# Adaptive Testing for Alphas in Conditional Factor Models with High
Dimensional Assets

Huifang Ma
Nankai University

Long Feng
Nankai University

Zhaojun Wang
Nankai University

###### Abstract

This paper focuses on testing for the presence of alpha in time-varying factor
pricing models, specifically when the number of securities $N$ is larger than
the time dimension of the return series $T$. We introduce a maximum-type test
that performs well in scenarios where the alternative hypothesis is sparse. We
establish the limit null distribution of the proposed maximum-type test
statistic and demonstrate its asymptotic independence from the sum-type test
statistics proposed by Ma et al. (2020). Additionally, we propose an adaptive
test by combining the maximum-type test and sum-type test, and we show its
advantages under various alternative hypotheses through simulation studies and
two real data applications.

Keywords: Alpha tests, Conditional factor model, High dimensionality, Sparse
alternatives, Spline estimator

##  1 Introduction

Tests of alpha have attracted much attention in the econometrics literature.
In the traditional linear pricing factor models, e.g. CAPM (Sharpe, 1964;
Lintner, 1965) and the Fama-French three-factor model (abbreviated as FF model
hereafter) (Fama and French, 1993, 2015), Gibbons et al. (1989) proposed an
exact multivariate F-test under the joint normality assumption. Several
extended methods include MacKinlay and Richardson (1991), Zhou (1993) and
Beaulieu et al. (2007), etc. However, their application has been confined to a
relatively small number of portfolios. With the development of modern
financial markets, thousands of securities are traded. So the assumption of
fixed dimension is not appropriate. Consequently, recent efforts have focused
on developing tests that allow the number of securities $N$ are larger than
the time periods $T$. For instance, Pesaran and Yamagata (2012, 2017) propose
a sum-type test statistic by replacing the sample covariance matrix with
identity matrix in traditional F-test, which performs well under the dense
alternatives. For sparse alternatives, Feng et al. (2023) proposed a max-type
test statistics. Yu et al. (2023) employed the thresholding covariance
estimator of Fan et al. (2013) and proposed a novel Wald-type test. They also
proved the asymptotic independence between the new Wald-type test and the
maximum-type test and proposed a new Fisher combination test which performs
very well under neither dense nor sparse alternatives.

Although these tests have addressed the limitation of $T>N$, they still
require the factor loadings be constant over time. This assumption can be
quite restrictive in empirical finance. Much empirical evidence indicates that
the factor loadings in CAPM and the FF model vary substantially over time even
at the portfolio level (Lewellen and Nagel, 2006; Ang and Chen, 2007). As a
result, the aforementioned tests can lead to inaccurate conclusions when the
factor loadings are time-varying. To remove the limiting of time-invariant
factor loadings, Li and Yang (2011) and Ang and Kristensen (2012) proposed
nonparametric Wald-type tests under the case that $N$ is fixed and $T$ goes to
infinity, and Gagliardini et al. (2016) developed an econometric methodology
under high dimensional case. Furthermore, Ma et al. (2020) proposed a sum-type
statistic based on the residuals obtained from the null model, which is
asymptotically normal-distributed.

The above methods are all sum-of-squares types, generally having good power
performance against dense alternatives. However, inefficient market pricing is
more likely to occur in exceptional assets rather than systematic mispricing
of the entire market, i.e. $\bm{\alpha}$ has few nonzero elements with a large
$||\bm{\alpha}||_{\infty}$. In this case, it is problematic to use the sum-of-
squares type tests, since summarizing information through averaging will
weaken the signals carried in the few securities with strong signals.
Therefore, it is desirable to develop a test that has good power against
sparse alternatives.

Recently, the maximum-type tests are wildly used in high dimensional testing
problems. Cai et al. (2014) proposed three-types maximum-type tests for high
dimensional two-sample location problems. Chang et al. (2017) consider testing
for high-dimensional white noise using maximum cross-correlations. In the
traditional linear factor pricing models, many studies also show that the
maximum-type test statistic performs very well under the sparse alternatives,
such as Gungor and Luger (2013); Feng et al. (2023); Yu et al. (2023). So, we
construct an maximum-type test statistic for the time-varying factor pricing
models and establish its theoretical properties. Theoretical results and
simulation studies show the proposed maximum-type test statistic also has good
performance under sparse alternatives.

It’s worth notable that the underlying truth is usually unknown in real
applications, as whether it is dense or sparse depends on the properties of
the involved securities. Recently, many literatures showed that the maximum-
type test statistic is asymptotically independent with the sum-type test
statistic and proposed a corresponding combination test in many high
dimensional problems, such as He et al. (2021), Feng et al. (2023) for high
dimensional mean testing problems, Feng et al. (2022) for cross-sectional
independence test in high dimensinal panel data models, Yu et al. (2022) for
testing of high dimensional covariance matrix, Wang and Feng (2023) for high
dimensional change point inference. In the time-varying factor pricing model,
we also demonstrate the asymptotic independence between the maximum-type
statistic and the sum-type statistic proposed in Ma et al. (2020). Then, we
construct an adaptive testing procedure by combining the information from
these two statistics, which would yield high power against various
alternatives. We demonstrate the advantages of the proposed tests over
existing methods through extensive Monte Carlo experiments and two empirical
applications.

The rest of the paper is organized as follows. In Section 2, we introduce the
maximum-type statistic and establish its theoretical properties. In Section 3,
we integrate the proposed maximum-type test with an existing sum-type test to
obtain an adaptive test. Monte Carlo experiment results are presented in
Section 4 to evaluate the finite sample performance of the proposed tests in
comparison with the main competitors. Two empirical applications to the
security return data from the Chinese and the U.S. financial markets are
presented in Section 5. Finally, we conclude the paper with some discussions
in Section 6 and relegate the technical proofs to the Appendix.

Finally we introduce some notation. For any vector
$\bm{z}=(z_{1},...,z_{m})^{\prime}\in\mathbb{R}^{m}$, let
$||\bm{z}||=(\sum_{i=1}^{m}z_{i}^{2})^{1/2}$ and
$||\bm{z}||_{\infty}=\max_{1\leq i\leq m}|z_{i}|$. Let $\bm{1}_{m}$ be the
$m\times 1$ vector of ones. For any positive numbers $a_{n}$ and $b_{n}$, let
$a_{n}\ll b_{n}$ denote $a_{n}b_{n}^{-1}=o(1)$, $a_{n}\sim b_{n}$ denote
$\lim_{n\to\infty}a_{n}b_{n}^{-1}=1$, and $a_{n}\asymp b_{n}$ denote
$\lim_{n\to\infty}a_{n}b_{n}^{-1}=c$ for some finite positive constant $c$.
For an $m\times n$ matrix $\bm{A}=(a_{ij})$, let $\text{Tr}(\bm{A})$ denote
the trace of $\bm{A}$,
$P_{\bm{A}}=\bm{A}(\bm{A}^{\prime}\bm{A})^{-1}\bm{A}^{\prime}$ and
$M_{\bm{A}}=\bm{I}_{m}-P_{\bm{A}}$, where $\bm{I}_{m}$ is the $m\times m$
identity matrix. Moreover, denote
$||\bm{A}||=\max_{\bm{a}\in\mathbb{R}^{n},||\bm{a}||=1}||\bm{Aa}||$ and
$||\bm{A}||_{\infty}=\max_{1\leq i\leq m}\sum_{j=1}^{n}|a_{ij}|$. For any
symmetric matrix $\bm{A}\in\mathbb{R}^{n\times n}$, let
$\lambda_{\min}(\bm{A})$ and $\lambda_{\max}(\bm{A})$ denote the smallest and
largest eigenvalues of $\bm{A}$, respectively. $(N,T)\to\infty$ denotes that
$N$ and $T$ go to infinity jointly. The operators $\stackrel{{\scriptstyle
d}}{{\to}}$ and $\stackrel{{\scriptstyle p}}{{\to}}$ denote convergence in
distribution and in probability, respectively.

##  2 The Maximum-type Test

###  2.1 Econometric Framework and B-spline Approximation

We consider the following conditional time-varying factor model under high-
dimensional case,

$R_{it}=\alpha_{it}+\bm{\beta}_{it}^{\prime}\bm{f}_{t}+e_{it}=\alpha_{it}+\sum_{j=1}^{d}\beta_{ijt}f_{jt}+e_{it},\quad
i=1,...,N;t=1,...,T,$ (1)

where $R_{it}$ denotes the excess returns of the $i$-th asset at time $t$,
$\alpha_{it}$ is the conditional alpha of asset $i$ at time $t$,
$\bm{\beta}_{it}=(\beta_{i1t},...,\beta_{idt})^{\prime}$ is a $d\times 1$
vector of time-variant factor loadings,
$\bm{f}_{t}=(f_{it},...,f_{dt})^{\prime}$ stands for the risk premium on
$d$-dimensional tradable systematic risks at time $t$, and $e_{it}$ is the
idiosyncratic error term with
$\text{Cov}(\bm{e}_{t})=\bm{\Sigma}=(\sigma_{ij})_{N\times N}$, where
$\bm{e}_{t}=(e_{1t},...,e_{Nt})^{\prime}\in\mathbb{R}^{N\times 1}$.

