuous bounded distribution is geometrically $\alpha $%
-mixing and therefore geometrically NED. This extends to linear or nonlinear
GARCH errors \citep[see,
e.g.,][]{Doukhan94,MeitzSaikkonen2008}.

By construction $G_{t}$ $=$ $x_{t-1}$ and $s_{t}$ $=$ $0$ hence $\mathcal{D}%
(h)$ $\equiv $ $E[u_{t-h}x_{t-1}]$. If $\hat{\phi}_{n}$ is the least squares
estimator then Assumptions \ref{assum:dgp} and \ref{assum:plug} are
satisfied, with $\mathcal{A}$ $=$ $(E[x_{t}x_{t}^{\prime }])^{-1}$ and $%
m_{t} $ $=$ $u_{t}x_{t-1}$. Therefore:%
\begin{equation*}
\sqrt{n}\hat{\rho}_{n}(h)=\frac{1}{E\left[ u_{t}^{2}\right] }\frac{1}{\sqrt{n%
}}\sum_{t=1+h}^{n}u_{t}\left\{ u_{t-h}-E[u_{t-h}x_{t-1}]^{\prime
}(E[x_{t}x_{t}^{\prime }])^{-1}x_{t-1}\right\} +o_{p}(1).
\end{equation*}%
If $u_{t}^{2}$ $-$ $E[u_{t}^{2}]$ is an adapted mds then $E[m_{i,t}^{2}]$ $=$
$E[u_{t}^{2}]E[x_{i,t}^{2}]$ $<$ $\infty $ and $E[(u_{t}u_{t-h})^{2}]$ $=$ $%
(E[u_{t}^{2}])^{2}$ $<$ $\infty $, hence we only need $\{x_{t},u_{t}\}$ to
be $L_{r}$-bounded, $r$ $>$ $2$, and $\alpha $-mixing.

\subsection{Mean Model with Sample Mean}

The mean model is $y_{t}$ $=$ $E[y_{t}]$ $+$ $u_{t}$, hence $f(x_{t-1},\phi
_{0})$ $=$ $E[y_{t}]$, $\sigma _{t}$ $=$ $1$ and $E[u_{t}]$ $=$ $0$. Assume $%
\{y_{t}\}$ is stationary, ergodic, $L_{r}$-bounded for some $r$ $>$ $4$, and
$L_{4}$-NED on an $\alpha$-mixing base with decay $O(h^{-r/(r-4)-\iota })$.
Then $G_{t}$ $=$ $1$ and $s_{t}$ $=$ $0$, hence $\mathcal{D}(h)$ $\equiv $ $%
E[u_{t}u_{t-h}]$ $+$ $E[u_{t}]$ $=$ $E[(y_{t}$ $-$ $E[y_{t}])(y_{t-h}$ $-$ $%
E[y_{t-h}])]$ hence $\mathcal{D}(h)$ $=$ $0$ under $H_{0}$ for $h$ $\geq $ $%
1 $. The plug-in estimator is the sample mean $\hat{\theta}_{n}$ $=$ $%
1/n\sum_{t=1}^{n}y_{t}$, so that $m_{t}$ $=$ $u_{t}$ and $\mathcal{A}$ $=$ $%
1 $. Assumptions \ref{assum:dgp} and \ref{assum:plug} are satisfied, hence
from (\ref{rho_ee_eGAm}):%
\begin{equation*}
\sqrt{n}\hat{\rho}_{n}(h)=\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}\frac{\left(
y_{t}-E[y_{t}]\right) \left( y_{t-h}-E[y_{t-h}]\right) }{E\left[ \left(
y_{t}-E[y_{t}]\right) ^{2}\right] }+o_{p}(1).
\end{equation*}

\subsection{GARCH(1,1) with QML}

The model is GARCH(1,1) $y_{t}$ $=$ $\sigma _{t}\epsilon _{t}$ with $\sigma
_{t}^{2}$ $=$ $\omega _{0}$ $+$ $\alpha _{0}y_{t-1}^{2}$ $+$ $\beta
_{0}\sigma _{t-1}^{2}$, $\omega _{0},\alpha _{0},\beta _{0}$ $>$ $0$, $%
E[\epsilon _{t}]$ $=$ $0$ and $E[\epsilon _{t}^{2}]$ $=$ $1$. We ignore
boundary cases by assuming $\alpha _{0},\beta _{0}$ $>$ $0$. The model
includes weak, semi-strong or strong GARCH \citep[see][]{DrostNijman1993},
in which case the model is correct in some sense since $\epsilon _{t}$ is
assumed to be serially uncorrelated. Conditions for strict stationarity in
the case of iid or mds $\epsilon _{t}$ are given in \cite{Nelson1990} and
\cite{LeeHansen1994}, and \cite{Boussama2006} proves geometric ergodicity.

Let $\theta $ $\equiv $ $[\omega ,\alpha ,\beta ]^{\prime }$, and $\Theta $ $%
=$ $[\iota _{\omega },u_{\omega }]$ $\times $ $[0,1$ $-$ $\iota ]$ $\times $
$[0,1$ $-$ $\iota ]$, where $u_{\omega }$ $>$ $\iota _{\omega }$ $>$ $0$ and
$\iota $ $\in $ $(0,1)$. Define the unobserved volatility process $\sigma
_{t}^{2}(\theta )$ $=$ $\omega $ $+$ $\alpha y_{t-1}^{2}$ $+$ $\beta \sigma
_{t-1}^{2}(\theta )$ on $\Theta $, and define the iterated process used for
estimation: $\tilde{\sigma}_{0}^{2}(\theta )$ $=$ $\omega $ and $\tilde{%
\sigma}_{t}^{2}(\theta )$ $=$ $\omega $ $+$ $\alpha y_{t-1}^{2}$ $+$ $\beta
\tilde{\sigma}_{t-1}^{2}(\theta )$ for $t$ $\geq $ $1$. Let $\theta _{0}$ be
the unique interior point of $\Theta $ such that $\sigma _{t}^{2}(\theta
_{0})$ $=$ $\sigma _{t}^{2}$ and $E[(y_{t}^{2}/\sigma _{t}^{2}(\theta _{0})$
$-$ $1)(\partial /\partial \theta )\ln (\sigma _{t}^{2}(\theta _{0}))]$ $=$ $%
0$, the QML first order moment condition. The feasible QML estimator is $%
\hat{\theta}_{n}$ $\equiv $ $\arg \inf_{\theta \in \Theta
}\{\sum_{t=1}^{n}\{\ln \tilde{\sigma}_{t}^{2}(\theta )$ $+$ $%
\sum_{t=1}^{n}y_{t}^{2}/\tilde{\sigma}_{t}^{2}(\theta )\}\}$.\footnote{%
Since we assume the start condition $\tilde{\sigma}_{0}^{2}(\theta )$ $=$ $%
\omega $ we avoid the case where $\alpha _{0}$ $=$ $0$, $\beta _{0}$ is not
identified and is therefore a nuisance parameter, and there are no GARCH
effects \citep[see][]{Andrews2001}. We do not allow nuisance parameters for
brevity, but their inclusion is straightforward, although beyond this
paper's scope.} See \cite{FZ04} for refined QML asymptotics when $\epsilon
_{t}$ is iid, and see \cite{LeeHansen1994} for the semi-strong case.

Since our assumptions must hold whether $\epsilon _{t}$ is white noise or
not, under potentially much weaker conditions than weak-GARCH %
\citep{DrostNijman1993}, we assume $\{y_{t},\epsilon _{t}\}$ are stationary
and ergodic, $(E|y_{t}|^{\iota }$,$E|\sigma _{t}^{2}|^{\iota })$ $<$ $\infty
$ for some $\iota $ $>$ $0$, $\inf_{\theta \in \Theta }|\sigma
_{t}^{2}(\theta )|$ $\geq $ $\iota $ $>$ $0$ $a.s.$, and $\{\epsilon
_{t},(\partial /\partial \theta )^{i}\ln (\sigma _{t}^{2}(\theta _{0}))$ $:$
$i$ $=$ $0,1,2,3\}$ are stationary geometrically $\alpha $-mixing. Further,
for each $\theta $ $\in $ $\Theta $ unique stationary and ergodic solutions
exist for the iterated process and its derivatives $\{(\partial /\partial
\theta )^{j}\tilde{\sigma}_{i}^{2}(\theta )$ $:$ $j$ $=$ $%
0,1,2,3\}_{i=0}^{t} $ as $t$ $\rightarrow $ $\infty $ at a geometric rate.\
We also require for some compact subset $\mathcal{N}_{\theta _{0}}$ $%
\subseteq $ $\Theta $ containing $\theta _{0}$:
\begin{eqnarray}
&&E\left[ s_{t}(\theta _{0})s_{t}^{\prime }(\theta _{0})\right] -E\left[
\left( \frac{y_{t}^{2}}{\sigma _{t}^{2}(\theta _{0})}-1\right) \frac{%
\partial ^{2}}{\partial \theta \partial \theta ^{\prime }}\ln (\sigma
_{t}^{2}(\theta _{0}))\right] \text{ is non-singular}  \label{Ess-E(e2-1)ds}
\\
&&  \notag \\
&&E\left[ \sup_{\theta \in \mathcal{N}_{\theta _{0}}}\left\vert \left( \frac{%
\partial }{\partial \theta }\right) ^{j}\ln \sigma _{t}^{2}(\theta
)\right\vert ^{4}\right] <\infty \text{ for each }j=1,2,3.
\label{dlnsigma_mom}
\end{eqnarray}

If $\epsilon _{t}$ is iid, or $\{\epsilon _{t},\epsilon _{t}^{2}$ $-$ $1\}$
are martingale differences adapted to some sequence of sigma fields $\{%
\mathcal{G}_{t}\}$, then stationary solutions exist respectively when $E[\ln
(\omega _{0}$ $+$ $\alpha _{0}\epsilon _{t}^{2})]$ $<$ $0$ and $E[\ln
(\omega _{0}$ $+$ $\alpha _{0}\epsilon _{t}^{2})|\mathcal{G}_{t}]$ $<$ $0$ $%
a.s.$ \citep{Nelson1990,LeeHansen1994}. Write $s_{t}(\theta )$ $\equiv $ $%
0.5\times (\partial /\partial \theta )\ln (\sigma _{t}^{2}(\theta ))$. If $%
\epsilon _{t}$ is iid or $\{\epsilon _{t},\epsilon _{t}^{2}$ $-$ $1\}$ are
martingale differences then (\ref{Ess-E(e2-1)ds}) holds, and (\ref%
{dlnsigma_mom}) holds by arguments in \citet[Section 4.2]{FZ04}. The latter
assume an iid error, but their proofs of (\ref{dlnsigma_mom}) do not make
use of independence. See, e.g., their equation (4.28).

Under the above conditions, Assumptions \ref{assum:dgp} and \ref{assum:plug}
hold. Assuming $\theta _{0}$ does not lie on the boundary of $\Theta $,
plug-in estimator Assumption \ref{assum:plug}.c holds with $m_{t}$ $=$ $%
(\epsilon _{t}^{2}$ $-$ $1)s_{t}$ and $\mathcal{A}$ $=$ $\{2E[\epsilon
_{t}^{2}s_{t}s_{t}^{\prime }]$ $-$ $E[(\epsilon _{t}^{2}$ $-$ $1)(\partial
/\partial \theta )s_{t}(\theta _{0})]\}$ $=$ $\{2E[s_{t}s_{t}^{\prime }]$ $-$
$E[(\epsilon _{t}^{2}$ $-$ $1)(\partial /\partial \theta )s_{t}(\theta
_{0})]\}^{-1}$. Finally, $G_{t}$ $=$ $0$ hence $\mathcal{D}(h)\equiv
E[\epsilon _{t}\epsilon _{t-h}(s_{t}$ $+$ $s_{t-h})]$, and expansion (\ref%
{Zn_H0}) holds.

\setcounter{equation}{0}

\section{Local Asymptotic Power\label{app:local_power}}

\cite{Hong1996} shows that his standardized periodogram yields non-trivial
asymptotic power against a sequence of local alternatives applied to the
spectrum, with a slower than $\sqrt{n}$ drift. Ultimately the reduced rate
arises from an increasing bandwidth parameter. Using a similar spectrum
local alternative, but with $\sqrt{n}$ drift, \cite{Shao2011_JoE} proves the
Cram\'{e}r-von Mises test achieves non-trivial local power. \cite%
{DelgadoVelasco2011} impose local $\sqrt{n}$ drift to the correlations, and
show that a weighted average portmanteau statistic yields non-trivial local
power.

The proposed bootstrap p-value test operates on $\sqrt{n}\hat{\rho}_{n}(h)$
and $\sqrt{n}\hat{\rho}_{n}^{(dw)}(h)$. The test therefore achieves the
parametric rate of local asymptotic power against the sequence of
alternatives:%
\begin{equation*}
H_{1}^{L}:\rho (h)=\rho _{n}(h)=\frac{r(h)}{\sqrt{n}}\text{ for each }h\text{%
, where }\left\vert r(h)\right\vert \leq \sqrt{n}.
\end{equation*}%
Note that $r(h)$ is a fixed constant for each $h$, where $|r(h)|$ $\leq $ $%
\sqrt{n}$ ensures $|\rho _{n}(h)|$ $\leq $ $1$.

\begin{theorem}
\label{th:local_power}Let Assumptions \ref{assum:dgp} and \ref{assum:plug}, $%
\widehat{\mathcal{A}}_{n}$ $\overset{p}{\rightarrow }$ $\mathcal{A}$, and $M$
$=$ $M_{n}$ $\rightarrow $ $\infty $ hold. There exists a non-unique
sequence of maximum lags $\{\mathcal{L}_{n}\}$, $\mathcal{L}_{n}$ $%
\rightarrow $ $\infty $ and $\mathcal{L}_{n}$ $=$ $o(n)$, such that under $%
H_{1}^{L}$, $\lim_{n\rightarrow \infty }P(\hat{p}_{n,M}^{(dw)}$ $<$ $\alpha
) $ $>$ $\alpha $ if $r(h)$ $\neq $ $0$ for some $h$ $\in $ $\mathbb{N}$.
Further $\lim_{n\rightarrow \infty }P(\hat{p}_{n,M}^{(dw)}$ $<$ $\alpha )$ $%
\nearrow $ $1$ monotonically in $|r(h)|$ $\nearrow $ $\infty $.
\end{theorem}

\noindent \textbf{Proof.}\medskip \newline
\textbf{Step 1.}\qquad Recall $\bar{F}_{n}^{(0)}(c)$ $\equiv $ $P(\vartheta
([\mathcal{\mathring{Z}}(h)]_{h=1}^{\mathcal{L}_{n}})$ $>$ $c)$, where $\{%
\mathcal{\mathring{Z}}(h)$ $:$ $h$ $\in $ $\mathbb{N}\}$ is an independent
copy of $\{\mathcal{Z}(h)$ $:$ $h$ $\in $ $\mathbb{N}\}$, the Lemma 2.2 zero
mean Gaussian process with variance $E[\mathcal{Z}(h)^{2}]$ $<$ $\infty $.
Recall $\rho _{n}(h)$ $=$ $r(h)/\sqrt{n}$ under $H_{1}^{L}$. We prove in
Step 2 that%
\begin{equation}
\mathcal{P}_{n,1}^{(L)}(\alpha )+o_{p}(1)\leq \hat{p}_{n,M_{n}}^{(dw)}\leq
\mathcal{P}_{n,2}^{(L)}(\alpha )+o_{p}(1),  \label{p_val_bound}
\end{equation}%
where%
\begin{eqnarray*}
&&\mathcal{P}_{n,1}^{(L)}(\alpha )=\bar{F}_{n}^{(0)}\left( \vartheta \left( %
\left[ \mathcal{Z}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) +\vartheta
\left( \left[ r(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) \right) \\
&&\mathcal{P}_{n,2}^{(L)}(\alpha )=\bar{F}_{n}^{(0)}\left( -\vartheta \left( %
\left[ \mathcal{Z}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) +\vartheta
\left( \left[ r(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) \right) .
\end{eqnarray*}%
By construction
\begin{equation*}
P\left( \bar{F}_{n}^{(0)}\left( \vartheta \left( \lbrack \mathcal{\mathring{Z%
}}(h)]_{h=1}^{\mathcal{L}_{n}}\right) \right) <\alpha \right) =\alpha \quad
\text{and} \quad P\left( -\bar{F}_{n}^{(0)}\left( \vartheta \left( \lbrack
\mathcal{\mathring{Z}}(h)]_{h=1}^{\mathcal{L}_{n}}\right) \right) <\alpha
\right) =1-\alpha .
\end{equation*}%
Therefore, if $r(h)$ $\neq $ $0$ for some $h$ $\in $ $\mathbb{N}$:
\begin{equation*}
\lim_{n\rightarrow \infty }P\left( \hat{p}_{n,M}^{(dw)}<\alpha \right)
>\alpha
\end{equation*}

Moreover, monotonically $P(\mathcal{P}_{n,1}^{(L)}(\alpha )$ $<$ $\alpha )$ $%
\searrow $ $0$ and $P(\mathcal{P}_{n,2}^{(L)}(\alpha )$ $<$ $\alpha )$ $%
\searrow $ $0$ as $\vartheta ([r(h)]_{h=1}^{\mathcal{L}_{n}})$ $\rightarrow $
$\infty $ and therefore as $\max_{1\leq h\leq \mathcal{L}_{n}}|r(h)|$ $%
\rightarrow $ $\infty $ for each $n$. Therefore%
\begin{eqnarray*}
&&P\left( \hat{p}_{n,M}^{(dw)}<\alpha \right) \leq P\left( o_{p}(1)<\alpha -%
\mathcal{P}_{n,2}^{(L)}(\alpha )\right) \nearrow 1\text{ as }\left\vert
r(h)\right\vert \nearrow \infty \\
&&P\left( \hat{p}_{n,M}^{(dw)}<\alpha \right) \geq P\left( o_{p}(1)<\alpha -%
\mathcal{P}_{n,1}^{(L)}(\alpha )\right) \nearrow 1\text{ as }\left\vert
r(h)\right\vert \nearrow \infty
\end{eqnarray*}%
hence $\lim_{n\rightarrow \infty }P(\hat{p}_{n,M}^{(dw)}$ $<$ $\alpha )$ $%
\nearrow $ $1$ monotonically in $|r(h)|$ $\nearrow $ $\infty $.\medskip
\newline
\textbf{Step 2.}\qquad We now prove (\ref{p_val_bound}). Recall $\vartheta $
satisfies the triangle inequality. Thus%
\begin{eqnarray*}
\vartheta \left( \left[ \sqrt{n}\hat{\rho}_{n}(h)\right] _{h=1}^{\mathcal{L}%
_{n}}\right) -\vartheta \left( \left[ \sqrt{n}\rho _{n}(h)\right] _{h=1}^{%
\mathcal{L}_{n}}\right) &\leq &\vartheta \left( \left[ \sqrt{n}\left\{ \hat{%
\rho}_{n}(h)-\rho _{n}(h)\right\} \right] _{h=1}^{\mathcal{L}_{n}}\right) \\
\vartheta \left( \left[ \sqrt{n}\rho _{n}(h)\right] _{h=1}^{\mathcal{L}%
_{n}}\right) -\vartheta \left( \left[ \sqrt{n}\hat{\rho}_{n}(h)\right]
_{h=1}^{\mathcal{L}_{n}}\right) &\leq &\vartheta \left( \left[ \sqrt{n}%
\left\{ \hat{\rho}_{n}(h)-\rho _{n}(h)\right\} \right] _{h=1}^{\mathcal{L}%
_{n}}\right)
\end{eqnarray*}%
and therefore%
\begin{equation*}
\left\vert \vartheta \left( \left[ \sqrt{n}\hat{\rho}_{n}(h)\right] _{h=1}^{%
\mathcal{L}_{n}}\right) -\vartheta \left( \left[ \sqrt{n}\rho _{n}(h)\right]
_{h=1}^{\mathcal{L}_{n}}\right) \right\vert \leq \vartheta \left( \left[
\sqrt{n}\left\{ \hat{\rho}_{n}(h)-\rho _{n}(h)\right\} \right] _{h=1}^{%
\mathcal{L}_{n}}\right) .
\end{equation*}%
Now apply arguments used in the proof of Theorem 2.5 under $H_{0}$, and the
triangle inequality, to yield:%
\begin{eqnarray*}
\vartheta \left( \left[ \sqrt{n}\hat{\rho}_{n}(h)\right] _{h=1}^{\mathcal{L}%
_{n}}\right) &\leq &\left\vert \vartheta \left( \left[ \sqrt{n}\hat{\rho}%
_{n}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) -\vartheta \left( \left[
\sqrt{n}\rho _{n}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) \right\vert
+\vartheta \left( \left[ \sqrt{n}\rho _{n}(h)\right] _{h=1}^{\mathcal{L}%
_{n}}\right) \\
&\leq &\vartheta \left( \left[ \sqrt{n}\left\{ \hat{\rho}_{n}(h)-\rho
_{n}(h)\right\} \right] _{h=1}^{\mathcal{L}_{n}}\right) +\vartheta \left( %
\left[ \sqrt{n}\rho _{n}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) \\
&=&\vartheta \left( \left[ \mathcal{Z}(h)\right] _{h=1}^{\mathcal{L}%
_{n}}\right) +\vartheta \left( \left[ \sqrt{n}\rho _{n}(h)\right] _{h=1}^{%
\mathcal{L}_{n}}\right) +o_{p}(1)
\end{eqnarray*}%
hence
\begin{eqnarray*}
\hat{p}_{n,M_{n}}^{(dw)} &=&P\left( \vartheta \left( \left[ \sqrt{n}\hat{\rho%
}_{n}^{(dw)}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) \geq \vartheta \left( %
\left[ \sqrt{n}\hat{\rho}_{n}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) |%
\mathfrak{X}_{n}\right) +o_{p}(1) \\
&\geq &P\left( \vartheta \left( \left[ \sqrt{n}\hat{\rho}_{n}^{(dw)}(h)%
\right] _{h=1}^{\mathcal{L}_{n}}\right) \geq \left\vert \vartheta \left( %
\left[ \sqrt{n}\hat{\rho}_{n}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right)
-\vartheta \left( \left[ \sqrt{n}\rho _{n}(h)\right] _{h=1}^{\mathcal{L}%
_{n}}\right) \right\vert \right. \\
&&\text{ \ \ \ \ \ \ \ \ \ }\left. +\vartheta \left( \left[ \sqrt{n}\rho
_{n}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) |\mathfrak{X}_{n}\right)
+o_{p}(1) \\
&\geq &\bar{F}_{n}^{(0)}\left( \vartheta \left( \left[ \mathcal{Z}(h)\right]
_{h=1}^{\mathcal{L}_{n}}\right) +\vartheta \left( \left[ \sqrt{n}\rho _{n}(h)%
\right] _{h=1}^{\mathcal{L}_{n}}\right) \right) +o_{p}(1).
\end{eqnarray*}%
Similarly:
\begin{eqnarray*}
\hat{p}_{n,M_{n}}^{(dw)} &=&P\left( \vartheta \left( \left[ \sqrt{n}\hat{\rho%
}_{n}^{(dw)}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) \geq \vartheta \left( %
\left[ \sqrt{n}\hat{\rho}_{n}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) |%
\mathfrak{X}_{n}\right) +o_{p}(1) \\
&=&P\left( \vartheta \left( \left[ \sqrt{n}\hat{\rho}_{n}^{(dw)}(h)\right]
_{h=1}^{\mathcal{L}_{n}}\right) \geq \vartheta \left( \left[ \sqrt{n}\hat{%
\rho}_{n}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) \right. \\
&&\text{ \ \ \ \ \ \ \ }\left. -\vartheta \left( \left[ \sqrt{n}\rho _{n}(h)%
\right] _{h=1}^{\mathcal{L}_{n}}\right) +\vartheta \left( \left[ \sqrt{n}%
\rho _{n}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) |\mathfrak{X}_{n}\right)
+o_{p}(1) \\
&\leq &P\left( \vartheta \left( \left[ \sqrt{n}\hat{\rho}_{n}^{(dw)}(h)%
\right] _{h=1}^{\mathcal{L}_{n}}\right) \geq -\left\vert \vartheta \left( %
\left[ \sqrt{n}\hat{\rho}_{n}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right)
-\vartheta \left( \left[ \sqrt{n}\rho _{n}(h)\right] _{h=1}^{\mathcal{L}%
_{n}}\right) \right\vert \right. \\
&&\text{ \ \ \ \ \ \ \ }\left. +\vartheta \left( \left[ \sqrt{n}\rho _{n}(h)%
\right] _{h=1}^{\mathcal{L}_{n}}\right) |\mathfrak{X}_{n}\right) +o_{p}(1) \\
&\leq &P\left( \vartheta \left( \left[ \sqrt{n}\hat{\rho}_{n}^{(dw)}(h)%
\right] _{h=1}^{\mathcal{L}_{n}}\right) \geq -\left\vert \vartheta \left( %
\left[ \sqrt{n}\left\{ \hat{\rho}_{n}(h)-\rho _{n}(h)\right\} \right]
_{h=1}^{\mathcal{L}_{n}}\right) \right\vert \right. \\
&&\text{ \ \ \ \ \ \ \ }\left. +\vartheta \left( \left[ \sqrt{n}\rho _{n}(h)%
\right] _{h=1}^{\mathcal{L}_{n}}\right) |\mathfrak{X}_{n}\right) +o_{p}(1) \\
&=&\bar{F}_{n}^{(0)}\left( -\vartheta \left( \left[ \mathcal{Z}(h)\right]
_{h=1}^{\mathcal{L}_{n}}\right) +\vartheta \left( \left[ \sqrt{n}\rho _{n}(h)%
\right] _{h=1}^{\mathcal{L}_{n}}\right) \right) +o_{p}(1).
\end{eqnarray*}%
This completes the proof$.$ $\mathcal{QED}.$

\setcounter{equation}{0}

\section{Nonparametric Self-Standardization: Kernel Variance Estimation\label%
{app:kern_var}}

In order to control for variable dispersion across lags, a natural choice
for $\hat{\omega}_{n}(h)$ is an inverted standard deviation estimator. Based
on Corollary 2.4 we need to estimate the asymptotic variance under the null,
identically $\lim_{n\rightarrow \infty }E[(1/\sqrt{n}%
\sum_{t=1}^{n}r_{t}(h))^{2}]$. A now classic approach exploits a kernel
estimator, cf. \cite{NeweyWest1987}, \cite{AndrewsHAC91} and \cite%
{deJongDavidson2000}. See, e.g., \cite{Shao10} for a\ viable alternative to
kernel estimators.

Recall $m_{t}(\theta )$ are the estimating equations for $\hat{\theta}_{n}$,
let $\widehat{\mathcal{A}}_{n}$ be a consistent estimator of $\mathcal{A}$\
in Assumption \ref{assum:plug}.c, and define
\begin{equation}
\mathcal{\hat{D}}_{n}(h)\equiv \frac{1}{n}\sum_{t=h+1}^{n}\left\{ \left(
\epsilon _{t}(\hat{\theta}_{n})s_{t}(\hat{\theta}_{n})+\frac{G_{t}(\hat{%
\theta}_{n})}{\sigma _{t}(\hat{\theta}_{n})}\right) \epsilon _{t-h}(\hat{%
\theta}_{n})+\epsilon _{t}(\hat{\theta}_{n})\left( \epsilon _{t-h}(\hat{%
\theta}_{n})s_{t-h}(\hat{\theta}_{n})+\frac{G_{t-h}(\hat{\theta}_{n})}{%
\sigma _{t-h}(\hat{\theta}_{n})}\right) \right\}
\end{equation}%
and
\begin{equation*}
\widehat{\mathcal{E}}_{n,t,h}(\hat{\theta}_{n})\equiv \epsilon _{t}(\hat{%
\theta}_{n})\epsilon _{t-h}(\hat{\theta}_{n})-\mathcal{\hat{D}}%
_{n}(h)^{\prime }\widehat{\mathcal{A}}_{n}m_{t}(\hat{\theta}_{n}).
\end{equation*}%
The variance estimator and proposed weights are:%
\begin{equation*}
\hat{\nu}_{\mathcal{K},n}^{2}(h)\equiv \frac{1/n\sum_{s,t=1}^{n}\mathcal{K}%
\left( \left( s-t\right) /b_{n}\right) \widehat{\mathcal{E}}_{n,s,h}(\hat{%
\theta}_{n})\widehat{\mathcal{E}}_{n,t,h}(\hat{\theta}_{n})}{%
1/n\sum_{s=1}^{n}\epsilon _{t}^{2}(\hat{\theta}_{n})} \quad \text{and} \quad
\hat{\omega}_{n}(h)\equiv \frac{1}{\hat{\nu}_{\mathcal{K},n}(h)},
\end{equation*}%
where $\mathcal{K}$ is a kernel function and positive integer $b_{n}$ is
bandwidth satisfying $b_{n}$ $\rightarrow $ $\infty $ as $n$ $\rightarrow $ $%
\infty $. Recall%
\begin{equation*}
r_{t}(h)\equiv \frac{\epsilon _{t}\epsilon _{t-h}-E\left[ \epsilon
_{t}\epsilon _{t-h}\right] -\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}}{E\left[
\epsilon _{t}^{2}\right] }.
\end{equation*}

We need only show $\hat{\nu}_{\mathcal{K},n}^{2}(h)$ $\overset{p}{%
\rightarrow }$ $\lim_{n\rightarrow \infty }E[(1/\sqrt{n}%
\sum_{t=1}^{n}r_{t}(h))^{2}]$ since by Theorem 2.3 the latter is exactly the
asymptotic variance of $\hat{\rho}_{n}(h)$\ under the null. Let $i$ $\equiv $
$\sqrt{-1}$.

\begin{assumption}[kernel variance]
\label{assum:var} \ \ \ \medskip \newline
$a.$ $\mathcal{K}:\mathbb{R}\rightarrow \lbrack -1,1]$ satisfies $\mathcal{K}%
(0)$ $=$ $1$,$\mathcal{K}(x)$ $=$ $\mathcal{K}(-x)$ $\forall x$ $\in $ $%
\mathbb{R},$ $\mathcal{K}(\cdot )$ is continuous at $0$ and at all but a
finite number of points, $\int_{-\infty }^{\infty }|\mathcal{K}(x)|dx$ $<$ $%
\infty $, and $\int_{-\infty }^{\infty }|\psi (\xi )|d\xi $ $<$ $\infty $
where $\psi (\xi )$ $=$ $(2\pi )^{-1}\int_{-\infty }^{\infty }\mathcal{K}%
(x)e^{i\xi x}dx$ $<$ $\infty $.$\medskip $\newline
$b.$ $b_{n}/n$ $+$ $1/b_{n}$ $\rightarrow $ $0$.
\end{assumption}

\begin{lemma}
\label{lm:kernel_var}Under Assumptions \ref{assum:dgp}, \ref{assum:plug}%
.a,b,c$^{\prime }$,d and \ref{assum:var}, $\hat{\nu}_{\mathcal{K},n}^{2}(h)$
$\overset{p}{\rightarrow }$ $\lim_{n\rightarrow \infty }E[(1/\sqrt{n}%
\sum_{t=1}^{n}r_{t}(h))^{2}]$.
\end{lemma}

\noindent \textbf{Proof.}\qquad Define%
\begin{equation*}
\mathcal{E}_{t,h}(\theta )\equiv \epsilon _{t}(\theta )\epsilon
_{t-h}(\theta )-\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}(\theta ) \quad
\text{and} \quad \mathcal{E}_{t,h}\equiv \mathcal{E}_{t,h}(\theta _{0}).
\end{equation*}%
By the arguments leading to (A.27) in the main paper $1/n\sum_{s=1}^{n}%
\epsilon _{t}^{2}(\hat{\theta}_{n})$ $\overset{p}{\rightarrow }$ $E[\epsilon
_{t}^{2}]$ $\in $ $(0,\infty )$, hence we focus on $1/n\sum_{s,t=1}^{n}%
\mathcal{K}((s-t)/b_{n})\widehat{\mathcal{E}}_{n,s,h}(\hat{\theta}_{n})%
\widehat{\mathcal{E}}_{n,t,h}(\hat{\theta}_{n})$. Add and subtract terms to
yield:%
\begin{eqnarray*}
\hat{\nu}_{\mathcal{K},n}^{2}(h) &=&\frac{1}{n}\sum_{s,t=1}^{n}\mathcal{K}%
\left( \frac{s-t}{b_{n}}\right) \widehat{\mathcal{E}}_{n,s,h}(\hat{\theta}%
_{n})\widehat{\mathcal{E}}_{n,t,h}(\hat{\theta}_{n}) \\
&=&\frac{1}{n}\sum_{s,t=1}^{n}\mathcal{K}\left( \frac{s-t}{b_{n}}\right)
\mathcal{E}_{s,h}(\hat{\theta}_{n})\mathcal{E}_{t,h}(\hat{\theta}_{n}) \\
&&-\left\{ \mathcal{\hat{D}}_{n}(h)^{\prime }\widehat{\mathcal{A}}_{n}-%
\mathcal{D}(h)^{\prime }\mathcal{A}\right\} \frac{1}{n}\sum_{s,t=1}^{n}%
\mathcal{K}\left( \frac{s-t}{b_{n}}\right) \mathcal{E}_{s,h}(\hat{\theta}%
_{n})m_{t}(\hat{\theta}_{n}) \\
&&\text{\ }-\left\{ \mathcal{\hat{D}}_{n}(h)^{\prime }\widehat{\mathcal{A}}%
_{n}-\mathcal{D}(h)^{\prime }\mathcal{A}\right\} \frac{1}{n}\sum_{s,t=1}^{n}%
\mathcal{K}\left( \frac{s-t}{b_{n}}\right) m_{s}(\hat{\theta}_{n})\mathcal{E}%
_{t,h}(\hat{\theta}_{n}) \\
&&\text{\ }+\left\{ \mathcal{\hat{D}}_{n}(h)^{\prime }\widehat{\mathcal{A}}%
_{n}-\mathcal{D}(h)^{\prime }\mathcal{A}\right\} \frac{1}{n}\sum_{s,t=1}^{n}%
\mathcal{K}\left( \frac{s-t}{b_{n}}\right) m_{s}(\hat{\theta}_{n})m_{t}(\hat{%
\theta}_{n})^{\prime }\left\{ \mathcal{\hat{D}}_{n}(h)^{\prime }\widehat{%
\mathcal{A}}_{n}-\mathcal{D}(h)^{\prime }\mathcal{A}\right\} ^{\prime }.
\end{eqnarray*}%
We will prove
\begin{equation}
\frac{1}{n}\sum_{s,t=1}^{n}\mathcal{K}\left( \frac{s-t}{b_{n}}\right)
\mathcal{E}_{s,h}(\hat{\theta}_{n})\mathcal{E}_{t,h}(\hat{\theta}_{n})%
\overset{p}{\rightarrow }\lim_{n\rightarrow \infty }E\left[ \left( \frac{1}{%
\sqrt{n}}\sum_{t=1}^{n}\mathcal{E}_{t,h}\right) ^{2}\right] <\infty .
\label{v_lim_E}
\end{equation}%
Proofs that the remaining summands converge in probability to finite
constants is similar. Furthermore, $\mathcal{\hat{D}}_{n}(h)$ $\overset{p}{%
\rightarrow }$ $\mathcal{D}(h)$ by arguments in the proof of Lemma 2.1, and $%
\widehat{\mathcal{A}}_{n}$ $\overset{p}{\rightarrow }$ $\mathcal{A}$ by
supposition. Therefore $\hat{\nu}_{\mathcal{K},n}^{2}(h)$ $\overset{p}{%
\rightarrow }$ $\lim_{n\rightarrow \infty }E[(1/\sqrt{n}\sum_{t=1}^{n}%
\mathcal{E}_{t,h})^{2}]/E[\epsilon _{t}^{2}]$ $=$ $\lim_{n\rightarrow \infty
}E[(1/\sqrt{n}\sum_{t=1}^{n}r_{t}(h))^{2}]$.