To identify the parameters, we follow Li and Yang (2011) and others in
imposing a smoothness condition. That is we assume that the sequence of alphas
and betas are generated from two smoothing functions of time such that
$\alpha_{it}=\alpha_{i}(t/T)$ and $\beta_{ijt}=\beta_{ij}(t/T)$. Accordingly,
we can rewrite equation (1) as

$R_{it}=\delta_{i}^{0}+\delta_{i}(t/T)+\sum_{j=1}^{d}\beta_{ij}(t/T)f_{jt}+e_{it},$
(2)

where $\delta_{i}^{0}=T^{-1}\sum_{t=1}^{T}\alpha_{it}$ and
$\delta_{i}(t/T)=\alpha_{i}(t/T)-T^{-1}\sum_{i=1}^{T}\alpha_{i}(t/T)$. To test
whether the average pricing error for $N$ assets across all time periods is
equal to zero, we then focus on testing

$H_{0}:\delta_{i}^{0}=0\text{ for all }i=1,...,N\quad\text{v.s.}\quad
H_{1}:\delta_{i}^{0}\neq 0\text{ for some }i=1,...,N.$ (3)

For testing hypothesis in equation (3), we employ the polynomial spline
approach to estimating the unknown parameters $\delta_{i}(t/T)$ and
$\beta_{ij}(t/T)$ under $H_{0}$. Consider $p$ interior knots on $[0,1]$,
$0=\ell_{0}<\ell_{1}<...<\ell_{p}<\ell_{p+1}=1$ that satisfy

$\max_{0\leq i\leq p}|\ell_{i+1}-\ell_{i}|/\min_{0\leq l\leq
p}|\ell_{i+1}-\ell_{i}|\leq c$

for some finite positive constant $c$, where $p=p(N,T)\to\infty$ as
$(N,T)\to\infty$. For any $t$, define its location as $l(t)$ satisfying
$\ell_{l(t)}\leq t/T<\ell_{l(t)+1}$. Consider the space of polynomial splines
of order $q$ and denote the normalized B spline basis of this space as
$\bm{B}(t/T)=\\{B_{1}(t/T),...,B_{L}(t/T)\\}^{\prime}$, where $L=p+q$ (de
Boor, 1978). To estimate $\delta_{i}(\cdot)$, we consider the centered spline
basis functions,
$\widetilde{B}_{l}(t/T)=B_{l}(t/T)-T^{-1}\sum_{t=1}^{T}B_{l}(t/T)$, and denote
$\widetilde{\bm{B}}(t/T)=\\{\widetilde{B}_{1}(t/T),...,\widetilde{B}_{L}(t/T)\\}^{\prime}$.
Then, the unknown functions $\delta_{i}(\cdot)$ and $\beta_{ij}(\cdot)$ can be
well approximated by the B-spline functions (Schumaker, 2007) such that

$\delta_{i}(t/T)\approx\bm{\lambda}_{i0}^{\prime}\widetilde{\bm{B}}(t/T)\quad\text{and}\quad\beta_{ij}(t/T)\approx\bm{\lambda}_{ij}^{\prime}\bm{B}(t/T),$

where $\bm{\lambda}_{ij}\in\mathbb{R}^{L\times 1}$, $j=0,1,...,d$ are the
coefficients of the B-spline functions.

Denote $\bm{R}_{i.}=(R_{i1},...,R_{iT})^{\prime}\in\mathbb{R}^{T\times 1}$,
$\bm{e}_{i.}=(e_{i1},...,e_{iT})^{\prime}\in\mathbb{R}^{T\times 1}$,
$\bm{\lambda}_{i}=(\bm{\lambda}_{ij}^{\prime},0\leq j\leq
d)^{\prime}\in\mathbb{R}^{(1+d)L\times 1}$,
$\bm{Z}_{t}=\\{\widetilde{\bm{B}}(t/T)^{\prime},f_{jt}\bm{B}(t/T)^{\prime},1\leq
j\leq d\\}^{\prime}\in\mathbb{R}^{(1+d)L\times 1}$ and
$\bm{Z}=(\bm{Z}_{1},...,\bm{Z}_{T})^{\prime}\in\mathbb{R}^{T\times(1+d)L}$. We
then choose estimators
$\widehat{\bm{\lambda}}_{i}=(\widehat{\bm{\lambda}}_{ij}^{\prime},0\leq j\leq
d)^{\prime}$ to minimize the following sum of squared residuals:

$\widehat{\bm{\lambda}}_{i}=\arg\min_{\bm{\lambda}_{i}}||\bm{R}_{i.}-\bm{Z}\bm{\lambda}_{i}||^{2}=(\bm{Z}^{\prime}\bm{Z})^{-1}\bm{Z}^{\prime}\bm{R}_{i.}.$

Consequently, the resulting residuals are

$\widehat{\bm{e}}_{i.}=\bm{R}_{i.}-\bm{Z}\widehat{\bm{\lambda}}_{i}=\bm{M_{Z}R}_{i.}.$
(4)

###  2.2 The Proposed Test

Under $H_{0}$, it can be shown that
$\delta_{i}(t/T)-\widehat{\delta}_{i}(t/T)\stackrel{{\scriptstyle p}}{{\to}}0$
and $\beta_{ij}(t/T)-\widehat{\beta}_{ij}(t/T)\stackrel{{\scriptstyle
p}}{{\to}}0$. This motivates us to consider the max-type test, with the test
statistic constructed as,

$M_{NT}=\max_{1\leq i\leq N}t_{i}^{2},$ (5)

where

$t_{i}^{2}=T^{-1}\widehat{\sigma}_{ii}^{-1}(\widehat{\bm{e}}_{i.}^{\prime}\bm{1}_{T})^{2},$
(6)

with
$\widehat{\sigma}_{ij}=\widehat{\bm{e}}_{i.}^{\prime}\widehat{\bm{e}}_{j.}/(T-d-1)$.

We will establish that when $(N,T)\to\infty$,
$M_{NT}-2\log(N)+\log\\{\log(N)\\}$ has a type I extreme value distribution.
To proceed, we first introduce some assumptions.

Let $\mathcal{H}_{r}$ denote the collection of all functions on $[0,1]$ such
that the $m$-th order derivative satisfies the Hölder condition of order $n$
with $r=m+n$, i.e. there exists a constant $C\in(0,\infty)$ such that for each
$f\in\mathcal{H}_{r}$,

$|f^{(m)}(x_{1})-f^{(m)}(x_{2})|\leq C|x_{1}-x_{2}|^{n}$

for any $0\leq x_{1},x_{2}\leq 1$. Let $\mathcal{F}_{NT,t}$ denote the
$\sigma$-algebra generated from
$\\{\bm{f},\\{e_{it},e_{i,t-1},...\\}_{i=1}^{N}\\}$, where
$\bm{f}=(\bm{f}_{1}^{\prime},...,\bm{f}_{T}^{\prime})^{\prime}$. Let
$\bm{e}_{-t}=\\{e_{i1},...,e_{i,t-1},e_{i,t+1},...,e_{iT}\\}_{i=1}^{N}$. Let
$\bm{D}$ denote the diagonal matrix of $\bm{\Sigma}$ and
$\bm{R}=(r_{ij})=\bm{D}^{-1/2}\bm{\Sigma}\bm{D}^{-1/2}$ denote the correlation
matrix.

######  Assumption 2.1.

$\delta_{i}(\cdot)\in\mathcal{H}_{r}$ and
$\beta_{ij}(\cdot)\in\mathcal{H}_{r}$ for some $r>3/2$.

######  Assumption 2.2.

(i) There exist constants $0<c_{0}<C_{0}<\infty$ such that

$c_{0}\leq\lambda_{\min}(\mathbb{E}\\{(1,\bm{f}_{t}^{\prime})^{\prime}(1,\bm{f}_{t}^{\prime})\\})\leq\lambda_{\max}(\mathbb{E}\\{(1,\bm{f}_{t}^{\prime})^{\prime}(1,\bm{f}_{t}^{\prime})\\})\leq
C_{0}$

holds uniformly for $t\in[1,T]$; (ii) There exist a constant $0<K<\infty$ such
that $\mathbb{E}||\bm{f}_{t}||^{4(2+\kappa)}\leq K$ for some $\kappa>0$; (iii)
The process $\\{\bm{f}_{t}\\}_{t\geq 1}$ is strong mixing with mixing
coefficient $\alpha(\cdot)$ satisfying
$\sum_{k=0}^{\infty}\alpha(k)^{\kappa/(2+\kappa)}<\infty$; (iv)
$\\{\bm{f}_{t}\\}_{t=1}^{T}$ and $\\{\bm{e}_{t}\\}_{t=1}^{T}$ are independent.

######  Assumption 2.3.

(i) $\bm{e}_{1},...,\bm{e}_{T}$ are independently and identically distributed
with for each $i$, $\mathbb{E}(e_{it}|\mathcal{F}_{NT,t-1})=0$ and
$\mathbb{E}(\bm{e}_{t}\bm{e}_{t}^{\prime}|\bm{e}_{-t})=\bm{\Sigma}$, where
$\bm{\Sigma}$ is positive definite and $\sigma_{ii}\in(0,\infty)$ for every
$1\leq i\leq N$; (ii) $e_{it}$’s have sub-Gaussian-type tails, i.e. there
exist $\eta>0$ and $K>0$ such that $\mathbb{E}\\{\exp(\eta
e_{it}^{2}/\sigma_{ii})\\}\leq K$ for $1\leq i\leq N$.

######  Assumption 2.4.

(i) $TL^{-2r}=o(1)$; (ii) There exists $c_{1}>0$ such that
$c_{1}^{-1}\leq\lambda_{\min}(\bm{R})\leq\lambda_{\max}(\bm{R})\leq c_{1}$;
(iii) There exists $k>0$ such that $\max_{1\leq i,j\leq N}|r_{ij}|\leq k<1$.

Assumption 2.1 is the common smoothness assumption on the unknown functions
(He and Shi, 1996). Assumption 2.2 is the typical condition for the regression
design matrix. Specially, Assumption 2.2(i) follows Condition (C2) in Wang et
al. (2008), and Assumption 2.2(iv) follows from Assumption 3.1(ii) in Fan et
al. (2011). Moreover, Assumptions 2.2(ii)-(iii) weaken Assumption 3.2 and
3.3(ii) in Fan et al. (2011). Assumption 2.3 is the moment conditions on the
distribution of the error terms. Following Pesaran and Yamagata (2012) and
others, we assume that $\bm{e}_{t}$’s are independently and identically
distributed. Moreover, Assumption 2.3(i) contains the assumption for a
martingale difference sequence, and the homogeneity assumption on the
covariance of the error terms. Assumption 2.3(ii) is equivalent to Condition
(C6) in Cai et al. (2014), which allows the theoretical results to hold for
error distributions more general than the Gaussian type. Assumption 2.4(i) is
the condition on the number of spline basis functions $L$. Assumption 2.4(ii)
is a common assumption on the eigenvalues in the high dimensional setting,
which is equivalent to Assumption 4.1(i) in Fan et al. (2015) when
$\sigma_{ii}$’s are bounded. Assumption 2.4(iii) is also mild. For example, if
$\max_{1\leq i<j\leq N}|r_{ij}|=1$, then $\bm{\Sigma}$ is singular.

We now state our first main result, which is about the asymptotic property of
$M_{NT}$.

######  Theorem 2.1.

Suppose that Assumptions 2.1-2.4 hold. Assuming $L^{3}T^{-1}=o(1)$ and
$\log(N)=o(T^{1/4})$, we have as $(N,T)\to\infty$, for any $x\in\mathbb{R}$,

$\mathbb{P}_{H_{0}}\left(M_{NT}-2\log(N)+\log\\{\log(N)\\}\leq x\right)\to
F(x)\equiv\exp\left\\{-\frac{1}{\sqrt{\pi}}\exp\left(-\frac{x}{2}\right)\right\\}.$

Here, $\mathbb{P}_{H_{0}}$ denotes the probability measure under the null
hypothesis $H_{0}$. According to the limiting null distribution derived in, we
can easily obtain the $p$-value associated with $M_{NT}$, namely,

$p_{M}=1-F(M_{NT}-2\log(N)+\log\\{\log(N)\\}).$

If the $p$-value is below some pre-specified significant level, say
$\gamma\in(0,1)$, then we rejected the null hypothesis that the traded factors
are sufficient to price all assets. Next, we turn to analyze the power of the
maximum-type testing procedure.

######  Proposition 2.1.

Suppose Assumptions 2.1-2.4 hold. Assuming $L^{3}T^{-1}=o(1)$ and
$\log(N)=o(T^{1/4})$, we have as $(N,T)\to\infty$,

$\inf_{\bm{\alpha}\in\mathcal{A}(C)}\mathbb{P}(p_{M}<\gamma)\to 1,$

for some large enough constant $C>0$, where

$\mathcal{A}(C)=\left\\{\bm{\alpha}=(\alpha_{it})_{N\times T}:\max_{1\leq
i\leq N}\left|\frac{1}{T}\sum_{t=1}^{T}\alpha_{it}\right|\geq
C\sqrt{\frac{\log(N)}{T}}\right\\}.$

Proposition 2.1 shows that the proposed max-type test is effective in
detecting sparse alternatives.

##  3 The Adaptive Test

###  3.1 The Existing sum-type Test

To detect dense alternatives, we consider the sum-type test statistic proposed
by Ma et al. (2020). To wit,

$S_{NT}=N^{-1}T^{-1}\sum_{i=1}^{N}(\widehat{\bm{e}}_{i.}^{\prime}\bm{1}_{T})^{2},$
(7)

whose mean and variance under $H_{0}$ are given below,

$\mu_{NT}^{0}=N^{-1}T^{-1}\sum_{i=1}^{N}\sum_{t=1}^{T}\mathbb{E}(e_{it}^{2})\mathbb{E}(h_{t}^{2})\text{
and }\sigma_{NT}^{2}=2N^{-2}T^{-2}\text{Tr}(\bm{\Sigma}^{2})\sum_{1\leq t\neq
s\leq T}\mathbb{E}(h_{t}^{2}h_{s}^{2}),$

where $\bm{h}=(h_{1},...,h_{T})^{\prime}=M_{\bm{Z}}\bm{1}_{T}$. Let
$\mu_{NT}=N^{-1}T^{-1}\sum_{i=1}^{N}\sum_{t=1}^{T}e_{it}^{2}h_{t}^{2}$ be an
empirical approximation of the mean. For standardization, they considered the
consistent estimator of $\mu_{NT}^{0}$ as

$\widehat{\mu}_{NT}=N^{-1}T^{-1}\sum_{i=1}^{N}\sum_{t=1}^{T}\widehat{e}_{it}^{2}h_{t}^{2}.$
(8)

As for $\sigma_{NT}^{2}$, following Lan et al. (2014), they proposed the
following estimator

$\widehat{\sigma}_{NT}^{2}=2N^{-2}T^{-2}\widehat{\text{Tr}(\bm{\Sigma}^{2})}\sum_{1\leq
t\neq s\leq T}h_{t}^{2}h_{s}^{2},$ (9)

where
$\widehat{\text{Tr}(\bm{\Sigma}^{2})}=T^{2}\\{T+(1+d)L-1\\}^{-1}\\{T-(1+d)L\\}^{-1}[\text{Tr}(\widehat{\bm{\Sigma}}^{2})-\text{Tr}^{2}(\widehat{\bm{\Sigma}})/\\{T-(1+d)L\\}]$,
with
$\widehat{\bm{\Sigma}}=T^{-1}\sum_{t=1}^{T}(\widehat{\bm{e}}_{t}-\overline{\bm{e}})(\widehat{\bm{e}}_{t}-\overline{\bm{e}})^{T}$
and $\overline{\bm{e}}=T^{-1}\sum_{t=1}^{T}\widehat{\bm{e}}_{t}$.

The following Lemma restates the asymptotic null distribution of $S_{NT}$
derived in Ma et al. (2020).

######  Assumption 3.1.

(i) $TL^{-2r}N\text{Tr}^{-1/2}(\bm{\Sigma}^{2})=o(1)$; (ii)
$\text{Tr}^{-1/2}(\bm{\Sigma}^{2})\max_{i}\sum_{j=1}^{N}|\sigma_{ij}|=o(1)$.

######  Assumption 3.2.

(i) $tr(\bm{\Sigma}^{4})=o\\{\text{Tr}^{2}(\bm{\Sigma}^{2})\\}$ as
$N\to\infty$; (ii)
$T^{-2}\sum_{t=1}^{T}\mathbb{(}\bm{e}_{t}^{\prime}\bm{\Sigma}\bm{e}_{t})^{2}=o\\{\text{Tr}^{2}(\bm{\Sigma}^{2})\\}$;
(iii) $T^{-1+\varrho}N^{1+\varrho}L\text{Tr}^{-1/2}(\bm{\Sigma}^{2})=O(1)$ for
an arbitratily small $\varrho>0$.

######  Lemma 3.1.

Suppose Assumptions 2.1-2.3 and 3.1-3.2 hold. Assume that $L^{3}T^{-1}=o(1)$
and $L^{r}T^{3/2}=o(1)$. Then, under $H_{0}$, as $(N,T)\to\infty$,

$(S_{NT}-\widehat{\mu}_{NT})/\widehat{\sigma}_{NT}\stackrel{{\scriptstyle
d}}{{\to}}\mathcal{N}(0,1).$

By Lemma 3.1, the $p$-value associated with $S_{NT}$ is

$p_{S}=1-\Phi\left((S_{NT}-\widehat{\mu}_{NT})/\widehat{\sigma}_{NT}\right),$

where $\Phi(\cdot)$ is the cumulative distribution function (CDF) of
$\mathcal{N}(0,1)$. Again, small values of $p_{S}$ indicate rejecting the null
hypothesis.

###  3.2 Adaptive Strategy

In practice, we seldom know whether the vector of intercepts is sparse or
dense. In order to adapt to various alternative behaviors, we combine the
maximum- and sum-type testing procedures. The key message is that both test
statistics are asymptotically independent under some mild conditions if
$H_{0}$ holds.

######  Theorem 3.1.

Suppose Assumptions 2.1-2.4 and 3.1-3.2 hold. Assuming $L^{3}T^{-1}=o(1)$,
$\log(N)=o(T^{1/4})$ and $L^{r}T^{3/2}=o(1)$, we have as $(N,T)\to\infty$,
under $H_{0}$,

$\mathbb{P}_{H_{0}}\left(\frac{S_{NT}-\widehat{\mu}_{NT}}{\widehat{\sigma}_{NT}}\leq
x,M_{NT}-2\log(N)+\log\\{\log(N)\\}\leq y\right)\to\Phi(x)F(y).$

According to Theorem 3.1, we suggest combining the corresponding $p$-values by
using Fisher’s method (Littell and Folks, 1971), to wit,

$p_{adp}=1-G(-2\\{\log(p_{M})+\log(p_{S})\\}),$

where $G(\cdot)$ is the CDF of the chi-squared distribution with 4 degrees of
freedom. The rationality is that $-2\\{\log(p_{M})+\log(p_{S})\\}$ converges
in distribution to $G$ under $H_{0}$ due to Theorem 3.1. If the final
$p$-value is less than some pre-specified significant level $\gamma\in(0,1)$,
then we reject $H_{0}$.

Next, we analyze the power of the adaptive testing procedure. We consider the
following sequence of alternative hypotheses, to wit,

$H_{1,NT}:||\bm{\delta}^{0}||_{0}=o[N/\log^{2}\\{\log(N)\\}]\text{ and
}||\bm{\delta}^{0}||=O\\{T^{-1/2}\text{Tr}^{1/4}(\bm{\Sigma}^{2})\\},$ (10)

where $\bm{\delta}^{0}=(\delta_{1}^{0},...,\delta_{N}^{0})^{T}$ with
$\delta_{i}^{0}=T^{-1}\sum_{t=1}^{T}\alpha_{it}$. In fact, the asymptotic
independence between the maximum-type and sum-type statistics still hold under
the hypotheses given in equation (10).

######  Theorem 3.2.

Under the same condition as Theorem 3.1, we have as $(N,T)\to\infty$, under
$H_{1,NT}$,

$\displaystyle\mathbb{P}\left(\frac{S_{NT}-\widehat{\mu}_{NT}}{\widehat{\sigma}_{NT}}\leq
x,M_{NT}-2\log(N)+\log\\{\log(N)\\}\leq y\right)$
$\displaystyle\qquad\to\mathbb{P}\left(\frac{S_{NT}-\widehat{\mu}_{NT}}{\widehat{\sigma}_{NT}}\leq
x\right)\mathbb{P}\left(M_{NT}-2\log(N)+\log\\{\log(N)\\}\leq y\right).$

Simulation studies show that the power of Fisher’s p-value combination-based
test would be comparable to that of the test based on $\min\\{p_{M},p_{S}\\}$
(referred to as the minimal p-value combination), say $\beta_{M\wedge
S,\alpha}=P(\min\\{{\rm p}_{M},{\rm p}_{S}\\}\leq 1-\sqrt{1-\alpha})$.
Obviously,

$\displaystyle\beta_{M\wedge S,\alpha}$ $\displaystyle\geq P(\min\\{{\rm
p}_{M},{\rm p}_{S}\\}\leq\alpha/2)$
$\displaystyle=\beta_{M,\alpha/2}+\beta_{S,\alpha/2}-P({\rm
p}_{M}\leq\alpha/2,{\rm p}_{S}\leq\alpha/2)$
$\displaystyle\geq\max\\{\beta_{M,\alpha/2},\beta_{S,\alpha/2}\\}.$ (11)

On the other hand, under $H_{1}$ in (10), we have

$\displaystyle\beta_{M\wedge
S,\alpha}\geq\beta_{M,\alpha/2}+\beta_{S,\alpha/2}-\beta_{M,\alpha/2}\beta_{S,\alpha/2}+o(1),$
(12)

due to the asymptotic independence entailed by Theorem 3.2. For a small
$\alpha$, the difference between $\beta_{M,\alpha}$ and $\beta_{M,\alpha/2}$
should be small, and the same fact applies to $\beta_{S,\alpha}$.
Consequently, by (3.2)–(12), the power of the adaptive test would be no
smaller than or even significantly larger than that of either max-type or sum-
type test.

##  4 Monte Carlo Experiments

###  4.1 Experiment Settings

We conduct Monte Carlo experiments to evaluate the performance of the proposed
tests, and investigate the relationship between the power and sparsity levels
or signal sizes. For the conditional model discussed in Section 2, we consider
the following two examples.