In order to prove (\ref{v_lim_E}) we will verify the conditions of Theorem
2.2 in \cite{deJongDavidson2000}, which requires demonstrating their
Assumptions 1-4 hold, which we label A1-A4. A1 holds by our stated kernel
properties. Under Assumptions \ref{assum:dgp} and \ref{assum:plug}, and
Theorem 17.8 in \cite{Davidson1994}, $\{\mathcal{E}_{t,h}\}$ satisfies A2.
A3 holds by our stated bandwidth properties.\footnote{%
Assumption 2 in \cite{deJongDavidson2000} requires two sets of constants $%
c_{nt}$ and $d_{nt}$. We have stationarity and standard asymptotics, hence $%
c_{nt}$ $=$ $1/\sqrt{n}$. Further, $d_{nt}$ $=$ $K$ since $\mathcal{E}_{t,h}$
is stationary and does not depend on $n$. Their Assumption 3 states $1/b_{n}$
$+$ $b_{n}\max_{1\leq t\leq n}c_{nt}$ $\rightarrow $ $0$ which holds under
our assumptions when $c_{nt}$ $=$ $1/\sqrt{n}$.} A4.a holds since $\sqrt{n}(%
\hat{\theta}_{n}$ $-$ $\theta _{0})$ $=$ $O_{p}(1)$ by our Assumption \ref%
{assum:plug}.c$^{\prime }$ and (\ref{NED_var}). A4.b holds by the continuous
differentiability properties under Assumption \ref{assum:plug}.a,b,c$%
^{\prime }$. A4 holds sufficiently if for some open neighborhood $\mathcal{N}%
_{0}$ $\subset $ $\Theta $ of $\theta _{0}$, and every $\xi $ $\in $ $%
\mathbb{R}$:
\begin{equation*}
\limsup_{n\rightarrow \infty }\frac{1}{n}\sum_{t=1}^{n}E\left[ \sup_{\theta
\in \mathcal{N}_{0}}\left\Vert \frac{\partial }{\partial \theta }\mathcal{E}%
_{t,h}(\theta )\right\Vert ^{2}\right] <\infty \text{ and }\sup_{\theta \in
\mathcal{N}_{0}}\left\vert \frac{1}{n}\sum_{t=1}^{n}e^{i\xi t/b_{n}}\left(
\frac{\partial }{\partial \theta }\mathcal{E}_{t,h}(\theta )-E\left[ \frac{%
\partial }{\partial \theta }\mathcal{E}_{t,h}(\theta )\right] \right)
\right\vert \overset{p}{\rightarrow }0.
\end{equation*}%
The first holds by the moment envelope bounds in Assumption \ref{assum:plug}%
.a,b. Consider the second, and note that the term inside $|\cdot |$ has real
and imaginary components $\mathcal{R}_{n}(\theta )$ and $\mathcal{I}%
_{n}(\theta )$. Assumption \ref{assum:plug}a,b,c$^{\prime }$ ensures $%
(\partial /\partial \theta )\mathcal{E}_{t,h}(\theta )$ is $\mathcal{F}_{t}$%
-measurable and therefore stationary and ergodic, while, $|e^{i\xi t/b_{n}}|$
$\leq $ $1$ for any $t$ $=$ $1,...,n$. Therefore the same arguments used to
prove the uniform laws in Step 2 of the proof of Lemma 2.1 carry over here
to prove $\sup_{\theta \in \mathcal{N}_{0}}|\mathcal{R}_{n}(\theta )|$ $%
\overset{p}{\rightarrow }$ $0$ and $\sup_{\theta \in \mathcal{N}_{0}}|%
\mathcal{I}_{n}(\theta )|$ $\overset{p}{\rightarrow }$ $0$. $\mathcal{QED}$.

\setcounter{equation}{0}

\section{Omitted Proofs: Lemmas A.4-A.6\label{app:proofs}}

We now present omitted proofs.

\subsection{Lemma A.4 (Expansion)}

Let $h$ $\geq $ $0$. Recall $\rho (h)$ $\equiv $ $E[\epsilon _{t}\epsilon
_{t-h}]/E[\epsilon _{t}^{2}]$ and%
\begin{eqnarray*}
&&G_{t}(\phi )\equiv \left[ \frac{\partial }{\partial \phi ^{\prime }}%
f(x_{t-1},\phi ),\boldsymbol{0}_{k_{\delta }}^{\prime }\right] ^{\prime }\in
\mathbb{R}^{k_{\theta }}\text{ \ and \ }s_{t}(\theta )\equiv \frac{1}{2}%
\frac{\partial }{\partial \theta }\ln \sigma _{t}^{2}(\theta ) \\
&&\mathcal{D}(h)\equiv E\left[ \left( \epsilon _{t}s_{t}+G_{t}/\sigma
_{t}\right) \epsilon _{t-h}\right] +E\left[ \epsilon _{t}\left( \epsilon
_{t-h}s_{t-h}+G_{t-h}/\sigma _{t-h}\right) \right] \in \mathbb{R}^{k_{\theta
}} \\
&&z_{t}(h)\equiv r_{t}(h)-\rho (h)r_{t}(0)\text{ where }r_{t}(h)\equiv \frac{%
\epsilon _{t}\epsilon _{t-h}-E\left[ \epsilon _{t}\epsilon _{t-h}\right] -%
\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}}{E\left[ \epsilon _{t}^{2}\right] },
\end{eqnarray*}%
where $m_{t}$ and $\mathcal{A}$\ appear in plug-in expansion Assumption \ref%
{assum:plug}.c: $\sqrt{n}(\hat{\theta}_{n}$ $-$ $\theta _{0})$ $=$ $\mathcal{%
A}n^{-1/2}\sum_{t=1}^{n}m_{t}(\theta _{0})$ $+$ $O_{p}(n^{-\zeta })$ for
some $\zeta $ $>$ $0$.

\begin{lemmaA4}
Under Assumptions \ref{assum:dgp} and \ref{assum:plug}: for some $\zeta $ $>$
$0$ that appears in Assumption \ref{assum:plug}.c, and each $h$ $\geq $ $0$:
\begin{equation}
\mathcal{X}_{n}(h)\equiv \left\vert \sqrt{n}\left\{ \hat{\rho}_{n}(h)-\rho
(h)\right\} -\frac{1}{n^{1/2}}\sum_{t=1+h}^{n}\left\{ r_{t}(h)-\rho
(h)r_{t}(0)\right\} \right\vert =O_{p}\left( \frac{1}{n^{\min \{\zeta ,1/2\}}%
}\right) .  \label{expans_h}
\end{equation}
\end{lemmaA4}

\noindent \textbf{Proof.}\qquad There exists $\theta _{n}^{\ast }$, $%
||\theta _{n}^{\ast }$ $-$ $\theta _{0}||$ $\leq $ $||\hat{\theta}_{n}$ $-$ $%
\theta _{0}||$, that may be different in different places, such that
\begin{equation}
\sqrt{n}\hat{\gamma}_{n}(h)=\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}\epsilon
_{t}\epsilon _{t-h}+\sqrt{n}\mathcal{A}_{n}(h)+\sqrt{n}\mathcal{A}_{n}(-h),
\label{0A}
\end{equation}%
where{\small
\begin{eqnarray}
&&\mathcal{A}_{n}(h)  \label{A} \\
&&\text{ }=-\left( \hat{\theta}_{n}-\theta _{0}\right) ^{\prime }\frac{1}{n}%
\sum_{t=1+h}^{n}\left( \epsilon _{t}s_{t}+\frac{G_{t}}{\sigma _{t}}\right)
\epsilon _{t-h}  \notag \\
&&\text{ \ }-\left( \hat{\theta}_{n}-\theta _{0}\right) ^{\prime }\frac{1}{n}%
\sum_{t=1+h}^{n}\epsilon _{t}s_{t}\frac{G_{t-h}}{\sigma _{t-h}}\left( \hat{%
\theta}_{n}-\theta _{0}\right)  \notag \\
&&\text{ \ }+\left( \hat{\theta}_{n}-\theta _{0}\right) ^{\prime }\left\{
\frac{1}{n}\sum_{t=1+h}^{n}\epsilon _{t-h}\frac{\partial }{\partial \theta
^{\prime }}\left( \frac{G_{t}(\phi _{n}^{\ast })}{\sigma _{t}(\theta
_{n}^{\ast })}\right) -\frac{1}{n}\sum_{t=1+h}^{n}\frac{G_{t}(\phi
_{n}^{\ast })}{\sigma _{t}(\theta _{n}^{\ast })}\frac{G_{t-h}(\phi
_{n}^{\ast })^{\prime }}{\sigma _{t-h}(\theta _{n}^{\ast })}\right\} \left(
\hat{\theta}_{n}-\theta _{0}\right)  \notag \\
&&\text{ \ }-\left( \hat{\theta}_{n}-\theta _{0}\right) ^{\prime }\left\{
\frac{1}{n}\sum_{t=1+h}^{n}\epsilon _{t-h}(\theta _{n}^{\ast })\frac{%
G_{t}(\phi _{n}^{\ast })}{\sigma _{t}(\theta _{n}^{\ast })}s_{t-h}(\theta
_{n}^{\ast })^{\prime }+\frac{1}{n}\sum_{t=1+h}^{n}\epsilon _{t}(\theta
_{n}^{\ast })s_{t}(\theta _{n}^{\ast })\epsilon _{t-h}(\theta _{n}^{\ast
})s_{t-h}(\theta _{n}^{\ast })^{\prime }\right\} \left( \hat{\theta}%
_{n}-\theta _{0}\right)  \notag \\
&&\text{ \ }+\left( \hat{\theta}_{n}-\theta _{0}\right) ^{\prime }\frac{1}{n}%
\sum_{t=1+h}^{n}\epsilon _{t}\epsilon _{t-h}\frac{\partial }{\partial \theta
^{\prime }}s_{t}(\theta _{n}^{\ast })\left( \hat{\theta}_{n}-\theta
_{0}\right)  \notag \\
&&\text{ \ }+\left( \hat{\theta}_{n}-\theta _{0}\right) ^{\prime }\left\{
\frac{1}{n}\sum_{t=1+h}^{n}\epsilon _{t-h}s_{t}(\theta _{n}^{\ast })\frac{%
\partial }{\partial \theta ^{\prime }}\epsilon _{t}(\theta _{n}^{\ast })-%
\frac{1}{n}\sum_{t=1+h}^{n}s_{t}(\theta _{n}^{\ast })\frac{G_{t-h}(\phi
_{n}^{\ast })}{\sigma _{t-h}(\theta _{n}^{\ast })}\frac{\partial }{\partial
\theta ^{\prime }}\epsilon _{t}(\theta _{n}^{\ast })\right\} \left( \hat{%
\theta}_{n}-\theta _{0}\right)  \notag \\
&&\text{ \ }-\left( \hat{\theta}_{n}-\theta _{0}\right) ^{\prime }\left\{
\frac{1}{n}\sum_{t=1+h}^{n}\epsilon _{t}s_{t}\frac{\partial }{\partial
\theta ^{\prime }}\left( \frac{G_{t-h}(\phi _{n}^{\ast })}{\sigma
_{t-h}(\theta _{n}^{\ast })}\right) +\frac{1}{n}\sum_{t=1+h}^{n}\epsilon _{t}%
\frac{G_{t-h}(\phi _{n}^{\ast })}{\sigma _{t-h}(\theta _{n}^{\ast })}\frac{%
\partial }{\partial \theta ^{\prime }}s_{t}(\theta _{n}^{\ast })\right\}
\left( \hat{\theta}_{n}-\theta _{0}\right)  \notag \\
&&  \notag \\
&&\text{ }=-\left( \hat{\theta}_{n}-\theta _{0}\right) ^{\prime }\mathcal{B}%
_{n}(h)+\left( \hat{\theta}_{n}-\theta _{0}\right) ^{\prime }\sum_{i=1}^{6}%
\mathcal{C}_{n,i}(h,\theta _{n}^{\ast })\left( \hat{\theta}_{n}-\theta
_{0}\right) .  \notag
\end{eqnarray}%
}

In the following we will show each
\begin{equation*}
\sup_{\theta \in \Theta }\left\Vert \mathcal{C}_{n,i}(h,\theta )\right\Vert
=O_{p}\left( 1\right) ,
\end{equation*}%
while Assumption \ref{assum:plug}.c implies $\sqrt{n}(\hat{\theta}_{n}$ $-$ $%
\theta _{0})$ $=$ $O_{p}(1/\sqrt{n})$. This yields%
\begin{equation*}
\sqrt{n}\left( \hat{\theta}_{n}-\theta _{0}\right) ^{\prime }\sum_{i=1}^{6}%
\mathcal{C}_{n,i}(h,\theta _{n}^{\ast })\left( \hat{\theta}_{n}-\theta
_{0}\right) =O_{p}\left( 1/\sqrt{n}\right) ,
\end{equation*}%
hence%
\begin{equation*}
\sqrt{n}\hat{\gamma}_{n}(h)=\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}\epsilon
_{t}\epsilon _{t-h}-\sqrt{n}\left( \hat{\theta}_{n}-\theta _{0}\right)
^{\prime }\left\{ \mathcal{B}_{n}(h)+\mathcal{B}_{n}(-h)\right\}
+O_{p}\left( 1/\sqrt{n}\right) .
\end{equation*}%
We set the convention that
\begin{equation*}
\left\{ \hat{\gamma}_{n}(h),\mathcal{A}_{n}(h),\mathcal{B}_{n}(h),\mathcal{C}%
_{n,i}(h,\theta )\right\} =0\text{ }\forall h\geq n-1.
\end{equation*}

Stationarity, ergodicity, and the Assumptions \ref{assum:dgp} and \ref%
{assum:plug}\ moment bounds imply $\mathcal{B}_{n}(h)$ $\overset{p}{%
\rightarrow }$ $E[(\epsilon _{t}s_{t}$ $+$ $G_{t}/\sigma _{t})\epsilon
_{t-l}]$ and $\mathcal{B}_{n}(-h)$ $\overset{p}{\rightarrow }$ $E[\epsilon
_{t}(\epsilon _{t-h}s_{t-h}$ $+$ $G_{t-h}/\sigma _{t-h})]$ for each $h$,
hence%
\begin{equation}
\mathcal{B}_{n}(h)+\mathcal{B}_{n}(-h)\overset{p}{\rightarrow }\mathcal{D}%
(h)\equiv E\left[ \left( \epsilon _{t}s_{t}+G_{t}/\sigma _{t}\right)
\epsilon _{t-h}\right] +E\left[ \epsilon _{t}\left( \epsilon
_{t-h}s_{t-h}+G_{t-h}/\sigma _{t-h}\right) \right] \in \mathbb{R}^{k_{\theta
}}.  \label{BBD}
\end{equation}%
Furthermore,
\begin{equation*}
\mathcal{C}_{n,1}(h,\theta _{n}^{\ast })=\mathcal{C}_{n,1}(h,\theta _{0})%
\overset{p}{\rightarrow }E\left[ \mathcal{C}_{n,i}(h,\theta _{0})\right]
\text{ and }E\left[ \left\vert \mathcal{C}_{n,1}(h,\theta _{0})\right\vert %
\right] =O(1).
\end{equation*}

Now let $i$ $=$ $2,...,6$. Stationarity, ergodicity, and the Assumptions \ref%
{assum:dgp} and \ref{assum:plug}\ moment bounds imply pointwise $\mathcal{C}%
_{n,i}(h,\theta )$ $\overset{p}{\rightarrow }$ $E[\mathcal{C}_{n,i}(h,\theta
)]$ and $E[\sup_{\theta \in \mathcal{N}_{\theta _{0}}}|\mathcal{C}%
_{n,i}(h,\theta )|]$ $=$ $O(1)$ for some compact set $\mathcal{N}_{\theta
_{0}}$ $\subseteq $ $\Theta $ containing $\theta _{0}$.

Moreover, $\mathcal{C}_{n,i}(\cdot ,\theta )$ are stochastically
equicontinuous on $\mathcal{N}_{\theta _{0}}$. Consider $i$ $=$ $2$, the
remaining terms being similar. Use $\inf_{\theta \in \Theta }\sigma
_{t}^{2}(\theta )$ $\geq $ $\iota $ $>$ $0$ $a.s.$ under Assumption \ref%
{assum:plug}, the mean value theorem and properties of the matrix norm to
yield $|\mathcal{C}_{n,i}(h,\theta )$ $-$ $\mathcal{C}_{n,i}(h,\tilde{\theta}%
)|$ $\leq $ $\mathcal{E}_{t}$ $\times $ $|\theta -\tilde{\theta}|$ for any $%
\{\theta ,\tilde{\theta}\}$ $\in $ $\mathcal{N}_{\theta _{0}}$, where{\small
\begin{eqnarray*}
\mathcal{E}_{t} &\equiv &K\left\vert \epsilon _{t-h}\right\vert \times
\sup_{\theta \in \mathcal{N}_{\theta _{0}}}\left\vert \frac{\partial ^{2}}{%
\partial \phi \partial \phi ^{\prime }}G_{t}(\phi )\right\vert \times
\left\vert \theta -\tilde{\theta}\right\vert \\
&&+K\left\vert \epsilon _{t-h}\right\vert \times \sup_{\theta \in \mathcal{N}%
_{\theta _{0}}}\left\vert G_{t}(\phi )\right\vert \times \sup_{\theta \in
\mathcal{N}_{\theta _{0}}}\left\vert \frac{\partial ^{2}}{\partial \theta
\partial \theta ^{\prime }}\ln \left( \sigma _{t}(\theta )\right)
\right\vert \times \left\vert \theta -\tilde{\theta}\right\vert \\
&&+K\left\vert \epsilon _{t-h}\right\vert \times \sup_{\theta \in \mathcal{N}%
_{\theta _{0}}}\left\{ \left( K\left\vert \frac{\partial }{\partial \phi }%
G_{t}(\phi )\right\vert +\left\vert G_{t}(\phi )\right\vert \times
\left\vert \frac{\partial }{\partial \theta }\ln \left( \sigma _{t}(\theta
)\right) \right\vert \right) \times \sup_{\theta \in \mathcal{N}_{\theta
_{0}}}\left\vert \frac{\partial }{\partial \theta }\ln \left( \sigma _{t}(%
\tilde{\theta})\right) \right\vert \right\} \times \left\vert \theta -\tilde{%
\theta}\right\vert \\
&&+2\sup_{\theta \in \mathcal{N}_{\theta _{0}}}\left\{ \left\vert G_{t}(\phi
)\right\vert \times \left( K\left\vert \frac{\partial }{\partial \phi }%
G_{t}(\phi )\right\vert +\left\vert G_{t}(\phi )\right\vert \times
\left\vert \frac{\partial }{\partial \theta }\ln \left( \sigma _{t}(\theta
)\right) \right\vert \right) \right\} \times \left\vert \theta -\tilde{\theta%
}\right\vert .
\end{eqnarray*}%
} Note that, e.g., $\sup_{\theta \in \mathcal{N}_{\theta _{0}}}|G_{t}(\phi
)| $ $=$ $\sup_{\phi \in \mathcal{N}_{\phi _{0}}}\left\vert G_{t}(\phi
)\right\vert $ for some compact $\mathcal{N}_{\phi _{0}}$ $\subseteq $ $%
\Delta $ containing $\phi _{0}$. Under Assumptions \ref{assum:dgp} and \ref%
{assum:plug}, and multiple applications of the Cauchy-Schwartz inequality, $%
E[\mathcal{E}_{t}]$ $<$ $\infty $. Markov's inequality now yields for any $%
(\epsilon ,\eta )$ $>$ $0$ there exists $0$ $<$ $\delta $ $<$ $\epsilon \eta
/E[\mathcal{E}_{t}]$ sufficiently small such that
\begin{eqnarray*}
\lim_{n\rightarrow \infty }P\left( \sup_{\theta ,\theta \in \mathcal{N}%
_{\theta _{0}}:|\theta -\tilde{\theta}|<\delta }\left\vert \mathcal{C}%
_{n,i}(h,\theta )-\mathcal{C}_{n,i}(h,\tilde{\theta})\right\vert >\eta
\right) &\leq &\lim_{n\rightarrow \infty }P\left( \sup_{\theta ,\theta \in
\mathcal{N}_{\theta _{0}}:|\theta -\tilde{\theta}|<\delta }\mathcal{E}%
_{t}\times \left\vert \theta -\tilde{\theta}\right\vert >\eta \right) \\
&\leq &\lim_{n\rightarrow \infty }P\left( \delta \mathcal{E}_{t}>\eta
\right) \leq \frac{\delta }{\eta }E\left[ \mathcal{E}_{t}\right] <\epsilon .
\end{eqnarray*}%
Therefore $\{\mathcal{C}_{n,i}(\cdot ,\theta )$ $:$ $\theta $ $\in $ $%
\mathcal{N}_{\theta _{0}}\}$ is stochastically equicontinuous. Hence, in
conjunction with pointwise probability convergence
\citep[see
e.g.][Corollary 3.1]{Newey1991}:%
\begin{equation}
\sup_{\theta \in \mathcal{N}_{\theta _{0}}}\left\vert \mathcal{C}%
_{n,i}(h,\theta )-E\left[ \mathcal{C}_{n,i}(h,\theta )\right] \right\vert
\overset{p}{\rightarrow }0.  \label{supC}
\end{equation}

Further, $||\theta _{n}^{\ast }$ $-$ $\theta _{0}||$ $\leq $ $||\hat{\theta}%
_{n}$ $-$ $\theta _{0}||$ $\overset{p}{\rightarrow }$ $0$ under Assumption %
\ref{assum:plug}.c. Therefore $\theta _{n}^{\ast }$ $\in $ $\mathcal{N}%
_{\theta _{0}}$ asymptotically with probability approaching one. Hence, by (%
\ref{supC}) and continuity of $\mathcal{C}_{n,i}(\cdot ,\theta )$:%
\begin{equation}
\left\vert \mathcal{C}_{n,i}(h,\theta _{n}^{\ast })-E[\mathcal{C}%
_{n,i}(h,\theta _{0})]\right\vert \overset{p}{\rightarrow }0.
\label{C_theta*}
\end{equation}%
Finally, under Assumption \ref{assum:plug}.c%
\begin{equation*}
\sqrt{n}\left( \hat{\theta}_{n}-\theta _{0}\right) =\mathcal{A}\frac{1}{%
\sqrt{n}}\sum_{t=1}^{n}m_{t}+O_{p}\left( \frac{1}{n^{\zeta }}\right) ,
\end{equation*}%
where $1/\sqrt{n}\sum_{t=1}^{n}m_{t}$ $=$ $O_{p}(1)$ by square integrability
and the NED property for $m_{t}$. Hence:
\begin{equation}
\hat{\theta}_{n}-\theta _{0}=O_{p}\left( 1/\sqrt{n}\right).
\label{theta_hat_rootn}
\end{equation}%
\

Combine (\ref{A})-(\ref{theta_hat_rootn})\ to yield:%
\begin{equation}
\sqrt{n}\hat{\gamma}_{n}(h)=\frac{1}{\sqrt{n}}\sum\limits_{t=1+h}^{n}%
\epsilon _{t}\epsilon _{t-h}-\sqrt{n}\left( \hat{\theta}_{n}-\theta
_{0}\right) ^{\prime }\mathcal{D}(h)+O_{p}\left( 1/\sqrt{n}\right).
\label{root(n)g}
\end{equation}%
Now use $\sqrt{n}(\hat{\theta}_{n}$ $-$ $\theta _{0})$ $=$ $\mathcal{A}%
n^{-1/2}\sum_{t=1}^{n}m_{t}$ $+$ $O_{p}\left( n^{-\zeta }\right) $, $\zeta
>0 $, to yield%
\begin{eqnarray*}
\sqrt{n}\left\{ \hat{\gamma}_{n}(h)-\gamma (h)\right\} &=&\frac{1}{\sqrt{n}}%
\sum\limits_{t=1+h}^{n}\epsilon _{t}\epsilon _{t-h}-\sqrt{n}E\left[ \epsilon
_{t}\epsilon _{t-h}\right] \\
&&-\mathcal{D}(h)^{\prime }\left\{ \mathcal{A}\frac{1}{\sqrt{n}}%
\sum_{t=1}^{n}m_{t}+O_{p}\left( 1/n^{\zeta }\right) \right\} +O_{p}\left( 1/%
\sqrt{n}\right) \\
&& \\
&=&\frac{1}{\sqrt{n}}\sum\limits_{t=1+h}^{n}\left( \epsilon _{t}\epsilon
_{t-h}-E\left[ \epsilon _{t}\epsilon _{t-h}\right] -\mathcal{D}(h)^{\prime }%
\mathcal{A}m_{t}\right) +O_{p}\left( 1/n^{\min \{\zeta ,1/2\}}\right) .
\end{eqnarray*}%
If we set $h$ $=$ $0$ then similarly:%
\begin{equation}
\sqrt{n}\left\{ \hat{\gamma}_{n}(0)-\gamma (0)\right\} =\frac{1}{\sqrt{n}}%
\sum\limits_{t=1+h}^{n}\left( \epsilon _{t}^{2}-E\left[ \epsilon _{t}^{2}%
\right] -\mathcal{D}(0)^{\prime }\mathcal{A}m_{t}\right) +O_{p}\left(
1/n^{\min \{\zeta ,1/2\}}\right) .  \label{g0_expand}
\end{equation}%
By the assumed NED properties under Assumption \ref{assum:dgp} and \ref%
{assum:plug}.c, (\ref{NED_var}) applies to $\epsilon _{t}^{2}$ $-$ $%
E[\epsilon _{t}^{2}]$ $-$ $\mathcal{D}(0)^{\prime }\mathcal{A}m_{t}$ %
\citep[see][Chap. 17]{Davidson1994}, hence:
\begin{equation*}
\hat{\gamma}_{n}(0)-\gamma (0)=\frac{1}{n}\sum\limits_{t=1+h}^{n}\left(
\epsilon _{t}^{2}-E\left[ \epsilon _{t}^{2}\right] -\mathcal{D}(0)^{\prime }%
\mathcal{A}m_{t}\right) +O_{p}\left( \frac{1}{n^{1/2+\min \{\zeta ,1/2\}}}%
\right) =O_{p}\left( \frac{1}{\sqrt{n}}\right) .
\end{equation*}

Next we tackle the key term
\begin{equation*}
r_{t}(h)\equiv \frac{\left\{ \epsilon _{t}\epsilon _{t-h}-E[\epsilon
_{t}\epsilon _{t-h}]-\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}\right\} }{E%
\left[ \epsilon _{t}^{2}\right] }.
\end{equation*}%
Under Assumptions \ref{assum:dgp} and \ref{assum:plug}.c, $r_{t}(h)$ is zero
mean, stationary, $L_{p}$-bounded for some $p$ $>$ $2$, and $L_{2}$-NED with
size $1/2$, on an $\alpha $-mixing base with decay $O(h^{-p/(p-2)-\iota })$
\citep[Theorems 17.8 and
17.9]{Davidson1994}. Therefore $E[(1/\sqrt{n}\sum_{t=1}^{n}r_{t}(h))^{2}]$ $%
= $ $O(1)$ by (\ref{NED_var}). This implies
\begin{equation*}
\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}r_{t}(h)(1+o_{p}(1))=\frac{1}{\sqrt{n}}%
\sum_{t=1+h}^{n}r_{t}(h)+o_{p}(1).
\end{equation*}%
Additionally, $E[r_{t}^{2}(0)]$ $<$ $\infty $ implies
\begin{equation*}
\frac{1}{\sqrt{n}}\sum_{t=1}^{n}r_{t}(0)=\frac{1}{\sqrt{n}}%
\sum_{t=1+h}^{n}r_{t}(0)+O_{p}\left( 1/\sqrt{n}\right) .
\end{equation*}

Combine the above derivations to yield:%
\begin{eqnarray*}
\sqrt{n}\left\{ \hat{\rho}_{n}(h)-\rho (h)\right\} &=&\sqrt{n}\left\{ \frac{%
\hat{\gamma}_{n}(h)}{\hat{\gamma}_{n}(0)}-\frac{\gamma (h)}{\gamma (0)}%
\right\} \\
&=&\frac{1}{\hat{\gamma}_{n}(0)}\sqrt{n}\left\{ \hat{\gamma}_{n}(h)-\gamma
(h)\right\} -\frac{\gamma (h)}{\hat{\gamma}_{n}(0)\gamma (0)}\sqrt{n}\left\{
\hat{\gamma}_{n}(0)-\gamma (0)\right\} \\
&=&\frac{1}{\gamma (0)+O_{p}\left( 1/\sqrt{n}\right) }\frac{1}{\sqrt{n}}%
\sum\limits_{t=1+h}^{n}\left( \epsilon _{t}\epsilon _{t-h}-E\left[ \epsilon
_{t}\epsilon _{t-h}\right] -\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}\right)
\\
&&-\frac{\gamma (h)}{\left( \gamma (0)+O_{p}\left( 1/\sqrt{n}\right) \right)
\gamma (0)}\frac{1}{\sqrt{n}}\sum\limits_{t=1+h}^{n}\left( \epsilon
_{t}^{2}-E\left[ \epsilon _{t}^{2}\right] -\mathcal{D}(0)^{\prime }\mathcal{A%
}m_{t}\right) +O_{p}\left( \frac{1}{n^{\min \{\zeta ,1/2\}}}\right) \\
&=&\frac{1}{\gamma (0)}\frac{1}{\sqrt{n}}\sum\limits_{t=1}^{n}\left(
\epsilon _{t}\epsilon _{t-h}-E\left[ \epsilon _{t}\epsilon _{t-h}\right] -%
\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}\right) \\
&&-\frac{\rho (h)}{\gamma (0)}\frac{1}{\sqrt{n}}\sum\limits_{t=1}^{n}\left(
\epsilon _{t}^{2}-E\left[ \epsilon _{t}^{2}\right] -\mathcal{D}(0)^{\prime }%
\mathcal{A}m_{t}\right) +O_{p}\left( \frac{1}{n^{\min \{\zeta ,1/2\}}}\right)
\\
&=&\frac{1}{\sqrt{n}}\sum\limits_{t=1+h}^{n}\left\{ r_{t}(h)-\rho
(h)r_{t}(0)\right\} +O_{p}\left( \frac{1}{n^{\min \{\zeta ,1/2\}}}\right).
\end{eqnarray*}%
This proves (\ref{expans_h}) as required. $\mathcal{QED}$.

\subsection{Lemma A.5 (Convergence in Finite Dimensional Distributions)}

Define%
\begin{equation*}
z_{t}(h)\equiv r_{t}(h)-\rho (h)r_{t}(0)\text{ where }r_{t}(h)\equiv \frac{%
\epsilon _{t}\epsilon _{t-h}-E\left[ \epsilon _{t}\epsilon _{t-h}\right] -%
\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}}{E\left[ \epsilon _{t}^{2}\right] }.
\end{equation*}

\begin{lemmaA5}
\label{lm:conv_fdd}Let Assumptions \ref{assum:dgp} and \ref{assum:plug}
hold, and write $\mathcal{Z}_{n}(h)$ $\equiv $ $1/\sqrt{n}%
\sum_{t=1+h}^{n}z_{t}(h)$. For each $\mathcal{L}$\ $\in $ $\mathbb{N}$:%
\begin{equation}
\left\{ \mathcal{Z}_{n}(h):1\leq h\leq \mathcal{L}\right\} \overset{d}{%
\rightarrow }\left\{ \mathcal{Z}(h):1\leq h\leq \mathcal{L}\right\} ,
\label{ZR}
\end{equation}%
where $\{\mathcal{Z}(h)$ $:$ $1$ $\leq $ $h$ $\leq $ $\mathcal{L}\}$ is a
zero mean Gaussian process with variance $\lim_{n\rightarrow \infty
}n^{-1}\sum_{s,t=1}^{n}E[z_{s}(h)z_{t}(h)]$ $\in $ $(0,\infty )$, and
covariance function $\lim_{n\rightarrow \infty
}n^{-1}\sum_{s,t=1}^{n}E[z_{s}(h)z_{t}(\tilde{h})]$.
\end{lemmaA5}

\noindent \textbf{Proof.}\qquad For arbitrary $\mathcal{L}$ $\in $ $\mathbb{N%
}$, and $\lambda $ $\in $ $\mathbb{R}^{\mathcal{L}}$, $\lambda ^{\prime
}\lambda $ $=$ $1$:%
\begin{eqnarray*}
\sum_{h=1}^{\mathcal{L}}\lambda _{h}\mathcal{Z}_{n}(h) &=&\frac{1}{\sqrt{n}}%
\sum_{h=1}^{\mathcal{L}}\lambda _{h}\sum_{t=1+h}^{n}z_{t}(h)=\frac{1}{\sqrt{n%
}}\sum_{h=1}^{\mathcal{L}}\lambda _{h}\sum_{t=1}^{n}z_{t}(h)I\left( t\in
\left\{ 1,...,n-h\right\} \right) \\
&=&\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\sum_{h=1}^{\mathcal{L}}\lambda
_{h}z_{t}(h)I\left( t\in \left\{ 1,...,n-h\right\} \right) =\frac{1}{\sqrt{n}%
}\sum_{t=1}^{n}\lambda ^{\prime }\boldsymbol{z}_{t}(\mathcal{L}),
\end{eqnarray*}%
where $\boldsymbol{z}_{t}(\mathcal{L})$ $\equiv $ $[z_{t}(h)I(t$ $\in $ $%
\{1,...,n$ $-$ $h\})]_{h=1}^{\mathcal{L}}$ $\in $ $\mathbb{R}^{\mathcal{L}}$%
. Define $\sigma _{n}^{2}(\lambda )$ $\equiv $ $E[(\sum_{h=1}^{\mathcal{L}%
}\lambda _{h}\mathcal{Z}_{n}(h))^{2}]$. We need:
\begin{equation}
\frac{1}{\sqrt{n}}\sum_{t=1}^{n}\lambda ^{\prime }\boldsymbol{z}_{t}(%
\mathcal{L})\overset{d}{\rightarrow }N\left( 0,\lim_{n\rightarrow \infty
}\sigma _{n}^{2}(\lambda )\right) \text{ where }\lim_{n\rightarrow \infty
}\sigma _{n}^{2}(\lambda )\in \left( 0,\infty \right) .  \label{clt_z(L)}
\end{equation}%
The claim (\ref{ZR}) then follows by the Cram\'{e}r-Wold theorem.