Example 1. Following Ma et al. (2020), we generate data from the well known
conditional CAPM:

$R_{it}=\alpha_{it}+\beta_{it}f_{t}+e_{it},\quad i=1,...N;t=1,...,T,$ (13)

where $f_{t}$ is the Market factor. Specifically, we generate the factor from
the following AR(1)-GARCH(1,1) processes:

$f_{t}=0.34+0.05(f_{t-1}-0.34)+h_{t}^{1/2}\zeta_{t},$

where $\zeta_{t}$ is simulated from a standard normal distribution, the
variance terms $h_{t}$ follows from the process

$h_{t}=0.23+0.67h_{t-1}+0.13h_{t-1}\zeta_{t-1}^{2},$

and the above coefficients are obtained by fitting the model to the U.S. stock
market data.

As for the error terms, $\bm{e}_{t}=(e_{1t},...,e_{Nt})^{T}\in\mathbb{R}^{N}$
is generated from $\bm{e}_{t}\sim\bm{\Sigma}^{1/2}\bm{z}_{t}$, where
$\bm{z}_{t}$ has $N$ i.i.d. entries of $N(0,1)$ and $\exp(1)$, respectively.
Following Fan et al. (2011),
$\bm{\Sigma}=(\sigma_{i_{1}i_{2}})\in\mathbb{R}^{N\times N}$ with
$\sigma_{i_{1}i_{2}}=0.5^{|i_{1}-i_{2}|}$, which implies that $e_{i_{1}t}$ and
$e_{i_{2}t}$ are approximately uncorrelated when the difference
$|i_{1}-i_{2}|$ is sufficiently large.

To assess the robustness of the proposed test for the random factor loadings,
we set the conditional factor loadings to be $\beta_{it}=1+0.5\xi_{t}$ for
$i=1,...,N$ and $t=1,...,T$, where the unobservable state variable $\xi_{t}$
follows an AR(1)-ARCH(1) process, $\xi_{t}=0.8\xi_{t-1}+u_{t}$ with
$u_{t}=v_{t}\varepsilon_{t}$, $\varepsilon_{t}\sim N(0,1)$ and
$v_{t}^{2}=0.1+0.6v_{t-1}^{2}$.

Finally, we consider the conditional alphas. We set $\bm{\alpha}=\bm{0}$ under
the null hypothesis. For the alternative hypothesis, we set
$\alpha_{it}=\alpha_{i}t/T$ for $i\in S\subset\\{1,...,N\\}$ and $t=1,...,T$,
where each element in $S$ is uniformly and randomly drawn from $\\{1,...,N\\}$
with $|S|=s$, and $\alpha_{i}$’s are independently generated from
$U(0,c\sqrt{\log(N)/(Ts)})$. And we keep the remaining $\alpha_{i}$ with
$i\notin S$ zero. To examine how the power changes accordingly, we let the
signal strength $c$ and sparsity level $s$ vary.

The above processes are simulated over the period $t=-24,...,0,1,...,T$ with
the initial values $R_{i,-25}=0$, $f_{-25}=0$, $h_{-25}=1$, $z_{-25}=0$ and
$v_{-25}^{2}=1$. To offset the start-up effects, we drop the first $25$
simulated observations and use $t=1,...,T$ for our final experiments.

Example 2. To mimic the commonly used conditional FF model, where the factors
$\bm{f}_{t}$ have strong serial correlation and heterogeneous variance, we
generate $R_{it}$ according to the following model with $d=3$:

$R_{it}=\alpha_{it}+\sum_{j=1}^{d}\beta_{ijt}f_{jt}+e_{it},\quad
i=1,...N;t=1,...,T,$ (14)

where $f_{1t}$, $f_{2t}$ and $f_{3t}$ are the market factor, SMB and HML,
respectively. These factors are correspondingly simulated from the following
AR(1)-GARCH(1,1) processes,

$\displaystyle\text{Market factor:
}f_{1t}=0.34+0.05(f_{1,t-1}-0.34)+h_{1t}^{1/2}\zeta_{1t},$
$\displaystyle\text{SMB factor:
}f_{2t}=0.04+0.07(f_{2,t-1}-0.04)+h_{2t}^{1/2}\zeta_{2t},$
$\displaystyle\text{HML factor:
}f_{3t}=0.06+0.04(f_{3,t-1}-0.06)+h_{3t}^{1/2}\zeta_{3t},$

where $\zeta_{jt}$’s are simulated from a standard normal distribution,
$h_{jt}$’s are generated through the following processes,

$\displaystyle\text{Market factor:
}h_{1t}=0.32+0.67h_{1,t-1}+0.13h_{1,t-1}\zeta_{1,t-1}^{2},$
$\displaystyle\text{SMB factor:
}h_{2t}=0.33+0.51h_{1,t-1}+0.03h_{2,t-1}\zeta_{2,t-1}^{2},$
$\displaystyle\text{HML factor:
}h_{3t}=0.26+0.72h_{1,t-1}+0.05h_{3,t-1}\zeta_{3,t-1}^{2},$

and all the coefficients are the same as that in Ma et al. (2020).

The three groups of conditional factor loadings are
$\beta_{ijt}=a_{j}+b_{j}z_{t}$ for $i=1,...,N;j=1,2,3$ and $t=1,...,T$ with
$(a_{1},b_{1})=(1,0.5)$, $(a_{2},b_{2})=(0.1,0.5)$ and
$(a_{3},b_{3})=(0.2,0.4)$, respectively. Additionally, the conditional alphas,
the error terms, initial values, and the simulated observations have the same
settings as in Example 1.

###  4.2 Experiment Results

We now present the Monte Carlo experiment results of the proposed maximum-type
test and adaptive test, the sum-type test in Ma et al. (2020) and the LY test
in Li and Yang (2011), which are denoted as Max, Adp, Sum and LY,
respectively. All the results are based on 1000 replications at the 5% nominal
significance level. In addition, the number of interior knots $n$ is
determined by the BIC criterion and the order of B-splines is set as $3$.

Table 1: Empirical size comparison of various tests from Examples 1-2 for
testing conditional alphas with a nominal level 5% and normal or nonnormal
errors.

|  |  | Normal errors | Nonnormal errors
---|---|---|---|---
Example |  T | N |  Max | Adp |  Sum | LY |  Max | Adp |  Sum | LY
1 |  500 | 200 |  0.025 | 0.053 |  0.050 | 1 |  0.040 | 0.060 |  0.050 | 1
500 |  0.031 | 0.052 |  0.049 | 1 |  0.059 | 0.063 |  0.047 | 1
1000 |  0.029 | 0.050 |  0.048 | 1 |  0.057 | 0.059 |  0.045 | 1
|  |  | Normal errors | Nonnormal errors
Example |  T | N |  Max | Adp |  Sum | LY |  Max | Adp |  Sum | LY
2 |  500 | 200 |  0.018 | 0.053 |  0.053 | 1 |  0.037 | 0.063 |  0.053 | 1
500 |  0.025 | 0.057 |  0.050 | 1 |  0.040 | 0.054 |  0.050 | 1
1000 |  0.027 | 0.055 |  0.060 | 1 |  0.044 | 0.064 |  0.057 | 1

Table 1 summarizes the empirical sizes of four tests under the settings of
Example 1-2 over $N\in\\{200,500,1000\\}$ and $T=500$. It indicates that Max
and Adp perform very well regardless of $N<T$ or $N>T$ and the error
distribution being normal or nonnormal, which demonstrates the validity of
Theorems 2.1 and 3.1. Sum can roughly maintain the nominal significance level
due to Gaussian approximations. In contrast, LY exhibits serious size
distortion, since it is not designed for $N>T$. In conclusion, Max, Adp and
Sum have a satisfactory performance under the null hypothesis, and LY will be
abandoned in the evaluation of power.

To compare the power performance under different sparsity levels of alphas, we
present the empirical power of each test under different $s$’s. For a better
visualization, we set the signal strength $c=4$ if the sparsity level
$s\in\\{4,8,12,16\\}$, $c=7$ if $s\in\\{18,21,24,27\\}$ and $c=10$ if
$s\in\\{30,60,90,120\\}$, which can be roughly regarded as a sparse, a
moderately sparse and a dense regime, respectively. To illustrate, Figure 1
summarizes the results for Example 1 when $(N,T)=(500,500)$. The results of
Example 2 are similar. It suggests that under each setting, Adp has the best
power performance from the global point of view, since its power performance
is always in the first camp in the whole range of $s$. Max outperforms Sum in
very sparse case, i.e. $s<12$, while it falls behind Sum in moderately sparse
or dense case, i.e. $s>18$.

Furthermore, we demonstrate the relationship between the signal sizes and the
power of the tests. For the alternative hypothesis, we also consider three
cases, i.e. the sparsity level $s\in\\{2,16,100\\}$. In each case, the signal
strength $c$ ranges from 0 to 10 with an increment of 0.5. Figure 2 presents
the results for Example 2 with $(N,T)=(500,500)$. The results of Example 1 are
similar. It shows that the type I error rates of Max and Adp are well
controlled under $H_{0}$, and the empirical power of each test increases with
the signal strength. Specifically, Max is not powerful when the alternative is
dense but becomes more powerful when the alternative gets sparser, while Sum
performs much better than Max for dense alternative. Importantly, Adp
maintains high power across different signal strengths under all settings of
sparsity level.

In summary, the above experiment results confirm the theoretical conclusions
in Section 2 and 3. Notably, the proposed adaptive testing procedure is
powerful against a wide range of alternatives, and thus advantageous in
practice when the true alternative is unknown.

Figure 1: Power of alpha tests with different sparsity levels for Example 1
over $(N,T)=(500,500)$, where the panels of the first row depict the powers
for normal errors, while the panels of the second row depict the powers for
nonnormal errors.  Figure 2: Power of alpha tests with different signal
strengths for Example 2 over $(N,T)=(500,500)$, where the panels of the first
row depict the powers for normal errors, while the panels of the second row
depict the powers for nonnormal errors.

##  5 Empirical Applications

In this section, we employ the proposed tests to analyze the stocks in Chinese
and U.S. stock markets. To demonstrate the performance of the Max and Adp
tests, we compare with the results from their main competitors Sum.

###  5.1 Data Description

First, we consider the stocks in Chinese stock market. We collected the daily
returns of all the stocks in A-shares from 05/12/2021 to 03/09/2022, i.e.
$T=200$. After eliminating the stocks with missing observations to avoid
analyzing an unbalanced panel, we randomly selected $N=600$ stocks in our
final experiment. The time series data on the safe rate of return, and the
market factors are calculated according to the corresponding formulas. The
risk-free rate $r_{ft}$ is $0.000041$ (from China Asset Management Center).
The value-weighted return on all the stocks of Shanghai Stock Exchange and
Shenzhen Stock Exchange is used as a proxy for the market return $r_{mt}$. The
average return on the three small portfolios minus the average return on the
three big portfolios $\text{SMB}_{t}$, and the average return on two value
portfolios minus the average return on two growth portfolios $\text{HML}_{t}$
are calculated based on the stocks listed on Shanghai Stock Exchange and
Shenzhen Stock Exchange. We use $r_{it}$ to denote the return rate of security
$i$ on time $t$.