Under Assumptions \ref{assum:dgp} and \ref{assum:plug}.c, $\lambda ^{\prime }%
\boldsymbol{z}_{t}(\mathcal{L})$ is zero mean, $L_{p}$-bounded for some $p$ $%
>$ $2$, and $L_{2}$-NED with size $1/2$, on an $\alpha $-mixing base with
decay $O(h^{-p/(p-2)-\iota })$ \citep[Theorems 17.8,
17.9]{Davidson1994}. Therefore $\sigma _{n}^{2}(\lambda )$ $=$ $O(1)$ by (%
\ref{NED_var}). By Assumption \ref{assum:plug}.d, $\lim \inf_{n\rightarrow
\infty }\inf_{\lambda ^{\prime }\lambda }\sigma _{n}^{2}(\lambda )$ $>$ $0$.
Now (\ref{clt_z(L)}) follows by Theorem 2 in \cite{deJong1997}. $\mathcal{QED%
}$

\subsection{Lemma A.6 (ULLN's)}

Set a block size $b_{n}$ such that $1\leq b_{n}<n$, $b_{n}$ $\rightarrow $ $%
\infty $ and $b_{n}/n$ $\rightarrow $ $0$. Denote the blocks by $\mathcal{B}%
_{s}=\{(s-1)b_{n}+1,\dots ,sb_{n}\}$ with $s=1,\dots ,n/b_{n}$. Assume for
simplicity that the number of blocks $n/b_{n}$ is an integer. Generate iid
random numbers $\{\xi _{1},\dots ,\xi _{n/b_{n}}\}$ with $E[\xi _{i}]$ $=$ $%
0 $, $E[\xi _{i}^{2}]$ $=$ $1$, and $E[\xi _{i}^{4}]$ $<$ $\infty $. Define
an auxiliary variable $\omega _{t}=\xi _{s}$ if $t$ $\in $ $\mathcal{B}_{s}$.

\begin{lemmaA6}
Under Assumptions \ref{assum:dgp} and \ref{assum:plug}.a,b,c$^{\prime }$,d$:$%
\begin{eqnarray}
&&\sup_{\theta \in \Theta }\left\Vert \frac{1}{n}\sum_{t=1}^{n}\omega _{t}%
\frac{\partial }{\partial \theta }m_{t}(\theta )\right\Vert \overset{p}{%
\rightarrow }0  \label{domulln} \\
&&\sup_{\theta \in \Theta }\left\Vert \frac{1}{n}\sum_{t=1}^{n}\frac{%
\partial }{\partial \theta }m_{t}(\theta )-E\left[ \frac{\partial }{\partial
\theta }m_{t}(\theta )\right] \right\Vert \overset{p}{\rightarrow }0
\label{domEdom} \\
&&\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}\omega _{t}m_{t}=O_{p}(1).  \label{omm}
\end{eqnarray}
\end{lemmaA6}

\noindent \textbf{Proof.}\qquad \medskip \newline
\textbf{Step 1.}\qquad Consider (\ref{domulln}). The proof for (\ref{domEdom}%
) is similar, although simpler. For iid $\xi _{s}$ that is distributed $%
N(0,1)$ and independent of the data:
\begin{equation*}
\frac{1}{n}\sum_{t=1}^{n}\omega _{t}\frac{\partial }{\partial \theta }%
m_{t}(\theta )=\frac{1}{n/b_{n}}\sum_{s=1}^{n/b_{n}}\xi _{s}\frac{1}{b_{n}}%
\sum_{t=(s-1)b_{n}+1}^{sb_{n}}\frac{\partial }{\partial \theta }m_{t}(\theta
),
\end{equation*}%
where $(\partial /\partial \theta )m_{t}(\theta )$ is integrable uniformly
on $\Theta $ under Assumption \ref{assum:plug}.c$^{\prime }$. Therefore, for
each $i$ $=$ $1,...,k_{\theta }$ and $j=1,...,k_{m}$, stationarity and
Minkowski's inequality yield:%
\begin{eqnarray}
E\left[ \left( \frac{1}{n}\sum_{t=1}^{n}\omega _{t}\frac{\partial }{\partial
\theta _{i}}m_{j,t}(\theta )\right) ^{2}\right] &=&\frac{1}{n/b_{n}}E\left[
\left( \frac{1}{b_{n}}\sum_{t=(s-1)b_{n}+1}^{sb_{n}}\frac{\partial }{%
\partial \theta _{i}}m_{j,t}(\theta )\right) ^{2}\right]  \label{Ewdm} \\
&\leq &\frac{b_{n}}{n}\sup_{\theta \in \Theta }E\left[ \left( \frac{\partial
}{\partial \theta _{i}}m_{j,t}(\theta )\right) ^{2}\right] \rightarrow 0.
\notag
\end{eqnarray}%
Therefore pointwise $1/n\sum_{t=1}^{n}\omega _{t}(\partial /\partial \theta
)m_{t}(\theta )$ $\overset{p}{\rightarrow }$ $0$. Moreover, $%
1/n\sum_{t=1}^{n}\omega _{t}(\partial /\partial \theta )m_{t}(\theta )$ is
stochastically equicontinuous. This follows by using a first order expansion
of $(\partial /\partial \theta )m_{t}(\theta )$ and integrability of $%
\sup_{\theta \in \Theta }|1/n\sum_{t=1}^{n}\omega _{t}(\partial
^{2}/\partial \theta \partial \theta )m_{i,t}(\theta )|$ under Assumption %
\ref{assum:plug}.c$^{\prime }$. For each $i$, by the mean-value-theorem:
\begin{eqnarray*}
\sup_{|\theta -\tilde{\theta}|<\delta }\left\vert \frac{1}{n}%
\sum_{t=1}^{n}\omega _{t}\left( \frac{\partial }{\partial \theta }%
m_{i,t}(\theta )-\frac{\partial }{\partial \theta }m_{i,t}(\tilde{\theta}%
)\right) \right\vert &\leq &\sup_{|\theta -\tilde{\theta}|<\delta }\left\{
\left\vert \frac{1}{n}\sum_{t=1}^{n}\omega _{t}\frac{\partial ^{2}}{\partial
\theta \partial \theta }m_{i,t}(\theta )\right\vert \times \left\vert \theta
-\tilde{\theta}\right\vert \right\} \\
&\leq &\sup_{\theta \in \Theta }\left\vert \frac{1}{n}\sum_{t=1}^{n}\omega
_{t}\frac{\partial ^{2}}{\partial \theta \partial \theta }m_{i,t}(\theta
)\right\vert \times \delta .
\end{eqnarray*}%
Hence, for any $(\epsilon ,\eta )$ $>$ $0$ there exists
\begin{equation*}
0<\delta <\left\{
\begin{array}{cc}
\infty & \text{if }E\left[ \sup_{\theta \in \Theta }\left\vert \frac{%
\partial ^{2}}{\partial \theta \partial \theta }m_{i,t}(\theta )\right\vert %
\right] =0 \\
\frac{\eta \epsilon }{E\left[ \sup_{\theta \in \Theta }\left\vert \frac{%
\partial ^{2}}{\partial \theta \partial \theta }m_{i,t}(\theta )\right\vert %
\right] } & \text{if }0<E\left[ \sup_{\theta \in \Theta }\left\vert \frac{%
\partial ^{2}}{\partial \theta \partial \theta }m_{i,t}(\theta )\right\vert %
\right] <\infty%
\end{array}%
\right.
\end{equation*}%
such that:
\begin{eqnarray*}
&&\lim_{n\rightarrow \infty }P\left( \sup_{\theta ,\theta \in \mathcal{N}%
_{\theta _{0}}:|\theta -\tilde{\theta}|<\delta }\left\vert \frac{1}{n}%
\sum_{t=1}^{n}\omega _{t}\left( \frac{\partial }{\partial \theta }%
m_{t}(\theta )-\frac{\partial }{\partial \theta }m_{t}(\theta )\right)
\right\vert >\eta \right) \\
&&\text{ \ \ \ \ \ \ \ }\leq \lim_{n\rightarrow \infty }P\left( \sup_{\theta
\in \Theta }\left\vert \frac{1}{n}\sum_{t=1}^{n}\omega _{t}\frac{\partial
^{2}}{\partial \theta \partial \theta }m_{i,t}(\theta )\right\vert >\frac{%
\eta }{\delta }\right) \leq \frac{\delta }{\eta }\lim_{n\rightarrow \infty }E%
\left[ \sup_{\theta \in \Theta }\left\vert \frac{1}{n}\sum_{t=1}^{n}\omega
_{t}\frac{\partial ^{2}}{\partial \theta \partial \theta }m_{i,t}(\theta
)\right\vert \right] \\
&&\text{ \ \ \ \ \ \ \ }=\frac{\delta }{\eta }\lim_{n\rightarrow \infty }E%
\left[ \sup_{\theta \in \Theta }\left\vert \frac{1}{n/b_{n}}%
\sum_{s=1}^{n/b_{n}}\xi _{s}\frac{1}{b_{n}}\sum_{t=(s-1)b_{n}+1}^{sb_{n}}%
\frac{\partial ^{2}}{\partial \theta \partial \theta }m_{i,t}(\theta
)\right\vert \right] \leq \frac{\delta }{\eta }E\left[ \sup_{\theta \in
\Theta }\left\vert \frac{\partial ^{2}}{\partial \theta \partial \theta }%
m_{i,t}(\theta )\right\vert \right] <\epsilon .
\end{eqnarray*}%
This proves stochastic equicontinuity
\citep[see,
e.g.,][]{Newey1991,Andrews92}. Therefore (\ref{domulln}) holds by %
\citet[Corollary 3.1]{Newey1991}.\medskip \newline
\textbf{Step 2.}\qquad Next we show (\ref{omm}). Under Assumption \ref%
{assum:plug}.c$^{\prime }$, $m_{t}$\ is zero mean, stationary, $L_{p}$%
-bounded for some $p$ $>$ $2$, and $L_{2}$-NED with size $1/2$, on an $%
\alpha $-mixing base with decay $O(h^{-p/(p-2)-\iota })$
\citep[Theorems
17.8 and 17.9]{Davidson1994}. Therefore $E[(1/\sqrt{b_{n}}%
\sum_{t=1}^{b_{n}}m_{i,t})^{2}]$ $=$ $O(1)$ by Theorem 17.5 in \cite%
{Davidson1994} and Theorem 1.6 in \cite{McLeish1975}. By construction of $%
\omega _{t}$ this yields:
\begin{eqnarray*}
E\left[ \left( \frac{1}{\sqrt{n}}\sum_{t=1}^{n}\omega _{t}m_{i,t}\right) ^{2}%
\right] &=&E\left[ \left( \frac{1}{\sqrt{n/b_{n}}}\sum_{s=1}^{n/b_{n}}\xi
_{s}\frac{1}{\sqrt{b_{n}}}\sum_{t=(s-1)b_{n}+1}^{sb_{n}}m_{i,t}\right) ^{2}%
\right] \\
&=&E\left[ \left( \frac{1}{\sqrt{b_{n}}}%
\sum_{t=(s-1)b_{n}+1}^{sb_{n}}m_{i,t}\right) ^{2}\right] \leq K,
\end{eqnarray*}%
hence $1/\sqrt{n}\sum_{t=1}^{n}\omega _{t}m_{i,t}$ $=$ $O_{p}(1)$. $\mathcal{%
QED}$.

\setcounter{equation}{0}

\section{Maximum Lag Upper Bound\label{app:max_L}}

In this appendix we present conditions that lead to an upper bound on the
maximum lag rate of increase $\mathcal{L}_{n}$ $\rightarrow $ $\infty $, cf.
Lemma 2.1.

Recall the Lemma 2.1 expansion result. This requires $\kappa $ $>$ $0$ in $%
\hat{\omega}_{n}(h)$ $=$ $\omega (h)+O_{p}(1/n^{\kappa })$ and $\zeta $ $>$ $%
0$ in$\ \sqrt{n}(\hat{\theta}_{n}$ $-$ $\theta _{0})$ $=$ $\mathcal{A}%
n^{-1/2}\sum_{t=1}^{n}m_{t}(\theta _{0})$ $+$ $\mathcal{R}_{m}(n)$ where $%
\mathcal{R}_{m}(n)$ $=$ $O_{p}(n^{-\zeta })$.

\begin{lemma21}
Let Assumptions \ref{assum:dgp} and \ref{assum:plug} hold. Then%
\begin{equation}
\mathcal{\tilde{X}}_{n}(h)\equiv \left\vert \sqrt{n}\hat{\omega}%
_{n}(h)\left\{ \hat{\rho}_{n}(h)-\rho (h)\right\} -\omega (h)\frac{1}{\sqrt{n%
}}\sum_{t=1+h}^{n}\left\{ r_{t}(h)-\rho (h)r_{t}(0)\right\} \right\vert
=O_{p}\left( 1/n^{\min \{\zeta ,\kappa ,1/2\}}\right) .  \label{expans_rate}
\end{equation}%
Moreover, for some non-unique monotonic sequence of positive integers $\{%
\mathcal{L}_{n}\}$, $\mathcal{L}_{n}$ $\rightarrow $ $\infty $ and $\mathcal{%
L}_{n}$ $=$ $o(n)$, we have: $|\vartheta (\sqrt{n}[\hat{\omega}_{n}(h)\{\hat{%
\rho}_{n}(h)$ $-$ $\rho (h)\}]_{h=1}^{\mathcal{L}_{n}})$ $-$ $\vartheta
([\omega (h)\mathcal{Z}_{n}(h)]_{h=1}^{\mathcal{L}_{n}})|$ $\leq $ $%
\vartheta ([\sqrt{n}\hat{\omega}_{n}(h)\{\hat{\rho}_{n}(h)$ $-$ $\rho (h)\}$
$-$ $\omega (h)\mathcal{Z}_{n}(h)]_{h=1}^{\mathcal{L}_{n}})$ $\overset{p}{%
\rightarrow }0$. Therefore, under the null hypothesis:
\begin{equation}
\left\vert \vartheta \left( \left[ \sqrt{n}\hat{\omega}_{n}(h)\hat{\rho}%
_{n}(h)\right] _{h=1}^{\mathcal{L}_{n}}\right) -\vartheta \left( \left[
\omega (h)\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}\left\{ \frac{\epsilon
_{t}\epsilon _{t-h}-\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}}{E\left[
\epsilon _{t}^{2}\right] }\right\} \right] _{h=1}^{\mathcal{L}_{n}}\right)
\right\vert \overset{p}{\rightarrow }0.  \label{maxmax}
\end{equation}%
Finally, if $\vartheta (\cdot )$ is the maximum transform, and $(n^{\min
\{\zeta ,\kappa ,1/2\}}/\ln (n))\mathcal{\tilde{X}}_{n}(h)$ for all $h$\ is
uniformly integrable, then $\mathcal{L}_{n}$ $=$ $O(n^{\min \{\zeta ,\kappa
,1/2\}}/\ln (n))$ must be satisfied.
\end{lemma21}

In this appendix we seek conditions such that $(n^{\min \{\zeta ,\kappa
,1/2\}}/\ln (n))\mathcal{\tilde{X}}_{n}(h)$ is uniformly integrable. In
order to simplify notation we assume $\hat{\omega}_{n}(h)$ $=$ $\omega (h)$ $%
=$ $1$, but the main results carry over under straightforward modifications
(see also Remark \ref{rm:om} below). Hence, from Assumption \ref{assum:dgp}%
.c $\hat{\omega}_{n}(h)$ $=$ $\omega (h)+O_{p}(1/n^{\kappa })$ for some $%
\kappa $ $>$ $0$, we may take $\kappa $ $=$ $\infty $. Thus:%
\begin{equation}
\mathcal{\tilde{X}}_{n}(h)\equiv \left\vert \sqrt{n}\left\{ \hat{\rho}%
_{n}(h)-\rho (h)\right\} -\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}\left\{
r_{t}(h)-\rho (h)r_{t}(0)\right\} \right\vert =O_{p}\left( 1/n^{\min \{\zeta
,1/2\}}\right)  \label{expans_error}
\end{equation}%
and%
\begin{eqnarray}
&&\left\vert \max_{1\leq i\leq \mathcal{L}_{n}}\sqrt{n}\left\vert \hat{\rho}%
_{n}(h)-\rho (h)\right\vert \right. \overset{p}{\rightarrow }0  \label{L21}
\\
&&\text{ \ \ \ \ \ \ \ }\left. -\max_{1\leq i\leq \mathcal{L}_{n}}\left\vert
\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}\left\{ \frac{\epsilon _{t}\epsilon
_{t-h}-\rho (h)\epsilon _{t}^{2}-\left( \mathcal{D}(h)-\rho (h)\mathcal{D}%
(0)\right) ^{\prime }\mathcal{A}m_{t}}{E\left[ \epsilon _{t}^{2}\right] }%
\right\} \right\vert \text{ \ }\right\vert \overset{p}{\rightarrow }0.
\notag
\end{eqnarray}

Demonstrating $(n^{\min \{\zeta ,1/2\}}/\ln (n))\mathcal{\tilde{X}}_{n}(h)$
is uniformly integrable requires several additional technical conditions. In
the following $\iota $ $>$ $0$ is a tiny number that may be different in
different places.

\begin{assumption}[maximum lag conditions]
\label{assum:Ln} \ \ \medskip \newline
$a.$ $\epsilon _{t}$ is $L_{r}$-bounded for some $r$ $>$ $6$.$\medskip $%
\newline
$b.$ $||1/\hat{\gamma}_{n}(0)||_{p}$ $=$ $O(1)$ for some $p$ $>$ $1$, $\sqrt{%
n}||\hat{\theta}_{n}$ $-$ $\theta _{0}||_{4}$ $=$ $O(1)$, and $||n^{\lambda }%
\mathcal{R}_{m}(n)||_{q}$ $=$ $O(1)$ for some $\lambda $ $>$ $0$ and $q$ $>$
$2$.\medskip \newline
$c.$ $\{(\epsilon _{t}s_{t}$ $+$ $G_{t}/\sigma _{t})\epsilon _{t-h},$ $%
(\epsilon _{t-h}s_{t-h}$ $+$ $G_{t-h}/\sigma _{t-h})\epsilon _{t}\}$ are $%
L_{p}$-bounded, $p$ $>$ $2$, $L_{r}$-NED, $2$ $<$ $r$ $\leq $ $p$, with size
$1$, on an $\alpha $-mixing base $\{\upsilon _{t}\}$ with $\sigma $-fields $%
\mathfrak{V}_{t}$ $\equiv $ $\sigma (\upsilon _{\tau }$ $:$ $\tau $ $\leq $ $%
t)$ and decay $O(h^{-\eta p/(p-2)-\iota })$\ where $\eta $ $=$ $(1/(2+\iota
) $ $-$ $1/p)^{-1}$.
\end{assumption}

\begin{remark}
\normalfont The higher error moment ($a$) extends $||\epsilon _{t}||_{r}$ $<$
$\infty $ for some $r$ $>$ $4$ under Assumption \ref{assum:dgp}.b. This is
required since we demonstrate uniform integrability by establishing $%
L_{1+\iota }$-boundedness for $(n^{\min \{\zeta ,1/2\}}/\ln (n))\mathcal{%
\tilde{X}}_{n}(h)$.
\end{remark}

\begin{remark}
\normalfont All conditions save $||1/\hat{\gamma}_{n}(0)||_{p}$ $=$ $O(1)$
are easily verified for most of the simulation study processes.\footnote{%
Some processes in the simulation study likely fail to have the required
higher moments, but are used to demonstrate the sensitivity of the proposed
test to moment condition failure. See Section 4 in the main paper, and see
Appendix \ref{app:sim_all_supp_mat}.}
\end{remark}

\begin{remark}
\normalfont NED condition $(c)$ is imposed because we require a moment bound
similar to (\ref{NED_var}) for $\{(\epsilon _{t}s_{t}$ $+$ $G_{t}/\sigma
_{t})\epsilon _{t-h},$ $(\epsilon _{t-h}s_{t-h}$ $+$ $G_{t-h}/\sigma
_{t-h})\epsilon _{t}\}$. In view of the higher moment conditions in
Assumption \ref{assum:plug}.a,b, and the NED definition, it always suffices
to assume $\{\epsilon _{t},x_{t}\}$ are $L_{p}$-bounded, $p$ $>$ $6$,
stationary $\alpha $-mixing with mixing coefficients $\alpha _{h}$ $=$ $%
O(h^{-\eta p/(p-2)-\iota })$.
\end{remark}

\begin{remark}
\label{rm:om}\normalfont In the general case for $\hat{\omega}_{n}(h)$ $=$ $%
\omega (h)+O_{p}(1/n^{\kappa })$ we additionally need $n^{\min \{\zeta
,\kappa ,1/2\}}||\hat{\omega}_{n}(h)-\omega (h)||_{q}$ $=$ $O(1)$ for some $%
q $ $>$ $2$.
\end{remark}

We now have the main result in this appendix. Under the appropriate
conditions $(n^{\min \{\zeta ,1/2\}}/\ln (n))\mathcal{\tilde{X}}_{n}(h)$ is
uniformly integrable, hence the Lemma 2.1 upper bound on $\mathcal{L}_{n}$
applies.

\begin{lemma}
\label{lm:suffXui}Let Assumptions \ref{assum:dgp}, \ref{assum:plug} and \ref%
{assum:Ln} hold, and assume $H_{0}$ is true. Then $(n^{\min \{\zeta
,1/2\}}/\ln (n))\mathcal{\tilde{X}}_{n}(h)$ is uniformly integrable for each
$h$.
\end{lemma}

Before we prove the claim, we verify Assumption \ref{assum:Ln} for a
stationary AR(1) process.

\begin{example}
\normalfont Let $y_{t}$ $=$ $\theta _{0}y_{t-1}$ $+$ $\epsilon _{t},$ $%
|\theta _{0}|$ $<$ $1$, where $E[\epsilon _{t}]$ $=$ $0$, $E\left\vert
\epsilon _{t}\right\vert ^{p}$ $<$ $\infty $ for some $p$ $>$ $12$, and $%
E[\epsilon _{t}y_{t-1}]$ $=$ $0$. Assume $\{\epsilon _{t},y_{t}\}$ are
stationary geometrically $\alpha $-mixing. Let $||(1/n%
\sum_{t=1}^{n}y_{t}^{2})^{-1}||_{12+\iota }$ $=$ $O(1)$. The plug-in
estimator $\hat{\theta}_{n}$ is least squares. It is easily verified that $%
\mathcal{D}(h)$ $=$ $E[y_{t-1}\epsilon _{t-h}]$ and $s_{t}(\theta )$ $=$ $0$
for any $\theta $.

If $\epsilon _{t}$ is iid with a bounded density function, then the mixing
condition follows. See \cite{Gorodetskii1977} and \citet[p. 77]{Doukhan94}.
If $\epsilon _{t}$ follows a stationary GARCH process, then the mixing
condition holds under mild regularity conditions %
\citep[see][]{MeitzSaikkonen2008}. Conversely, the stationary solution $%
y_{t} $ $=$ $\sum_{i=0}^{\infty }\theta _{0}^{i}\epsilon _{t-i}$ $a.s.$ is
an infinite order lag function of a stationary $\alpha $-mixing process $%
\{\epsilon _{t}\}$. See \citet[Chapt. 2.3.2]{Doukhan94} for deep theory and
conditions under which such a process is also $\alpha $-mixing.

\textit{\textbf{Assumption \ref{assum:Ln}.a}}. $E|\epsilon _{t}|^{r}$ $<$ $%
\infty $ for some $r$ $>$ $6$ holds by construction.

\textit{\textbf{Assumption \ref{assum:Ln}.b}}. The first condition $||1/\hat{%
\gamma}_{n}(0)||_{p}$ $=$ $O(1)$ for some $p$ $>$ $1$\ holds by supposition.

For the second condition $\sqrt{n}||\hat{\theta}_{n}$ $-$ $\theta _{0}||_{4}$
$=$ $O(1)$, by H\"{o}lder's inequality and $||1/\hat{\gamma}%
_{n}(0)||_{12+\iota }$ $=$ $O(1)$:
\begin{equation*}
\left\Vert \sqrt{n}\left( \hat{\theta}_{n}-\theta _{0}\right) \right\Vert
_{4}\leq \left\Vert \frac{1}{1/n\sum_{t=2}^{n}y_{t-1}^{2}}\right\Vert
_{12}\left\Vert \frac{1}{\sqrt{n}}\sum_{t=2}^{n}\epsilon
_{t}y_{t-1}\right\Vert _{6}\leq K\left( E\left[ \left( \frac{1}{\sqrt{n}}%
\sum_{t=2}^{n}\epsilon _{t}y_{t-1}\right) ^{6}\right] \right) ^{1/6}.
\end{equation*}%
Under the stated conditions and Cauchy-Schwartz inequality, $\epsilon
_{t}y_{t-1}$ is stationary $L_{p}$-bounded, $p$ $>$ $6$, and geometrically $%
\alpha $-mixing. Hence $E|1/\sqrt{n}\sum_{t=2}^{n}\epsilon _{t}y_{t-1}|^{6}$
$\leq $ $K$ by Corollary 3 in \cite{Hansen1991}. Therefore $||\sqrt{n}(\hat{%
\theta}_{n}$ $-$ $\theta _{0})||_{4}$ $=$ $O(1).$

\ Next, for the third condition $||n^{\lambda }\mathcal{R}_{m}(n)||_{q}$ $=$
$O(1)$ for some $\lambda $ $>$ $0$ and$\ q$ $>$ $2$, the least squares
expansion satisfies:%
\begin{eqnarray*}
&&\sqrt{n}\left( \hat{\theta}_{n}-\theta _{0}\right) =\frac{1}{E\left[
y_{t-1}^{2}\right] }\frac{1}{\sqrt{n}}\sum_{t=2}^{n}\epsilon _{t}y_{t-1}+%
\mathcal{R}_{m}(n) \\
&&\text{where }\mathcal{R}_{m}(n)=-\frac{\left(
1/n\sum_{t=2}^{n}y_{t-1}^{2}-E\left[ y_{t-1}^{2}\right] \right) }{E\left[
y_{t-1}^{2}\right] n^{-1}\sum_{t=2}^{n}y_{t-1}^{2}}\frac{1}{\sqrt{n}}%
\sum_{t=2}^{n}\epsilon _{t}y_{t-1}.
\end{eqnarray*}%
The geometric $\alpha $-mixing property and $E[\epsilon _{t}y_{t-1}]$ $=$ $0$
yields $1/\sqrt{n}\sum_{t=2}^{n}\epsilon _{t}y_{t-1}$ $=$ $O_{p}(1)$ by
Theorem 1.6 in \cite{McLeish1975}. Similarly, $y_{t}^{2}$ is $L_{2}$%
-bounded, geometrically $\alpha $-mixing hence:
\begin{equation*}
\frac{1}{n}\sum_{t=2}^{n}y_{t-1}^{2}-E\left[ y_{t-1}^{2}\right] =\frac{1}{n}%
\sum_{t=2}^{n}\left( y_{t-1}^{2}-E\left[ y_{t-1}^{2}\right] \right) -\frac{1%
}{n}E\left[ y_{t-1}^{2}\right] =O_{p}(1/\sqrt{n}).
\end{equation*}%
Therefore
\begin{equation*}
\sqrt{n}\mathcal{R}_{m}(n)=-\frac{\sqrt{n}\left(
1/n\sum_{t=2}^{n}y_{t-1}^{2}-E\left[ y_{t-1}^{2}\right] \right) }{E\left[
y_{t-1}^{2}\right] n^{-1}\sum_{t=2}^{n}y_{t-1}^{2}}\frac{1}{\sqrt{n}}%
\sum_{t=2}^{n}\epsilon _{t}y_{t-1}=O_{p}(1)
\end{equation*}%
so that $\mathcal{R}_{m}(n)$ $=$ $O_{p}(n^{-\lambda })$ with $\lambda $ $=$ $%
1/2$.

H\"{o}lder's inequality, $||1/\sqrt{n}\sum_{t=2}^{n}\epsilon
_{t}y_{t-1}||_{6}\leq K$ from above, and $||1/\hat{\gamma}%
_{n}(0)||_{12+\iota }$ $=$ $O(1)$ yield:%
\begin{eqnarray*}
\left\Vert \sqrt{n}\mathcal{R}_{m}(n)\right\Vert _{2+\iota } &\leq
&\left\Vert \frac{\sqrt{n}\left( 1/n\sum_{t=2}^{n}y_{t-1}^{2}-E\left[
y_{t-1}^{2}\right] \right) }{E\left[ y_{t-1}^{2}\right] n^{-1}%
\sum_{t=2}^{n}y_{t-1}^{2}}\right\Vert _{3+\iota }\left\Vert \frac{1}{\sqrt{n}%
}\sum_{t=2}^{n}\epsilon _{t}y_{t-1}\right\Vert _{6} \\
&\leq &K\left\Vert \frac{\sqrt{n}\left( 1/n\sum_{t=2}^{n}y_{t-1}^{2}-E\left[
y_{t-1}^{2}\right] \right) }{1/n\sum_{t=2}^{n}y_{t-1}^{2}}\right\Vert
_{3+\iota } \\
&\leq &K\left\Vert \frac{1}{1/n\sum_{t=2}^{n}y_{t-1}^{2}}\right\Vert
_{12+\iota }\left\Vert \sqrt{n}\left( \frac{1}{n}\sum_{t=2}^{n}y_{t-1}^{2}-E%
\left[ y_{t-1}^{2}\right] \right) \right\Vert _{4} \\
&\leq &K\left\Vert \sqrt{n}\left( \frac{1}{n}\sum_{t=2}^{n}y_{t-1}^{2}-E%
\left[ y_{t-1}^{2}\right] \right) \right\Vert _{4}.
\end{eqnarray*}%
Coupled with $L_{p}$-boundedness for $\epsilon _{t}$ for some $p$ $>$ $12$,
and the $\alpha $-mixing property, $y_{t-1}^{2}$ is stationary $L_{p/2}$%
-bounded geometrically $\alpha $-mixing. Hence $||\sqrt{n}%
(1/n\sum_{t=2}^{n}y_{t-1}^{2}$ $-$ $E\left[ y_{t-1}^{2}\right] )||_{4}$ $=$ $%
O(1)$ by Corollary 3 in \cite{Hansen1991}. Therefore $||\sqrt{n}\mathcal{R}%
_{m}(n)||_{2+\iota }$ $=$ $O(1)$ which verifies the third condition.

\textit{\textbf{Assumption \ref{assum:Ln}.c}}. By construction $G_{t}(\phi )$
$=$ $y_{t-1}$, $\sigma _{t}$ $=$ $1$\ and $s_{t}(\theta )$ $=$ $0$, hence $%
\{(\epsilon _{t}s_{t}$ $+$ $G_{t}/\sigma _{t})\epsilon _{t-h},$ $(\epsilon
_{t-h}s_{t-h}$ $+$ $G_{t-h}/\sigma _{t-h})\epsilon _{t}\}$ $=$ $%
\{y_{t-1t}\epsilon _{t-h},$ $y_{t-1-h}\epsilon _{t}\}$. From the above
arguments, $\{y_{t-1t}\epsilon _{t-h},$ $y_{t-1-h}\epsilon _{t}\}$ is $L_{p}$%
-bounded, $p$ $>$ $6$, geometrically $\alpha $-mixing. Therefore they are $%
L_{r}$-NED, $2$ $<$ $r$ $\leq $ $p$, with arbitrary size, on a geometrically
$\alpha $-mixing base which therefore has arbitrarily fast decay, i.e. $%
\alpha _{h}$ $=$ $O(h^{-\lambda })$ for any $\lambda $ $>$ $0$. Hence all
components of Assumption \ref{assum:Ln}.c hold
\citep[see, e.g.,][Chapt.
7]{Davidson1994}.
\end{example}

\noindent\textbf{Proof of Lemma \ref{lm:suffXui}.}\qquad In order to reduce
notation, let $H_{0}$ $:$ $\rho (h)$ $=$ $0$ $\forall h$ $\geq $ $1$ be
true. Then:%
\begin{equation*}
\mathcal{\tilde{X}}_{n}(h)\equiv \left\vert \sqrt{n}\hat{\rho}_{n}(h)-\frac{1%
}{\sqrt{n}}\sum_{t=1+h}^{n}r_{t}(h)\right\vert =\left\vert \sqrt{n}\hat{\rho}%
_{n}(h)-\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}\left( \frac{\epsilon _{t}\epsilon
_{t-h}-\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}}{E\left[ \epsilon _{t}^{2}%
\right] }\right) \right\vert .
\end{equation*}%
Hence:%
\begin{equation*}
\frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}\mathcal{\tilde{X}}_{n}(h)=\frac{%
n^{\min \{\zeta ,1/2\}}}{\ln (n)}\left\vert \sqrt{n}\hat{\rho}_{n}(h)-\frac{1%
}{\sqrt{n}}\sum_{t=1+h}^{n}\left( \frac{\epsilon _{t}\epsilon _{t-h}-%
\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}}{E\left[ \epsilon _{t}^{2}\right] }%
\right) \right\vert .
\end{equation*}

By the triangle inequality:%
\begin{eqnarray*}
\frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}\mathcal{\tilde{X}}_{n}(h) &=&\frac{%
n^{\min \{\zeta ,1/2\}}}{\ln (n)}\left\vert \sqrt{n}\hat{\rho}_{n}(h)-\frac{1%
}{\sqrt{n}}\sum_{t=1+h}^{n}\left( \frac{\epsilon _{t}\epsilon _{t-h}-%
\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}}{E\left[ \epsilon _{t}^{2}\right] }%
\right) \right\vert \\
&& \\
&\leq &\frac{1}{\gamma (0)}\frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}\left\vert
\sqrt{n}\hat{\gamma}_{n}(h)-\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}\left(
\epsilon _{t}\epsilon _{t-h}-\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}\right)
\right\vert \\
&&+\frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}\sqrt{n}\hat{\gamma}%
_{n}(h)\left\vert \frac{1}{\hat{\gamma}_{n}(0)}-\frac{1}{\gamma (0)}%
\right\vert \\
&& \\
&\leq &\frac{1}{\gamma (0)}\frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}\left\vert
\sqrt{n}\hat{\gamma}_{n}(h)-\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}\left(
\epsilon _{t}\epsilon _{t-h}-\mathcal{D}(h)^{\prime }\mathcal{A}m_{t}\right)
\right\vert \\
&&+\frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}\frac{\hat{\gamma}_{n}(h)}{\hat{%
\gamma}_{n}(0)\gamma (0)}\sqrt{n}\left\vert \hat{\gamma}_{n}(0)-\gamma
(0)\right\vert \\
&& \\
&\equiv &\mathcal{M}_{n,1}(h)+\mathcal{M}_{n,2}(h),
\end{eqnarray*}%
say. It suffices to prove each $\mathcal{M}_{n,i}(h)$ is $L_{p}$-bounded for
some $p$ $>$ $1$ \citep[eq. (3.18)]{Billingsley1999}. In the following $p$ $%
> $ $1$ and tiny $\iota $ $>$ $0$\ may be different in different
places.\medskip \newline
\textbf{Step 1 (}$M_{n,1}(h)$\textbf{):}\qquad Using notation from the proof
of Lemma A.3, by (\ref{0A}) and (\ref{A}):%
\begin{eqnarray}
\mathcal{M}_{n,1}(h) &=&\frac{1}{\gamma (0)}\frac{n^{\min \{\zeta ,1/2\}}}{%
\ln (n)}\left\vert \sqrt{n}\hat{\gamma}_{n}(h)-\frac{1}{\sqrt{n}}%
\sum_{t=1+h}^{n}\left( \epsilon _{t}\epsilon _{t-h}-\mathcal{D}(h)^{\prime }%
\mathcal{A}m_{t}\right) \right\vert  \notag \\
&=&\frac{1}{\gamma (0)}\frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}\left\vert
\sqrt{n}\mathcal{A}_{n}(h)+\sqrt{n}\mathcal{A}_{n}(-h)+\mathcal{D}%
(h)^{\prime }\mathcal{A}m_{t}\right\vert  \notag \\
&\leq &\frac{1}{\gamma (0)}\frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}\left\vert
-\sqrt{n}\left( \hat{\theta}_{n}-\theta _{0}\right) ^{\prime }\left\{
\mathcal{B}_{n}(h)+\mathcal{B}_{n}(-h)\right\} +\mathcal{D}(h)^{\prime }%
\mathcal{A}\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}m_{t}\right\vert  \label{M1n} \\
&&+\frac{1}{\gamma (0)}\left\vert \sqrt{n}\left( \hat{\theta}_{n}-\theta
_{0}\right) ^{\prime }\sum_{i=1}^{6}\left\{ \mathcal{C}_{n,i}(h,\theta
_{n}^{\ast })+\mathcal{C}_{n,i}(-h,\theta _{n}^{\ast })\right\} \frac{\sqrt{n%
}}{\ln (n)}\left( \hat{\theta}_{n}-\theta _{0}\right) \right\vert .  \notag
\end{eqnarray}%
By multiple applications of H\"{o}lder and Minkowsky inequalities, for some $%
\iota $ $>$ $0$ the second term in (\ref{M1n}) satisfies:%
\begin{eqnarray*}
&&\left\Vert \sqrt{n}\left( \hat{\theta}_{n}-\theta _{0}\right) ^{\prime
}\sum_{i=1}^{6}\left\{ \mathcal{C}_{n,i}(h,\theta _{n}^{\ast })+\mathcal{C}%
_{n,i}(-h,\theta _{n}^{\ast })\right\} \frac{\sqrt{n}}{\ln (n)}\left( \hat{%
\theta}_{n}-\theta _{0}\right) \right\Vert _{1+\iota } \\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\leq \left\Vert \sqrt{n}\left( \hat{%
\theta}_{n}-\theta _{0}\right) \right\Vert _{3+\iota }\left\Vert
\sum_{i=1}^{6}\left\{ \mathcal{C}_{n,i}(h,\theta _{n}^{\ast })+\mathcal{C}%
_{n,i}(-h,\theta _{n}^{\ast })\right\} \right\Vert _{3/2+\iota }\left\Vert
\frac{\sqrt{n}}{\ln (n)}\left( \hat{\theta}_{n}-\theta _{0}\right)
\right\Vert _{3+\iota } \\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\leq \frac{1}{\ln (n)}\left\Vert
\sqrt{n}\left( \hat{\theta}_{n}-\theta _{0}\right) \right\Vert _{3+\iota
}^{2}\sup_{\theta \in \Theta }\sum_{i=1}^{6}\left\Vert \mathcal{C}%
_{n,i}(h,\theta )+\mathcal{C}_{n,i}(-h,\theta )\right\Vert _{3/2+\iota } \\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }=O\left( 1/\ln (n)\right) =o(1).
\end{eqnarray*}%
The last line follows from Assumptions \ref{assum:dgp} and \ref{assum:plug}:
each $\sup_{\theta \in \Theta }||\mathcal{C}_{n,i}(h,\theta )||$ satisfies $%
E|\sup_{\theta \in \Theta }||\mathcal{C}_{n,i}(h,\theta )|||^{4+\iota }$ $=$
$O(1)$, and by Assumption \ref{assum:Ln}.b $||\sqrt{n}(\hat{\theta}_{n}$ $-$
$\theta _{0})||_{4}$ $=$ $O(1)$.

Use $\sqrt{n}(\hat{\theta}_{n}$ $-$ $\theta _{0})$ $=$ $\mathcal{A}%
n^{-1/2}\sum_{t=1}^{n}m_{t}$ $+$ $\mathcal{R}_{m}(n)$ under Assumption \ref%
{assum:plug}, and the definitions of $\mathcal{D}(h)$ and $\mathcal{B}%
_{n}(h) $ in (\ref{Dh}) and (\ref{A}) to deduce for the first term in (\ref%
{M1n}):%
\begin{eqnarray*}
&&\frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}\left\vert \sqrt{n}\left( \hat{%
\theta}_{n}-\theta _{0}\right) ^{\prime }\left\{ \mathcal{B}_{n}(h)+\mathcal{%
B}_{n}(-h)\right\} -\mathcal{D}(h)^{\prime }\mathcal{A}\frac{1}{\sqrt{n}}%
\sum_{t=1+h}^{n}m_{t}\right\vert \\
&& \\
&&\text{ \ \ \ \ \ \ \ }=\frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}\left\vert
\left\{ \mathcal{B}_{n}(h)+\mathcal{B}_{n}(-h)\right\} ^{\prime }\sqrt{n}%
\left( \hat{\theta}_{n}-\theta _{0}\right) -\mathcal{D}(h)^{\prime }\mathcal{%
A}\frac{1}{\sqrt{n}}\sum_{t=1+h}^{n}m_{t}\right\vert \\
&& \\
&&\text{ \ \ \ \ \ \ \ }=\frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}\left\vert
\left\{ \frac{1}{n}\sum_{t=1+h}^{n}\left( \epsilon _{t}s_{t}+\frac{G_{t}}{%
\sigma _{t}}\right) \epsilon _{t-h}+\frac{1}{n}\sum_{t=1+h}^{n}\left(
\epsilon _{t-h}s_{t-h}+\frac{G_{t-h}}{\sigma _{t-h}}\right) \epsilon
_{t}\right\} ^{\prime }\right. \\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left.
\times \left\{ \mathcal{A}\frac{1}{\sqrt{n}}\sum_{t=1}^{n}m_{t}+\mathcal{R}%
_{m}(n)\right\} -\mathcal{D}(h)^{\prime }\mathcal{A}\frac{1}{\sqrt{n}}%
\sum_{t=1+h}^{n}m_{t}\right\vert \\
&& \\
&&\text{ \ \ \ \ \ \ \ }\leq \frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}%
\left\vert \left\{ \frac{1}{n}\sum_{t=1+h}^{n}\left( \epsilon _{t}s_{t}+%
\frac{G_{t}}{\sigma _{t}}\right) \epsilon _{t-h}+\frac{1}{n}%
\sum_{t=1+h}^{n}\left( \epsilon _{t-h}s_{t-h}+\frac{G_{t-h}}{\sigma _{t-h}}%
\right) \epsilon _{t}\right. \right. \\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left.
\left. -E\left[ \left( \epsilon _{t}s_{t}+\frac{G_{t}}{\sigma _{t}}\right)
\epsilon _{t-h}+\left( \epsilon _{t-h}s_{t-h}+\frac{G_{t-h}}{\sigma _{t-h}}%
\right) \epsilon _{t}\right] \right\} ^{\prime }\mathcal{A}\frac{1}{\sqrt{n}}%
\sum_{t=1+h}^{n}m_{t}\right\vert \\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }+\frac{1}{\ln (n)}\left\vert
\frac{1}{n}\sum_{t=1+h}^{n}\left( \epsilon _{t}s_{t}+\frac{G_{t}}{\sigma _{t}%
}\right) \epsilon _{t-h}+\frac{1}{n}\sum_{t=1+h}^{n}\left( \epsilon
_{t-h}s_{t-h}+\frac{G_{t-h}}{\sigma _{t-h}}\right) \epsilon _{t}\right\vert
\times \left\vert n^{\min \zeta }\mathcal{R}_{m}(n)\right\vert .
\end{eqnarray*}%
Now apply H\"{o}lder and Minkowsky inequalities to yield for some $\iota $ $%
> $ $0$:
\begin{eqnarray*}
&&\frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}\left\Vert \sqrt{n}\left( \hat{%
\theta}_{n}-\theta _{0}\right) ^{\prime }\left\{ \mathcal{B}_{n}(h)+\mathcal{%
B}_{n}(-h)\right\} -\mathcal{D}(h)^{\prime }\mathcal{A}\frac{1}{\sqrt{n}}%
\sum_{t=1+h}^{n}m_{t}\right\Vert _{1+\iota } \\
&&\text{ \ \ \ \ \ \ \ }\leq \frac{1}{\ln (n)}\sqrt{n}\left\Vert \frac{1}{n}%
\sum_{t=1+h}^{n}\left( \epsilon _{t}s_{t}+\frac{G_{t}}{\sigma _{t}}\right)
\epsilon _{t-h}+\frac{1}{n}\sum_{t=1+h}^{n}\left( \epsilon _{t-h}s_{t-h}+%
\frac{G_{t-h}}{\sigma _{t-h}}\right) \epsilon _{t}\right. \\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left. -E\left[
\left( \epsilon _{t}s_{t}+\frac{G_{t}}{\sigma _{t}}\right) \epsilon
_{t-h}+\left( \epsilon _{t-h}s_{t-h}+\frac{G_{t-h}}{\sigma _{t-h}}\right)
\epsilon _{t}\right] \right\Vert _{2+i}\times \left\Vert \mathcal{A}\frac{1}{%
\sqrt{n}}\sum_{t=1+h}^{n}m_{t}\right\Vert _{2} \\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ }+\frac{1}{\ln (n)}\left\Vert \frac{1}{n}%
\sum_{t=1+h}^{n}\left( \epsilon _{t}s_{t}+\frac{G_{t}}{\sigma _{t}}\right)
\epsilon _{t-h}+\frac{1}{n}\sum_{t=1+h}^{n}\left( \epsilon _{t-h}s_{t-h}+%
\frac{G_{t-h}}{\sigma _{t-h}}\right) \epsilon _{t}\right\Vert _{2}\times
\left\Vert n^{\min \zeta }\mathcal{R}_{m}(n)\right\Vert _{2+i}.
\end{eqnarray*}%
By variance property (\ref{NED_var}) $||\mathcal{A}n^{-1/2}%
\sum_{t=1+h}^{n}m_{t}||_{2}$ $=$ $O(1)$, and by supposition $||n^{\min \zeta
}\mathcal{R}_{m}(n)||_{2+i}=O(1)$. Under Assumptions \ref{assum:dgp}, \ref%
{assum:plug} and \ref{assum:Ln}, applications of H\"{o}lder and Minkowsky
inequalities yield:
\begin{equation*}
\left\Vert \frac{1}{n}\sum_{t=1+h}^{n}\left( \epsilon _{t}s_{t}+\frac{G_{t}}{%
\sigma _{t}}\right) \epsilon _{t-h}+\frac{1}{n}\sum_{t=1+h}^{n}\left(
\epsilon _{t-h}s_{t-h}+\frac{G_{t-h}}{\sigma _{t-h}}\right) \epsilon
_{t}\right\Vert _{2}\leq K.
\end{equation*}

Finally, we show%
\begin{equation}
\sqrt{n}\left\Vert \frac{1}{n}\sum_{t=1+h}^{n}\left( \epsilon _{t}s_{t}+%
\frac{G_{t}}{\sigma _{t}}\right) \epsilon _{t-h}-E\left[ \left( \epsilon
_{t}s_{t}+\frac{G_{t}}{\sigma _{t}}\right) \epsilon _{t-h}\epsilon _{t}%
\right] \right\Vert _{2+\iota }=O(1).  \label{esG_var}
\end{equation}%
The same argument extends to $(\epsilon _{t-h}s_{t-h}$ $+$ $G_{t-h}/\sigma
_{t-h})\epsilon _{t}$. Observe that:
\begin{eqnarray}
&&\text{ \ \ \ \ }\sqrt{n}\left\Vert \frac{1}{n}\sum_{t=1+h}^{n}\left(
\epsilon _{t}s_{t}+\frac{G_{t}}{\sigma _{t}}\right) \epsilon _{t-h}-E\left[
\left( \epsilon _{t}s_{t}+\frac{G_{t}}{\sigma _{t}}\right) \epsilon
_{t-h}\epsilon _{t}\right] \right\Vert _{2+\iota }  \label{esG_exp} \\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\leq \sqrt{n}\left\Vert \frac{1}{n}%
\sum_{t=1}^{n}\left\{ \left( \epsilon _{t}s_{t}+\frac{G_{t}}{\sigma _{t}}%
\right) \epsilon _{t-h}-E\left[ \left( \epsilon _{t}s_{t}+\frac{G_{t}}{%
\sigma _{t}}\right) \epsilon _{t-h}\epsilon _{t}\right] \right\} \right\Vert
_{2+\iota }  \notag \\
&&\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }+%
\frac{h}{\sqrt{n}}\left\Vert \left( \epsilon _{t}s_{t}+\frac{G_{t}}{\sigma
_{t}}\right) \epsilon _{t-h}\right\Vert _{2+\iota }.  \notag
\end{eqnarray}%
H\"{o}lder and Minkowsky inequalities combined with Assumptions \ref%
{assum:plug} and \ref{assum:Ln}.a imply $||(\epsilon _{t}s_{t}$ $+$ $%
G_{t}/\sigma _{t})\epsilon _{t-h}||_{2+\iota }$ $<$ $\infty $.

By Assumption \ref{assum:Ln}.c$\ \{(\epsilon _{t}s_{t}$ $+$ $G_{t}/\sigma
_{t})\epsilon _{t-h}\}$ is $L_{p}$-bounded, $p$ $>$ $2$, $L_{r}$-NED, $2$ $<$
$r$ $\leq $ $p$, with size $1$, on an $\alpha $-mixing base $\{\upsilon
_{t}\}$ with $\sigma $-fields $\mathfrak{V}_{t}$ $\equiv $ $\sigma (\upsilon
_{\tau }$ $:$ $\tau $ $\leq $ $t)$ and decay $O(h^{-\eta p/(p-2)-\iota })$
where $\eta $ $=$ $(\frac{1}{2+\iota }$ $-$ $\frac{1}{p})^{-1}$. Since $%
\iota $ $>$ $0$ in (\ref{esG_exp})\ is an arbitrary tiny number, we can
always choose it small enough such that $2$ $+$ $\iota <r\leq p$. By H\"{o}%
lder's inequality $\{(\epsilon _{t}s_{t}$ $+$ $G_{t}/\sigma _{t})\epsilon
_{t-h}\}$ is therefore also $L_{2+\iota }$-NED. If $\{(\epsilon _{t}s_{t}$ $%
+ $ $G_{t}/\sigma _{t})\epsilon _{t-h},\mathfrak{V}_{t}\}$ subsequently
forms an $L_{2+\iota }$-mixingale sequence with size $1$, then by Corollary
1 in \cite{Hansen1991}:
\begin{equation*}
\sqrt{n}\left\Vert \frac{1}{n}\sum_{t=1}^{n}\left\{ \left( \epsilon
_{t}s_{t}+\frac{G_{t}}{\sigma _{t}}\right) \epsilon _{t-h}-E\left[ \left(
\epsilon _{t}s_{t}+\frac{G_{t}}{\sigma _{t}}\right) \epsilon _{t-h}\epsilon
_{t}\right] \right\} \right\Vert _{2+\iota }=O(1),
\end{equation*}%
and the proof is complete.

Apply Theorem 17.6 of \cite{Davidson1994} to deduce that $\{(\epsilon
_{t}s_{t}$ $+$ $G_{t}/\sigma _{t})\epsilon _{t-h},\mathfrak{V}_{t}\}$ forms
an $L_{2+\iota }$-mixingale sequence with size%
\begin{equation*}
\min \left\{ 1,\frac{1}{\frac{1}{2+\iota }-\frac{1}{p}}\frac{p}{p-2}\left(
\frac{1}{2+\iota }-\frac{1}{p}\right) \right\} =\min \left\{ 1,\frac{1}{%
\frac{1}{2+\iota }-\frac{1}{p}}\left( \frac{1}{2+\iota }-\frac{1}{p}\right)
\right\} =\min \{1,1\}=1.
\end{equation*}%
This completes the proof. $\mathcal{QED}$.

\setcounter{equation}{0}

\section{Complete Monte Carlo Study \label{app:sim_all_supp_mat}}

In this appendix we perform a large scale Monte Carlo study by presenting
all tests noted in the main paper
\citet[][Section
4]{HillMotegi_maxcorr_main_paper}.

\subsection{Simulation Design \label{app:sim_design_supp_mat}}

We first construct an error term $e_{t}$ that drives an observed variable $%
y_{t}$. Let $\nu _{t}$ be iid $N(0,1)$. We consider iid $e_{t}=\nu _{t}$;
GARCH(1,1) $e_{t}=\nu _{t}w_{t}$ with random volatility process $w_{1}^{2}=1$
and $w_{t}^{2}=1+0.2e_{t-1}^{2}+0.5w_{t-1}^{2}$ for $t\geq 2$; MA(2) $%
e_{t}=\nu _{t}+0.5\nu _{t-1}+0.25\nu _{t-2}$ for $t$ $\geq $ $3$, with
initial values $e_{1}$ $=$ $0$ and $e_{2}$ $=$ $\nu _{2}+0.5\nu _{1}$; and
AR(1) $e_{t}=0.7e_{t-1}+\nu _{t}$ for $t$ $\geq $ $2$ with initial $e_{1}$ $%
= $ $0$. Each error process is strictly stationary and ergodic.\footnote{%
Ergodicity follows since each error process is stationary $\alpha $-mixing.
See, e.g., \cite{KolmogorovRozanov1960} for processes with continuous
bounded spectral densities (e.g. stationary Gaussian AR, Gaussian MA(2));
\cite{Nelson1990} for GARCH process stationarity; and \cite{CarrascoChen2002}
for mixing properties of stationary GARCH processes.}

Now recall Scenarios \#1--\#9 from the main paper:

\renewcommand\labelitemi{}

\begin{itemize}
\item \textbf{Scenario \#1: Simple} $y_{t}=e_{t}$; mean filter $\epsilon
_{t}=y_{t}-E[y_{t}]$; $\hat{\phi}_{n}=1/n\sum_{t=1}^{n}y_{t}$.

\item \textbf{Scenario \#2: Bilinear} $y_{t}=0.5e_{t-1}y_{t-2}+e_{t}$; mean
filter $\epsilon _{t}=y_{t}-E[y_{t}]$; $\hat{\phi}_{n}=1/n%
\sum_{t=1}^{n}y_{t} $.

\item \textbf{Scenario \#3: AR(2)} $y_{t}=0.3y_{t-1}-0.15y_{t-2}+e_{t}$;
AR(2) filter $\epsilon _{t}=y_{t}-\phi _{1}y_{t-1}-\phi _{2}y_{t-2}$; least
squares.

\item \textbf{Scenario \#4: AR(2)} $y_{t}=0.3y_{t-1}-0.15y_{t-2}+e_{t}$;
AR(1) filter $\epsilon _{t}=y_{t}-\phi _{1}y_{t-1}$; least squares.

\item \textbf{Scenario \#5: GARCH(1,1)} $y_{t}=\sigma _{t}e_{t}$, $\sigma
_{t}^{2}=1+0.2y_{t-1}^{2}+0.5\sigma _{t-1}^{2}$; no filter.

\item \textbf{Scenario \#6: GARCH(1,1)} $y_{t}=\sigma _{t}e_{t}$, $\sigma
_{t}^{2}=1+0.2y_{t-1}^{2}+0.5\sigma _{t-1}^{2}$; GARCH(1,1) filter $\epsilon
_{t}=y_{t}/\sigma _{t}$ with $\sigma _{t}^{2}=\omega +\alpha
y_{t-1}^{2}+\beta \sigma _{t-1}^{2}$; quasi-maximum likelihood.\footnote{%
QML is performed using the iterated process $\tilde{\sigma}_{1}^{2}(\theta )$
$=$ $\omega $ and $\tilde{\sigma}_{t}^{2}(\theta )$ $=$ $\omega $ $+$ $%
\alpha y_{t-1}^{2}$ $+$ $\beta \tilde{\sigma}_{t-1}^{2}(\theta )$ for $t$ $=$
$2,\dots ,n$. We impose $(\omega ,\alpha ,\beta )$ $>$ $0$ and $\alpha $ $+$
$\beta $ $\leq $ $1$ during estimation.}

\item \textbf{Scenario \#7: Remote MA(6)} $y_{t}=e_{t}+0.25e_{t-6}$; mean
filter $\epsilon _{t}=y_{t}-E[y_{t}]$; $\hat{\phi}_{n}=1/n%
\sum_{t=1}^{n}y_{t} $.

\item \textbf{Scenario \#8: Remote MA(12)} $y_{t}=e_{t}+0.25e_{t-12}$; mean
filter $\epsilon _{t}=y_{t}-E[y_{t}]$; $\hat{\phi}_{n}=1/n%
\sum_{t=1}^{n}y_{t} $.

\item \textbf{Scenario \#9: Remote MA(24)} $y_{t}=e_{t}+0.25e_{t-24}$; mean
filter $\epsilon _{t}=y_{t}-E[y_{t}]$; $\hat{\phi}_{n}=1/n%
\sum_{t=1}^{n}y_{t} $.
\end{itemize}

\subsubsection{White Noise Tests}

In the main paper, we study the max-correlation test with the dependent wild
bootstrap and automatic lag selection, and Shao's \citeyearpar{Shao2011_JoE}
Cram\'{e}r-von Mises test with the dependent wild bootstrap and $\mathcal{L}%
_{n}=n-1$. In this appendix, we consider several additional white noise
tests and a non-random lag length.

We consider fourteen total tests: eight tests with a non-random $\mathcal{L}%
_{n}=o(n)$ and five tests with $\mathcal{L}_{n}=n-1$, including $CvM^{dw}$;
and the max-correlation test with automatic lag $\mathcal{L}_{n}^{\ast }$.

Tests with $\mathcal{L}_{n}=o(n)$ require a specific selection of lag
length. We use a fixed length at $\mathcal{L}_{n}=5$ and sample-size
dependent length $\mathcal{L}_{n}$ $=$ $[\delta n/\ln (n)]$ with $\delta $ $%
\in $ $\{.5,1\}$, where $[\cdot ]$ truncates to an integer value. In this
set-up, we have $\mathcal{L}_{n}\in \{5,10,21\}$ for $n=100$; $\mathcal{L}%
_{n}\in \{5,22,45\}$ for $n=250$; $\mathcal{L}_{n}\in \{5,40,80\}$ for $%
n=500 $; and $\mathcal{L}_{n}\in \{5,72,144\}$ for $n=1000$. We summarize
the choices in Table \ref{table:Ln}.

\begin{table}[h!]
\caption{Non-Stochastic Lag Values: $\mathcal{L}_{n} \in \{5,[\protect\delta %
n/\ln(n)]\}$, $\protect\delta \in \{.5,1\}$}
\label{table:Ln}
\begin{center}
{\fontsize{11pt}{18pt} \selectfont
\begin{tabular}{l|c|c|c|c}
\hline
$n$ & $100$ & $250$ & $500$ & $1000$ \\ \hline
$\mathcal{L}_{n}$ & $\{5,10,21\}$ & $\{5,22,45\}$ & $\{5,40,80\}$ & $%
\{5,72,144\}$ \\ \hline
\end{tabular}
}
\end{center}
\end{table}

The following tests are summarized in Table \ref{table:list_tests}.

\paragraph{Max-Correlation with Fixed $\mathcal{L}_{n}$ or Automatic Lag $%
\mathcal{L}_{n}^{\ast }$}

We perform both a max-correlation test with automatic lag $\mathcal{L}%
_{n}^{\ast }$ and dependent wild bootstrap, and a max-correlation test with
a non-random lag $\mathcal{L}_{n}$ and wild bootstrap or dependent wild
bootstrap.

The dependent wild bootstrap is always valid asymptotically under our
assumptions, but the wild bootstrap requires independence or a martingale
difference assumption under the null \citep[see][]{Wu1986,Liu1988,Hansen1996}%
. We omit results for the max-correlation test with $\mathcal{L}_{n}^{\ast }$
and wild bootstrap because the max-correlation test with $\mathcal{L}%
_{n}^{\ast }$ and dependent wild bootstrap is both valid under the null and
our assumptions, and overall dominates all other tests in this study.

The test with non-random $\mathcal{L}_{n}$ from Table \ref{table:Ln} is
based on $\mathcal{\hat{T}}_{n}(\mathcal{L}_{n})$ $\equiv $ $\sqrt{n}%
\max_{1\leq h\leq \mathcal{L}_{n}}|\hat{\omega}_{n}(h)\hat{\rho}_{n}(h)|$
with weight $\hat{\omega}_{n}(h)$ $=$ $1$.

The test statistic with the automatic lag is similarly $\mathcal{\hat{T}}%
_{n}(\mathcal{L}_{n}^{\ast })$ $\equiv $ $\sqrt{n}\max_{1\leq h\leq \mathcal{%
L}_{n}^{\ast }}|\hat{\rho}_{n}(h)|$. The automatic lag selection $\mathcal{L}%
_{n}^{\ast }$ is chosen from a set $\{1,...,\mathcal{\bar{L}}_{n}\}$ for
some pre-chosen upper-bound $\mathcal{\bar{L}}_{n}$ $\rightarrow $ $\infty $%
. In the case of the maximum and $\hat{\omega}_{n}(h)$ $=$ $\omega (h)$ $=$ $%
1$, we have from expansion Lemma 2.1 and dependent wild bootstrap Theorem
2.5 that
\begin{equation*}
\mathcal{\bar{L}}_{n}=o\left( \frac{n^{\min \{\zeta ,1/2\}}}{\ln (n)}\right)
\end{equation*}%
must hold, where $\zeta $ $>$ $0$ appears in the Assumption \ref{assum:plug}%
.c or \ref{assum:plug}.c$^{\prime }$ plug-in expansion $\sqrt{n}(\hat{\theta}%
_{n}$ $-$ $\theta _{0})$ $=$ $\mathcal{A}n^{-1/2}\sum_{t=1}^{n}m_{t}(\theta
_{0})$ $+$ $O_{p}(n^{-\zeta })$. Under standard regularity conditions many
plug-in estimators will satisfy $\zeta $ $=$ $1/2$, hence $\mathcal{\bar{L}}%
_{n}$ $=$ $O(\sqrt{n}/\ln (n))$. We use
\begin{equation*}
\bar{\mathcal{L}}_{n}=\left[ \delta \times \frac{\sqrt{n}}{\ln n}\right]
\text{ with }\delta =10,
\end{equation*}%
where $[\cdot ]$ denotes the integer part, hence
\begin{equation*}
\bar{\mathcal{L}}_{n}\in \{21,28,35,45\}\text{ for }n\in
\{100,250,500,1000\}.
\end{equation*}

We also require the tuning parameter $q$ (cf. (12) and (13) in the main
paper). In order to choose a plausible value of $q$, in the main paper we
performed a preliminary simulation study that computes empirical size and
size-adjusted power for the max-correlation test with $\hat{\mathcal{T}}_{n}(%
\mathcal{L}_{n}^{\ast })$\ across $q\in \{1.50,1.75,\dots ,4.50\}$. We
considered two cases. In Case 1, size is computed under Scenario \#1 with an
iid error; and size-adjusted power is computed under \#4 with an iid error.
In Case 2, size is computed under \#5 with an iid error; and size-adjusted
power is computed under \#5 with MA(2) error. For each case, sample size is $%
\{100,500\}$; nominal size is $0.05$; and $1000$ Monte Carlo samples and $%
500 $ bootstrap samples are generated.

See Figure \ref{fig:size_power_maxcorr_automatic_lag} for results. Variation
of empirical size and size-adjusted power for the test based on $\hat{%
\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast })$ across the values of $q$ is
fairly small in each experiment, implying that a choice of $q$ should not
have a critical impact on the test performance. For each case and sample
size, we obtain relatively accurate size and high power around $q$ $=$ $3$.
We therefore use $q$ $=$ $3$ throughout.

\paragraph{Andrews and Ploberger's Sup-LM Test}

Andrews and Ploberger's (\citeyear{AndrewsPloberger1996}) sup-LM test is
based on the representation \citep[see][]{NankervisSavin2010}:
\begin{equation*}
\mathcal{AP}_{n}\equiv \sup_{\lambda \in \Lambda }\text{LM}_{n}(\lambda ,n-1)%
\text{ \, where \, LM}_{n}(\lambda ,n-1)\equiv n(1-\lambda ^{2})\left\{
\sum_{h=1}^{n-1}\lambda ^{h-1}\hat{\rho}_{n}(h)\right\} ^{2}.
\end{equation*}%
\cite{AndrewsPloberger1996} compute LM$_{n}(\lambda ,n-1)$ but use a
truncated maximum lag $\mathcal{L}$ $=$ $50$ for simulating critical values.
\cite{NankervisSavin2010} compute LM$_{n}(\lambda ,\mathcal{L})$ for $%
\mathcal{L}$ $=$ $20$, and use the same $\mathcal{L}$ for simulating
critical values. Thus, the test proposed in \cite{NankervisSavin2010} is,
strictly speaking, not the same test as in \cite{AndrewsPloberger1996}, but
a truncated variant that is therefore inconsistent against some deviations
from the white noise null hypothesis.

We use either $\mathcal{L}_{n} = o(n)$ or $\mathcal{L}_{n} = n - 1$ in order
to accomplish test consistency and fair comparison. Hence we compute $\text{%
LM}_{n} (\lambda, \mathcal{L}_{n}) = n(1 -\lambda ^{2})\{\sum_{h=1}^{%
\mathcal{L}_{n}}\lambda ^{h-1}\hat{\rho}_{n}(h)\}^{2}$. Since the lags vary
with $n$, we do not simulate critical values. We only bootstrap a p-value
via the wild bootstrap or the dependent wild bootstrap.

The parameter space $\Lambda $ is discretized to $\{-.800,-.795,\dots
,.795,.800\}$. We use the same end-points $\pm .8$ as in \cite%
{AndrewsPloberger1996} and \cite{NankervisSavin2010}, but a twice finer grid
(their increment is $.010$). \cite{NankervisSavin2012} use endpoints $\pm
.95 $ with increment $.010$. The distance from the maximum endpoints $\pm 1$
is due to their placing the test in a stationary ARMA framework. We perform
the sup-LM test without an ARMA interpretation and therefore also relax the
endpoints to $\pm .95$ for the sake of comparison. We only report results
based on $\pm .8$ because using $\pm .95$ yields similar rejection
frequencies.

We apply the wild bootstrap and dependent wild bootstrap procedures for the
sup-LM test.

\paragraph{Hong's Test}

Hong's (\citeyear{Hong1996}) test is based on a standardized periodogram. If
the periodogram is computed with a truncated kernel, then the statistic is
just a standardized Box-Pierce statistic. We use a standardized Ljung-Box
statistic $\mathcal{N}_{n}$ $\equiv $ $(2\mathcal{L}_{n})^{-1/2}\sum_{h=1}^{%
\mathcal{L}_{n}}\hat{\omega}_{n}(h)\{n\hat{\rho}_{n}^{2}(h)$ $-$ $1\}$ with $%
\hat{\omega}_{n}(h)$ $=$ $(n$ $+$ $2)/(n$ $-$ $h)$, cf
\citet[eq.
(3)]{Hong1996}. If the null hypothesis of white noise is true \textit{and} $%
\{\sqrt{n}\hat{\rho}_{n}^{2}(h)\}_{h=1}^{\mathcal{L}_{n}}$ are
asymptotically independent, then under Hong's Assumptions 1.a, 2 and 3 we
have $\mathcal{N}_{n}\overset{d}{\rightarrow }$ $N(0,1)$, else $\mathcal{N}%
_{n}$ $\overset{p}{\rightarrow }$ $\infty $. This is a one-sided test where
the rejection region exists only at the upper tail of $N(0,1)$. The
asymptotic independence of $\{\sqrt{n}\hat{\rho}_{n}^{2}(h)\}_{h=1}^{%
\mathcal{L}_{n}}$ holds if tested variable $\epsilon _{t}$ is iid, but may
not hold if $\epsilon _{t}$ is only mds or white noise. Hence, the
asymptotic $N(0,1)$ test is not a white noise test in a strict sense.

We compute p-values via the asymptotic standard normal distribution, wild
bootstrap, and dependent wild bootstrap. Since $\mathcal{N}_{n}$ is just an
affine transformation of the Ljung-Box statistic, a bootstrapped test based
on $\mathcal{N}_{n}$ is identical to a bootstrapped Ljung-Box Q-test when
they are performed at the same lag $\mathcal{L}_{n}$. Bootstrapped p-values
are computed as $\hat{p}_{n, M} = 1/M \sum_{i = 1}^{M} I(\mathcal{N}_{n,
i}^{(boot)} \ge \mathcal{N}_{n})$.

\paragraph{Cram\'{e}r-von Mises Test}

In the main paper, we run the Cram\'{e}r-von Mises test with the dependent
wild bootstrap. In this study we use three bootstrap procedures: the wild
bootstrap, dependent wild bootstrap, and Zhu and Li's \citeyearpar{ZhuLi2015}
blockwise random weighting bootstrap [BRWB].

The BRWB algorithm is performed as follows (we ignore the first-order
correlation expansion for notational clarity). Suppose that the objective
function to be minimized is written as $1/n\sum_{t=1}^{n}l_{t}(\theta )$.
Set a block size $b_{n}$, $1\leq b_{n}<n$, and denote the blocks by $%
\mathcal{B}_{s}=\{(s-1)b_{n}+1,\dots ,sb_{n}\}$ with $s=1,\dots ,n/b_{n}$.
Assume $n/b_{n}$ is an integer for simplicity. Generate positive iid random
numbers $\{\delta _{1},\dots ,\delta _{n/b_{n}}\}$ from a common
distribution with mean $1$ and variance $1$. Following \cite{ZhuLi2015}, we
use the Bernoulli distribution with $P[\delta _{t}$ $=$ $0.5$ $\times $ $(3$
$-$ $\sqrt{5})]$ $=$ $(2\sqrt{5})^{-1}$ $\times $ $(1$ $+$ $\sqrt{5})$ and $%
P[\delta _{t}$ $=$ $0.5$ $\times $ $(3$ $+$ $\sqrt{5})]$ $=$ $1$ $-$ $(2%
\sqrt{5})^{-1}$ $\times $ $(1$ $+$ $\sqrt{5})$. Define an auxiliary variable
$\omega _{t}^{\ast }=\delta _{s}$ if $t\in \mathcal{B}_{s}$. Calculate $\hat{%
\theta}_{n}^{\ast }=\text{argmin}_{\theta \in \Theta
}1/n\sum_{t=1}^{n}\omega _{t}^{\ast }l_{t}(\theta )$. Compute $\hat{\gamma}%
_{n}^{\ast }(h)=1/n\sum_{t=1+h}^{n}\omega _{t}^{\ast }\epsilon _{t}(\hat{%
\theta}_{n}^{\ast })\epsilon _{t-h}(\hat{\theta}_{n}^{\ast })$ and $%
S_{n}^{\ast }(\lambda )$ $=$ $\sum_{h=1}^{n-1}\sqrt{n}\hat{\gamma}_{n}^{\ast
}(h)\psi _{h}(\lambda )$, where $\psi _{h}(\lambda )=(h\pi )^{-1}\sin
(h\lambda )$. Define the bootstrapped process $\Delta _{n}(\lambda
)=S_{n}^{\ast }(\lambda )-S_{n}(\lambda )-Z_{n}(\lambda )$, where $%
Z_{n}(\lambda )=n\sum_{h=1}^{n-1}\{\sum_{t=1+h}^{n}\omega _{t}^{\ast }-n+h\}%
\hat{\gamma}_{n}(h)\psi _{h}(\lambda )$. Then compute the bootstrapped CvM
test statistic $\mathcal{C}_{n}^{\ast }=\int_{0}^{\pi }\{\Delta _{n}(\lambda
)\}^{2}d\lambda $. Repeat $M$ times, resulting in the sequence $\{\mathcal{C}%
_{n,i}^{\ast }\}_{i=1}^{M}$ and approximate p-value $1/M\sum_{i=1}^{M}I(%
\mathcal{C}_{n,i}^{\ast }\geq \mathcal{C}_{n})$.

\paragraph{Ljung-Box Q-test}

We perform the Ljung-Box Q-test with $\mathcal{L}_{n}$ from Table \ref%
{table:Ln}. P-values are computed from the $\chi ^{2}(\mathcal{L}_{n}$ $-$ $%
k_{\theta })$ distribution. We do not report bootstrap test because the
bootstrapped Q-test is arithmetically identical to Hong's (%
\citeyear{Hong1996}) bootstrapped standardized Q-test, as noted above.

\paragraph{Wild Bootstrap and Dependent Wild Bootstrap P-Value Computation
\label{app:p_values}}

The following details how sample autocorrelations are bootstrapped by the
wild bootstrap \citep{Wu1986,Liu1988} and dependent wild bootstrap %
\citep{Shao2011_JoE}. Recall $m_{t}(\theta )$ are the estimating equations
for $\hat{\theta}_{n}$, let $\widehat{\mathcal{A}}_{n}$ be a consistent
estimator of $\mathcal{A}$\ in Assumption \ref{assum:plug}.c, and define
\begin{equation}
\mathcal{\hat{D}}_{n}(h)\equiv \frac{1}{n}\sum_{t=h+1}^{n}\left\{ \left(
\epsilon _{t}(\hat{\theta}_{n})s_{t}(\hat{\theta}_{n})+\frac{G_{t}(\hat{%
\theta}_{n})}{\sigma _{t}(\hat{\theta}_{n})}\right) \epsilon _{t-h}(\hat{%
\theta}_{n})+\epsilon _{t}(\hat{\theta}_{n})\left( \epsilon _{t-h}(\hat{%
\theta}_{n})s_{t-h}(\hat{\theta}_{n})+\frac{G_{t-h}(\hat{\theta}_{n})}{%
\sigma _{t-h}(\hat{\theta}_{n})}\right) \right\}
\end{equation}%
and
\begin{equation*}
\widehat{\mathcal{E}}_{n,t,h}(\hat{\theta}_{n})\equiv \epsilon _{t}(\hat{%
\theta}_{n})\epsilon _{t-h}(\hat{\theta}_{n})-\mathcal{\hat{D}}%
_{n}(h)^{\prime }\widehat{\mathcal{A}}_{n}m_{t}(\hat{\theta}_{n}).
\end{equation*}

The dependent wild bootstrapped sample correlation $\hat{\rho}%
_{n,i}^{(dw)}(h)$ is computed as:
\begin{equation*}
\hat{\rho}_{n}^{(dw)}(h)\equiv \frac{1}{1/n\sum_{t=1}^{n}\epsilon _{t}^{2}(%
\hat{\theta}_{n})}\frac{1}{n}\sum_{t=1+h}^{n}\varphi _{t}\left\{ \widehat{%
\mathcal{E}}_{n,t,h}(\hat{\theta}_{n})-\frac{1}{n}\sum_{s=1+h}^{n}\widehat{%
\mathcal{E}}_{n,s,h}(\hat{\theta}_{n})\right\} .
\end{equation*}%
Notice $\hat{\rho}_{n}^{(dw)}(h)$ exploits the Lemma 2.1 correlation
expansion via $\widehat{\mathcal{E}}_{n,t,h}(\hat{\theta}_{n})$, which
correctly accounts for the first order (asymptotic) impact of the $i^{th}$
sample's plug-in $\hat{\theta}_{n,i}$.

In each case we generate $M$ $=$ $500$ bootstrap samples. The dependent wild
bootstrap requires a block size $b_{n}$, while \cite{Shao2011_JoE} uses $%
b_{n}$ $=$ $b\sqrt{n}$ with $b$ $\in $ $\{.5,1,2\}$, leading to
qualitatively similar results. We therefore simply use the middle value $%
b_{n}$ $=$ $\sqrt{n}$.\footnote{%
In simulations not reported here, we compared $b_{n}=b\sqrt{n}$ across $b$ $%
\in $ $\{.5,1,2\}$ and found there is little difference in test performance.}
The wild bootstrap has block size $b_{n}$ $=$ $1$ and no re-centering with $%
1/n\sum_{s=1+h}^{n}\widehat{\mathcal{E}}_{n,s,h}(\hat{\theta}_{n})$.

We perform the max-correlation test with fixed $\mathcal{L}_{n}$ and wild
bootstrap or dependent wild bootstrap, denoted $\mathcal{\hat{T}}_{n}^{w}(%
\mathcal{L}_{n})$ and $\mathcal{\hat{T}}_{n}^{dw}(\mathcal{L}_{n})$. We
perform the max-correlation test with automatic lag by dependent wild
bootstrap, $\mathcal{\hat{T}}_{n}^{dw}(\mathcal{L}_{n}^{\ast })$.

Consider $\mathcal{\hat{T}}_{n}^{dw}(\mathcal{L}_{n}^{\ast })$. We compute
the bootstrapped statistic $\mathcal{\hat{T}}_{n,i}^{(dw)}(\mathcal{L}%
_{n,i}^{\ast })$ $\equiv $ $\sqrt{n}\max_{1\leq h\leq \mathcal{L}%
_{n,i}^{\ast }}|\hat{\rho}_{n,i}^{(dw)}(h)|$ for each bootstrap sample $i\in
\{1,\dots ,M\}$ with $M=500$. Note that $\mathcal{L}_{n,i}^{\ast }$ is the
automatic lag for the $i^{th}$ bootstrap sample specifically. The
approximate p-value is computed as
\begin{equation*}
\hat{p}_{n,M}^{(dw)}=\frac{1}{M}\sum_{i=1}^{M}I\left( \mathcal{\hat{T}}%
_{n,i}^{(dw)}(\mathcal{L}_{n,i}^{\ast })\geq \hat{\mathcal{T}}_{n}(\mathcal{L%
}_{n}^{\ast })\right) .
\end{equation*}%
The test proposed rejects the null at nominal size $\alpha $ when $\hat{p}%
_{n,M}^{(dw)}$ $<$ $\alpha $.

The Q-statistic, Hong's normalized Q-statistic, the sup-LM statistic and the
CvM statistic are linear combinations of (squared) residual correlations.
Therefore, exactly as with the max-correlation test, both bootstrap
procedures are applied based on the Lemma 2.1 correlation expansion in order
to account for the impact of the estimator $\hat{\theta}_{n}$ on asymptotics.

\subsection{Simulation Results \label{app:sim_results_supp_mat}}

As in the main paper, sample size is $n\in \{100,250,500,1000\}$, and $%
J=1000 $ independent Monte Carlo samples and $M=500$ independent bootstrap
samples are generated. Rejection frequencies with respect to nominal size $%
\alpha \in \{.010,.050,.100\}$ are reported. See Tables \ref%
{table:supp_mat_scenario1_n100}-\ref{table:supp_mat_scenario9} for rejection
frequencies from Sup-LM, Hong, Q, and $\mathcal{\hat{T}}_{n}(\mathcal{L}%
_{n}) $ each based on $\mathcal{L}_{n} = o(n)$ from Table \ref{table:Ln}.
Tables \ref{table:supp_mat_AP_CvM_scenario123456_n100}-\ref%
{table:supp_mat_AP_CvM_scenario789} contain results for sup-LM with $%
\mathcal{L}_{n}$ $=$ $n$ $-$ $1$ and CvM with $\mathcal{L}_{n}$ $=$ $n$ $-$ $%
1$ and $\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$. The groupings
follow from the fact that the former group uses various $\mathcal{L}_{n}$,
while the latter group uses exactly one maximum lag.

\subsubsection{Empirical Size \label{app:sim_results_supp_mat_size}}

\paragraph{Max-Correlation test}

We focus on the max-correlation test with the dependent wild bootstrap and
fixed $\mathcal{L}_{n}$ (denoted $\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n})$).
We then compare results with $\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast
}) $ in order to see how our proposed lag selection approach improves
inference. We comment below on the wild bootstrap results as compared to the
dependent wild bootstrap.

Consider Scenario \#1, $n=100$, and iid error $e_{t}$ (see Table \ref%
{table:supp_mat_scenario1_n100}). The empirical size with respect to nominal
sizes $\alpha \in \{.010,.050,.100\}$ is $\{.009,.060,.135\}$ at $\mathcal{L}%
_{n}=5$, $\{.009,.047,.113\}$ at $\mathcal{L}_{n}=10$, and $%
\{.000,.026,.077\}$ at $\mathcal{L}_{n}=21$. The empirical size of $\hat{%
\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast})$, in contrast, is $\{.017, .068,
.128 \}$ (see Table \ref{table:supp_mat_AP_CvM_scenario123456_n100}). Those
results suggest that $\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n})$ with $%
\mathcal{L}_{n}=o(n)$ has a tendency of under-rejections when lag length is
large.