Next, we consider the stocks in the S&P 500 index, which is internationally
accepted as a leading indicator of the U.S. equities. Similarly, we compiled
returns on all the securities that constitute the S&P 500 index each week over
the period from 01/08/2010 to 10/25/2013, i.e. $T=200$. Because the securities
that make up the index change over time, we only consider $N=400$ securities
that were included in the S&P 500 index during the entire period. The time
series data on the safe rate of return, and the market factors are obtained
from Ken French’s data library web page. The one-month U.S. treasury bill rate
is chosen as the risk-free rate $r_{ft}$. The value-weighted return on all
NYSE, AMEX, and NASDAQ stocks from CRSP is used as a proxy for the market
return $r_{mt}$. The average return on the three small portfolios minus the
average return on the three big portfolios $\text{SMB}_{t}$, and the average
return on two value portfolios minus the average return on two growth
portfolios $\text{HML}_{t}$ are calculated based on the stocks listed on the
NYSE, AMEX and NASDAQ.

###  5.2 Conditional Alpha Test

We consider the following rolling window procedure with window length $h=100$.
For each $\tau\in\\{1,...,T-h\\}$, we separately estimate the conditional CAPM
and FF model using the data from period $\tau$ to $\tau+h-1$. As a result,

$\displaystyle\text{CAPM:
}R_{it}=r_{it}-r_{ft}=\widehat{\alpha}_{it}+\widehat{\beta}_{it}(r_{mt}-r_{ft})+\widehat{e}_{it},$
$\displaystyle\text{FF:
}R_{it}=r_{it}-r_{ft}=\widehat{\alpha}_{it}+\widehat{\beta}_{i1t}(r_{mt}-r_{ft})+\widehat{\beta}_{i2t}\text{SMB}_{t}+\widehat{\beta}_{i3t}\text{HML}_{t}+\widehat{e}_{it},$

for $1\leq t\leq\tau+h-1$. Based on the estimated residuals $\widehat{e}_{it}$
obtained by separately fitting CAPM and the FF model to the data in each
window, we calculate the Max, Sum and Adp test statistics and their
corresponding $p$-values. Here, the number of interior knots $n$ is determined
via BIC and the order of B-splines is set at 3 for all estimation windows.

Before applying the involved tests, it is necessary to examine whether alphas
and betas are time-varying, given the conclusion of Li and Yang (2011) that
the conditional CAPM and FF are not always superior to their unconditional
counterparts. We apply the constant coefficient test (henceforth the CC test)
proposed by Ma et al. (2020). The resulting $p$-values are presented in Figure
3, which shows that most of the $p$-values of the CC test for 600 (400) stocks
in the Chinese (U.S.) dataset is close to $0$, regardless of the model. This
provides strong evidence that alphas and betas are indeed time-varying in both
the Chinese and U.S. stock markets.

Now, we apply Max, Adp and Sum tests to the panel data of the securities in
the Chinese and U.S. stock markets under the conditional CAPM and FF. Figure 4
depicts the $p$-values across the 100 windows. The Box-plot of these
$p$-values are presented in Figures 5. In Chinese stock market, we note from
Figure 4 and Figures 5 that the $p$-values of Max and Adp are less than the 5%
significance level for the conditional CAPM, which indicates that the markets
are inefficient over these window periods. In contrast, the $p$-values
obtained from Sum in the corresponding window periods are greater than 5%. For
the conditional FF, the averaged $p$-values obtained from Max and Adp are
smaller than those from Sum. In addition, most of the $p$-values from FF are
larger than 5%, and they are also higher than those from CAPM. Accordingly, FF
is better than CAPM in explaining the Chinese stock market. In the U.S. stock
market, it is more prominent that the $p$-values are larger than 5% for both
CAPM and FF. This suggests that the U.S. stock market is more efficient than
the Chinese stock market.

|
---|---
|
Figure 3: Histogram of p-values of CC test for Chinese and U.S.’s datasets,
respectively. Figure 4: P-values of alpha tests for Chinese and U.S.’s
datasets, respectively. Figure 5: Box-plot of p-values of alpha tests for
Chinese and U.S.’s datasets, respectively.

##  6 Conclusion

In this paper, we propose the Max test for alpha testing under high
dimensional case, which aims to simultaneously alleviate the difficulties of
the time-variation in the risk-exposure coefficients and the sparse
alternatives. Considering the good performance of Sum test under dense
alternative, we suggest an adaptive test, which integrates the power
advantages of both Max and Sum and is applicable for various alternatives.
Moreover, the usefulness of the proposed Max and Adp tests is illustrated by
two empirical examples.

To further broaden the usefulness of our proposed tests, we conclude this
article by identifying the following possible research avenues. First, if the
goal is to identify the significance of alphas for all possible assets, then
one can apply our Adp test in a multiple testing procedure to control the
false discovery rate (Giglio et al., 2020). Second, the assumption of sub-
Gaussian-type tails may be too restrictive for stock data. Zhao (2023)
proposed a robust high-dimensional alpha test based on spatial-sign for
conditional time-varying factor models. How to construct an adaptive test for
time-varying factor pricing models with heavy-tailed errors deserves some
further studies. Finally, for sparse alternative, one may consider an
L-statistic which combines the first several largest signals together, see
more information in Remark 1 in Chang et al. (2022).

##  7 Appendix

###  7.1 Proof of Theorem 2.1

###### Proof.

Define
$\rho_{i0t}=\delta_{i}(t/T)-\bm{\lambda}_{0,i0}^{\prime}\widetilde{\bm{B}}(t/T)$
and $\rho_{ijt}=\beta_{ij}(t/T)-\lambda_{0,ij}^{\prime}B(t/T)$ for $1\leq
j\leq d$ and $1\leq i\leq N$. Denote
$\bm{\lambda}_{i}^{0}=(\bm{\lambda}_{0,ij}^{\prime},0\leq j\leq d)^{\prime}$,
$\rho_{it}=\rho_{i0t}+\sum_{j=1}^{d}\rho_{ijt}f_{jt}$ and
$\bm{\rho}_{i.}=(\rho_{i1},...,\rho_{iT})^{\prime}$. Then

$\bm{R}_{i.}=\delta_{i}^{0}\bm{1}_{T}+\bm{Z}\bm{\lambda}_{i}^{0}+\bm{e}_{i.}+\bm{\rho}_{i.},\text{
and
}\widehat{\bm{e}}_{i.}=M_{\bm{Z}}\bm{e}_{i.}+M_{\bm{Z}}\bm{\rho}_{i.}+\delta_{i}^{0}M_{\bm{Z}}\bm{1}_{T}.$

Accordingly, $t_{i}^{2}$ given in (6) can be written as

$\displaystyle t_{i}^{2}$
$\displaystyle=T^{-1}\widehat{\sigma}_{ii}^{-1}(M_{\bm{Z}}\bm{e}_{i.}+M_{\bm{Z}}\bm{\rho}_{i.}+\delta_{i}^{0}M_{\bm{Z}}\bm{1}_{T})^{\prime}\bm{1}_{T}\bm{1}_{T}^{\prime}(M_{\bm{Z}}\bm{e}_{i.}+M_{\bm{Z}}\bm{\rho}_{i.}+\delta_{i}^{0}M_{\bm{Z}}\bm{1}_{T})$
$\displaystyle=T^{-1}\widehat{\sigma}_{ii}^{-1}\\{\bm{e}_{i.}^{\prime}M_{\bm{Z}}\bm{1}_{T}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{e}_{i.}+\bm{\rho}_{i.}^{\prime}M_{\bm{Z}}\bm{1}_{T}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{\rho}_{i.}+2\bm{\rho}_{i.}^{\prime}M_{\bm{Z}}\bm{1}_{T}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{e}_{i.}$
$\displaystyle\qquad+2\delta_{i}^{0}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{1}_{T}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{e}_{i.}+2\delta_{i}^{0}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{1}_{T}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{\rho}_{i.}+(\delta_{i}^{0})^{2}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{1}_{T}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{1}_{T}\\}$
$\displaystyle=:\widehat{\sigma}_{ii}^{-1}(\varphi_{i}+\zeta_{i1}+\zeta_{i2}+\zeta_{i3}+\zeta_{i4}+\zeta_{i5}).$
(A.1)

That is, under $H_{0}$, $M_{NT}=\max_{1\leq i\leq
N}\widehat{\sigma}_{ii}^{-1}(\varphi_{i}+\zeta_{i1}+\zeta_{i2})$. Thus, it
suffices to show that

$\mathbb{P}\left(\max_{1\leq i\leq
N}\varphi_{i}/\widehat{\sigma}_{ii}-2\log(N)+\log\\{\log(N)\\}\leq x\right)\to
F(x),\text{ and }\max_{1\leq i\leq
N}\widehat{\sigma}_{ii}^{-1}\zeta_{ik}=o_{p}(1),$

for $k=1,2$.

Step 1. Show that

$\mathbb{P}\left(\max_{1\leq i\leq
N}\varphi_{i}/\widehat{\sigma}_{ii}-2\log(N)+\log\\{\log(N)\\}\leq x\right)\to
F(x).$ (A.2)

Define $\bm{h}=(h_{1},...,h_{T})^{\prime}=M_{\bm{Z}}\bm{1}_{T}$,
$V_{it}=e_{it}h_{t}/\sigma_{ii}^{1/2}$ and
$\widehat{V}_{it}=V_{it}I(V_{it}\leq\tau_{NT})$ for $i=1,...,N;t=1,...,T$,
where $\tau_{NT}=2\zeta_{NT}^{-1}\eta^{-1/2}\sqrt{\log(T+N)}$ under Assumption
2.3 (iii). Here $\zeta_{NT}\to 0$ will be specified later. Let
$W_{i}=\sum_{t=1}^{T}V_{it}/\sqrt{T}$ and
$\widehat{W}_{i}=\sum_{t=1}^{T}\widehat{V}_{it}/\sqrt{T}$. Notice that
$W_{i}^{2}=\sigma_{ii}^{-1}\varphi_{i}$.

Step 1.1. Show that for any $x\in\mathbb{R}$,

$\mathbb{P}\left(\max_{1\leq i\leq
N}\widehat{W}_{i}^{2}-2\log(N)+\log\\{\log(N)\\}\leq x\right)\to F(x).$ (A.3)

By the Bonferroni inequality in Lemma 7.1, for any fixed integer $k\leq N/2$,

$\displaystyle\sum_{t=1}^{2k}(-1)^{t-1}\sum_{1\leq i_{1}<...<i_{t}\leq
N}\mathbb{P}(|\widehat{W}_{i_{1}}|\geq x_{N}$
$\displaystyle,...|\widehat{W}_{i_{t}}|\geq x_{N})\leq\mathbb{P}(\max_{1\leq
i\leq N}|\widehat{W}_{i}|\geq x_{N})$ $\displaystyle\leq$
$\displaystyle\sum_{t=1}^{2k-1}(-1)^{t-1}\sum_{1\leq i_{1}<...<i_{t}\leq
N}\mathbb{P}(|\widehat{W}_{i_{1}}|\geq x_{N},...|\widehat{W}_{i_{t}}|\geq
x_{N}),$

where $x_{N}=\sqrt{2\log{N}-\log\\{\log(N)\\}+x}$. Define
$|\widehat{\bm{W}}|_{\min}=\min_{1\leq l\leq t}|\widehat{W}_{i_{l}}|$. Then,
by Theorem 1 in Zaitsev (1987), we have

$\displaystyle\mathbb{P}(|\widehat{\bm{W}}|_{\min}\geq x_{N})\leq$
$\displaystyle\mathbb{P}\left(|\bm{z}|_{\min}\geq
x_{N}-k_{n}\log^{-1/2}(N)\right)$
$\displaystyle\quad+c_{1}t^{5/2}\exp[-T^{1/2}k_{n}/\\{c_{2}t^{3}\tau_{NT}\log^{1/2}(N)\\}],$

where $c_{1},c_{2}>0$ are two constants, $k_{n}\to 0$ (specified later),
$|\bm{z}|_{\min}=\min_{1\leq l\leq t}|z_{l}|$,
$\bm{z}=(z_{1},...,z_{t})^{\prime}$ is a t-dimensional normal vector, which
has the same covariance as
$(\widehat{W}_{i_{1}},...,\widehat{W}_{i_{t}})^{\prime}$.

Because $\log(N)=o(T^{1/4})$, we let $k_{n}\to 0$ sufficiently slowly so that

$c_{1}t^{5/2}\exp[-T^{1/2}k_{n}/\\{c_{2}t^{3}\tau_{NT}\log^{1/2}(N)\\}]=O(N^{-\zeta})$

for any large $\zeta>0$. Thus

$\mathbb{P}\left(\max_{1\leq i\leq N}|\widehat{W}_{i}|\geq x_{N}\right)\\\
\leq\sum_{t=1}^{2k-1}(-1)^{t-1}\sum_{1\leq i_{1}<...<i_{t}\leq
N}\mathbb{P}\left(|\bm{z}|_{\min}\geq x_{N}-k_{n}\log^{-1/2}(N)\right)+o(1).$

Likewise,

$\mathbb{P}\left(\max_{1\leq i\leq N}|\widehat{W}_{i}|\geq x_{N}\right)\\\
\geq\sum_{t=1}^{2k}(-1)^{t-1}\sum_{1\leq i_{1}<...<i_{t}\leq
N}\mathbb{P}\left(|\bm{z}|_{\min}\geq x_{N}-k_{n}\log^{-1/2}(N)\right)+o(1).$

Define $V_{t}=\sum_{1\leq i_{1}<...<i_{t}\leq
N}P_{i_{1},...,i_{t}}=\sum_{1\leq i_{1}<...<i_{t}\leq
N}\mathbb{P}(|z_{i_{1}}|\geq\widetilde{x}_{N},...,|z_{i_{t}}|\geq\widetilde{x}_{N})$,
where $\widetilde{x}_{N}=x_{N}-k_{n}\log^{-1/2}(N)$. Write correlation matrix
$\bm{R}=\bm{Q}^{\prime}\bm{\Lambda}\bm{Q}$, where $\bm{Q}=(q_{ij})_{N\times
N}$ is an orthogonal matrix and
$\bm{\Lambda}=\text{Diag}(\lambda_{1},...,\lambda_{N})$, $\lambda_{i}$’s are
the eigenvalues of $\bm{R}$. Since $\sum_{1\leq j\leq N}r_{ij}^{2}$ is the
$i$th diagonal element of $\bm{R}^{2}=\bm{Q}^{T}\bm{\Lambda}^{2}\bm{Q}$, we
have $\sum_{1\leq j\leq
N}r_{ij}^{2}=\sum_{l=1}^{N}q_{li}^{2}\lambda_{l}^{2}\leq c_{3}^{2}$ for some
constant $c_{3}$ due to Assumption 2.4 (ii). Thus, together with Assumption
2.4 (iii), $\bm{R}$ satisfies the condition in Lemma 7.2.

Define $\mathcal{I}=\\{1\leq i_{1}<...<i_{t}\leq N:\max_{1\leq k<l\leq
t}|\text{Cov}(z_{i_{k}},z_{i_{l}})|\geq N^{-\gamma}\\}$, where $\gamma>0$ is a
sufficiently small number to be specified later. For $2\leq d\leq t-1$, define
$\mathcal{I}_{d}=\\{1\leq i_{1}<...<i_{t}\leq N:\text{Card}(S)=d,\text{ where
}S\text{ is the largest subset of }\\{i_{1},...,i_{t}\\}\text{ such that
}\forall i_{k},i_{t}\in S,i_{k}\neq
i_{t},|\text{Cov}(z_{i_{k}},z_{i_{t}})|<N^{-\gamma}\\}$. Here,
$\text{Card}(S)$ is the cardinality of $S$. For $d=1$, define
$\mathcal{I}_{1}=\\{1\leq i_{1}<...<i_{t}\leq
N:|\text{Cov}(z_{i_{k}},z_{i_{l}})|\geq N^{-\gamma},\forall 1\leq k<l\leq
t\\}$. Hence, $\mathcal{I}=\cap_{d=1}^{t-1}\mathcal{I}_{d}$. For a fixed
subset $S$ with $\text{Card}(S)=d$, the number of $i$ such that
$|\text{Cov}(z_{i},z_{j})|\geq N^{-\gamma}$ for some $j\in S$ is no more than
$c_{4}dN^{2\gamma}$, where $c_{4}$ is a constant. Indeed, if the number of
such $i$ had been larger than $c_{4}dN^{2\gamma}$, then

$\sum\nolimits_{1\leq i\leq
N}r_{ij}^{2}\geq\sum\nolimits_{\\{i:|\text{Cov}(z_{i},z_{j})|\geq
N^{-\gamma}\\}}r_{ij}^{2}\geq N^{-2\gamma}c_{4}dN^{2\gamma}=c_{4}d>c_{3}^{2},$

for $c_{4}>d^{-1}c_{3}^{2}$, which is a contradiction. Note that the total
number of $S$ with $\text{Card}(S)=d$ is $\binom{N}{d}$. This leads to

$\displaystyle\text{Card}(\mathcal{I}_{d})\leq\binom{N}{d}\binom{c_{4}dN^{2\gamma}}{t-d}\leq
N^{d}(c_{4}d)^{t-d}N^{2\gamma(t-d)}\leq(c_{4}t)^{t}N^{d+2\gamma t},\text{
and}$
$\displaystyle\text{Card}(\mathcal{I})\leq\sum_{d=1}^{t-1}(c_{4}t)^{t}N^{d+2\gamma
t}\leq(c_{4}t)^{t}tN^{2\gamma t+t-1}=O(N^{2\gamma t+t-1}).$

Define $\mathcal{I}^{c}=\\{1\leq i_{1}<...<i_{t}\leq
N\\}\backslash\mathcal{I}$. Then

$\text{Card}(\mathcal{I}^{c})=\binom{N}{t}-O(N^{2\gamma
t+t-1})=\\{1+o(1)\\}\binom{N}{t}$

as long as $\gamma<(2t)^{-1}$. Thus, by equation (20) in the proof of Lemma 6
in Cai et al. (2014), we have

$P_{i_{1},...,i_{t}}=\\{1+o(1)\\}\pi^{-t/2}N^{-t}\exp\left(-\frac{tx}{2}\right)$

uniformly in $(i_{1},...,i_{t})\in\mathcal{I}^{c}$. Similarly, by equation
(21) in the proof of Lemma 6 in Cai et al. (2014), we have for $1\leq d\leq
t-1$

$\sum_{(i_{1},...,i_{t})\in\mathcal{I}_{d}}P_{i_{1},...,i_{t}}\to 0.$

Thus,

$\displaystyle V_{t}$
$\displaystyle=\sum_{(i_{1},...,i_{t})\in\mathcal{I}^{c}}P_{i_{1},...,i_{t}}+\sum_{(i_{1},...,i_{t})\in\mathcal{I}}P_{i_{1},...,i_{t}}$
$\displaystyle=\sum_{(i_{1},...,i_{t})\in\mathcal{I}^{c}}P_{i_{1},...,i_{t}}+\sum_{d=1}^{t-1}\sum_{(i_{1},...,i_{t})\in\mathcal{I}_{d}}P_{i_{1},...,i_{t}}$
$\displaystyle=\\{1+o(1)\\}\pi^{-t/2}\frac{1}{t!}\exp\left(-\frac{tx}{2}\right).$

Combining all facts together, we get

$\displaystyle\sum_{t=1}^{2k}(-1)^{t-1}\pi^{-t/2}\frac{1}{t!}\exp\left(-\frac{tx}{2}\right)\\{1+o(1)\\}$
$\displaystyle\leq\mathbb{P}\left(\max_{1\leq i\leq N}|\widehat{W}_{i}|\geq
x_{N}\right)$
$\displaystyle\leq\sum_{t=1}^{2k-1}(-1)^{t-1}\pi^{-t/2}\frac{1}{t!}\exp\left(-\frac{tx}{2}\right)\\{1+o(1)\\}.$

Letting $k\to\infty$, we have (A.3).