The same implication holds under more complex scenarios which involve
filters. See, for example, Scenario \#6, $n=1000$, and iid error (Table \ref%
{table:supp_mat_scenario6_n1000}). The empirical size of $\hat{\mathcal{T}}%
^{dw}(\mathcal{L}_{n})$ is $\{.011,.052,.106\}$ at $\mathcal{L}_{n}=5$, $%
\{.002,.028,.083\}$ at $\mathcal{L}_{n}=72$, and $\{.004,.022,.066\}$ at $%
\mathcal{L}_{n}=144$. The empirical size of $\hat{\mathcal{T}}^{dw}(\mathcal{%
L}_{n}^{\ast })$, in contrast, is $\{ .011, .047, .101 \}$ (Table \ref%
{table:supp_mat_AP_CvM_scenario123456_n1000}). Those results highlight that
the automatic lag selection significantly stabilizes the empirical size of
the max-correlation test by trimming redundant lags.

\paragraph{Hong's Test with Dependent Wild Bootstrap}

Hong's test with the dependent wild bootstrap ($\mathcal{N}^{dw}$) is even
more conservative than $\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n})$ at large
lags. In Scenario \#1, $n=100$, and iid error $e_{t}$ (Table \ref%
{table:supp_mat_scenario1_n100}), the empirical size of $\mathcal{N}^{dw}$
is $\{.007,.054,.122\}$ at $\mathcal{L}_{n}=5$, $\{.002,.023,.073\}$ at $%
\mathcal{L}_{n}=10$, and $\{.000,.026,.075\}$ at $\mathcal{L}_{n}=21$. In
Scenario \#6, $n=1000$, and iid error (Table \ref%
{table:supp_mat_scenario6_n1000}), the empirical size of $\mathcal{N}^{dw}$
is $\{.018,.061,.137\}$ at $\mathcal{L}_{n}=5$, $\{.001,.006,.028\}$ at $%
\mathcal{L}_{n}=72$, and $\{.000,.001,.014\}$ at $\mathcal{L}_{n}=144$.
Given $\mathcal{L}_{n}=o(n)$, the max-correlation test statistic leads to a
more accurate size than Hong's statistic arguably because the former has a
simpler structure than the latter. The former picks only the largest sample
correlation, while the latter sums up all $\mathcal{L}_{n}$ sample
correlations into one test statistic.

\paragraph{Bootstrapped Sup-LM Test and CvM Test}

We now discuss the sup-LM tests assisted by the dependent wild bootstrap ($%
\mathcal{AP}^{dw}$) with $\mathcal{L}_{n}$ $=$ $o(n)$ from Table \ref%
{table:Ln} and $\mathcal{L}_{n}=n-1$; and the CvM test assisted by the
dependent wild bootstrap ($CvM^{dw}$) or BRWB ($CvM^{brw}$). Each has
similar empirical size. $\mathcal{AP}^{dw}$ with $\mathcal{L}_{n}$ $=$ $o(n)$
produces stable empirical size across various lag lengths, which is a
considerable advantage over $\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n})$ and $%
\mathcal{N}^{dw}$. See, for example, Scenario \#1, $n=100$, and iid error $%
e_{t}$ (Table \ref{table:supp_mat_scenario1_n100}). The empirical size of $%
\mathcal{AP}^{dw}$ is $\{.014,.071,.135\}$ at $\mathcal{L}_{n}=5$, $%
\{.020,.082,.154\}$ at $\mathcal{L}_{n}=10$, and $\{.020,.064,.134\}$ at $%
\mathcal{L}_{n}=21$.

A disadvantage of the sup-LM and CvM tests is that they sometimes lead to
over-rejections. Such a tendency is indeed observed from the rejection
frequencies of $\mathcal{AP}^{dw}$ reported above. Another example is $%
\mathcal{L}_{n}=n-1$, Scenario \#2, $n=100$, and iid error (Table \ref%
{table:supp_mat_AP_CvM_scenario123456_n100}). The empirical size is $%
\{.022,.083,.151\}$ for $\mathcal{AP}^{dw}$, $\{.018,.076,.149\}$ for $%
CvM^{dw}$, and $\{.045,.115,.186\}$ for $CvM^{brw}$. The empirical size of $%
\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast })$ is $\{ .008, .047, .090 \}$%
, which is much sharper than the size of sup-LM and CvM tests.

\paragraph{Asymptotic Q-Test and Asymptotic Hong Test}

The asymptotic Ljung-Box test ($Q_{\chi ^{2}}$) and the asymptotic Hong test
($\mathcal{N}^{N_{0,1}}$) produce fatal size distortions when the true
process is non-iid. See, for example, Scenario \#2, $n=1000$, and iid error $%
e_{t}$ (Table \ref{table:supp_mat_scenario2_n1000}), where the tested
variable is bilinear. The empirical size at $\mathcal{L}_{n}=5$ is $%
\{.064,.169,.254\}$ for $Q_{\chi ^{2}}$ and $\{.135,.207,.264\}$ for $%
\mathcal{N}^{N_{0,1}}$. Similar results are observed in Scenario \#5, $%
n=1000 $, and iid error $e_{t}$ (Table \ref{table:supp_mat_scenario5_n1000}%
), where the tested variable is GARCH. Those results confirm that the
asymptotic convergence of the Ljung-Box test statistic to $\chi ^{2}$ and
that of the Hong test statistic to $N(0,1)$ require more than serial
uncorrelatedness. Hence, those tests are not white noise tests in a strict
sense.

\paragraph{Wild Bootstrap Versus Dependent Wild Bootstrap}

Given our simulation design, the wild bootstrap often results in more
accurate size than the dependent wild bootstrap for each test under study.
Revisit Scenario \#6, $n=1000$, and iid error (Table \ref%
{table:supp_mat_scenario6_n1000}), where $\hat{\mathcal{T}}^{dw}(\mathcal{L}%
_{n})$ becomes more and more conservative as lag length increases as stated
above. The max-correlation test with the wild bootstrap ($\hat{\mathcal{T}}%
^{w}(\mathcal{L}_{n})$), in contrast, leads to stable empirical size of $%
\{.011,.048,.102\}$ at $\mathcal{L}_{n}=5$, $\{.006,.049,.090\}$ at $%
\mathcal{L}_{n}=72$, and $\{.005,.041,.103\}$ at $\mathcal{L}_{n}=144$.
Similar results arise for Hong's test, the sup-LM test, and the CvM test as
well. Interestingly, the wild bootstrap still operates well under non-mds
white noise cases like bilinear. Those results, however, call for some
caution since there is not a theoretical guarantee that the wild bootstrap
is valid under non-mds white noise.\footnote{%
It is left as a future task to find a numerical example of non-mds white
noise where the wild bootstrap fails and the dependent wild bootstrap
operates well.}

\paragraph{Summary}

The empirical size of $\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast })$ is
overall more accurate than the size of any other test in this study. $\hat{%
\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast })$ is free of an ad-hoc selection
of lag length, and has sharp size even in small samples with relatively
challenging scenarios like bilinear or GARCH.

\subsubsection{Empirical Power \label{app:sim_results_supp_mat_power}}

We now discuss empirical power of each test. We do not discuss $Q_{\chi
^{2}} $ or $\mathcal{N}^{N_{0,1}}$ since they produce fatal size distortions
under non-iid scenarios. We do not discuss $\hat{\mathcal{T}}^{dw}(\mathcal{L%
}_{n}) $ or $\mathcal{N}^{dw}$ with $\mathcal{L}_{n}=o(n)$ since they tend
to be conservative at large lags. Since $\mathcal{AP}^{dw}$ with $\mathcal{L}%
_{n}=o(n)$ and $\mathcal{L}_{n}=n-1$ produce virtually identical results, we
discuss the latter only. Thus, we compare $\hat{\mathcal{T}}^{dw}(\mathcal{L}%
_{n}^{\ast })$ with $\mathcal{AP}^{dw}$, $CvM^{dw}$, and $CvM^{brw}$ with $%
\mathcal{L}_{n}=n-1$.

In Scenarios \#1--\#6, there is not an obvious ranking among $\{\hat{%
\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast }),\mathcal{AP}%
^{dw},CvM^{dw},CvM^{brw}\}$. $\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast
}) $ is more powerful than the other tests in some cases, but not in other
cases. Overall we do not observe a drastic difference in power. See, for
example, Scenario \#2, $n=1000$, and AR(1) error (Table \ref%
{table:supp_mat_AP_CvM_scenario123456_n1000}). The empirical power with
respect to nominal sizes $\alpha \in \{.010,.050,.100\}$ is $%
\{.723,.823,.864\}$ for $\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast })$, $%
\{.408,.661,.774\}$ for $\mathcal{AP}^{dw}$, $\{.474,.697,.810\}$ for $%
CvM^{dw}$, and $\{.615,.761,.833\}$ for $CvM^{brw}$. In this specific case, $%
\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast })$ is most powerful; $%
CvM^{brw} $ is the second; $CvM^{dw}$ is the third; $\mathcal{AP}^{dw}$ is
least powerful.

Another example is Scenario \#3, $n=1000$, and AR(1) error (Table \ref%
{table:supp_mat_AP_CvM_scenario123456_n1000}). The empirical power is $%
\{.599,.847,.922\}$ for $\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast })$, $%
\{.459,.681,.787\}$ for $\mathcal{AP}^{dw}$, $\{.688,.876,.923\}$ for $%
CvM^{dw}$, and $\{.700,.878,.913\}$ for $CvM^{brw}$. In this specific case, $%
\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast })$ is more powerful than $%
\mathcal{AP}^{dw}$, but slightly less powerful than $\{CvM^{dw},CvM^{brw}\}$
when the nominal size is 1\%.

In Scenarios \#7--\#9, $\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast })$
dominates $\{\mathcal{AP}^{dw},CvM^{dw},CvM^{brw}\}$ completely. The former
can detect remote autocorrelations given a large enough sample size $n$,
while the latter cannot. The power of $\hat{\mathcal{T}}^{dw}(\mathcal{L}%
_{n}^{\ast })$ under Scenario \#8 (Remote MA(12)), for instance, is $%
\{.013,.067,.117\}$ for $n=100$, $\{.024,.134,.244\}$ for $n=250$, $%
\{.371,.673,.770\}$ for $n=500$, and $\{.983,.997,.997\}$ for $n=1000$
(Table \ref{table:supp_mat_AP_CvM_scenario789}). We observe that $\hat{%
\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast })$ captures remote autocorrelation
with increasing probability as $n$ grows. The reason for this desired result
is that, as confirmed in \citet[][Table
2]{HillMotegi_maxcorr_main_paper}, $\mathcal{L}_{n}^{\ast }$ converges to $%
h^{\ast }=12$ sufficiently fast under Remote MA(12). See Section 4.2.3 in
the main paper for further discussion.

The power of $CvM^{brw}$, by contrast, is $\{.056,.125,.189\}$ for $n=100$, $%
\{.020,.094,.156\}$ for $n=250$, $\{.020,.068,.128\}$ for $n=500$, and $%
\{.026,.081,.152\}$ for $n=1000$ (Table \ref%
{table:supp_mat_AP_CvM_scenario789}). $CvM^{brw}$ has no power against the
remote autocorrelation even when sample size is as large as $n=1000$. The
power of $\mathcal{AP}^{dw}$ and $CvM^{dw}$ is virtually identical to the
power of $CvM^{brw}$. Hence, $\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast
}) $ is the only test under study that has power against remote
autocorrelations.

The reason why the sup-LM and CvM tests fail to capture remote
autocorrelations is that those test statistics assign the largest weights to
small lags. That feature delivers sharp size and high power against adjacent
correlations like those in Scenarios \#1--\#6, but critically low power
against remote correlations like those in Scenarios \#7--\#9. The present
max-correlation test statistic, in contrast, assigns equal weights to all
lags. That feature itself delivers under-rejections and low power against
adjacent correlations when $\mathcal{L}_{n}$ is large, but such a
shortcoming is addressed by our proposed automatic lag selection mechanism.%
\footnote{%
As mentioned in the main paper, a weighted max-correlation test where the
weights are inverted standard errors may provide necessary improvements when
the data are extremely volatile (e.g. bilinear with GARCH error). The
difficulty there, however, is the need for a non-parametric variance
estimator that will be sensitive to choice of tuning parameter. These issues
are left for a future study.}

\paragraph{Summary}

Taking both size and power into account, $\hat{\mathcal{T}}^{dw}(\mathcal{L}%
_{n}^{\ast })$ generally dominates each white noise test studied here.
First, $\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast })$ achieves the most
accurate size. Second, under $H_{1}$ with adjacent correlations, the
relative performance of $\{\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast }),%
\mathcal{AP}^{dw},CvM^{dw},CvM^{brw}\}$ varies across cases, but in any case
there is not a large difference in power. Third, $\hat{\mathcal{T}}^{dw}(%
\mathcal{L}_{n}^{\ast })$ has much higher power against remote
autocorrelations than the strongest competitors $\{\mathcal{AP}%
^{dw},CvM^{dw},CvM^{brw}\}$. The latter tests cannot detect remote
autocorrelations even when sample size is as large as $n=1000$. Thus, $\hat{%
\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast })$ is indeed the \textit{only}
white noise test in this study that achieves \textit{both} accurate size
under each null \textit{and} high power under each alternative.