Step 1.2. Show that

$\mathbb{P}\left(\max_{1\leq i\leq
N}|W_{i}-\widehat{W}_{i}|\geq\frac{1}{\log(N)}\right)\to 0.$ (A.4)

Notice that

$\mathbb{P}\left(\max_{1\leq i\leq
N}|W_{i}-\widehat{W}_{i}|\geq\frac{1}{\log(N)}\right)\leq\mathbb{P}\left(\max_{1\leq
i\leq N}\max_{1\leq t\leq T}|V_{it}|\geq\tau_{NT}\right)\leq\sum_{1\leq i\leq
N}\sum_{1\leq t\leq T}\mathbb{P}(|V_{it}|\geq\tau_{NT}).$

For any $\zeta_{t}\to 0$, we have

$\displaystyle\mathbb{P}(|V_{it}|\geq\tau_{NT})$
$\displaystyle=\mathbb{P}(|e_{it}/\sigma_{ii}^{1/2}|\geq|h_{t}^{-1}|\tau_{NT})$
$\displaystyle\leq\mathbb{P}(|e_{it}/\sigma_{ii}^{1/2}|\geq|h_{t}^{-1}|\tau_{NT},|h_{t}^{-1}|\geq\zeta_{NT})+\mathbb{P}(|e_{it}/\sigma_{ii}^{1/2}|\geq|h_{t}^{-1}|\tau_{NT},|h_{t}^{-1}|<\zeta_{NT})$
$\displaystyle\leq\mathbb{P}(|e_{it}/\sigma_{ii}^{1/2}|\geq\tau_{NT}\zeta_{NT})+\mathbb{P}(|h_{t}|\geq\zeta_{NT}^{-1}).$

Under Assumption 2.3 (iii), by the Markov inequality, we have

$\mathbb{P}(|e_{it}/\sigma_{ii}^{1/2}|\geq\tau_{NT}\zeta_{NT})\leq
K\exp(-\eta\zeta_{NT}^{2}\tau_{NT}^{2}).$

Next, we consider $\mathbb{P}(|h_{t}|\geq\zeta_{NT}^{-1})$. By Bernstein’s
inequality in Bosq (1996) and the same proof for Lemma A.8 of Ma and Yang
(2011), under $L^{3}T^{-1}=o(1)$, we have

$||\bm{Z}^{\prime}\bm{1}_{T}/T-\mathbb{E}(\bm{Z}^{\prime}\bm{1}_{T}/T)||_{\infty}=O_{\text{a.s.}}\\{\log(T)/\sqrt{TL}\\}.$

Further, under Assumption 2.2 (ii), we have

$\displaystyle||\mathbb{E}(\bm{Z}^{\prime}\bm{1}_{T}/T)||_{\infty}$
$\displaystyle\leq\max_{1\leq l\leq
L}T^{-1}\sum_{t=1}^{T}|B_{l}(t/T)|\left(1+\sum_{j=1}^{d}\mathbb{E}|f_{jt}|\right)$
$\displaystyle\leq MT^{-1}\max_{1\leq l\leq L}\sum_{t\in\\{t:|l(t)-l|\leq
q-1\\}}|B_{l}(t/T)|$ $\displaystyle\leq ML^{-1}$

for some constant $0<M<\infty$, which leads to
$||\bm{Z}^{\prime}\bm{1}_{T}/T||_{\infty}=O_{\text{a.s.}}(L^{-1})$. By Lemma
7.4 and the result in Demko (1986), we have with probability one,

$||(\bm{Z}^{\prime}\bm{Z}/T)^{-1}||_{\infty}\leq c_{5}L,$

for some constant $0<c_{5}<\infty$, as $T\to\infty$. Then, using the fact that
$\sum_{l=1}^{L}B_{l}(t/T)$ is bounded, we have

$\displaystyle|h_{t}|$
$\displaystyle=|1-\bm{Z}_{t}^{\prime}(\bm{Z}^{\prime}\bm{Z})^{-1}\bm{Z}^{\prime}\bm{1}_{T}|$
$\displaystyle\leq
1+\sum_{k=1}^{(1+d)L}|Z_{tk}|\cdot||(\bm{Z}^{\prime}\bm{Z}/T)^{-1}||_{\infty}\cdot||\bm{Z}^{\prime}\bm{1}_{T}/T||_{\infty}$
$\displaystyle\leq dc_{6}(1+||\bm{f}_{t}||),$ (A.5)

for some constant $0<c_{6}<\infty$, which leads to

$\displaystyle\mathbb{P}(|h_{t}|\geq\zeta_{NT}^{-1})\leq\mathbb{P}\left(||\bm{f}_{t}||>d^{-1}c_{6}^{-1}\zeta_{NT}^{-1}-1\right)\leq
M(d^{-1}c_{6}^{-1}\zeta_{NT}^{-1}-1)^{-4(2+\kappa)},$

where the last inequality comes from Assumption 2.2 (ii) and the Markov
inequality. Combining all facts above and setting
$\zeta_{t}=o((NT)^{-1/\\{4(2+\kappa)\\}})$, we obtain

$\displaystyle\mathbb{P}\left(\max_{1\leq i\leq N}|W_{i}-\widehat{W}_{i}|\geq
1/\log(N)\right)\leq$ $\displaystyle
NT\left\\{M(d^{-1}c_{6}^{-1}\zeta_{NT}^{-1}-1)^{-4(2+\kappa)}+K\exp(-\eta\zeta_{NT}^{2}\tau_{NT}^{2})\right\\}$
$\displaystyle\leq$ $\displaystyle
NT\left\\{M(d^{-1}c_{6}^{-1}\zeta_{NT}^{-1}-1)^{-4(2+\kappa)}+K(N+T)^{-4}\right\\}\to
0.$

Hence, the proof of (A.4) is complete.

Step 1.3. Show that

$\left|\max_{1\leq i\leq N}\varphi_{i}\widehat{\sigma}_{ii}^{-1}-\max_{1\leq
i\leq N}W_{i}^{2}\right|=o_{p}(1).$ (A.6)

By (A.3) and letting $x=1/2\log\\{\log(N)\\}$, we have

$\mathbb{P}\left(\max_{1\leq i\leq N}\widehat{W}_{i}^{2}\leq
2\log(N)-\frac{1}{2}\log\\{\log(N)\\}\right)\to 1,$

which, together with (A.4), leads to

$\displaystyle\left|\max_{1\leq i\leq N}W_{i}^{2}-\max_{1\leq i\leq
N}\widehat{W}_{i}^{2}\right|\leq 2\max_{1\leq i\leq N}|W_{i}|\cdot\max_{1\leq
i\leq N}|W_{i}-\widehat{W}_{i}|+\max_{1\leq i\leq
N}|W_{i}-\widehat{W}_{i}|^{2}=o_{p}(1),$

and $\max_{1\leq i\leq N}W_{i}^{2}=O_{p}\\{\log(N)\\}$. By Lemma E.2 and
Proposition 3.1 in Fan et al. (2015), for some $c>0$,

$\mathbb{P}\left(\max_{1\leq i\leq N}|\sigma_{ii}-\widehat{\sigma}_{ii}|\geq
c\sqrt{\frac{\log(N)}{T}}\right)\to 0\text{ and
}\mathbb{P}\left(\frac{4}{9}\leq\frac{\widehat{\sigma}_{ii}}{\sigma_{ii}}\leq\frac{9}{4},i=1,...,N\right)\to
1.$ (A.7)

Combining the results above, with probability tending to one, we have

$\displaystyle\left|\max_{1\leq i\leq
N}\varphi_{i}\widehat{\sigma}_{ii}^{-1}-\max_{1\leq i\leq
N}W_{i}^{2}\right|\leq\max_{1\leq i\leq N}W_{i}^{2}\cdot\max_{1\leq i\leq
N}\left|\frac{\sigma_{ii}}{\widehat{\sigma}_{ii}}-1\right|=O_{p}\\{\log^{3/2}(N)T^{-1/2}\\}\to
0,$

due to $\log(N)=o(T^{1/3})$. Then (A.6) follows, which together with (A.3)
implies (A.2).

Step 2. Show that

$\max_{1\leq i\leq N}\widehat{\sigma}_{ii}^{-1}\zeta_{ik}=o_{p}(1),k=1,2.$
(A.8)

By Lemma 7.3 and Assumption 2.2, for each $1\leq i\leq N$, we have

$\sup_{1\leq t\leq T}|\rho_{it}|=\sup_{1\leq t\leq
T}\left|\rho_{i0t}+\sum_{j=1}^{d}\rho_{ijt}f_{jt}\right|=O(L^{-r}).$

This, together with the fact that
$\lambda_{\max}(M_{\bm{Z}}\bm{1}_{T}\bm{1}_{T}^{\prime}M_{\bm{Z}})\leq T$,
leads to

$\zeta_{i1}=T^{-1}(\bm{\rho}_{i.}^{\prime}M_{\bm{Z}}\bm{1}_{T})^{2}=O(TL^{-2r}),$

for each $1\leq i\leq N$. Thus, by Assumption 2.4 (i) and (A.7), we have
$\max_{1\leq i\leq N}\widehat{\sigma}_{ii}^{-1}\zeta_{i1}=o_{p}(1)$.

Define $\varpi_{i}=\sum_{s=1}^{T}h_{s}\rho_{is}$, then
$\zeta_{i2}=2T^{-1}\sum_{t=1}^{T}e_{it}h_{t}\varpi_{i}$. By (7.1) and
Assumption 2.2 (ii), we obtain
$\mathbb{E}(h_{t}^{2})\leq\mathbb{E}\\{dc_{6}(1+||\bm{f}_{t}||)\\}^{2}\leq
c_{7}$, for some constant $0<c_{7}<\infty$. In addition, by Lemma 7.3, we
obtain $|\varpi_{i}|=O(TL^{-r})$. Furthermore, we have

$\text{Var}(\zeta_{i2})=4T^{-2}\sum_{t=1}^{T}\mathbb{E}(e_{it}h_{t}\varpi_{i})^{2}=4T^{-2}\sum_{t=1}^{T}\sigma_{ii}\mathbb{E}(h_{t}\varpi_{i})^{2}=O(TL^{-2r}).$

Hence, $|\zeta_{i2}|=O_{p}(T^{1/2}L^{-r})$, which, together with Assumption
2.4 (i) and (A.7), implies that $\max_{1\leq i\leq
N}\widehat{\sigma}_{ii}^{-1}\zeta_{i2}=o_{p}(1)$.