\clearpage

\begin{table}[th]
\caption{List of All White Noise Tests }
\label{table:list_tests}
\begin{center}
{\fontsize{10pt}{18pt} \selectfont%
\begin{tabular}{c|c|c|c}
\multicolumn{4}{c}{Tests with non-random $\mathcal{L}_{n}=o(n)$ (cf. Table %
\ref{table:Ln})} \\ \hline
Symbol & Test Statistic & P-Value Computation & Reference \\ \hline
$\hat{\mathcal{T}}^{w}(\mathcal{L}_{n})$ & \multicolumn{1}{|l|}{max-corr
with fixed $\mathcal{L}_{n}$} & \multicolumn{1}{|l|}{wild bootstrap} &
\multicolumn{1}{|l}{\cite{HillMotegi_maxcorr_main_paper}} \\
$\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n})$ & \multicolumn{1}{|l|}{max-corr
with fixed $\mathcal{L}_{n}$} & \multicolumn{1}{|l|}{dependent wild bootstrap
} & \multicolumn{1}{|l}{\cite{HillMotegi_maxcorr_main_paper}} \\
$\mathcal{AP}^{w}$ & \multicolumn{1}{|l|}{sup-LM} & \multicolumn{1}{|l|}{
wild bootstrap} & \multicolumn{1}{|l}{\cite{AndrewsPloberger1996}} \\
$\mathcal{AP}^{dw}$ & \multicolumn{1}{|l|}{sup-LM} & \multicolumn{1}{|l|}{
dependent wild bootstrap} & \multicolumn{1}{|l}{\cite{AndrewsPloberger1996}}
\\
$Q_{\chi ^{2}}$ & \multicolumn{1}{|l|}{Ljung-Box Q-statistic} &
\multicolumn{1}{|l|}{asymptotic $\chi ^{2}$} & \multicolumn{1}{|l}{\cite%
{LjungBox1978}} \\
$\mathcal{N}^{N_{0,1}}$ & \multicolumn{1}{|l|}{std. Ljung-Box Q-statistic} &
\multicolumn{1}{|l|}{asymptotic $N(0,1)$} & \multicolumn{1}{|l}{\cite%
{Hong1996}} \\
$\mathcal{N}^{w}$ & \multicolumn{1}{|l|}{std. Ljung-Box Q-statistic} &
\multicolumn{1}{|l|}{wild bootstrap} & \multicolumn{1}{|l}{\cite{Hong1996}}
\\
$\mathcal{N}^{dw}$ & \multicolumn{1}{|l|}{std. Ljung-Box Q-statistic} &
\multicolumn{1}{|l|}{dependent wild bootstrap} & \multicolumn{1}{|l}{\cite%
{Hong1996}} \\ \hline
\multicolumn{4}{c}{} \\
\multicolumn{4}{c}{Tests with $\mathcal{L}_{n}=n-1$} \\ \hline
Symbol & Test Statistic & P-Value Computation & Reference \\ \hline
$\mathcal{AP}^{w}$ & \multicolumn{1}{|l|}{sup-LM} & \multicolumn{1}{|l|}{
wild bootstrap} & \multicolumn{1}{|l}{\cite{AndrewsPloberger1996}} \\
$\mathcal{AP}^{dw}$ & \multicolumn{1}{|l|}{sup-LM} & \multicolumn{1}{|l|}{
dependent wild bootstrap} & \multicolumn{1}{|l}{\cite{AndrewsPloberger1996}}
\\
$CvM^{w}$ & \multicolumn{1}{|l|}{Cram\'{e}r-von Mises} &
\multicolumn{1}{|l|}{wild bootstrap} & \multicolumn{1}{|l}{-} \\
$CvM^{dw}$ & \multicolumn{1}{|l|}{Cram\'{e}r-von Mises} &
\multicolumn{1}{|l|}{dependent wild bootstrap} & \multicolumn{1}{|l}{\cite%
{Shao2011_JoE}} \\
$CvM^{brw}$ & \multicolumn{1}{|l|}{Cram\'{e}r-von Mises} &
\multicolumn{1}{|l|}{blockwise random weighting bootstrap} &
\multicolumn{1}{|l}{\cite{ZhuLi2015}} \\ \hline
\multicolumn{4}{c}{} \\
\multicolumn{4}{c}{Test with automatic lag $\mathcal{L}_{n}^{*}$} \\ \hline
Symbol & Test Statistic & P-Value Computation & Reference \\ \hline
$\hat{\mathcal{T}}^{dw}(\mathcal{L}_{n}^{\ast })$ & \multicolumn{1}{|l|}{
max-corr with automatic $\mathcal{L}_{n}^{\ast }$} & \multicolumn{1}{|l|}{
dependent wild bootstrap} & \multicolumn{1}{|l}{\cite%
{HillMotegi_maxcorr_main_paper}} \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont std. $=$ standardized. For $Q_{\chi ^{2}}$%
, the degrees of freedom of the asymptotic $\chi ^{2}$-distribution are $%
\mathcal{L}_{n}-2$ in Scenario \#3 since an AR(2) filter is used; $\mathcal{L%
}_{n}-1$ in Scenario \#4 since an AR(1) filter is used; and $\mathcal{L}_{n}$
in the other scenarios. The automatic $\mathcal{L}_{n}^{\ast }$ is computed
such that $\mathcal{L}_{n}^{\ast }/\mathcal{\bar{L}}_{n}$ $\rightarrow $ $%
[0,1]$ for a non-random upper bound $\mathcal{\bar{L}}_{n}$ $=$ $[10 \times
\sqrt{n} / (\ln n)] = o(\sqrt{n} / \ln (n))$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#1 ($n = 100$) }
\label{table:supp_mat_scenario1_n100}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 10 & Lag = 21 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .006, .039, .089 & .004, .038, .096 & .005, .028,
.066 & .005, .041, .090 & .003, .043, .082 & .003, .044, .089 \\
$\hat{\mathcal{T}}^{dw}$ & .009, .060, .135 & .009, .047, .113 & .000, .026,
.077 & .002, .048, .093 & .005, .044, .099 & .001, .021, .057 \\ \hline
$\mathcal{AP}^{w}$ & .002, .047, .097 & .008, .026, .087 & .001, .033, .071
& .007, .050, .087 & .003, .038, .090 & .007, .035, .073 \\
$\mathcal{AP}^{dw}$ & .014, .071, .135 & .020, .082, .154 & .020, .064, .134
& .014, .062, .134 & .013, .063, .145 & .012, .058, .125 \\ \hline
$Q_{\chi^{2}}$ & .010, .045, .094 & .020, .057, .113 & .029, .095, .157 &
.033, .098, .172 & .036, .091, .150 & .049, .103, .152 \\
$\mathcal{N}^{N_{0,1}}$ & .021, .041, .087 & .019, .055, .096 & .023, .050,
.070 & .087, .139, .191 & .042, .094, .144 & .032, .058, .092 \\
$\mathcal{N}^{w}$ & .002, .032, .077 & .005, .036, .092 & .003, .040, .086 &
.008, .042, .086 & .008, .031, .088 & .005, .044, .081 \\
$\mathcal{N}^{dw}$ & .007, .054, .122 & .002, .023, .073 & .000, .026, .075
& .003, .040, .103 & .000, .027, .082 & .000, .012, .047 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 10 & Lag = 21 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .918, .984, .991 & .907, .974, .984 & .902, .970,
.985 & 1.00, 1.00, 1.00 & .999, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\hat{\mathcal{T}}^{dw}$ & .788, .956, .990 & .706, .950, .984 & .623, .922,
.974 & .955, .998, 1.00 & .941, 1.00, 1.00 & .936, .995, .999 \\ \hline
$\mathcal{AP}^{w}$ & .922, .992, 1.00 & .900, .986, .996 & .904, .982, .995
& 1.00, 1.00, 1.00 & .999, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{AP}^{dw}$ & .743, .956, .991 & .697, .934, .979 & .677, .935, .985
& .831, .981, .998 & .756, .965, .992 & .758, .956, .998 \\ \hline
$Q_{\chi^{2}}$ & .937, .977, .991 & .870, .960, .970 & .762, .880, .929 &
.998, 1.00, 1.00 & 1.00, 1.00, 1.00 & .996, 1.00, 1.00 \\
$\mathcal{N}^{N_{0,1}}$ & .971, .991, .994 & .913, .963, .978 & .774, .856,
.899 & 1.00, 1.00, 1.00 & .998, .998, .998 & .996, 1.00, 1.00 \\
$\mathcal{N}^{w}$ & .844, .975, .986 & .644, .894, .950 & .434, .771, .884 &
.996, 1.00, 1.00 & .987, 1.00, 1.00 & .964, .998, 1.00 \\
$\mathcal{N}^{dw}$ & .492, .839, .950 & .168, .608, .841 & .048, .253, .529
& .703, .961, .996 & .482, .897, .978 & .172, .594, .853 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#1: Simple $y_{t} = e_{t}$;
mean filter $\epsilon_{t} = y_{t} - E[y_{t}]$; $\hat{\phi}_{n} = 1/n
\sum_{t=1}^{n} y_{t}$. In this table we report rejection frequencies with
the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the proposed
max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#1 ($n = 250$) }
\label{table:supp_mat_scenario1_n250}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 22 & Lag = 45 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .008, .043, .086 & .007, .039, .090 & .007, .039,
.072 & .005, .039, .084 & .008, .050, .086 & .004, .041, .082 \\
$\hat{\mathcal{T}}^{dw}$ & .013, .065, .135 & .003, .046, .094 & .001, .032,
.080 & .007, .041, .108 & .001, .033, .080 & .001, .019, .052 \\ \hline
$\mathcal{AP}^{w}$ & .006, .040, .093 & .001, .037, .092 & .008, .047, .093
& .004, .038, .077 & .007, .042, .082 & .006, .039, .073 \\
$\mathcal{AP}^{dw}$ & .008, .060, .123 & .013, .062, .134 & .010, .064, .109
& .018, .064, .108 & .007, .055, .118 & .011, .050, .109 \\ \hline
$Q_{\chi^{2}}$ & .015, .056, .109 & .022, .057, .097 & .026, .086, .133 &
.049, .114, .188 & .039, .108, .164 & .035, .091, .151 \\
$\mathcal{N}^{N_{0,1}}$ & .024, .049, .085 & .028, .057, .092 & .020, .053,
.083 & .080, .149, .201 & .045, .096, .132 & .016, .054, .082 \\
$\mathcal{N}^{w}$ & .010, .051, .095 & .005, .039, .088 & .006, .031, .073 &
.006, .044, .097 & .008, .040, .084 & .002, .030, .080 \\
$\mathcal{N}^{dw}$ & .004, .046, .116 & .000, .015, .050 & .001, .007, .033
& .003, .044, .103 & .000, .005, .036 & .000, .009, .031 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 22 & Lag = 45 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\hat{\mathcal{T}}^{dw}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\ \hline
$\mathcal{AP}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00
& 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{AP}^{dw}$ & .995, 1.00, 1.00 & .974, .999, 1.00 & .971, .999, 1.00
& .991, .999, 1.00 & .973, 1.00, 1.00 & .980, 1.00, 1.00 \\ \hline
$Q_{\chi^{2}}$ & 1.00, 1.00, 1.00 & .999, 1.00, 1.00 & .990, .998, .998 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{N_{0,1}}$ & 1.00, 1.00, 1.00 & .999, 1.00, 1.00 & .991, .998,
.999 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{w}$ & 1.00, 1.00, 1.00 & .998, 1.00, 1.00 & .949, .996, .998 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{dw}$ & .979, 1.00, 1.00 & .435, .917, .987 & .091, .534, .824
& .987, 1.00, 1.00 & .823, .986, .998 & .365, .904, .990 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#1: Simple $y_{t} = e_{t}$;
mean filter $\epsilon_{t} = y_{t} - E[y_{t}]$; $\hat{\phi}_{n} = 1/n
\sum_{t=1}^{n} y_{t}$. In this table we report rejection frequencies with
the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the proposed
max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#1 ($n = 500$) }
\label{table:supp_mat_scenario1_n500}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 40 & Lag = 80 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .006, .048, .109 & .010, .043, .098 & .004, .033,
.081 & .005, .056, .103 & .006, .048, .081 & .005, .039, .095 \\
$\hat{\mathcal{T}}^{dw}$ & .012, .055, .111 & .001, .029, .062 & .000, .021,
.060 & .007, .043, .102 & .001, .031, .070 & .000, .014, .052 \\ \hline
$\mathcal{AP}^{w}$ & .005, .038, .082 & .009, .050, .087 & .006, .050, .103
& .010, .051, .102 & .014, .053, .094 & .008, .046, .095 \\
$\mathcal{AP}^{dw}$ & .014, .065, .121 & .008, .050, .120 & .007, .061, .115
& .008, .044, .100 & .011, .045, .089 & .009, .065, .122 \\ \hline
$Q_{\chi^{2}}$ & .012, .044, .104 & .014, .057, .099 & .026, .073, .110 &
.042, .135, .221 & .028, .081, .158 & .028, .098, .145 \\
$\mathcal{N}^{N_{0,1}}$ & .023, .059, .122 & .019, .049, .079 & .014, .028,
.058 & .089, .157, .213 & .040, .084, .135 & .019, .053, .096 \\
$\mathcal{N}^{w}$ & .008, .053, .108 & .003, .045, .094 & .005, .035, .085 &
.010, .048, .096 & .005, .037, .105 & .007, .028, .075 \\
$\mathcal{N}^{dw}$ & .004, .051, .121 & .001, .010, .030 & .000, .001, .018
& .007, .043, .104 & .000, .004, .030 & .000, .002, .015 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 40 & Lag = 80 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\hat{\mathcal{T}}^{dw}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\ \hline
$\mathcal{AP}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00
& 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{AP}^{dw}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00
& 1.00, 1.00, 1.00 & .998, 1.00, 1.00 & .999, 1.00, 1.00 \\ \hline
$Q_{\chi^{2}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{N_{0,1}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{dw}$ & 1.00, 1.00, 1.00 & .876, .997, 1.00 & .269, .898, .991
& 1.00, 1.00, 1.00 & .974, 1.00, 1.00 & .770, .994, 1.00 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#1: Simple $y_{t} = e_{t}$;
mean filter $\epsilon_{t} = y_{t} - E[y_{t}]$; $\hat{\phi}_{n} = 1/n
\sum_{t=1}^{n} y_{t}$. In this table we report rejection frequencies with
the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the proposed
max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#1 ($n = 1000$) }
\label{table:supp_mat_scenario1_n1000}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 72 & Lag = 144 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .006, .043, .090 & .003, .053, .104 & .015, .059,
.110 & .005, .055, .098 & .009, .047, .089 & .008, .039, .094 \\
$\hat{\mathcal{T}}^{dw}$ & .013, .068, .128 & .004, .033, .072 & .003, .019,
.069 & .006, .054, .122 & .003, .025, .068 & .002, .025, .069 \\ \hline
$\mathcal{AP}^{w}$ & .008, .044, .094 & .010, .044, .096 & .011, .051, .094
& .001, .035, .085 & .005, .042, .094 & .007, .037, .086 \\
$\mathcal{AP}^{dw}$ & .015, .063, .120 & .013, .061, .106 & .007, .056, .111
& .012, .059, .100 & .006, .040, .101 & .005, .044, .101 \\ \hline
$Q_{\chi^{2}}$ & .013, .050, .105 & .014, .050, .122 & .021, .067, .127 &
.048, .137, .213 & .024, .089, .148 & .024, .075, .128 \\
$\mathcal{N}^{N_{0,1}}$ & .020, .053, .097 & .017, .064, .102 & .008, .022,
.057 & .093, .162, .208 & .020, .056, .112 & .007, .028, .044 \\
$\mathcal{N}^{w}$ & .015, .050, .114 & .013, .064, .115 & .005, .037, .078 &
.009, .050, .096 & .010, .048, .095 & .008, .041, .082 \\
$\mathcal{N}^{dw}$ & .007, .041, .103 & .000, .003, .021 & .000, .004, .014
& .005, .051, .099 & .000, .006, .026 & .000, .000, .011 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 72 & Lag = 144 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\hat{\mathcal{T}}^{dw}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\ \hline
$\mathcal{AP}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00
& 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{AP}^{dw}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00
& 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\ \hline
$Q_{\chi^{2}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{N_{0,1}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{dw}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & .842, 1.00, 1.00
& 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & .987, 1.00, 1.00 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#1: Simple $y_{t} = e_{t}$;
mean filter $\epsilon_{t} = y_{t} - E[y_{t}]$; $\hat{\phi}_{n} = 1/n
\sum_{t=1}^{n} y_{t}$. In this table we report rejection frequencies with
the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the proposed
max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#2 ($n = 100$) }
\label{table:supp_mat_scenario2_n100}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 10 & Lag = 21 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .007, .047, .108 & .006, .040, .090 & .006, .041,
.080 & .005, .035, .098 & .004, .032, .078 & .000, .020, .069 \\
$\hat{\mathcal{T}}^{dw}$ & .016, .066, .147 & .002, .035, .105 & .003, .032,
.080 & .000, .022, .076 & .001, .015, .052 & .000, .008, .035 \\ \hline
$\mathcal{AP}^{w}$ & .010, .062, .125 & .011, .055, .126 & .008, .040, .103
& .004, .026, .066 & .006, .040, .080 & .007, .036, .091 \\
$\mathcal{AP}^{dw}$ & .028, .093, .167 & .018, .080, .167 & .024, .084, .156
& .008, .038, .095 & .002, .040, .084 & .006, .038, .092 \\ \hline
$Q_{\chi^{2}}$ & .045, .138, .229 & .037, .104, .165 & .057, .114, .181 &
.295, .440, .531 & .238, .376, .462 & .145, .234, .298 \\
$\mathcal{N}^{N_{0,1}}$ & .084, .151, .211 & .058, .097, .153 & .046, .080,
.107 & .366, .471, .522 & .275, .367, .428 & .113, .173, .213 \\
$\mathcal{N}^{w}$ & .007, .058, .120 & .005, .040, .110 & .007, .050, .105 &
.005, .030, .069 & .002, .027, .078 & .004, .025, .062 \\
$\mathcal{N}^{dw}$ & .003, .045, .128 & .004, .028, .090 & .002, .015, .051
& .002, .024, .060 & .000, .007, .039 & .000, .005, .020 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 10 & Lag = 21 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .614, .819, .885 & .548, .765, .867 & .576, .805,
.869 & .685, .891, .942 & .701, .891, .947 & .689, .893, .945 \\
$\hat{\mathcal{T}}^{dw}$ & .328, .657, .803 & .319, .628, .794 & .235, .577,
.762 & .262, .538, .695 & .258, .498, .682 & .234, .489, .677 \\ \hline
$\mathcal{AP}^{w}$ & .471, .809, .901 & .440, .817, .900 & .451, .793, .900
& .612, .874, .944 & .596, .874, .944 & .619, .870, .939 \\
$\mathcal{AP}^{dw}$ & .313, .629, .783 & .319, .608, .770 & .338, .613, .770
& .247, .497, .674 & .218, .476, .661 & .221, .511, .694 \\ \hline
$Q_{\chi^{2}}$ & .811, .929, .967 & .726, .846, .905 & .621, .742, .809 &
.965, .986, .996 & .939, .971, .987 & .912, .947, .966 \\
$\mathcal{N}^{N_{0,1}}$ & .863, .910, .934 & .769, .834, .877 & .586, .707,
.765 & .985, .993, .996 & .957, .977, .988 & .896, .925, .938 \\
$\mathcal{N}^{w}$ & .365, .718, .842 & .320, .644, .794 & .224, .532, .702 &
.565, .831, .919 & .546, .839, .921 & .481, .783, .875 \\
$\mathcal{N}^{dw}$ & .151, .453, .666 & .062, .284, .546 & .012, .121, .297
& .166, .416, .619 & .078, .342, .561 & .042, .209, .453 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#2: Bilinear $y_{t} = 0.5
e_{t-1} y_{t-2} + e_{t}$; mean filter $\epsilon_{t} = y_{t} - E[y_{t}]$; $%
\hat{\phi}_{n} = 1/n \sum_{t=1}^{n} y_{t}$. In this table we report
rejection frequencies with the nominal sizes 1\%, 5\%, and 10\%. $\hat{%
\mathcal{T}}$ is the proposed max-correlation test. $\mathcal{AP}$ is
Andrews and Ploberger's (\citeyear{AndrewsPloberger1996}) sup-LM test. $%
Q_{\chi^{2}}$ is the asymptotic Ljung-Box test based on the $\chi^{2}$
distribution. $\mathcal{N}^{N_{0,1}}$ is Hong's (\citeyear{Hong1996})
asymptotic test based on $N(0,1)$. Superscript ``$w$'' means the wild
bootstrap while ``$dw$'' means Shao's (\citeyear{Shao2011_JoE}) dependent
wild bootstrap. Lag length is $\mathcal{L}_{n} \in \{ 5, [.5n/\ln(n)],
[n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#2 ($n = 250$) }
\label{table:supp_mat_scenario2_n250}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 22 & Lag = 45 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .015, .073, .140 & .008, .056, .108 & .007, .044,
.100 & .000, .029, .084 & .006, .039, .082 & .002, .026, .071 \\
$\hat{\mathcal{T}}^{dw}$ & .006, .046, .099 & .002, .034, .091 & .002, .026,
.071 & .008, .030, .064 & .008, .018, .052 & .007, .018, .038 \\ \hline
$\mathcal{AP}^{w}$ & .014, .073, .128 & .015, .063, .117 & .024, .075, .128
& .005, .030, .091 & .005, .032, .084 & .005, .026, .073 \\
$\mathcal{AP}^{dw}$ & .019, .081, .152 & .014, .068, .141 & .012, .058, .126
& .010, .039, .084 & .007, .024, .071 & .008, .037, .069 \\ \hline
$Q_{\chi^{2}}$ & .065, .155, .226 & .031, .099, .164 & .051, .114, .168 &
.564, .698, .755 & .432, .522, .584 & .244, .325, .386 \\
$\mathcal{N}^{N_{0,1}}$ & .113, .191, .249 & .037, .094, .129 & .039, .066,
.091 & .611, .672, .715 & .439, .530, .581 & .247, .307, .342 \\
$\mathcal{N}^{w}$ & .014, .072, .138 & .011, .044, .100 & .011, .042, .090 &
.005, .028, .072 & .001, .020, .064 & .000, .011, .060 \\
$\mathcal{N}^{dw}$ & .006, .050, .096 & .001, .012, .041 & .000, .004, .032
& .005, .013, .049 & .003, .011, .031 & .001, .005, .010 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 22 & Lag = 45 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .946, .980, .991 & .945, .985, .994 & .951, .981,
.989 & .856, .956, .980 & .863, .951, .977 & .855, .952, .971 \\
$\hat{\mathcal{T}}^{dw}$ & .743, .915, .964 & .717, .907, .964 & .711, .898,
.957 & .344, .601, .744 & .343, .584, .738 & .383, .626, .753 \\ \hline
$\mathcal{AP}^{w}$ & .917, .986, .995 & .911, .990, .997 & .900, .977, .995
& .846, .946, .969 & .841, .955, .982 & .813, .947, .975 \\
$\mathcal{AP}^{dw}$ & .582, .839, .920 & .601, .810, .901 & .572, .811, .909
& .290, .562, .717 & .281, .538, .698 & .298, .563, .720 \\ \hline
$Q_{\chi^{2}}$ & .997, .998, 1.00 & .968, .987, .992 & .925, .969, .983 &
.999, .999, .999 & 1.00, 1.00, 1.00 & .998, .999, .999 \\
$\mathcal{N}^{N_{0,1}}$ & .999, .999, .999 & .979, .992, .995 & .923, .956,
.973 & 1.00, 1.00, 1.00 & .999, 1.00, 1.00 & .995, .997, .998 \\
$\mathcal{N}^{w}$ & .847, .962, .979 & .746, .924, .957 & .623, .866, .924 &
.799, .956, .983 & .817, .947, .974 & .777, .939, .967 \\
$\mathcal{N}^{dw}$ & .489, .797, .907 & .123, .523, .778 & .025, .227, .508
& .260, .525, .691 & .139, .442, .642 & .074, .337, .586 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#2: Bilinear $y_{t} = 0.5
e_{t-1} y_{t-2} + e_{t}$; mean filter $\epsilon_{t} = y_{t} - E[y_{t}]$; $%
\hat{\phi}_{n} = 1/n \sum_{t=1}^{n} y_{t}$. In this table we report
rejection frequencies with the nominal sizes 1\%, 5\%, and 10\%. $\hat{%
\mathcal{T}}$ is the proposed max-correlation test. $\mathcal{AP}$ is
Andrews and Ploberger's (\citeyear{AndrewsPloberger1996}) sup-LM test. $%
Q_{\chi^{2}}$ is the asymptotic Ljung-Box test based on the $\chi^{2}$
distribution. $\mathcal{N}^{N_{0,1}}$ is Hong's (\citeyear{Hong1996})
asymptotic test based on $N(0,1)$. Superscript ``$w$'' means the wild
bootstrap while ``$dw$'' means Shao's (\citeyear{Shao2011_JoE}) dependent
wild bootstrap. Lag length is $\mathcal{L}_{n} \in \{ 5, [.5n/\ln(n)],
[n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#2 ($n = 500$) }
\label{table:supp_mat_scenario2_n500}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 40 & Lag = 80 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .010, .062, .118 & .015, .053, .116 & .015, .055,
.114 & .003, .015, .073 & .002, .027, .061 & .002, .024, .065 \\
$\hat{\mathcal{T}}^{dw}$ & .015, .054, .112 & .001, .028, .069 & .001, .019,
.056 & .007, .012, .052 & .005, .023, .047 & .006, .014, .036 \\ \hline
$\mathcal{AP}^{w}$ & .012, .069, .121 & .019, .084, .147 & .013, .077, .132
& .002, .025, .080 & .004, .028, .078 & .002, .028, .068 \\
$\mathcal{AP}^{dw}$ & .020, .063, .132 & .012, .074, .126 & .013, .069, .130
& .002, .023, .063 & .008, .017, .047 & .003, .016, .055 \\ \hline
$Q_{\chi^{2}}$ & .052, .136, .215 & .034, .089, .151 & .039, .085, .147 &
.742, .817, .862 & .534, .613, .663 & .362, .438, .480 \\
$\mathcal{N}^{N_{0,1}}$ & .120, .187, .240 & .041, .086, .134 & .022, .055,
.085 & .781, .838, .863 & .550, .619, .665 & .321, .398, .433 \\
$\mathcal{N}^{w}$ & .016, .073, .145 & .010, .047, .110 & .004, .037, .082 &
.001, .021, .065 & .001, .018, .060 & .001, .011, .052 \\
$\mathcal{N}^{dw}$ & .006, .040, .095 & .000, .006, .039 & .000, .003, .018
& .007, .023, .043 & .002, .008, .017 & .002, .003, .006 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 40 & Lag = 80 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .992, .996, .998 & .993, .998, .999 & .991, .993,
.996 & .907, .979, .989 & .891, .967, .987 & .889, .969, .984 \\
$\hat{\mathcal{T}}^{dw}$ & .907, .971, .991 & .894, .976, .990 & .897, .977,
.987 & .402, .648, .764 & .372, .629, .761 & .411, .621, .770 \\ \hline
$\mathcal{AP}^{w}$ & .977, .994, .996 & .984, 1.00, 1.00 & .976, .993, .999
& .913, .975, .988 & .892, .971, .985 & .891, .974, .986 \\
$\mathcal{AP}^{dw}$ & .767, .923, .967 & .767, .921, .966 & .767, .936, .970
& .378, .628, .784 & .347, .623, .751 & .343, .590, .740 \\ \hline
$Q_{\chi^{2}}$ & 1.00, 1.00, 1.00 & .999, .999, 1.00 & .999, .999, .999 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{N_{0,1}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & .994, .998,
.998 & 1.00, 1.00, 1.00 & .998, .998, .999 & .999, .999, .999 \\
$\mathcal{N}^{w}$ & .969, .993, .999 & .959, .991, .996 & .924, .984, .996 &
.872, .967, .988 & .856, .962, .984 & .875, .966, .982 \\
$\mathcal{N}^{dw}$ & .744, .917, .965 & .315, .788, .924 & .055, .460, .777
& .317, .570, .721 & .193, .483, .681 & .094, .411, .667 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#2: Bilinear $y_{t} = 0.5
e_{t-1} y_{t-2} + e_{t}$; mean filter $\epsilon_{t} = y_{t} - E[y_{t}]$; $%
\hat{\phi}_{n} = 1/n \sum_{t=1}^{n} y_{t}$. In this table we report
rejection frequencies with the nominal sizes 1\%, 5\%, and 10\%. $\hat{%
\mathcal{T}}$ is the proposed max-correlation test. $\mathcal{AP}$ is
Andrews and Ploberger's (\citeyear{AndrewsPloberger1996}) sup-LM test. $%
Q_{\chi^{2}}$ is the asymptotic Ljung-Box test based on the $\chi^{2}$
distribution. $\mathcal{N}^{N_{0,1}}$ is Hong's (\citeyear{Hong1996})
asymptotic test based on $N(0,1)$. Superscript ``$w$'' means the wild
bootstrap while ``$dw$'' means Shao's (\citeyear{Shao2011_JoE}) dependent
wild bootstrap. Lag length is $\mathcal{L}_{n} \in \{ 5, [.5n/\ln(n)],
[n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#2 ($n = 1000$) }
\label{table:supp_mat_scenario2_n1000}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 72 & Lag = 144 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .017, .087, .137 & .013, .066, .125 & .009, .052,
.107 & .002, .026, .083 & .001, .028, .076 & .002, .026, .064 \\
$\hat{\mathcal{T}}^{dw}$ & .012, .062, .112 & .008, .026, .072 & .004, .016,
.057 & .002, .012, .040 & .003, .007, .030 & .000, .005, .018 \\ \hline
$\mathcal{AP}^{w}$ & .015, .075, .138 & .015, .079, .149 & .016, .080, .147
& .001, .030, .082 & .003, .024, .073 & .003, .036, .065 \\
$\mathcal{AP}^{dw}$ & .014, .063, .109 & .015, .076, .137 & .012, .066, .119
& .001, .014, .039 & .004, .019, .047 & .000, .010, .042 \\ \hline
$Q_{\chi^{2}}$ & .064, .169, .254 & .030, .087, .153 & .020, .081, .145 &
.855, .907, .931 & .632, .715, .765 & .436, .509, .550 \\
$\mathcal{N}^{N_{0,1}}$ & .135, .207, .264 & .033, .087, .127 & .009, .024,
.044 & .877, .901, .917 & .632, .698, .718 & .411, .469, .518 \\
$\mathcal{N}^{w}$ & .029, .085, .147 & .009, .053, .107 & .007, .037, .076 &
.001, .031, .069 & .001, .016, .053 & .000, .007, .047 \\
$\mathcal{N}^{dw}$ & .007, .045, .102 & .000, .003, .036 & .000, .000, .005
& .001, .010, .024 & .001, .001, .006 & .000, .000, .001 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 72 & Lag = 144 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .999, 1.00, 1.00 & .997, .997, .997 & .995, 1.00,
1.00 & .929, .974, .987 & .924, .977, .986 & .901, .968, .980 \\
$\hat{\mathcal{T}}^{dw}$ & .966, .994, .995 & .966, .992, .999 & .961, .989,
.997 & .486, .678, .789 & .482, .705, .811 & .480, .680, .791 \\ \hline
$\mathcal{AP}^{w}$ & .996, .998, .998 & .996, 1.00, 1.00 & .996, 1.00, 1.00
& .924, .972, .988 & .933, .977, .989 & .924, .984, .989 \\
$\mathcal{AP}^{dw}$ & .891, .973, .991 & .902, .973, .991 & .895, .968, .981
& .396, .659, .789 & .375, .627, .786 & .415, .677, .807 \\ \hline
$Q_{\chi^{2}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{N_{0,1}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & .998, .998, .999 \\
$\mathcal{N}^{w}$ & .997, .999, 1.00 & .990, .998, 1.00 & .990, .996, .997 &
.926, .977, .984 & .918, .977, .992 & .917, .973, .986 \\
$\mathcal{N}^{dw}$ & .911, .971, .990 & .674, .933, .972 & .218, .793, .953
& .392, .643, .775 & .254, .548, .706 & .145, .492, .669 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#2: Bilinear $y_{t} = 0.5
e_{t-1} y_{t-2} + e_{t}$; mean filter $\epsilon_{t} = y_{t} - E[y_{t}]$; $%
\hat{\phi}_{n} = 1/n \sum_{t=1}^{n} y_{t}$. In this table we report
rejection frequencies with the nominal sizes 1\%, 5\%, and 10\%. $\hat{%
\mathcal{T}}$ is the proposed max-correlation test. $\mathcal{AP}$ is
Andrews and Ploberger's (\citeyear{AndrewsPloberger1996}) sup-LM test. $%
Q_{\chi^{2}}$ is the asymptotic Ljung-Box test based on the $\chi^{2}$
distribution. $\mathcal{N}^{N_{0,1}}$ is Hong's (\citeyear{Hong1996})
asymptotic test based on $N(0,1)$. Superscript ``$w$'' means the wild
bootstrap while ``$dw$'' means Shao's (\citeyear{Shao2011_JoE}) dependent
wild bootstrap. Lag length is $\mathcal{L}_{n} \in \{ 5, [.5n/\ln(n)],
[n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#3 ($n = 100$) }
\label{table:supp_mat_scenario3_n100}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 10 & Lag = 21 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .008, .048, .099 & .008, .049, .090 & .005, .034,
.093 & .007, .037, .088 & .010, .036, .072 & .006, .038, .080 \\
$\hat{\mathcal{T}}^{dw}$ & .025, .106, .181 & .015, .068, .142 & .006, .047,
.118 & .019, .075, .164 & .005, .053, .133 & .002, .034, .088 \\ \hline
$\mathcal{AP}^{w}$ & .010, .045, .102 & .005, .036, .107 & .006, .047, .095
& .005, .045, .100 & .006, .042, .106 & .004, .029, .075 \\
$\mathcal{AP}^{dw}$ & .036, .110, .185 & .037, .120, .198 & .028, .103, .182
& .033, .109, .177 & .034, .112, .194 & .037, .121, .204 \\ \hline
$Q_{\chi^{2}}$ & .008, .056, .107 & .009, .050, .098 & .019, .069, .128 &
.023, .062, .118 & .019, .050, .111 & .014, .049, .085 \\
$\mathcal{N}^{N_{0,1}}$ & .005, .013, .021 & .010, .020, .041 & .008, .023,
.042 & .010, .020, .035 & .008, .024, .037 & .006, .014, .024 \\
$\mathcal{N}^{w}$ & .007, .050, .119 & .006, .031, .063 & .005, .035, .070 &
.007, .043, .084 & .003, .037, .087 & .002, .025, .073 \\
$\mathcal{N}^{dw}$ & .025, .102, .191 & .006, .044, .116 & .000, .023, .084
& .022, .082, .155 & .002, .044, .122 & .000, .018, .078 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 10 & Lag = 21 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .008, .068, .139 & .006, .049, .114 & .007, .045,
.099 & .028, .101, .192 & .005, .058, .129 & .010, .040, .092 \\
$\hat{\mathcal{T}}^{dw}$ & .041, .143, .235 & .012, .078, .174 & .007, .063,
.141 & .039, .143, .239 & .011, .070, .170 & .004, .061, .141 \\ \hline
$\mathcal{AP}^{w}$ & .015, .068, .115 & .022, .061, .111 & .014, .050, .108
& .023, .115, .191 & .028, .113, .203 & .030, .112, .188 \\
$\mathcal{AP}^{dw}$ & .030, .108, .194 & .031, .108, .192 & .032, .118, .196
& .091, .218, .308 & .098, .211, .299 & .109, .227, .298 \\ \hline
$Q_{\chi^{2}}$ & .020, .084, .153 & .010, .063, .131 & .024, .069, .116 &
.030, .113, .185 & .027, .100, .168 & .035, .084, .142 \\
$\mathcal{N}^{N_{0,1}}$ & .021, .036, .057 & .017, .037, .062 & .019, .035,
.058 & .028, .054, .078 & .017, .047, .079 & .015, .034, .055 \\
$\mathcal{N}^{w}$ & .013, .085, .170 & .015, .070, .134 & .009, .041, .091 &
.026, .101, .184 & .013, .069, .151 & .009, .052, .114 \\
$\mathcal{N}^{dw}$ & .033, .129, .219 & .007, .064, .142 & .007, .045, .109
& .046, .148, .234 & .013, .086, .171 & .003, .040, .111 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#3: AR(2) $y_{t} = 0.3 y_{t-1}
- 0.15 y_{t-2} + e_{t}$; AR(2) filter $\epsilon_{t} = y_{t} - \phi_{1}
y_{t-1} - \phi_{2} y_{t-2}$; least squares estimation. In this table we
report rejection frequencies with the nominal sizes 1\%, 5\%, and 10\%. $%
\hat{\mathcal{T}}$ is the proposed max-correlation test. $\mathcal{AP}$ is
Andrews and Ploberger's (\citeyear{AndrewsPloberger1996}) sup-LM test. $%
Q_{\chi^{2}}$ is the asymptotic Ljung-Box test based on the $\chi^{2}$
distribution. $\mathcal{N}^{N_{0,1}}$ is Hong's (\citeyear{Hong1996})
asymptotic test based on $N(0,1)$. Superscript ``$w$'' means the wild
bootstrap while ``$dw$'' means Shao's (\citeyear{Shao2011_JoE}) dependent
wild bootstrap. Lag length is $\mathcal{L}_{n} \in \{ 5, [.5n/\ln(n)],
[n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#3 ($n = 250$) }
\label{table:supp_mat_scenario3_n250}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 22 & Lag = 45 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .011, .054, .104 & .005, .039, .083 & .007, .045,
.099 & .011, .050, .088 & .010, .044, .092 & .007, .035, .092 \\
$\hat{\mathcal{T}}^{dw}$ & .020, .063, .125 & .005, .033, .082 & .004, .033,
.082 & .015, .065, .139 & .005, .029, .084 & .006, .032, .080 \\ \hline
$\mathcal{AP}^{w}$ & .014, .053, .118 & .007, .049, .096 & .010, .044, .096
& .003, .044, .085 & .006, .035, .087 & .007, .034, .092 \\
$\mathcal{AP}^{dw}$ & .019, .092, .149 & .015, .079, .144 & .022, .080, .139
& .018, .078, .152 & .014, .075, .146 & .023, .082, .169 \\ \hline
$Q_{\chi^{2}}$ & .006, .045, .103 & .014, .058, .089 & .016, .069, .113 &
.023, .079, .138 & .017, .081, .134 & .026, .068, .110 \\
$\mathcal{N}^{N_{0,1}}$ & .003, .006, .012 & .011, .032, .056 & .010, .024,
.041 & .016, .031, .049 & .014, .034, .064 & .012, .024, .043 \\
$\mathcal{N}^{w}$ & .005, .049, .111 & .008, .037, .080 & .006, .039, .080 &
.011, .045, .104 & .004, .039, .081 & .007, .036, .076 \\
$\mathcal{N}^{dw}$ & .007, .062, .135 & .002, .022, .072 & .001, .007, .038
& .012, .065, .136 & .001, .014, .060 & .000, .007, .034 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 22 & Lag = 45 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .035, .137, .227 & .013, .065, .132 & .008, .046,
.098 & .052, .187, .309 & .014, .063, .142 & .007, .054, .107 \\
$\hat{\mathcal{T}}^{dw}$ & .038, .143, .246 & .005, .045, .110 & .004, .038,
.104 & .059, .190, .309 & .007, .053, .148 & .006, .039, .109 \\ \hline
$\mathcal{AP}^{w}$ & .028, .098, .184 & .032, .085, .156 & .021, .097, .164
& .082, .220, .341 & .058, .195, .303 & .085, .208, .306 \\
$\mathcal{AP}^{dw}$ & .030, .114, .184 & .022, .110, .196 & .024, .089, .162
& .136, .298, .424 & .169, .310, .393 & .154, .307, .397 \\ \hline
$Q_{\chi^{2}}$ & .040, .149, .239 & .025, .081, .139 & .042, .107, .172 &
.108, .255, .373 & .047, .137, .207 & .049, .127, .198 \\
$\mathcal{N}^{N_{0,1}}$ & .033, .077, .106 & .020, .050, .078 & .019, .039,
.054 & .087, .126, .172 & .034, .075, .123 & .023, .058, .093 \\
$\mathcal{N}^{w}$ & .048, .157, .258 & .011, .066, .128 & .014, .066, .117 &
.074, .210, .331 & .032, .105, .192 & .014, .062, .136 \\
$\mathcal{N}^{dw}$ & .032, .134, .253 & .001, .029, .089 & .001, .012, .054
& .084, .256, .377 & .007, .050, .143 & .002, .016, .075 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#3: AR(2) $y_{t} = 0.3 y_{t-1}
- 0.15 y_{t-2} + e_{t}$; AR(2) filter $\epsilon_{t} = y_{t} - \phi_{1}
y_{t-1} - \phi_{2} y_{t-2}$; least squares estimation. In this table we
report rejection frequencies with the nominal sizes 1\%, 5\%, and 10\%. $%
\hat{\mathcal{T}}$ is the proposed max-correlation test. $\mathcal{AP}$ is
Andrews and Ploberger's (\citeyear{AndrewsPloberger1996}) sup-LM test. $%
Q_{\chi^{2}}$ is the asymptotic Ljung-Box test based on the $\chi^{2}$
distribution. $\mathcal{N}^{N_{0,1}}$ is Hong's (\citeyear{Hong1996})
asymptotic test based on $N(0,1)$. Superscript ``$w$'' means the wild
bootstrap while ``$dw$'' means Shao's (\citeyear{Shao2011_JoE}) dependent
wild bootstrap. Lag length is $\mathcal{L}_{n} \in \{ 5, [.5n/\ln(n)],
[n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#3 ($n = 500$) }
\label{table:supp_mat_scenario3_n500}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 40 & Lag = 80 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .011, .055, .116 & .012, .041, .084 & .012, .054,
.106 & .005, .043, .094 & .010, .054, .101 & .005, .035, .088 \\
$\hat{\mathcal{T}}^{dw}$ & .009, .067, .120 & .002, .027, .095 & .003, .026,
.081 & .020, .076, .131 & .002, .030, .078 & .001, .021, .055 \\ \hline
$\mathcal{AP}^{w}$ & .007, .059, .110 & .005, .043, .090 & .011, .051, .098
& .014, .049, .102 & .007, .048, .106 & .006, .044, .087 \\
$\mathcal{AP}^{dw}$ & .015, .063, .119 & .019, .076, .149 & .024, .074, .132
& .015, .060, .115 & .012, .058, .125 & .012, .062, .128 \\ \hline
$Q_{\chi^{2}}$ & .011, .054, .111 & .009, .049, .091 & .019, .070, .121 &
.018, .076, .136 & .023, .069, .125 & .023, .071, .122 \\
$\mathcal{N}^{N_{0,1}}$ & .004, .015, .030 & .009, .034, .062 & .005, .016,
.029 & .010, .023, .048 & .009, .038, .069 & .007, .023, .041 \\
$\mathcal{N}^{w}$ & .012, .051, .091 & .004, .046, .103 & .007, .032, .085 &
.006, .050, .104 & .009, .038, .085 & .004, .036, .079 \\
$\mathcal{N}^{dw}$ & .018, .053, .114 & .001, .008, .046 & .000, .002, .023
& .016, .066, .126 & .001, .006, .036 & .000, .001, .012 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 40 & Lag = 80 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .087, .251, .379 & .014, .080, .150 & .015, .069,
.116 & .149, .372, .539 & .026, .091, .171 & .008, .084, .165 \\
$\hat{\mathcal{T}}^{dw}$ & .098, .284, .390 & .005, .054, .117 & .005, .041,
.099 & .133, .357, .505 & .006, .061, .137 & .004, .038, .107 \\ \hline
$\mathcal{AP}^{w}$ & .042, .158, .232 & .034, .126, .205 & .037, .115, .199
& .177, .407, .539 & .128, .342, .492 & .143, .339, .488 \\
$\mathcal{AP}^{dw}$ & .048, .139, .220 & .026, .118, .211 & .018, .085, .161
& .276, .489, .613 & .263, .441, .550 & .257, .442, .572 \\ \hline
$Q_{\chi^{2}}$ & .119, .294, .413 & .029, .102, .180 & .037, .114, .184 &
.267, .459, .596 & .069, .175, .264 & .064, .163, .252 \\
$\mathcal{N}^{N_{0,1}}$ & .085, .155, .212 & .035, .073, .117 & .019, .037,
.065 & .191, .287, .364 & .058, .139, .198 & .033, .079, .109 \\
$\mathcal{N}^{w}$ & .108, .293, .398 & .031, .090, .168 & .025, .077, .144 &
.217, .415, .548 & .052, .144, .232 & .025, .097, .192 \\
$\mathcal{N}^{dw}$ & .097, .258, .410 & .003, .032, .092 & .001, .008, .048
& .178, .425, .552 & .001, .042, .130 & .000, .010, .054 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#3: AR(2) $y_{t} = 0.3 y_{t-1}
- 0.15 y_{t-2} + e_{t}$; AR(2) filter $\epsilon_{t} = y_{t} - \phi_{1}
y_{t-1} - \phi_{2} y_{t-2}$; least squares estimation. In this table we
report rejection frequencies with the nominal sizes 1\%, 5\%, and 10\%. $%
\hat{\mathcal{T}}$ is the proposed max-correlation test. $\mathcal{AP}$ is
Andrews and Ploberger's (\citeyear{AndrewsPloberger1996}) sup-LM test. $%
Q_{\chi^{2}}$ is the asymptotic Ljung-Box test based on the $\chi^{2}$
distribution. $\mathcal{N}^{N_{0,1}}$ is Hong's (\citeyear{Hong1996})
asymptotic test based on $N(0,1)$. Superscript ``$w$'' means the wild
bootstrap while ``$dw$'' means Shao's (\citeyear{Shao2011_JoE}) dependent
wild bootstrap. Lag length is $\mathcal{L}_{n} \in \{ 5, [.5n/\ln(n)],
[n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#3 ($n = 1000$) }
\label{table:supp_mat_scenario3_n1000}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 72 & Lag = 144 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .011, .049, .104 & .009, .048, .091 & .011, .045,
.089 & .013, .056, .113 & .010, .056, .093 & .009, .040, .085 \\
$\hat{\mathcal{T}}^{dw}$ & .013, .064, .110 & .003, .034, .091 & .003, .025,
.052 & .008, .060, .132 & .002, .026, .071 & .004, .022, .059 \\ \hline
$\mathcal{AP}^{w}$ & .010, .044, .104 & .012, .053, .100 & .008, .057, .097
& .010, .051, .109 & .004, .028, .084 & .005, .045, .094 \\
$\mathcal{AP}^{dw}$ & .012, .065, .121 & .020, .073, .138 & .018, .065, .106
& .020, .059, .114 & .011, .067, .119 & .016, .064, .133 \\ \hline
$Q_{\chi^{2}}$ & .008, .049, .098 & .012, .055, .085 & .021, .081, .133 &
.033, .100, .164 & .019, .066, .123 & .020, .072, .130 \\
$\mathcal{N}^{N_{0,1}}$ & .009, .028, .040 & .007, .027, .058 & .007, .018,
.036 & .017, .032, .050 & .011, .037, .068 & .006, .021, .033 \\
$\mathcal{N}^{w}$ & .010, .047, .093 & .013, .044, .094 & .009, .036, .088 &
.007, .056, .103 & .006, .033, .081 & .009, .036, .092 \\
$\mathcal{N}^{dw}$ & .010, .054, .111 & .000, .012, .034 & .000, .003, .014
& .006, .046, .114 & .000, .006, .028 & .000, .001, .010 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 72 & Lag = 144 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .236, .496, .630 & .038, .117, .186 & .042, .105,
.171 & .405, .685, .798 & .060, .205, .316 & .053, .159, .250 \\
$\hat{\mathcal{T}}^{dw}$ & .223, .485, .656 & .016, .070, .156 & .009, .055,
.131 & .390, .673, .800 & .029, .125, .255 & .015, .088, .184 \\ \hline
$\mathcal{AP}^{w}$ & .079, .240, .366 & .065, .208, .322 & .056, .190, .288
& .426, .684, .797 & .351, .627, .756 & .355, .632, .745 \\
$\mathcal{AP}^{dw}$ & .071, .231, .363 & .047, .179, .281 & .042, .171, .298
& .502, .751, .854 & .493, .707, .801 & .506, .735, .819 \\ \hline
$Q_{\chi^{2}}$ & .304, .543, .677 & .065, .163, .254 & .058, .159, .234 &
.547, .757, .832 & .119, .265, .364 & .106, .261, .366 \\
$\mathcal{N}^{N_{0,1}}$ & .270, .390, .477 & .040, .098, .159 & .019, .062,
.092 & .492, .613, .685 & .072, .189, .272 & .037, .097, .144 \\
$\mathcal{N}^{w}$ & .294, .550, .680 & .036, .118, .209 & .026, .113, .189 &
.541, .759, .859 & .067, .212, .316 & .057, .170, .267 \\
$\mathcal{N}^{dw}$ & .256, .527, .674 & .000, .022, .063 & .000, .002, .018
& .473, .730, .845 & .004, .043, .130 & .001, .005, .034 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#3: AR(2) $y_{t} = 0.3 y_{t-1}
- 0.15 y_{t-2} + e_{t}$; AR(2) filter $\epsilon_{t} = y_{t} - \phi_{1}
y_{t-1} - \phi_{2} y_{t-2}$; least squares estimation. In this table we
report rejection frequencies with the nominal sizes 1\%, 5\%, and 10\%. $%
\hat{\mathcal{T}}$ is the proposed max-correlation test. $\mathcal{AP}$ is
Andrews and Ploberger's (\citeyear{AndrewsPloberger1996}) sup-LM test. $%
Q_{\chi^{2}}$ is the asymptotic Ljung-Box test based on the $\chi^{2}$
distribution. $\mathcal{N}^{N_{0,1}}$ is Hong's (\citeyear{Hong1996})
asymptotic test based on $N(0,1)$. Superscript ``$w$'' means the wild
bootstrap while ``$dw$'' means Shao's (\citeyear{Shao2011_JoE}) dependent
wild bootstrap. Lag length is $\mathcal{L}_{n} \in \{ 5, [.5n/\ln(n)],
[n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#4 ($n = 100$) }
\label{table:supp_mat_scenario4_n100}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 10 & Lag = 21 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .048, .204, .340 & .021, .111, .213 & .021, .118,
.198 & .047, .193, .327 & .031, .138, .224 & .031, .130, .210 \\
$\hat{\mathcal{T}}^{dw}$ & .082, .227, .343 & .023, .131, .265 & .019, .098,
.204 & .055, .203, .329 & .026, .137, .259 & .005, .080, .181 \\ \hline
$\mathcal{AP}^{w}$ & .051, .214, .347 & .053, .190, .313 & .040, .179, .311
& .043, .175, .302 & .031, .163, .298 & .041, .150, .277 \\
$\mathcal{AP}^{dw}$ & .199, .349, .465 & .185, .377, .494 & .195, .364, .483
& .131, .302, .417 & .121, .297, .430 & .131, .328, .472 \\ \hline
$Q_{\chi^{2}}$ & .070, .200, .281 & .068, .172, .245 & .076, .177, .263 &
.106, .252, .361 & .077, .184, .279 & .068, .147, .212 \\
$\mathcal{N}^{N_{0,1}}$ & .090, .164, .207 & .072, .122, .161 & .063, .095,
.125 & .114, .194, .246 & .081, .138, .196 & .058, .100, .143 \\
$\mathcal{N}^{w}$ & .060, .195, .335 & .043, .149, .251 & .029, .094, .169 &
.050, .183, .310 & .033, .139, .237 & .019, .090, .160 \\
$\mathcal{N}^{dw}$ & .045, .195, .323 & .015, .098, .220 & .006, .048, .148
& .048, .211, .338 & .010, .100, .222 & .005, .043, .104 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 10 & Lag = 21 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .259, .580, .728 & .177, .424, .576 & .124, .341,
.473 & .147, .436, .618 & .079, .286, .447 & .041, .198, .315 \\
$\hat{\mathcal{T}}^{dw}$ & .229, .533, .705 & .089, .350, .546 & .032, .182,
.360 & .135, .404, .605 & .035, .239, .443 & .009, .102, .265 \\ \hline
$\mathcal{AP}^{w}$ & .455, .703, .818 & .455, .702, .814 & .437, .706, .803
& .441, .753, .851 & .440, .709, .832 & .413, .722, .821 \\
$\mathcal{AP}^{dw}$ & .547, .757, .850 & .516, .734, .834 & .529, .758, .857
& .558, .782, .860 & .564, .787, .875 & .556, .786, .868 \\ \hline
$Q_{\chi^{2}}$ & .540, .750, .829 & .397, .609, .714 & .345, .519, .626 &
.296, .558, .681 & .227, .455, .569 & .194, .367, .483 \\
$\mathcal{N}^{N_{0,1}}$ & .575, .691, .755 & .442, .554, .633 & .316, .437,
.500 & .340, .489, .574 & .258, .373, .455 & .143, .233, .289 \\
$\mathcal{N}^{w}$ & .426, .694, .817 & .256, .547, .695 & .157, .404, .563 &
.214, .517, .669 & .129, .380, .557 & .067, .249, .406 \\
$\mathcal{N}^{dw}$ & .288, .640, .797 & .086, .350, .568 & .022, .165, .340
& .202, .496, .684 & .051, .237, .439 & .014, .097, .229 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#4: AR(2) $y_{t} = 0.3 y_{t-1}
- 0.15 y_{t-2} + e_{t}$; AR(1) filter $\epsilon_{t} = y_{t} - \phi_{1}
y_{t-1}$; least squares estimation. In this table we report rejection
frequencies with the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$
is the proposed max-correlation test. $\mathcal{AP}$ is Andrews and
Ploberger's (\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is
the asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $%
\mathcal{N}^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based
on $N(0,1)$. Superscript ``$w$'' means the wild bootstrap while ``$dw$''
means Shao's (\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length
is $\mathcal{L}_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#4 ($n = 250$) }
\label{table:supp_mat_scenario4_n250}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 22 & Lag = 45 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .221, .443, .574 & .101, .266, .377 & .086, .228,
.350 & .174, .385, .512 & .107, .282, .396 & .085, .208, .312 \\
$\hat{\mathcal{T}}^{dw}$ & .190, .455, .611 & .047, .200, .345 & .026, .153,
.280 & .153, .365, .522 & .043, .192, .308 & .033, .135, .253 \\ \hline
$\mathcal{AP}^{w}$ & .202, .451, .597 & .171, .395, .564 & .172, .406, .556
& .151, .386, .523 & .141, .356, .522 & .131, .329, .476 \\
$\mathcal{AP}^{dw}$ & .314, .564, .672 & .280, .523, .656 & .293, .535, .652
& .242, .512, .649 & .263, .484, .618 & .243, .466, .598 \\ \hline
$Q_{\chi^{2}}$ & .257, .449, .578 & .103, .248, .358 & .082, .221, .299 &
.302, .521, .631 & .141, .293, .382 & .124, .260, .346 \\
$\mathcal{N}^{N_{0,1}}$ & .251, .373, .445 & .108, .204, .268 & .069, .141,
.207 & .308, .438, .507 & .157, .250, .319 & .087, .161, .211 \\
$\mathcal{N}^{w}$ & .223, .462, .600 & .064, .205, .321 & .039, .152, .259 &
.172, .384, .531 & .054, .172, .288 & .037, .150, .250 \\
$\mathcal{N}^{dw}$ & .171, .428, .590 & .007, .078, .201 & .000, .039, .104
& .127, .369, .525 & .001, .067, .172 & .000, .028, .098 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 22 & Lag = 45 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .784, .944, .985 & .577, .810, .889 & .505, .759,
.843 & .699, .910, .962 & .390, .685, .811 & .307, .601, .736 \\
$\hat{\mathcal{T}}^{dw}$ & .665, .932, .980 & .284, .650, .805 & .193, .514,
.724 & .567, .875, .963 & .163, .521, .734 & .073, .366, .590 \\ \hline
$\mathcal{AP}^{w}$ & .938, .986, .996 & .912, .978, .988 & .910, .987, .995
& .953, .988, .996 & .958, .992, .997 & .938, .985, .995 \\
$\mathcal{AP}^{dw}$ & .933, .983, .993 & .901, .975, .992 & .899, .975, .988
& .945, .988, .999 & .944, .993, 1.00 & .941, .987, .994 \\ \hline
$Q_{\chi^{2}}$ & .961, .991, .994 & .794, .908, .955 & .661, .823, .875 &
.853, .957, .975 & .504, .729, .825 & .430, .637, .740 \\
$\mathcal{N}^{N_{0,1}}$ & .970, .987, .994 & .823, .898, .929 & .581, .726,
.788 & .874, .932, .962 & .523, .678, .761 & .323, .459, .548 \\
$\mathcal{N}^{w}$ & .952, .987, .993 & .643, .879, .929 & .426, .718, .831 &
.778, .956, .982 & .365, .693, .828 & .237, .526, .659 \\
$\mathcal{N}^{dw}$ & .851, .974, .990 & .133, .550, .802 & .029, .215, .466
& .639, .916, .975 & .041, .325, .576 & .009, .100, .280 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#4: AR(2) $y_{t} = 0.3 y_{t-1}
- 0.15 y_{t-2} + e_{t}$; AR(1) filter $\epsilon_{t} = y_{t} - \phi_{1}
y_{t-1}$; least squares estimation. In this table we report rejection
frequencies with the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$
is the proposed max-correlation test. $\mathcal{AP}$ is Andrews and
Ploberger's (\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is
the asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $%
\mathcal{N}^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based
on $N(0,1)$. Superscript ``$w$'' means the wild bootstrap while ``$dw$''
means Shao's (\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length
is $\mathcal{L}_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#4 ($n = 500$) }
\label{table:supp_mat_scenario4_n500}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 40 & Lag = 80 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .570, .782, .871 & .336, .542, .637 & .268, .455,
.560 & .431, .657, .756 & .289, .492, .602 & .289, .466, .575 \\
$\hat{\mathcal{T}}^{dw}$ & .538, .783, .866 & .184, .442, .570 & .143, .376,
.516 & .412, .679, .787 & .180, .386, .534 & .135, .333, .475 \\ \hline
$\mathcal{AP}^{w}$ & .479, .776, .865 & .446, .728, .838 & .444, .721, .850
& .363, .656, .780 & .345, .614, .757 & .307, .593, .718 \\
$\mathcal{AP}^{dw}$ & .566, .794, .881 & .553, .751, .850 & .567, .781, .869
& .461, .701, .806 & .441, .673, .788 & .439, .675, .792 \\ \hline
$Q_{\chi^{2}}$ & .579, .776, .854 & .191, .372, .507 & .168, .322, .422 &
.587, .762, .839 & .248, .423, .527 & .166, .301, .411 \\
$\mathcal{N}^{N_{0,1}}$ & .614, .730, .781 & .226, .352, .440 & .098, .178,
.269 & .631, .755, .804 & .197, .332, .424 & .122, .213, .292 \\
$\mathcal{N}^{w}$ & .567, .795, .874 & .132, .345, .463 & .071, .229, .376 &
.439, .688, .779 & .108, .276, .409 & .076, .223, .348 \\
$\mathcal{N}^{dw}$ & .488, .768, .867 & .010, .076, .230 & .001, .027, .100
& .311, .625, .763 & .006, .071, .211 & .000, .022, .077 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 40 & Lag = 80 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .998, 1.00, 1.00 & .954, .994, .998 & .941, .983,
.992 & .992, 1.00, 1.00 & .913, .981, .994 & .872, .967, .988 \\
$\hat{\mathcal{T}}^{dw}$ & .979, 1.00, 1.00 & .834, .979, .992 & .751, .962,
.982 & .977, 1.00, 1.00 & .713, .958, .989 & .643, .919, .972 \\ \hline
$\mathcal{AP}^{w}$ & 1.00, 1.00, 1.00 & .999, 1.00, 1.00 & .999, 1.00, 1.00
& 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{AP}^{dw}$ & 1.00, 1.00, 1.00 & .997, 1.00, 1.00 & .995, 1.00, 1.00
& 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\ \hline
$Q_{\chi^{2}}$ & 1.00, 1.00, 1.00 & .970, .991, .998 & .921, .970, .986 &
.997, 1.00, 1.00 & .836, .944, .968 & .718, .870, .924 \\
$\mathcal{N}^{N_{0,1}}$ & 1.00, 1.00, 1.00 & .968, .993, .997 & .876, .926,
.951 & 1.00, 1.00, 1.00 & .840, .933, .960 & .625, .761, .826 \\
$\mathcal{N}^{w}$ & 1.00, 1.00, 1.00 & .945, .991, .996 & .798, .946, .978 &
.997, .998, 1.00 & .765, .920, .969 & .519, .794, .892 \\
$\mathcal{N}^{dw}$ & .998, 1.00, 1.00 & .294, .817, .949 & .036, .345, .663
& .981, 1.00, 1.00 & .105, .524, .789 & .017, .175, .419 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#4: AR(2) $y_{t} = 0.3 y_{t-1}
- 0.15 y_{t-2} + e_{t}$; AR(1) filter $\epsilon_{t} = y_{t} - \phi_{1}
y_{t-1}$; least squares estimation. In this table we report rejection
frequencies with the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$
is the proposed max-correlation test. $\mathcal{AP}$ is Andrews and
Ploberger's (\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is
the asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $%
\mathcal{N}^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based
on $N(0,1)$. Superscript ``$w$'' means the wild bootstrap while ``$dw$''
means Shao's (\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length
is $\mathcal{L}_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#4 ($n = 1000$) }
\label{table:supp_mat_scenario4_n1000}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 72 & Lag = 144 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .921, .979, .991 & .778, .904, .938 & .724, .878,
.918 & .833, .955, .980 & .705, .828, .874 & .655, .801, .856 \\
$\hat{\mathcal{T}}^{dw}$ & .914, .975, .994 & .658, .845, .914 & .573, .788,
.869 & .787, .921, .958 & .595, .795, .868 & .502, .733, .827 \\ \hline
$\mathcal{AP}^{w}$ & .854, .969, .990 & .825, .953, .980 & .788, .958, .983
& .733, .914, .955 & .678, .875, .939 & .659, .889, .941 \\
$\mathcal{AP}^{dw}$ & .893, .965, .985 & .881, .976, .988 & .864, .968, .995
& .771, .925, .958 & .726, .892, .940 & .724, .916, .951 \\ \hline
$Q_{\chi^{2}}$ & .937, .983, .996 & .338, .573, .705 & .265, .467, .599 &
.901, .956, .976 & .405, .615, .708 & .303, .495, .594 \\
$\mathcal{N}^{N_{0,1}}$ & .942, .969, .980 & .382, .558, .643 & .175, .319,
.415 & .941, .967, .979 & .387, .549, .633 & .175, .307, .387 \\
$\mathcal{N}^{w}$ & .934, .991, .997 & .279, .524, .662 & .140, .362, .502 &
.818, .935, .964 & .270, .525, .664 & .162, .358, .509 \\
$\mathcal{N}^{dw}$ & .900, .977, .991 & .016, .166, .372 & .000, .029, .113
& .716, .921, .970 & .003, .110, .291 & .001, .022, .099 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 72 & Lag = 144 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\hat{\mathcal{T}}^{dw}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & .999, 1.00,
1.00 & 1.00, 1.00, 1.00 & .999, 1.00, 1.00 & .999, 1.00, 1.00 \\ \hline
$\mathcal{AP}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00
& 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{AP}^{dw}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00
& 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\ \hline
$Q_{\chi^{2}}$ & 1.00, 1.00, 1.00 & .999, 1.00, 1.00 & 1.00, 1.00, 1.00 &
1.00, 1.00, 1.00 & .995, .999, 1.00 & .955, .991, .995 \\
$\mathcal{N}^{N_{0,1}}$ & 1.00, 1.00, 1.00 & .999, 1.00, 1.00 & .986, .996,
.998 & 1.00, 1.00, 1.00 & .988, .997, .998 & .899, .958, .975 \\
$\mathcal{N}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & .989, 1.00, 1.00 &
1.00, 1.00, 1.00 & .972, .995, .998 & .876, .978, .991 \\
$\mathcal{N}^{dw}$ & 1.00, 1.00, 1.00 & .761, .989, .999 & .129, .708, .919
& 1.00, 1.00, 1.00 & .380, .892, .981 & .028, .320, .638 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#4: AR(2) $y_{t} = 0.3 y_{t-1}
- 0.15 y_{t-2} + e_{t}$; AR(1) filter $\epsilon_{t} = y_{t} - \phi_{1}
y_{t-1}$; least squares estimation. In this table we report rejection
frequencies with the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$
is the proposed max-correlation test. $\mathcal{AP}$ is Andrews and
Ploberger's (\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is
the asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $%
\mathcal{N}^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based
on $N(0,1)$. Superscript ``$w$'' means the wild bootstrap while ``$dw$''
means Shao's (\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length
is $\mathcal{L}_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#5 ($n = 100$) }
\label{table:supp_mat_scenario5_n100}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 10 & Lag = 21 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .001, .037, .089 & .004, .036, .081 & .005, .031,
.065 & .001, .037, .096 & .003, .032, .074 & .003, .037, .079 \\
$\hat{\mathcal{T}}^{dw}$ & .005, .052, .108 & .004, .036, .093 & .004, .029,
.071 & .003, .034, .099 & .002, .024, .073 & .000, .016, .057 \\ \hline
$\mathcal{AP}^{w}$ & .006, .036, .087 & .006, .032, .072 & .001, .042, .080
& .002, .033, .077 & .007, .038, .082 & .003, .034, .077 \\
$\mathcal{AP}^{dw}$ & .009, .060, .127 & .006, .060, .136 & .012, .059, .129
& .004, .044, .095 & .008, .058, .111 & .010, .049, .122 \\ \hline
$Q_{\chi^{2}}$ & .037, .108, .162 & .032, .088, .140 & .021, .082, .134 &
.476, .620, .704 & .462, .617, .684 & .334, .476, .548 \\
$\mathcal{N}^{N_{0,1}}$ & .062, .128, .165 & .044, .082, .124 & .033, .063,
.088 & .566, .654, .710 & .515, .624, .685 & .317, .409, .453 \\
$\mathcal{N}^{w}$ & .005, .033, .072 & .009, .033, .076 & .004, .030, .069 &
.003, .022, .075 & .003, .034, .082 & .000, .014, .061 \\
$\mathcal{N}^{dw}$ & .003, .041, .112 & .000, .019, .068 & .001, .012, .049
& .002, .018, .066 & .000, .011, .045 & .000, .005, .029 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 10 & Lag = 21 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .893, .974, .991 & .848, .966, .981 & .823, .962,
.985 & .967, .998, 1.00 & .965, .998, .999 & .972, .998, .999 \\
$\hat{\mathcal{T}}^{dw}$ & .612, .889, .962 & .536, .857, .941 & .478, .849,
.941 & .594, .843, .933 & .579, .832, .933 & .580, .836, .929 \\ \hline
$\mathcal{AP}^{w}$ & .770, .965, .990 & .774, .967, .988 & .776, .957, .988
& .948, .994, .998 & .936, .989, 1.00 & .950, .994, .999 \\
$\mathcal{AP}^{dw}$ & .550, .844, .945 & .517, .830, .942 & .485, .799, .925
& .489, .785, .904 & .465, .793, .923 & .419, .768, .902 \\ \hline
$Q_{\chi^{2}}$ & .967, .988, .994 & .934, .979, .989 & .877, .956, .973 &
.996, .997, .999 & .997, .998, 1.00 & .995, .997, .999 \\
$\mathcal{N}^{N_{0,1}}$ & .987, .996, .998 & .958, .985, .988 & .865, .920,
.936 & .999, .999, .999 & .998, .998, .998 & .986, .991, .994 \\
$\mathcal{N}^{w}$ & .702, .946, .980 & .543, .871, .952 & .446, .790, .907 &
.929, .993, .999 & .923, .993, .999 & .893, .988, .998 \\
$\mathcal{N}^{dw}$ & .339, .704, .885 & .112, .496, .742 & .025, .229, .489
& .422, .775, .902 & .270, .678, .851 & .082, .422, .710 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#5: GARCH(1,1) $y_{t} =
\sigma_{t} e_{t}$, $\sigma_{t}^{2} = 1 + 0.2 y_{t-1}^{2} + 0.5
\sigma_{t-1}^{2}$; no filter. In this table we report rejection frequencies
with the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the
proposed max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#5 ($n = 250$) }
\label{table:supp_mat_scenario5_n250}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 22 & Lag = 45 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .009, .057, .119 & .006, .047, .091 & .008, .037,
.084 & .007, .040, .100 & .005, .027, .073 & .002, .029, .075 \\
$\hat{\mathcal{T}}^{dw}$ & .007, .054, .119 & .005, .032, .075 & .001, .021,
.075 & .005, .037, .083 & .007, .025, .054 & .005, .015, .048 \\ \hline
$\mathcal{AP}^{w}$ & .011, .048, .102 & .007, .042, .084 & .009, .036, .093
& .004, .032, .080 & .007, .025, .075 & .007, .041, .097 \\
$\mathcal{AP}^{dw}$ & .009, .063, .121 & .012, .065, .136 & .008, .063, .135
& .012, .054, .104 & .012, .036, .088 & .013, .042, .094 \\ \hline
$Q_{\chi^{2}}$ & .043, .122, .196 & .027, .080, .143 & .031, .090, .142 &
.808, .883, .916 & .801, .876, .903 & .652, .741, .781 \\
$\mathcal{N}^{N_{0,1}}$ & .089, .143, .208 & .029, .076, .117 & .024, .051,
.086 & .874, .909, .923 & .844, .894, .904 & .611, .693, .736 \\
$\mathcal{N}^{w}$ & .003, .036, .085 & .006, .040, .084 & .007, .045, .083 &
.005, .031, .069 & .003, .025, .086 & .001, .020, .061 \\
$\mathcal{N}^{dw}$ & .004, .041, .106 & .001, .008, .044 & .000, .007, .031
& .003, .021, .059 & .000, .006, .020 & .000, .002, .015 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 22 & Lag = 45 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .992, .999, .999 & .993, 1.00, 1.00 & .997, .999,
.999 & .981, .996, 1.00 & .985, .997, 1.00 & .982, 1.00, 1.00 \\
$\hat{\mathcal{T}}^{dw}$ & .902, .974, .994 & .905, .975, .993 & .913, .976,
.995 & .696, .861, .931 & .667, .848, .930 & .672, .850, .937 \\ \hline
$\mathcal{AP}^{w}$ & .982, .999, 1.00 & .985, .998, 1.00 & .983, .997, 1.00
& .983, .997, .999 & .979, .997, .999 & .978, .997, 1.00 \\
$\mathcal{AP}^{dw}$ & .829, .964, .987 & .801, .942, .980 & .800, .945, .979
& .631, .833, .914 & .608, .832, .912 & .563, .825, .904 \\ \hline
$Q_{\chi^{2}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & .997, 1.00, 1.00 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & .998, .998, .998 \\
$\mathcal{N}^{N_{0,1}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & .999, 1.00,
1.00 & .999, 1.00, 1.00 & .999, .999, .999 & .998, .998, .999 \\
$\mathcal{N}^{w}$ & .981, .998, .999 & .973, .997, 1.00 & .935, .993, .998 &
.975, 1.00, 1.00 & .970, .999, 1.00 & .971, 1.00, 1.00 \\
$\mathcal{N}^{dw}$ & .780, .945, .980 & .289, .781, .931 & .049, .450, .785
& .584, .826, .912 & .396, .733, .857 & .170, .645, .825 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#5: GARCH(1,1) $y_{t} =
\sigma_{t} e_{t}$, $\sigma_{t}^{2} = 1 + 0.2 y_{t-1}^{2} + 0.5
\sigma_{t-1}^{2}$; no filter. In this table we report rejection frequencies
with the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the
proposed max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#5 ($n = 500$) }
\label{table:supp_mat_scenario5_n500}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 40 & Lag = 80 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .006, .040, .097 & .005, .047, .089 & .009, .042,
.083 & .000, .028, .080 & .001, .027, .077 & .001, .033, .083 \\
$\hat{\mathcal{T}}^{dw}$ & .006, .046, .087 & .002, .019, .067 & .003, .019,
.058 & .006, .027, .058 & .003, .020, .049 & .006, .020, .041 \\ \hline
$\mathcal{AP}^{w}$ & .010, .042, .090 & .005, .054, .098 & .006, .044, .093
& .003, .035, .079 & .003, .047, .081 & .002, .031, .085 \\
$\mathcal{AP}^{dw}$ & .012, .053, .105 & .008, .053, .101 & .007, .065, .129
& .007, .031, .091 & .008, .029, .070 & .008, .031, .073 \\ \hline
$Q_{\chi^{2}}$ & .058, .155, .239 & .025, .088, .147 & .034, .085, .140 &
.933, .955, .972 & .940, .960, .972 & .801, .847, .878 \\
$\mathcal{N}^{N_{0,1}}$ & .083, .149, .209 & .032, .062, .105 & .014, .043,
.067 & .948, .958, .971 & .946, .966, .976 & .818, .855, .872 \\
$\mathcal{N}^{w}$ & .006, .039, .086 & .006, .039, .089 & .005, .041, .084 &
.004, .041, .079 & .004, .028, .060 & .001, .021, .043 \\
$\mathcal{N}^{dw}$ & .005, .041, .100 & .000, .001, .026 & .000, .004, .013
& .003, .024, .052 & .003, .007, .017 & .000, .001, .006 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 40 & Lag = 80 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & 1.00, 1.00, 1.00 & .998, .999, 1.00 & .999, 1.00,
1.00 & .983, 1.00, 1.00 & .995, 1.00, 1.00 & .986, .998, 1.00 \\
$\hat{\mathcal{T}}^{dw}$ & .970, .995, .998 & .955, .991, .997 & .956, .991,
.995 & .716, .868, .926 & .725, .874, .930 & .700, .850, .918 \\ \hline
$\mathcal{AP}^{w}$ & .998, 1.00, 1.00 & .996, 1.00, 1.00 & .994, .999, 1.00
& .984, .998, .999 & .982, .998, 1.00 & .979, .995, 1.00 \\
$\mathcal{AP}^{dw}$ & .930, .983, .995 & .910, .979, .994 & .905, .979, .992
& .672, .835, .901 & .623, .818, .895 & .658, .844, .916 \\ \hline
$Q_{\chi^{2}}$ & .999, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & .999, .999, .999 \\
$\mathcal{N}^{N_{0,1}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & .999, 1.00, 1.00 & .999, 1.00, 1.00 \\
$\mathcal{N}^{w}$ & .997, 1.00, 1.00 & .993, .999, 1.00 & .993, 1.00, 1.00 &
.986, .997, 1.00 & .974, .999, 1.00 & .981, .998, 1.00 \\
$\mathcal{N}^{dw}$ & .908, .974, .991 & .624, .923, .970 & .197, .772, .955
& .645, .824, .896 & .505, .753, .879 & .293, .694, .852 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#5: GARCH(1,1) $y_{t} =
\sigma_{t} e_{t}$, $\sigma_{t}^{2} = 1 + 0.2 y_{t-1}^{2} + 0.5
\sigma_{t-1}^{2}$; no filter. In this table we report rejection frequencies
with the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the
proposed max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#5 ($n = 1000$) }
\label{table:supp_mat_scenario5_n1000}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 72 & Lag = 144 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .013, .056, .107 & .004, .048, .082 & .011, .033,
.068 & .002, .041, .099 & .006, .029, .080 & .004, .032, .069 \\
$\hat{\mathcal{T}}^{dw}$ & .004, .044, .092 & .004, .038, .078 & .003, .017,
.056 & .006, .020, .050 & .000, .011, .044 & .006, .011, .028 \\ \hline
$\mathcal{AP}^{w}$ & .006, .059, .097 & .008, .050, .107 & .010, .045, .086
& .006, .045, .095 & .003, .032, .075 & .006, .038, .083 \\
$\mathcal{AP}^{dw}$ & .009, .045, .095 & .010, .043, .093 & .007, .056, .111
& .003, .025, .062 & .004, .017, .047 & .008, .019, .052 \\ \hline
$Q_{\chi^{2}}$ & .045, .138, .215 & .032, .085, .153 & .031, .094, .153 &
.988, .996, .996 & .974, .983, .986 & .921, .940, .946 \\
$\mathcal{N}^{N_{0,1}}$ & .103, .168, .221 & .025, .063, .121 & .016, .038,
.059 & .990, .993, .993 & .977, .982, .986 & .894, .917, .924 \\
$\mathcal{N}^{w}$ & .011, .044, .091 & .009, .043, .090 & .004, .037, .087 &
.005, .031, .077 & .003, .027, .065 & .000, .019, .047 \\
$\mathcal{N}^{dw}$ & .005, .045, .109 & .000, .002, .023 & .000, .000, .007
& .001, .017, .044 & .000, .001, .004 & .000, .001, .004 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 72 & Lag = 144 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & .988, .999, .999 & .987, .998, 1.00 & .986, 1.00, 1.00 \\
$\hat{\mathcal{T}}^{dw}$ & .985, .997, .999 & .987, .996, .996 & .991, 1.00,
1.00 & .751, .870, .917 & .768, .884, .935 & .762, .883, .933 \\ \hline
$\mathcal{AP}^{w}$ & .998, 1.00, 1.00 & .997, 1.00, 1.00 & .997, 1.00, 1.00
& .981, 1.00, 1.00 & .985, 1.00, 1.00 & .982, .998, 1.00 \\
$\mathcal{AP}^{dw}$ & .975, .996, .999 & .967, .987, .992 & .973, .993, .999
& .728, .864, .923 & .704, .841, .904 & .687, .848, .915 \\ \hline
$Q_{\chi^{2}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{N_{0,1}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{w}$ & .998, 1.00, 1.00 & .999, 1.00, 1.00 & .998, 1.00, 1.00 &
.981, .999, 1.00 & .981, .998, 1.00 & .977, .995, .998 \\
$\mathcal{N}^{dw}$ & .961, .988, .997 & .892, .983, .992 & .603, .958, .987
& .680, .838, .900 & .570, .780, .871 & .471, .747, .870 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#5: GARCH(1,1) $y_{t} =
\sigma_{t} e_{t}$, $\sigma_{t}^{2} = 1 + 0.2 y_{t-1}^{2} + 0.5
\sigma_{t-1}^{2}$; no filter. In this table we report rejection frequencies
with the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the
proposed max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#6 ($n = 100$) }
\label{table:supp_mat_scenario6_n100}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 10 & Lag = 21 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .006, .047, .089 & .009, .047, .084 & .005, .033,
.072 & .007, .047, .097 & .004, .040, .083 & .003, .035, .083 \\
$\hat{\mathcal{T}}^{dw}$ & .015, .095, .163 & .010, .051, .114 & .005, .031,
.070 & .016, .080, .157 & .010, .046, .128 & .003, .027, .093 \\ \hline
$\mathcal{AP}^{w}$ & .009, .050, .095 & .008, .036, .075 & .009, .041, .090
& .012, .054, .102 & .013, .044, .096 & .010, .044, .094 \\
$\mathcal{AP}^{dw}$ & .020, .075, .149 & .021, .084, .152 & .017, .065, .130
& .028, .097, .175 & .038, .100, .184 & .033, .091, .173 \\ \hline
$Q_{\chi^{2}}$ & 011, .055, .107 & .016, .051, .093 & .030, .082, .127 &
.033, .092, .151 & .040, .103, .169 & .035, .087, .146 \\
$\mathcal{N}^{N_{0,1}}$ & .025, .060, .088 & .021, .058, .092 & .015, .042,
.063 & .050, .116, .147 & .046, .090, .136 & .044, .077, .116 \\
$\mathcal{N}^{w}$ & .007, .050, .107 & .011, .044, .084 & .005, .030, .064 &
.008, .059, .114 & .008, .043, .089 & .013, .052, .093 \\
$\mathcal{N}^{dw}$ & .009, .063, .141 & .004, .039, .105 & .001, .016, .057
& .018, .085, .168 & .002, .045, .107 & .001, .029, .084 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 10 & Lag = 21 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .865, .946, .960 & .794, .923, .946 & .737, .850,
.885 & .970, .977, .983 & .949, .958, .962 & .927, .946, .954 \\
$\hat{\mathcal{T}}^{dw}$ & .799, .924, .950 & .655, .882, .926 & .471, .800,
.882 & .953, .975, .976 & .946, .971, .976 & .894, .948, .957 \\ \hline
$\mathcal{AP}^{w}$ & .902, .965, .974 & .873, .955, .968 & .873, .949, .962
& .964, .972, .980 & .967, .973, .975 & .964, .976, .979 \\
$\mathcal{AP}^{dw}$ & .821, .943, .960 & .782, .926, .947 & .801, .943, .966
& .959, .974, .979 & .923, .966, .970 & .939, .978, .981 \\ \hline
$Q_{\chi^{2}}$ & .903, .976, .992 & .794, .909, .948 & .670, .825, .888 &
.998, 1.00, 1.00 & .997, 1.00, 1.00 & .989, .997, .998 \\
$\mathcal{N}^{N_{0,1}}$ & .941, .976, .984 & .861, .925, .955 & .688, .798,
.838 & 1.00, 1.00, 1.00 & .997, 1.00, 1.00 & .990, .994, .998 \\
$\mathcal{N}^{w}$ & .760, .912, .942 & .572, .826, .895 & .355, .652, .778 &
.949, .965, .972 & .927, .959, .970 & .821, .916, .937 \\
$\mathcal{N}^{dw}$ & .546, .840, .931 & .199, .569, .770 & .056, .267, .494
& .909, .970, .978 & .589, .910, .961 & .212, .586, .816 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#6: GARCH(1,1) $y_{t} =
\sigma_{t} e_{t}$, $\sigma_{t}^{2} = 1 + 0.2 y_{t-1}^{2} + 0.5
\sigma_{t-1}^{2}$; GARCH(1,1) filter $\epsilon_{t} = y_{t} / \sigma_{t}$
with $\sigma_{t}^{2} = \omega + \alpha y_{t-1}^{2} + \beta \sigma_{t-1}^{2}$%
; QML estimation. In this table we report rejection frequencies with the
nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the proposed
max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#6 ($n = 250$) }
\label{table:supp_mat_scenario6_n250}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 22 & Lag = 45 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .014, .055, .115 & .008, .045, .102 & .008, .048,
.083 & .011, .055, .108 & .008, .046, .091 & .009, .052, .104 \\
$\hat{\mathcal{T}}^{dw}$ & .010, .073, .139 & .002, .034, .096 & .002, .026,
.068 & .010, .058, .127 & .005, .036, .094 & .003, .024, .083 \\ \hline
$\mathcal{AP}^{w}$ & .011, .045, .084 & .011, .042, .086 & .009, .048, .107
& .017, .057, .114 & .018, .052, .103 & .015, .056, .098 \\
$\mathcal{AP}^{dw}$ & .019, .075, .143 & .015, .064, .140 & .017, .072, .128
& .019, .074, .133 & .014, .072, .150 & .014, .065, .137 \\ \hline
$Q_{\chi^{2}}$ & .015, .053, .099 & .019, .073, .114 & .032, .077, .120 &
.030, .096, .156 & .027, .074, .139 & .035, .099, .155 \\
$\mathcal{N}^{N_{0,1}}$ & .031, .089, .124 & .031, .061, .095 & .022, .043,
.072 & .069, .132, .179 & .038, .078, .123 & .025, .057, .083 \\
$\mathcal{N}^{w}$ & .010, .045, .095 & .009, .049, .102 & .007, .031, .069 &
.014, .066, .127 & .009, .049, .093 & .010, .052, .113 \\
$\mathcal{N}^{dw}$ & .007, .056, .118 & .000, .026, .062 & .000, .015, .041
& .009, .061, .126 & .000, .023, .065 & .000, .011, .045 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 22 & Lag = 45 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .992, .992, .993 & .988, .989, .992 & .982, .988,
.989 & .995, .999, .999 & .991, .993, .993 & .991, .991, .992 \\
$\hat{\mathcal{T}}^{dw}$ & .994, .997, .997 & .989, .991, .993 & .967, .977,
.978 & .997, .997, .998 & .991, .993, .994 & .995, .997, .997 \\ \hline
$\mathcal{AP}^{w}$ & .993, .994, .997 & .993, .994, .995 & .994, .994, .997
& .999, 1.00, 1.00 & .999, .999, .999 & .993, .994, .994 \\
$\mathcal{AP}^{dw}$ & .990, .993, .994 & .988, .995, .995 & .992, .995, .996
& .994, .994, .996 & .994, .994, .996 & .998, .999, .999 \\ \hline
$Q_{\chi^{2}}$ & 1.00, 1.00, 1.00 & .997, 1.00, 1.00 & .979, .991, .998 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{N_{0,1}}$ & 1.00, 1.00, 1.00 & .999, 1.00, 1.00 & .975, .990,
.998 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{w}$ & .991, .992, .993 & .969, .986, .988 & .838, .948, .967 &
.993, .994, .994 & .995, .995, .996 & .991, .992, .993 \\
$\mathcal{N}^{dw}$ & .979, .992, .993 & .410, .894, .963 & .089, .470, .774
& .997, .997, .997 & .866, .986, .988 & .383, .903, .982 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#6: GARCH(1,1) $y_{t} =
\sigma_{t} e_{t}$, $\sigma_{t}^{2} = 1 + 0.2 y_{t-1}^{2} + 0.5
\sigma_{t-1}^{2}$; GARCH(1,1) filter $\epsilon_{t} = y_{t} / \sigma_{t}$
with $\sigma_{t}^{2} = \omega + \alpha y_{t-1}^{2} + \beta \sigma_{t-1}^{2}$%
; QML estimation. In this table we report rejection frequencies with the
nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the proposed
max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#6 ($n = 500$) }
\label{table:supp_mat_scenario6_n500}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 40 & Lag = 80 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .012, .057, .106 & .011, .051, .104 & .002, .040,
.086 & .012, .048, .089 & .010, .040, .093 & .009, .043, .089 \\
$\hat{\mathcal{T}}^{dw}$ & .007, .058, .113 & .006, .036, .107 & .001, .029,
.073 & .006, .050, .116 & .003, .029, .079 & .003, .021, .064 \\ \hline
$\mathcal{AP}^{w}$ & .006, .047, .096 & .008, .046, .087 & .017, .057, .104
& .010, .055, .110 & .015, .049, .102 & .009, .040, .094 \\
$\mathcal{AP}^{dw}$ & .013, .072, .125 & .014, .075, .145 & .016, .060, .111
& .016, .071, .126 & .021, .075, .130 & .016, .060, .133 \\ \hline
$Q_{\chi^{2}}$ & .013, .046, .092 & .018, .067, .105 & .029, .084, .135 &
.033, .104, .169 & .022, .078, .148 & .020, .083, .137 \\
$\mathcal{N}^{N_{0,1}}$ & .021, .064, .101 & .019, .063, .096 & .016, .039,
.069 & .065, .141, .192 & .032, .076, .120 & .026, .055, .080 \\
$\mathcal{N}^{w}$ & .009, .054, .105 & .004, .040, .098 & .010, .037, .071 &
.016, .059, .125 & .011, .043, .093 & .005, .033, .080 \\
$\mathcal{N}^{dw}$ & .014, .053, .109 & .000, .011, .047 & .000, .003, .025
& .011, .048, .113 & .001, .015, .048 & .000, .006, .035 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 40 & Lag = 80 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .999, 1.00, 1.00 & .998, .998, .998 & .998, .998,
.998 & .998, 1.00, 1.00 & .995, .995, .995 & .997, .997, .997 \\
$\hat{\mathcal{T}}^{dw}$ & .996, .996, .997 & .997, .998, .998 & .996, .996,
.996 & .996, .996, .996 & .999, .999, .999 & .998, .998, .999 \\ \hline
$\mathcal{AP}^{w}$ & .999, .999, .999 & .995, .995, .995 & 1.00, 1.00, 1.00
& .998, .998, .998 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{AP}^{dw}$ & 1.00, 1.00, 1.00 & .998, .998, .998 & 1.00, 1.00, 1.00
& .999, .999, 1.00 & .999, 1.00, 1.00 & .999, .999, .999 \\ \hline
$Q_{\chi^{2}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & .999, 1.00, 1.00 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{N_{0,1}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{w}$ & .999, .999, .999 & .997, .999, .999 & .991, .994, .994 &
.999, 1.00, 1.00 & .999, .999, .999 & .996, .996, .996 \\
$\mathcal{N}^{dw}$ & .999, 1.00, 1.00 & .821, .995, .996 & .210, .819, .962
& .997, .998, .998 & .990, .997, .998 & .774, .994, .995 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#6: GARCH(1,1) $y_{t} =
\sigma_{t} e_{t}$, $\sigma_{t}^{2} = 1 + 0.2 y_{t-1}^{2} + 0.5
\sigma_{t-1}^{2}$; GARCH(1,1) filter $\epsilon_{t} = y_{t} / \sigma_{t}$
with $\sigma_{t}^{2} = \omega + \alpha y_{t-1}^{2} + \beta \sigma_{t-1}^{2}$%
; QML estimation. In this table we report rejection frequencies with the
nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the proposed
max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#6 ($n = 1000$) }
\label{table:supp_mat_scenario6_n1000}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$e_{t}$ is iid: $e_{t} = \nu_{t}$} &
\multicolumn{3}{c}{$e_{t}$ is GARCH(1,1): $w_{t}^{2} = 1.0 + 0.2 e_{t-1}^{2}
+ 0.5 w_{t-1}^{2}$} \\ \hline
& Lag = 5 & Lag = 72 & Lag = 144 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .011, .048, .102 & .006, .049, .090 & .005, .041,
.103 & .011, .049, .097 & .010, .065, .109 & .009, .056, .098 \\
$\hat{\mathcal{T}}^{dw}$ & .011, .052, .106 & .002, .028, .083 & .004, .022,
.066 & .012, .068, .121 & .001, .038, .088 & .001, .034, .080 \\ \hline
$\mathcal{AP}^{w}$ & .011, .046, .096 & .010, .043, .093 & .011, .047, .106
& .013, .046, .091 & .009, .059, .102 & .007, .052, .101 \\
$\mathcal{AP}^{dw}$ & .021, .071, .123 & .013, .051, .108 & .020, .079, .129
& .011, .057, .137 & .012, .063, .121 & .008, .051, .114 \\ \hline
$Q_{\chi^{2}}$ & .013, .060, .100 & .023, .073, .123 & .019, .078, .148 &
.042, .115, .183 & .020, .075, .123 & .027, .088, .141 \\
$\mathcal{N}^{N_{0,1}}$ & .032, .073, .100 & .022, .053, .092 & .011, .030,
.047 & .069, .137, .190 & .030, .078, .121 & .013, .042, .068 \\
$\mathcal{N}^{w}$ & .011, .068, .110 & .008, .045, .098 & .004, .045, .094 &
.009, .046, .096 & .008, .047, .094 & .004, .036, .091 \\
$\mathcal{N}^{dw}$ & .018, .061, .137 & .001, .006, .028 & .000, .001, .014
& .006, .051, .118 & .000, .002, .020 & .000, .004, .017 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$e_{t}$ is MA(2): $e_{t} = \nu_{t} + 0.50 \nu_{t-1} +
0.25 \nu_{t-2}$} & \multicolumn{3}{c}{$e_{t}$ is AR(1): $e_{t} = 0.7e_{t-1}
+ \nu_{t}$} \\ \hline
& Lag = 5 & Lag = 72 & Lag = 144 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & .998, .998, .999 \\
$\hat{\mathcal{T}}^{dw}$ & 1.00, 1.00, 1.00 & .999, .999, 1.00 & .999, .999,
.999 & .998, .998, .998 & .999, .999, 1.00 & .999, .999, .999 \\ \hline
$\mathcal{AP}^{w}$ & .999, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00
& .999, .999, .999 & .999, .999, .999 & 1.00, 1.00, 1.00 \\
$\mathcal{AP}^{dw}$ & 1.00, 1.00, 1.00 & .998, .998, .998 & 1.00, 1.00, 1.00
& .999, .999, .999 & 1.00, 1.00, 1.00 & .999, .999, .999 \\ \hline
$Q_{\chi^{2}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 &
1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{N_{0,1}}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00,
1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{N}^{w}$ & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 &
1.00, 1.00, 1.00 & .999, .999, .999 & .997, .998, .998 \\
$\mathcal{N}^{dw}$ & .999, .999, .999 & .998, 1.00, 1.00 & .670, .992, 1.00
& 1.00, 1.00, 1.00 & .999, 1.00, 1.00 & .994, 1.00, 1.00 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#6: GARCH(1,1) $y_{t} =
\sigma_{t} e_{t}$, $\sigma_{t}^{2} = 1 + 0.2 y_{t-1}^{2} + 0.5
\sigma_{t-1}^{2}$; GARCH(1,1) filter $\epsilon_{t} = y_{t} / \sigma_{t}$
with $\sigma_{t}^{2} = \omega + \alpha y_{t-1}^{2} + \beta \sigma_{t-1}^{2}$%
; QML estimation. In this table we report rejection frequencies with the
nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the proposed
max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#7 (Remote MA(6)) }
\label{table:supp_mat_scenario7}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$n = 100$} & \multicolumn{3}{c}{$n = 250$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .014, .080, .146 & .068, .231, .381 & .057, .201,
.336 & .024, .077, .131 & .519, .732, .814 & .464, .666, .750 \\
$\hat{\mathcal{T}}^{dw}$ & .011, .063, .143 & .048, .226, .388 & .025, .147,
.293 & .005, .040, .098 & .319, .649, .789 & .269, .543, .678 \\ \hline
$\mathcal{AP}^{w}$ & .012, .066, .123 & .023, .110, .182 & .032, .104, .175
& .027, .086, .159 & .052, .147, .226 & .034, .147, .232 \\
$\mathcal{AP}^{dw}$ & .019, .075, .151 & .016, .086, .158 & .018, .084, .153
& .018, .069, .136 & .018, .093, .180 & .017, .072, .164 \\ \hline
$Q_{\chi^{2}}$ & .033, .107, .178 & .138, .290, .400 & .141, .283, .381 &
.044, .111, .188 & .342, .570, .703 & .302, .488, .599 \\
$\mathcal{N}^{N_{0,1}}$ & .071, .125, .175 & .194, .319, .399 & .132, .228,
.286 & .067, .124, .164 & .384, .545, .634 & .256, .381, .467 \\
$\mathcal{N}^{w}$ & .017, .065, .120 & .060, .229, .357 & .042, .154, .265 &
.019, .089, .157 & .219, .469, .634 & .131, .338, .482 \\
$\mathcal{N}^{dw}$ & .016, .074, .155 & .018, .149, .303 & .006, .056, .159
& .013, .050, .103 & .021, .173, .362 & .001, .051, .171 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$n = 500$} & \multicolumn{3}{c}{$n = 1000$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .019, .092, .161 & .942, .988, .993 & .952, .986,
.991 & .034, .092, .169 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\hat{\mathcal{T}}^{dw}$ & .010, .064, .122 & .878, .968, .985 & .823, .944,
.974 & .007, .059, .121 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\ \hline
$\mathcal{AP}^{w}$ & .020, .098, .171 & .102, .244, .343 & .092, .245, .363
& .025, .091, .162 & .165, .341, .455 & .178, .359, .481 \\
$\mathcal{AP}^{dw}$ & .008, .048, .115 & .026, .129, .236 & .027, .112, .211
& .009, .058, .124 & .063, .223, .347 & .064, .221, .357 \\ \hline
$Q_{\chi^{2}}$ & .037, .106, .186 & .661, .846, .910 & .495, .706, .801 &
.029, .097, .167 & .941, .983, .988 & .817, .926, .964 \\
$\mathcal{N}^{N_{0,1}}$ & .070, .137, .186 & .704, .833, .889 & .441, .589,
.680 & .083, .143, .189 & .935, .977, .987 & .692, .825, .893 \\
$\mathcal{N}^{w}$ & .034, .102, .171 & .488, .763, .869 & .295, .585, .720 &
.025, .108, .183 & .851, .960, .988 & .633, .869, .941 \\
$\mathcal{N}^{dw}$ & .007, .049, .115 & .037, .297, .571 & .004, .080, .243
& .009, .058, .113 & .177, .676, .877 & .010, .170, .433 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#7: Remote MA(6) $y_{t} = e_{t}
+ 0.25 e_{t-6}$ with $e_{t} \overset{i.i.d.}{\sim} N(0,1)$; mean filter $%
\epsilon_{t} = y_{t} - E[y_{t}]$; $\hat{\phi}_{n} = 1/n \sum_{t=1}^{n} y_{t}$%
. In this table we report rejection frequencies with the nominal sizes 1\%,
5\%, and 10\%. $\hat{\mathcal{T}}$ is the proposed max-correlation test. $%
\mathcal{AP}$ is Andrews and Ploberger's (\citeyear{AndrewsPloberger1996})
sup-LM test. $Q_{\chi^{2}}$ is the asymptotic Ljung-Box test based on the $%
\chi^{2}$ distribution. $\mathcal{N}^{N_{0,1}}$ is Hong's (%
\citeyear{Hong1996}) asymptotic test based on $N(0,1)$. Superscript ``$w$''
means the wild bootstrap while ``$dw$'' means Shao's (\citeyear{Shao2011_JoE}%
) dependent wild bootstrap. Lag length is $\mathcal{L}_{n} \in \{ 5,
[.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#8 (Remote MA(12)) }
\label{table:supp_mat_scenario8}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$n = 100$} & \multicolumn{3}{c}{$n = 250$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .010, .062, .126 & .008, .067, .132 & .053, .180,
.283 & .012, .060, .122 & .488, .715, .799 & .491, .705, .800 \\
$\hat{\mathcal{T}}^{dw}$ & .019, .067, .148 & .010, .067, .159 & .034, .163,
.280 & .019, .077, .155 & .383, .659, .779 & .320, .596, .729 \\ \hline
$\mathcal{AP}^{w}$ & .011, .048, .097 & .012, .054, .109 & .016, .062, .132
& .013, .055, .123 & .021, .087, .162 & .020, .079, .143 \\
$\mathcal{AP}^{dw}$ & .029, .115, .184 & .015, .085, .180 & .024, .097, .176
& .023, .083, .139 & .022, .098, .175 & .015, .084, .165 \\ \hline
$Q_{\chi^{2}}$ & .024, .075, .125 & .034, .100, .168 & .130, .252, .351 &
.029, .080, .156 & .327, .563, .692 & .261, .457, .570 \\
$\mathcal{N}^{N_{0,1}}$ & .046, .099, .139 & .062, .122, .167 & .120, .193,
.253 & .050, .098, .136 & .410, .572, .646 & .200, .309, .402 \\
$\mathcal{N}^{w}$ & .007, .061, .121 & .021, .072, .135 & .037, .159, .280 &
.015, .064, .119 & .224, .476, .641 & .117, .331, .483 \\
$\mathcal{N}^{dw}$ & .012, .073, .147 & .006, .060, .140 & .011, .082, .215
& .009, .067, .154 & .048, .287, .500 & .017, .128, .305 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$n = 500$} & \multicolumn{3}{c}{$n = 1000$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .014, .061, .128 & .939, .977, .989 & .901, .959,
.981 & .019, .082, .140 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\hat{\mathcal{T}}^{dw}$ & .009, .061, .121 & .883, .964, .976 & .854, .958,
.979 & .014, .051, .114 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\ \hline
$\mathcal{AP}^{w}$ & .012, .065, .135 & .023, .094, .161 & .027, .098, .170
& .007, .054, .110 & .027, .109, .168 & .029, .118, .198 \\
$\mathcal{AP}^{dw}$ & .010, .070, .131 & .005, .049, .119 & .013, .063, .145
& .016, .066, .125 & .011, .071, .149 & .014, .077, .137 \\ \hline
$Q_{\chi^{2}}$ & .010, .066, .118 & .621, .804, .897 & .500, .698, .785 &
.016, .075, .137 & .936, .978, .992 & .809, .924, .955 \\
$\mathcal{N}^{N_{0,1}}$ & .059, .108, .151 & .667, .814, .877 & .417, .574,
.674 & .042, .089, .130 & .930, .981, .992 & .710, .845, .890 \\
$\mathcal{N}^{w}$ & .014, .058, .132 & .497, .768, .855 & .296, .584, .733 &
.019, .086, .143 & .862, .966, .991 & .618, .864, .926 \\
$\mathcal{N}^{dw}$ & .006, .071, .139 & .094, .440, .697 & .015, .145, .372
& .011, .054, .115 & .229, .737, .926 & .020, .232, .527 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#8: Remote MA(12) $y_{t} =
e_{t} + 0.25 e_{t-12}$ with $e_{t} \overset{i.i.d.}{\sim} N(0,1)$; mean
filter $\epsilon_{t} = y_{t} - E[y_{t}]$; $\hat{\phi}_{n} = 1/n
\sum_{t=1}^{n} y_{t}$. In this table we report rejection frequencies with
the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the proposed
max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection Frequencies of Tests with Various Fixed $\mathcal{L}_{n}
= o(n)$ in Scenario \#9 (Remote MA(24)) }
\label{table:supp_mat_scenario9}
\begin{center}
{\fontsize{10pt}{20pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$n = 100$} & \multicolumn{3}{c}{$n = 250$} \\ \hline
& Lag = 5 & Lag = 10 & Lag = 21 & Lag = 5 & Lag = 22 & Lag = 45 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .014, .058, .115 & .008, .042, .092 & .012, .056,
.109 & .016, .060, .117 & .019, .077, .142 & .362, .586, .688 \\
$\hat{\mathcal{T}}^{dw}$ & .017, .094, .168 & .011, .073, .150 & .010, .068,
.143 & .014, .083, .160 & .011, .066, .167 & .258, .534, .670 \\ \hline
$\mathcal{AP}^{w}$ & .005, .044, .093 & .009, .044, .089 & .008, .045, .097
& .010, .064, .124 & .014, .078, .129 & .010, .056, .112 \\
$\mathcal{AP}^{dw}$ & .026, .084, .164 & .022, .083, .154 & .021, .094, .164
& .014, .073, .142 & .015, .075, .158 & .018, .069, .142 \\ \hline
$Q_{\chi^{2}}$ & .019, .074, .122 & .032, .097, .152 & .066, .135, .194 &
.016, .067, .123 & .048, .140, .213 & .229, .426, .533 \\
$\mathcal{N}^{N_{0,1}}$ & .034, .074, .105 & .053, .088, .124 & .055, .094,
.128 & .041, .088, .127 & .075, .141, .183 & .224, .337, .415 \\
$\mathcal{N}^{w}$ & .008, .063, .131 & .012, .059, .117 & .012, .059, .136 &
.017, .071, .142 & .028, .094, .160 & .106, .324, .457 \\
$\mathcal{N}^{dw}$ & .010, .065, .135 & .006, .056, .135 & .005, .050, .118
& .004, .053, .144 & .005, .052, .108 & .020, .130, .287 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$n = 500$} & \multicolumn{3}{c}{$n = 1000$} \\ \hline
& Lag = 5 & Lag = 40 & Lag = 80 & Lag = 5 & Lag = 72 & Lag = 144 \\ \hline
& 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%,
10\% & 1\%, \; 5\%, 10\% & 1\%, \; 5\%, 10\% \\ \hline
$\hat{\mathcal{T}}^{w}$ & .018, .069, .129 & .914, .974, .983 & .891, .965,
.977 & .008, .061, .119 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\hat{\mathcal{T}}^{dw}$ & .012, .059, .133 & .867, .959, .979 & .851, .949,
.979 & .012, .060, .128 & .999, 1.00, 1.00 & .999, 1.00, 1.00 \\ \hline
$\mathcal{AP}^{w}$ & .013, .051, .105 & .013, .063, .119 & .015, .064, .129
& .015, .067, .119 & .002, .049, .113 & .015, .065, .113 \\
$\mathcal{AP}^{dw}$ & .019, .079, .153 & .021, .075, .156 & .017, .064, .138
& .018, .073, .134 & .012, .072, .133 & .016, .070, .150 \\ \hline
$Q_{\chi^{2}}$ & .016, .083, .140 & .626, .818, .877 & .487, .696, .791 &
.016, .075, .126 & .936, .981, .989 & .821, .932, .955 \\
$\mathcal{N}^{N_{0,1}}$ & .058, .106, .162 & .673, .800, .860 & .404, .567,
.652 & .057, .101, .166 & .943, .979, .988 & .712, .842, .888 \\
$\mathcal{N}^{w}$ & .012, .073, .133 & .514, .772, .869 & .293, .559, .717 &
.009, .069, .147 & .867, .975, .989 & .646, .864, .928 \\
$\mathcal{N}^{dw}$ & .008, .069, .144 & .134, .516, .756 & .029, .247, .512
& .016, .076, .126 & .316, .787, .929 & .044, .354, .663 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont Scenario \#9: Remote MA(24) $y_{t} =
e_{t} + 0.25 e_{t-24}$ with $e_{t} \overset{i.i.d.}{\sim} N(0,1)$; mean
filter $\epsilon_{t} = y_{t} - E[y_{t}]$; $\hat{\phi}_{n} = 1/n
\sum_{t=1}^{n} y_{t}$. In this table we report rejection frequencies with
the nominal sizes 1\%, 5\%, and 10\%. $\hat{\mathcal{T}}$ is the proposed
max-correlation test. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $Q_{\chi^{2}}$ is the
asymptotic Ljung-Box test based on the $\chi^{2}$ distribution. $\mathcal{N}%
^{N_{0,1}}$ is Hong's (\citeyear{Hong1996}) asymptotic test based on $N(0,1)$%
. Superscript ``$w$'' means the wild bootstrap while ``$dw$'' means Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap. Lag length is $\mathcal{L}%
_{n} \in \{ 5, [.5n/\ln(n)], [n/\ln(n)] \}$. }
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection frequencies of $\hat{\mathcal{T}}_{n}(\mathcal{L}%
_{n}^{*}) $, $\mathcal{AP}$, and $CvM$ (Scenarios \#1--\#6, $n = 100$) }
\label{table:supp_mat_AP_CvM_scenario123456_n100}
\begin{center}
{\fontsize{8.8pt}{15.5pt}\selectfont
\begin{tabular}{l||c|c|c|c|c|c}
\multicolumn{7}{c}{IID error: $e_{t}=\nu _{t}$} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .017, .068, .128 & .008,
.047, .090 & .002, .061, .129 & .034, .197, .327 & .006, .031, .068 & .026,
.091, .140 \\ \hline
$\mathcal{AP}^{w}$ & .007, .038, .086 & .011, .072, .128 & .007, .042, .098
& .042, .195, .329 & .004, .028, .068 & .006, .039, .086 \\
$\mathcal{AP}^{dw}$ & .017, .079, .144 & .022, .083, .151 & .022, .102, .190
& .182, .376, .504 & .020, .072, .136 & .019, .078, .135 \\ \hline
$CvM^{w}$ & .012, .063, .132 & .013, .075, .135 & .009, .060, .124 & .106,
.287, .413 & .005, .035, .079 & .012, .043, .076 \\
$CvM^{dw}$ & .023, .081, .138 & .018, .076, .149 & .020, .086, .167 & .133,
.338, .483 & .021, .077, .141 & .034, .087, .144 \\
$CvM^{brw}$ & .028, .082, .142 & .045, .115, .186 & .009, .041, .076 & .123,
.276, .390 & .034, .076, .139 & .001, .024, .075 \\ \hline
\multicolumn{7}{c}{} \\
\multicolumn{7}{c}{GARCH(1,1) error: $e_{t}=\nu _{t}w_{t}$ with $%
w_{t}^{2}=1+0.2e_{t-1}^{2}+0.5w_{t-1}^{2}$} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .005, .026, .075 & .007,
.021, .040 & .004, .054, .109 & .027, .150, .248 & .001, .003, .012 & .026,
.090, .162 \\ \hline
$\mathcal{AP}^{w}$ & .003, .034, .083 & .004, .032, .078 & .003, .040, .091
& .049, .170, .287 & .006, .028, .075 & .013, .055, .102 \\
$\mathcal{AP}^{dw}$ & .010, .071, .141 & .006, .034, .077 & .034, .117, .218
& .148, .343, .480 & .005, .038, .109 & .028, .102, .175 \\ \hline
$CvM^{w}$ & .004, .048, .105 & .007, .045, .102 & .012, .053, .108 & .075,
.245, .364 & .005, .038, .099 & .018, .072, .117 \\
$CvM^{dw}$ & .017, .081, .149 & .002, .030, .070 & .026, .086, .168 & .118,
.287, .430 & .006, .049, .103 & .036, .100, .168 \\
$CvM^{brw}$ & .028, .079, .140 & .013, .056, .113 & .009, .032, .061 & .098,
.209, .300 & .015, .060, .113 & .006, .044, .091 \\ \hline
\multicolumn{7}{c}{} \\
\multicolumn{7}{c}{MA(2) error: $e_{t}=\nu _{t}+0.5\nu _{t-1}+0.25\nu _{t-2}$%
} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .693, .901, .951 & .582,
.769, .825 & .012, .068, .135 & .242, .601, .762 & .461, .707, .788 & .908,
.966, .979 \\ \hline
$\mathcal{AP}^{w}$ & .896, .989, .996 & .484, .816, .913 & .013, .052, .112
& .438, .693, .792 & .789, .965, .985 & .860, .932, .953 \\
$\mathcal{AP}^{dw}$ & .673, .916, .976 & .331, .627, .784 & .029, .115, .181
& .516, .751, .836 & .508, .828, .931 & .813, .922, .963 \\ \hline
$CvM^{w}$ & .963, .994, 1.00 & .681, .898, .943 & .015, .071, .156 & .604,
.829, .896 & .900, .989, .997 & .882, .917, .927 \\
$CvM^{dw}$ & .898, .984, .995 & .450, .743, .866 & .029, .113, .182 & .570,
.805, .898 & .681, .908, .969 & .878, .927, .940 \\
$CvM^{brw}$ & .929, .991, .995 & .712, .841, .899 & .018, .058, .110 & .587,
.802, .884 & .791, .932, .967 & .247, .739, .904 \\ \hline
\multicolumn{7}{c}{} \\
\multicolumn{7}{c}{AR(1) error: $e_{t}=0.7e_{t-1}+\nu _{t}$} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .531, .752, .847 & .477,
.637, .685 & .021, .128, .227 & .263, .636, .788 & .179, .345, .432 & .987,
.991, .991 \\ \hline
$\mathcal{AP}^{w}$ & 1.00, 1.00, 1.00 & .588, .872, .946 & .017, .078, .162
& .405, .735, .820 & .936, .996, .999 & .972, .987, .989 \\
$\mathcal{AP}^{dw}$ & .774, .972, .996 & .232, .500, .665 & .115, .243, .315
& .530, .756, .865 & .470, .758, .898 & .953, .987, .990 \\ \hline
$CvM^{w}$ & .998, 1.00, 1.00 & .743, .938, .975 & .035, .152, .250 & .434,
.727, .838 & .961, .997, .999 & .938, .948, .954 \\
$CvM^{dw}$ & .925, .996, 1.00 & .282, .567, .741 & .064, .193, .299 & .472,
.741, .849 & .564, .818, .923 & .958, .970, .973 \\
$CvM^{brw}$ & .981, 1.00, 1.00 & .564, .744, .830 & .044, .118, .192 & .520,
.748, .845 & .687, .862, .921 & .481, .863, .932 \\ \hline
\end{tabular}
}
\end{center}
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection frequencies of $\hat{\mathcal{T}}_{n}(\mathcal{L}%
_{n}^{*}) $, $\mathcal{AP}$, and $CvM$ (Scenarios \#1--\#6, $n = 250$) }
\label{table:supp_mat_AP_CvM_scenario123456_n250}
\begin{center}
{\fontsize{8.8pt}{15.5pt}\selectfont
\begin{tabular}{l||c|c|c|c|c|c}
\multicolumn{7}{c}{IID error: $e_{t}=\nu _{t}$} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .011, .045, .087 & .012,
.042, .087 & .005, .048, .093 & .178, .479, .634 & .005, .029, .063 & .019,
.064, .125 \\ \hline
$\mathcal{AP}^{w}$ & .008, .052, .105 & .011, .064, .134 & .011, .049, .098
& .162, .397, .537 & .012, .047, .114 & .007, .040, .085 \\
$\mathcal{AP}^{dw}$ & .018, .056, .114 & .016, .079, .145 & .023, .071, .126
& .298, .537, .666 & .012, .062, .114 & .013, .062, .136 \\ \hline
$CvM^{w}$ & .008, .047, .093 & .019, .076, .148 & .007, .042, .100 & .376,
.621, .730 & .011, .058, .104 & .005, .044, .091 \\
$CvM^{dw}$ & .016, .072, .144 & .030, .085, .154 & .011, .065, .127 & .370,
.615, .735 & .011, .058, .118 & .019, .065, .112 \\
$CvM^{brw}$ & .029, .075, .120 & .020, .064, .119 & .012, .054, .097 & .388,
.630, .728 & .016, .054, .107 & .002, .019, .061 \\ \hline
\multicolumn{7}{c}{} \\
\multicolumn{7}{c}{GARCH(1,1) error: $e_{t}=\nu _{t}w_{t}$ with $%
w_{t}^{2}=1+0.2e_{t-1}^{2}+0.5w_{t-1}^{2}$} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .004, .031, .069 & .008,
.023, .038 & .007, .040, .084 & .116, .312, .451 & .004, .010, .015 & .011,
.063, .107 \\ \hline
$\mathcal{AP}^{w}$ & .003, .039, .081 & .007, .033, .088 & .006, .030, .076
& .130, .355, .524 & .008, .037, .081 & .009, .043, .082 \\
$\mathcal{AP}^{dw}$ & .008, .045, .100 & .014, .036, .079 & .014, .066, .133
& .250, .461, .587 & .010, .030, .080 & .029, .098, .171 \\ \hline
$CvM^{w}$ & .007, .058, .118 & .007, .035, .095 & .004, .045, .097 & .291,
.528, .642 & .003, .047, .096 & .007, .047, .095 \\
$CvM^{dw}$ & .013, .059, .108 & .029, .048, .083 & .012, .058, .127 & .242,
.501, .648 & .009, .037, .080 & .020, .075, .132 \\
$CvM^{brw}$ & .013, .062, .108 & .006, .019, .052 & .009, .045, .086 & .299,
.509, .634 & .010, .039, .094 & .006, .055, .111 \\ \hline
\multicolumn{7}{c}{} \\
\multicolumn{7}{c}{MA(2) error: $e_{t}=\nu _{t}+0.5\nu _{t-1}+0.25\nu _{t-2}$%
} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .993, .998, 1.00 & .841,
.935, .962 & .006, .060, .114 & .677, .927, .982 & .707, .834, .868 & .992,
.993, .993 \\ \hline
$\mathcal{AP}^{w}$ & 1.00, 1.00, 1.00 & .919, .983, .997 & .021, .095, .166
& .928, .977, .991 & .982, .999, .999 & .992, .994, .995 \\
$\mathcal{AP}^{dw}$ & .969, .999, 1.00 & .592, .818, .921 & .016, .073, .144
& .911, .978, .989 & .801, .952, .982 & .982, .986, .989 \\ \hline
$CvM^{w}$ & 1.00, 1.00, 1.00 & .974, .994, .995 & .026, .103, .189 & .980,
.993, .999 & .998, 1.00, 1.00 & .975, .979, .984 \\
$CvM^{dw}$ & .999, 1.00, 1.00 & .769, .924, .968 & .019, .086, .189 & .951,
.996, .999 & .903, .979, .994 & .983, .989, .991 \\
$CvM^{brw}$ & 1.00, 1.00, 1.00 & .901, .946, .975 & .024, .088, .144 & .973,
.997, .999 & .939, .978, .990 & .616, .952, .981 \\ \hline
\multicolumn{7}{c}{} \\
\multicolumn{7}{c}{AR(1) error: $e_{t}=0.7e_{t-1}+\nu _{t}$} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .903, .979, .990 & .676,
.774, .812 & .041, .217, .355 & .714, .954, .984 & .156, .271, .334 & 1.00,
1.00, 1.00 \\ \hline
$\mathcal{AP}^{w}$ & 1.00, 1.00, 1.00 & .830, .958, .989 & .072, .217, .320
& .936, .984, .996 & .970, .995, 1.00 & .997, .998, .999 \\
$\mathcal{AP}^{dw}$ & .971, .996, 1.00 & .285, .535, .709 & .150, .296, .386
& .940, .989, .994 & .566, .788, .899 & .997, .999, .999 \\ \hline
$CvM^{w}$ & 1.00, 1.00, 1.00 & .888, .979, .988 & .136, .328, .464 & .953,
.996, .998 & .974, .999, 1.00 & .987, .989, .991 \\
$CvM^{dw}$ & .999, 1.00, 1.00 & .341, .572, .718 & .136, .341, .465 & .935,
.991, .999 & .680, .849, .912 & .984, .987, .988 \\
$CvM^{brw}$ & 1.00, 1.00, 1.00 & .575, .706, .792 & .127, .289, .419 & .948,
.991, .996 & .725, .860, .903 & .816, .983, .995 \\ \hline
\end{tabular}
}
\end{center}
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection frequencies of $\hat{\mathcal{T}}_{n}(\mathcal{L}%
_{n}^{*}) $, $\mathcal{AP}$, and $CvM$ (Scenarios \#1--\#6, $n = 500$) }
\label{table:supp_mat_AP_CvM_scenario123456_n500}
\begin{center}
{\fontsize{8.8pt}{15.5pt}\selectfont
\begin{tabular}{l||c|c|c|c|c|c}
\multicolumn{7}{c}{IID error: $e_{t}=\nu _{t}$} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .007, .047, .096 & .010,
.036, .077 & .004, .045, .094 & .462, .803, .894 & .005, .033, .083 & .016,
.053, .096 \\ \hline
$\mathcal{AP}^{w}$ & .006, .043, .088 & .008, .059, .117 & .008, .045, .095
& .440, .729, .828 & .008, .039, .079 & .014, .048, .096 \\
$\mathcal{AP}^{dw}$ & .012, .055, .118 & .016, .057, .111 & .010, .069, .139
& .561, .780, .870 & .007, .055, .120 & .025, .069, .128 \\ \hline
$CvM^{w}$ & .011, .048, .098 & .014, .081, .150 & .013, .052, .097 & .759,
.910, .944 & .009, .050, .108 & .014, .052, .102 \\
$CvM^{dw}$ & .010, .051, .102 & .014, .072, .124 & .012, .059, .132 & .710,
.882, .939 & .009, .053, .103 & .016, .072, .141 \\
$CvM^{brw}$ & .023, .064, .115 & .027, .070, .134 & .014, .057, .099 & .712,
.879, .934 & .009, .052, .103 & .005, .037, .087 \\ \hline
\multicolumn{7}{c}{} \\
\multicolumn{7}{c}{GARCH(1,1) error: $e_{t}=\nu _{t}w_{t}$ with $%
w_{t}^{2}=1+0.2e_{t-1}^{2}+0.5w_{t-1}^{2}$} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .001, .031, .063 & .017,
.029, .042 & .008, .032, .078 & .283, .588, .733 & .003, .005, .006 & .012,
.046, .089 \\ \hline
$\mathcal{AP}^{w}$ & .014, .053, .094 & .004, .037, .090 & .011, .048, .106
& .347, .609, .745 & .002, .030, .090 & .008, .052, .118 \\
$\mathcal{AP}^{dw}$ & .004, .049, .107 & .008, .021, .064 & .016, .066, .142
& .444, .687, .793 & .006, .030, .066 & .021, .080, .137 \\ \hline
$CvM^{w}$ & .009, .056, .107 & .006, .026, .084 & .010, .048, .092 & .579,
.813, .887 & .002, .045, .098 & .007, .044, .092 \\
$CvM^{dw}$ & .015, .066, .115 & .026, .038, .075 & .011, .051, .104 & .550,
.802, .881 & .013, .052, .111 & .026, .072, .143 \\
$CvM^{brw}$ & .013, .052, .105 & .003, .021, .056 & .011, .063, .116 & .583,
.802, .882 & .008, .029, .065 & .010, .055, .099 \\ \hline
\multicolumn{7}{c}{} \\
\multicolumn{7}{c}{MA(2) error: $e_{t}=\nu _{t}+0.5\nu _{t-1}+0.25\nu _{t-2}$%
} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & 1.00, 1.00, 1.00 & .932,
.965, .976 & .019, .086, .151 & .972, .999, .999 & .798, .874, .908 & 1.00,
1.00, 1.00 \\ \hline
$\mathcal{AP}^{w}$ & 1.00, 1.00, 1.00 & .982, .996, .999 & .040, .141, .234
& .999, 1.00, 1.00 & .996, .998, 1.00 & .998, .998, .999 \\
$\mathcal{AP}^{dw}$ & 1.00, 1.00, 1.00 & .767, .921, .970 & .025, .100, .189
& 1.00, 1.00, 1.00 & .920, .972, .990 & .997, .998, .998 \\ \hline
$CvM^{w}$ & 1.00, 1.00, 1.00 & .997, 1.00, 1.00 & .028, .150, .275 & 1.00,
1.00, 1.00 & .998, 1.00, 1.00 & .995, .996, .996 \\
$CvM^{dw}$ & 1.00, 1.00, 1.00 & .884, .966, .990 & .032, .144, .250 & 1.00,
1.00, 1.00 & .959, .994, .995 & .995, .998, .998 \\
$CvM^{brw}$ & 1.00, 1.00, 1.00 & .955, .980, .989 & .031, .131, .232 & 1.00,
1.00, 1.00 & .971, .988, .992 & .901, .993, .997 \\ \hline
\multicolumn{7}{c}{} \\
\multicolumn{7}{c}{AR(1) error: $e_{t}=0.7e_{t-1}+\nu _{t}$} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .998, 1.00, 1.00 & .715,
.819, .860 & .181, .512, .636 & .991, 1.00, 1.00 & .111, .176, .230 & 1.00,
1.00, 1.00 \\ \hline
$\mathcal{AP}^{w}$ & 1.00, 1.00, 1.00 & .891, .964, .984 & .162, .371, .501
& 1.00, 1.00, 1.00 & .986, 1.00, 1.00 & .997, .998, .998 \\
$\mathcal{AP}^{dw}$ & 1.00, 1.00, 1.00 & .338, .585, .743 & .270, .467, .559
& 1.00, 1.00, 1.00 & .626, .805, .896 & .998, .998, .999 \\ \hline
$CvM^{w}$ & 1.00, 1.00, 1.00 & .919, .985, .993 & .333, .597, .709 & 1.00,
1.00, 1.00 & .986, .999, 1.00 & .995, .995, .997 \\
$CvM^{dw}$ & 1.00, 1.00, 1.00 & .393, .630, .781 & .325, .592, .700 & .999,
1.00, 1.00 & .700, .852, .918 & .999, 1.00, 1.00 \\
$CvM^{brw}$ & 1.00, 1.00, 1.00 & .584, .730, .803 & .365, .603, .713 & 1.00,
1.00, 1.00 & .794, .876, .911 & .945, .999, 1.00 \\ \hline
\end{tabular}
}
\end{center}
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection frequencies of $\hat{\mathcal{T}}_{n}(\mathcal{L}%
_{n}^{*}) $, $\mathcal{AP}$, and $CvM$ (Scenarios \#1--\#6, $n = 1000$) }
\label{table:supp_mat_AP_CvM_scenario123456_n1000}
\begin{center}
{\fontsize{8.8pt}{15.5pt}\selectfont
\begin{tabular}{l||c|c|c|c|c|c}
\multicolumn{7}{c}{IID error: $e_{t}=\nu _{t}$} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .012, .050, .109 & .008,
.056, .103 & .006, .057, .097 & .929, .996, .998 & .008, .038, .077 & .011,
.047, .101 \\ \hline
$\mathcal{AP}^{w}$ & .008, .043, .085 & .015, .067, .134 & .007, .041, .089
& .822, .946, .977 & .010, .056, .106 & .012, .050, .110 \\
$\mathcal{AP}^{dw}$ & .012, .047, .095 & .011, .057, .123 & .016, .068, .124
& .870, .964, .984 & .013, .061, .123 & .006, .052, .115 \\ \hline
$CvM^{w}$ & .008, .053, .104 & .022, .074, .138 & .011, .050, .093 & .974,
.996, .998 & .012, .055, .111 & .014, .057, .102 \\
$CvM^{dw}$ & .008, .060, .108 & .016, .063, .106 & .010, .049, .102 & .974,
.991, .993 & .015, .058, .107 & .013, .057, .103 \\
$CvM^{brw}$ & .016, .057, .105 & .023, .072, .128 & .016, .051, .102 & .972,
.997, .999 & .017, .063, .104 & .003, .044, .081 \\ \hline
\multicolumn{7}{c}{} \\
\multicolumn{7}{c}{GARCH(1,1) error: $e_{t}=\nu _{t}w_{t}$ with $%
w_{t}^{2}=1+0.2e_{t-1}^{2}+0.5w_{t-1}^{2}$} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .006, .032, .071 & .014,
.026, .031 & .008, .033, .085 & .741, .925, .961 & .002, .002, .002 & .006,
.051, .102 \\ \hline
$\mathcal{AP}^{w}$ & .007, .045, .098 & .001, .028, .079 & .006, .057, .091
& .712, .907, .959 & .009, .034, .074 & .014, .052, .097 \\
$\mathcal{AP}^{dw}$ & .010, .058, .117 & .003, .008, .028 & .010, .054, .110
& .728, .900, .943 & .002, .032, .078 & .010, .058, .118 \\ \hline
$CvM^{w}$ & .007, .054, .101 & .002, .030, .092 & .009, .061, .102 & .901,
.980, .991 & .007, .043, .096 & .011, .018, .018 \\
$CvM^{dw}$ & .010, .060, .116 & .004, .014, .028 & .008, .056, .105 & .880,
.973, .993 & .006, .032, .065 & .049, .065, .073 \\
$CvM^{brw}$ & .015, .054, .104 & .002, .014, .051 & .014, .065, .109 & .911,
.967, .985 & .006, .035, .078 & .096, .122, .116 \\ \hline
\multicolumn{7}{c}{} \\
\multicolumn{7}{c}{MA(2) error: $e_{t}=\nu _{t}+0.5\nu _{t-1}+0.25\nu _{t-2}$%
} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & 1.00, 1.00, 1.00 & .968,
.982, .986 & .063, .184, .257 & 1.00, 1.00, 1.00 & .900, .934, .952 & 1.00,
1.00, 1.00 \\ \hline
$\mathcal{AP}^{w}$ & 1.00, 1.00, 1.00 & .995, .996, .998 & .083, .226, .350
& 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{AP}^{dw}$ & 1.00, 1.00, 1.00 & .898, .967, .989 & .053, .177, .296
& 1.00, 1.00, 1.00 & .971, .994, .997 & .999, 1.00, 1.00 \\ \hline
$CvM^{w}$ & 1.00, 1.00, 1.00 & .999, .999, .999 & .095, .287, .442 & 1.00,
1.00, 1.00 & 1.00, 1.00, 1.00 & .998, .999, .999 \\
$CvM^{dw}$ & 1.00, 1.00, 1.00 & .974, .994, .997 & .068, .295, .471 & 1.00,
1.00, 1.00 & .986, .997, 1.00 & .998, .998, .998 \\
$CvM^{brw}$ & 1.00, 1.00, 1.00 & .991, .994, .995 & .092, .270, .438 & 1.00,
1.00, 1.00 & .998, .998, .999 & .988, 1.00, 1.00 \\ \hline
\multicolumn{7}{c}{} \\
\multicolumn{7}{c}{AR(1) error: $e_{t}=0.7e_{t-1}+\nu _{t}$} \\ \hline
& \#1. Simple & \#2. Bilin & \#3. AR2/AR2 & \#4. AR2/AR1 & \#5. GARCH/wo &
\#6. GARCH/w \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & 1.00, 1.00, 1.00 & .723,
.823, .864 & .599, .847, .922 & 1.00, 1.00, 1.00 & .066, .126, .158 & 1.00,
1.00, 1.00 \\ \hline
$\mathcal{AP}^{w}$ & 1.00, 1.00, 1.00 & .923, .981, .992 & .369, .667, .792
& 1.00, 1.00, 1.00 & .976, .998, 1.00 & 1.00, 1.00, 1.00 \\
$\mathcal{AP}^{dw}$ & 1.00, 1.00, 1.00 & .408, .661, .774 & .459, .681, .787
& 1.00, 1.00, 1.00 & .677, .832, .901 & .998, .999, .999 \\ \hline
$CvM^{w}$ & 1.00, 1.00, 1.00 & .941, .984, .991 & .690, .886, .935 & 1.00,
1.00, 1.00 & .982, .998, 1.00 & .999, 1.00, 1.00 \\
$CvM^{dw}$ & 1.00, 1.00, 1.00 & .474, .697, .810 & .688, .876, .923 & 1.00,
1.00, 1.00 & .750, .877, .929 & .998, .999, .999 \\
$CvM^{brw}$ & 1.00, 1.00, 1.00 & .615, .761, .833 & .700, .878, .913 & 1.00,
1.00, 1.00 & .819, .882, .927 & .989, 1.00, 1.00 \\ \hline
\end{tabular}
}
\end{center}
\end{table}

\clearpage

\begin{table}[th]
\caption{Rejection frequencies of $\hat{\mathcal{T}}_{n}(\mathcal{L}%
_{n}^{*}) $, $\mathcal{AP}$, and $CvM$ (Scenarios \#7--\#9) }
\label{table:supp_mat_AP_CvM_scenario789}
\begin{center}
{\fontsize{8.8pt}{17pt} \selectfont
\begin{tabular}{l||c|c|c||c|c|c}
\hline
& \multicolumn{3}{c||}{$n=100$} & \multicolumn{3}{c}{$n=250$} \\ \hline
& \#7. MA(6) & \#8. MA(12) & \#9. MA(24) & \#7. MA(6) & \#8. MA(12) & \#9.
MA(24) \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .016, .084, .139 & .013,
.067, .117 & .017, .065, .118 & .155, .289, .352 & .024, .134, .244 & .012,
.042, .088 \\ \hline
$\mathcal{AP}^{w}$ & .036, .104, .163 & .013, .062, .133 & .014, .054, .107
& .050, .149, .233 & .020, .087, .148 & .006, .050, .110 \\
$\mathcal{AP}^{dw}$ & .009, .063, .139 & .026, .090, .171 & .022, .082, .171
& .020, .089, .162 & .017, .077, .139 & .015, .085, .149 \\ \hline
$CvM^{w}$ & .006, .065, .123 & .008, .048, .110 & .010, .057, .124 & .024,
.096, .163 & .010, .079, .144 & .010, .065, .121 \\
$CvM^{dw}$ & .040, .098, .171 & .034, .110, .179 & .029, .098, .186 & .026,
.080, .142 & .025, .087, .155 & .022, .088, .143 \\
$CvM^{brw}$ & .049, .112, .193 & .056, .125, .189 & .038, .091, .143 & .025,
.080, .161 & .020, .094, .156 & .017, .083, .137 \\ \hline
\multicolumn{7}{c}{} \\ \hline
& \multicolumn{3}{c||}{$n=500$} & \multicolumn{3}{c}{$n=1000$} \\ \hline
& \#7. MA(6) & \#8. MA(12) & \#9. MA(24) & \#7. MA(6) & \#8. MA(12) & \#9.
MA(24) \\ \hline
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\%
& 1\%, \ 5\%, 10\% & 1\%, \ 5\%, 10\% \\ \hline
$\mathcal{\hat{T}}_{n}(\mathcal{L}_{n}^{\ast })$ & .710, .812, .826 & .371,
.673, .770 & .024, .097, .192 & .999, 1.00, 1.00 & .983, .997, .997 & .578,
.833, .918 \\ \hline
$\mathcal{AP}^{w}$ & .096, .226, .311 & .025, .095, .168 & .015, .063, .129
& .162, .356, .485 & .037, .120, .195 & .012, .067, .119 \\
$\mathcal{AP}^{dw}$ & .022, .130, .226 & .015, .069, .142 & .018, .079, .158
& .060, .199, .332 & .011, .066, .152 & .008, .049, .115 \\ \hline
$CvM^{w}$ & .027, .112, .207 & .020, .086, .133 & .011, .060, .135 & .034,
.158, .344 & .021, .078, .151 & .020, .075, .146 \\
$CvM^{dw}$ & .014, .087, .175 & .026, .092, .161 & .024, .071, .133 & .038,
.160, .320 & .017, .083, .166 & .028, .079, .144 \\
$CvM^{brw}$ & .033, .101, .188 & .020, .068, .128 & .019, .078, .142 & .037,
.150, .321 & .026, .081, .152 & .027, .078, .138 \\ \hline
\end{tabular}
}
\end{center}
\par
{\fontsize{10pt}{13pt} \selectfont
\#7: Remote MA(6) $y_{t} = e_{t} + 0.25 e_{t-6}$ with a mean filter. \#8:
Remote MA(12) $y_{t} = e_{t} + 0.25 e_{t-12}$ with a mean filter. \#9:
Remote MA(24) $y_{t} = e_{t} + 0.25 e_{t-24}$ with a mean filter. For each
scenario, $e_{t} \overset{i.i.d.}{\sim} N(0,1)$. $\mathcal{\hat{T}}_{n}(%
\mathcal{L}_{n}^{\ast })$ is the max-correlation test based on Shao's (%
\citeyear{Shao2011_JoE}) dependent wild bootstrap, with automatic lag $%
\mathcal{L}_{n}^{\ast}$. $\mathcal{AP}$ is Andrews and Ploberger's (%
\citeyear{AndrewsPloberger1996}) sup-LM test. $CvM$ is the Cram\'{e}r-von
Mises test. The $\mathcal{AP}$ and $CvM$ tests use $\mathcal{L}_{n}$ $=$ $n$
$-$ $1$ lags. Superscript \textquotedblleft $w$\textquotedblright\ means the
wild bootstrap; \textquotedblleft $dw$\textquotedblright\ means the
dependent wild bootstrap; \textquotedblleft $brw$\textquotedblright means
Zhu and Li's (\citeyear{ZhuLi2015}) block-wise random weighting bootstrap. }
\end{table}

\clearpage

\begin{figure}[h!]
\caption{Empirical Size and Size-Adjusted Power of $\hat{\mathcal{T}}^{dw} (%
\mathcal{L}_{n}^{*})$ with $\protect\alpha = 0.05$ }
\label{fig:size_power_maxcorr_automatic_lag}
\begin{center}
\captionsetup[subfigure]{labelformat=empty}
\begin{minipage}[b]{72mm}
      \centering
      \includegraphics[width = 72mm, height = 55mm]{fig_case1_n100_050_v3.pdf}
      \vspace{0.0cm}
      \subcaption{Case 1, $n = 100$}
      \end{minipage}
\quad \quad \quad
\begin{minipage}[b]{72mm}
      \centering
      \includegraphics[width = 72mm, height = 55mm]{fig_case1_n500_050_v3.pdf}
      \vspace{0.0cm}
      \subcaption{Case 1, $n = 500$}
      \end{minipage} \\[0pt]
\par
\vspace{0.3cm}
\par
\begin{minipage}[b]{72mm}
      \centering
      \includegraphics[width = 72mm, height = 55mm]{fig_case2_n100_050_v3.pdf}
      \vspace{0.0cm}
      \subcaption{Case 2, $n = 100$}
      \end{minipage}
\quad \quad \quad
\begin{minipage}[b]{72mm}
      \centering
      \includegraphics[width = 72mm, height = 55mm]{fig_case2_n500_050_v3.pdf}
      \vspace{0.0cm}
      \subcaption{Case 2, $n = 500$}
      \end{minipage} \\[0pt]
\end{center}
\par
{\fontsize{9.5pt}{13pt} \selectfont
We plot empirical size and size-adjusted power of the bootstrapped
max-correlation test with automatic lag selection given nominal size 5\%. In
Case 1, the empirical size and empirical quantiles for size adjustment are
computed under Scenario \#1 (iid $y_{t}$ and mean filter) with i.i.d. error;
then the size-adjusted power is computed under Scenario \#4 (AR(2) $y_{t}$
and AR(1) filter) with i.i.d. error. In Case 2, the empirical size and
empirical quantiles for size adjustment are computed under Scenario \#5
(GARCH $y_{t}$ and no filter) with i.i.d. error; then the size-adjusted
power is computed under Scenario \#5 with MA(2) error. The tuning parameter
that affects the penalty term $\mathcal{P}_{n} (\mathcal{L}) $ is $q \in \{
1.50, 1.75, \dots, 4.50 \}$. The largest possible lag length is $\bar{%
\mathcal{L}}_{n} = [10 \sqrt{n} / (\ln n)]$, which implies that $\bar{%
\mathcal{L}}_{100} = 21$ and $\bar{\mathcal{L}}_{500} = 35$. $J = 1000$
Monte Carlo samples and $M = 500$ bootstrap samples are generated. }
\end{figure}

\clearpage

\singlespacing
\bibliographystyle{econometrica}
\bibliography{ref_max_corr}

\end{document}