Then the proof of Theorem 2.1 is complete. ∎

###  7.2 Proof of Proposition 2.1

###### Proof.

By (7.1) and the triangle inequality, we have

$\displaystyle M_{NT}$ $\displaystyle=\max_{1\leq i\leq
N}\widehat{\sigma}_{ii}^{-1}(\varphi_{i}+\zeta_{i1}+\zeta_{i2}+\zeta_{i3}+\zeta_{i4}+\zeta_{i5})$
$\displaystyle\geq\max_{1\leq i\leq
N}\widehat{\sigma}_{ii}^{-1}(\zeta_{i3}+\zeta_{i4}+\zeta_{i5})-\max_{1\leq
i\leq N}\widehat{\sigma}_{ii}^{-1}\varphi_{i}-\max_{1\leq i\leq
N}\widehat{\sigma}_{ii}^{-1}\zeta_{i1}-\max_{1\leq i\leq
N}\widehat{\sigma}_{ii}^{-1}\zeta_{i2}.$

According to the proof of Theorem 2.1, we have $\max_{1\leq i\leq
N}\widehat{\sigma}_{ii}^{-1}\zeta_{ik}=o_{p}(1)$ for $k=1,2$, and

$\mathbb{P}\left(\max_{1\leq i\leq N}\widehat{\sigma}_{ii}^{-1}\varphi_{i}\leq
2\log(N)-\frac{1}{2}\log\\{\log(N)\\}\right)\to 1.$

Define $\varpi_{i}^{*}=\delta_{i}^{0}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{1}_{T}$,
then $\zeta_{i3}=2T^{-1}\sum_{t=1}^{T}e_{it}h_{t}\varpi_{i}^{*}$ and
$|\varpi_{i}^{*}|\leq T|\delta_{i}^{0}|$. Furthermore, due to (7.1), we have

$\text{Var}(\zeta_{i3})=4T^{-2}\sum_{t=1}^{T}\mathbb{E}(e_{it}h_{t}\varpi_{i}^{*})^{2}=4T^{-2}\sum_{t=1}^{T}\sigma_{ii}\mathbb{E}(h_{t}\varpi_{i}^{*})^{2}=O(T|\delta_{i}^{0}|^{2}).$

That is $\zeta_{i3}=O_{p}(T^{1/2}|\delta_{i}^{0}|)$. In addition, by Lemma
7.3, we have

$\displaystyle\zeta_{i4}=2T^{-1}\delta_{i}^{0}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{1}_{T}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{\rho}_{i.}\leq
2T^{-1}|\delta_{i}^{0}|T\sum_{t=1}^{T}|h_{t}\rho_{it}|=O(|\delta_{i}^{0}|TL^{-r}),\text{
and}$
$\displaystyle\zeta_{i5}=T^{-1}(\delta_{i}^{0})^{2}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{1}_{T}\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{1}_{T}=T^{-1}(\delta_{i}^{0})^{2}\left(\sum_{t=1}^{T}h_{t}\right)^{2}=O(T|\delta_{i}^{0}|^{2}).$

These results above, together with (A.7), imply that the proposed
$M_{NT}$-based test is consistent provided that $\max_{1\leq i\leq
N}|\delta_{i}^{0}|\gtrsim\sqrt{\log(N)/T}$. ∎

###  7.3 Proof of Theorem 3.1

###### Proof.

Step 1. Investigate the asymptotic independence of $M_{NT}$ and
$(S_{NT}-\widehat{\mu}_{NT})/\widehat{\sigma}_{NT}$ under Gaussian case, i.e.
$\bm{e}_{t}\sim N(\bm{0},\bm{\Sigma})$. Using the same notations given in
(7.1), we have

$S_{NT}=N^{-1}\sum_{i=1}^{N}(\varphi_{i}+\zeta_{i1}+\zeta_{i2}+\zeta_{i3}+\zeta_{i4}+\zeta_{i5}).$

According to the proof of Theorem 1 and 2 in Ma et al. (2020), we have under
$H_{0}$,

$(S_{NT}-\mu_{NT})/\sigma_{NT}=\varphi_{NT}/\sigma_{NT}+o_{p}(1),$

where

$\varphi_{NT}=N^{-1}\sum_{i=1}^{N}\varphi_{i}-\mu_{NT}=2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{e}_{t}^{\prime}\bm{e}_{s}h_{t}h_{s}.$

According to the proof of Theorem 2.1, we have under $H_{0}$,

$M_{NT}=\max_{1\leq i\leq N}\sigma_{ii}^{-1}\varphi_{i}+o_{p}(1),$

where $\varphi_{i}=T^{-1}(\sum_{t=1}^{T}e_{it}h_{t})^{2}$. Hence, by Lemma
7.5, it suffices to show that $\varphi_{NT}/\sigma_{NT}$ and $\max_{1\leq
i\leq N}\varphi_{i}/\sigma_{ii}$ are asymptotically independent.

For any fixed $x,y\in\mathbb{R}$, define
$A_{N}=A_{N}(x)=\\{\varphi_{NT}/\sigma_{NT}\leq x\\}$ and
$B_{i}=B_{i}(y)=\\{\varphi_{i}/\sigma_{ii}>2\log(N)-\log\\{\log(N)\\}+y\\}$
for $i=1,...,N$. Then $\mathbb{P}(A_{N})\to\Phi(x)$ and
$\mathbb{P}(\cup_{i=1}^{N}B_{i})\to 1-F(y)$. Our goal is to prove that

$\mathbb{P}\left(\bigcup\limits_{i=1}^{N}A_{N}B_{i}\right)\to\Phi(x)\\{1-F(y)\\}.$

For each $d\geq 1$, define

$\displaystyle\zeta(N,d)=\sum_{1\leq i_{1}<...<i_{d}\leq
N}|\mathbb{P}(A_{N}B_{i_{1}}...B_{i_{d}})-\mathbb{P}(A_{N})\mathbb{P}(B_{i_{1}}...B_{i_{d}})|,$
$\displaystyle H(N,d)=\sum_{1\leq i_{1}<...<i_{d}\leq
N}|\mathbb{P}(B_{i_{1}}...B_{i_{d}})|.$

By the inclusion-exclusion principle, we observe that for any integer $k\geq
1$,

$\displaystyle\mathbb{P}\left(\bigcup\limits_{i=1}^{N}A_{N}B_{i}\right)$
$\displaystyle\leq$ $\displaystyle\sum_{1\leq i_{1}\leq
N}\mathbb{P}(A_{N}B_{i_{1}})-\sum_{1\leq i_{1}<i_{2}\leq
N}\mathbb{P}(A_{N}B_{i_{1}}B_{i_{2}})+...+\sum_{1\leq i_{1}<...<i_{2k}\leq
N}\mathbb{P}(A_{N}B_{i_{1}}...B_{i_{2k}})$ $\displaystyle\leq$
$\displaystyle\mathbb{P}(A_{N})\left\\{\sum_{1\leq i_{1}\leq
N}\mathbb{P}(B_{i_{1}})-\sum_{1\leq i_{1}<i_{2}\leq
N}\mathbb{P}(B_{i_{1}}B_{i_{2}})+...+\sum_{1\leq i_{1}<...<i_{2k}\leq
N}\mathbb{P}(B_{i_{1}}...B_{i_{2k}})\right\\}$
$\displaystyle\qquad+\sum_{d=1}^{2k}\zeta(N,d)+H(N,2k+1)$ $\displaystyle\leq$
$\displaystyle\mathbb{P}(A_{N})\mathbb{P}\left(\bigcup\limits_{i=1}^{N}B_{i}\right)+\sum_{d=1}^{2k}\zeta(N,d)+H(N,2k+1).$

According to the proof of Theorem 2.1, we have for each $d$,

$\lim\limits_{N\to\infty}H(N,d)=\pi^{-1/2}\frac{1}{d!}\exp\left(-\frac{dx}{2}\right).$

We claim that for each $d$,

$\lim\limits_{N\to\infty}\zeta(N,d)\to 0.$ (C.1)

Then, by letting $k\to\infty$, we have

$\limsup\limits_{N\to\infty}\mathbb{P}\left(\bigcup\limits_{i=1}^{N}A_{N}B_{i}\right)\leq\Phi(x)\\{1-F(y)\\}.$

Likewise, we have

$\liminf\limits_{N\to\infty}\mathbb{P}\left(\bigcup\limits_{i=1}^{N}A_{N}B_{i}\right)\geq\Phi(x)\\{1-F(y)\\}.$

Hence, the desired result follows.

It remains to prove that the claim (C.1) indeed holds.

For each $t$, let $\bm{e}_{(1),t}=(e_{i_{1},t},...,e_{i_{d},t})^{\prime}$,
$\bm{e}_{(2),t}=(e_{i_{d+1},t},...,e_{i_{N},t})^{\prime}$, and for
$k,l\in\\{1,2\\}$, let
$\bm{\Sigma}_{kl}=\text{Cov}(\bm{e}_{(k),t},\bm{e}_{(l),t})$. By Lemma 7.6,
$\bm{e}_{(2),t}$ can be decomposed as $\bm{e}_{(2),t}=\bm{U}_{t}+\bm{V}_{t}$,
where
$\bm{U}_{t}=\bm{e}_{(2),t}-\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}e_{(1),t}$ and
$\bm{V}_{t}=\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}\bm{e}_{(1),t}$ satisfying

$\bm{U}_{t}\sim
N(\bm{0},\bm{\Sigma}_{22}-\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}\bm{\Sigma}_{12}),\quad\bm{V}_{t}\sim
N(\bm{0},\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}\bm{\Sigma}_{12})\text{ and
}\bm{U}_{t}\text{ and }\bm{e}_{(1),t}\text{ are independent.}$

Thus, we have

$\displaystyle\varphi_{NT}$
$\displaystyle=2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\left(\bm{U}_{t}^{\prime}\bm{U}_{s}+\bm{e}_{(1),t}^{\prime}\bm{e}_{(1),s}+2\bm{V}_{t}^{\prime}\bm{U}_{s}+\bm{V}_{t}^{\prime}\bm{V}_{s}\right)h_{t}h_{s}$
$\displaystyle=:\varphi^{*}_{1}+\Theta_{1}+\Theta_{2}+\Theta_{3},$

where
$\varphi^{*}_{1}=2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{U}_{t}^{\prime}\bm{U}_{s}h_{t}h_{s}$,
$\Theta_{1}=2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{e}_{(1),t}^{\prime}\bm{e}_{(1),s}h_{t}h_{s}$,
$\Theta_{2}=2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}2\bm{V}_{t}^{\prime}\bm{U}_{s}h_{t}h_{s}$,
$\Theta_{2}=2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{V}_{t}^{\prime}\bm{V}_{s}h_{t}h_{s}$
and $\varphi^{*}_{2}=\Theta_{1}+\Theta_{2}+\Theta_{3}$.

We claim that for any $\epsilon>0$, $\exists$ a sequence of constants
$c:=c_{N}>0$ with $c_{N}\to\infty$ s.t.

$\mathbb{P}(|\Theta_{k}|\geq\epsilon\sigma_{NT})\leq N^{-c},k=1,2,3,$ (C.2)

for sufficiently large $N$. Consequently,
$\mathbb{P}(|\varphi^{*}_{2}/\sigma_{NT}|\geq\epsilon)\leq N^{-c}$.
Furthermore,

$\displaystyle\mathbb{P}\left(A_{N}(x)B_{i_{1}}...B_{i_{d}}\right)$
$\displaystyle\leq\mathbb{P}\left(A_{N}(x)B_{i_{1}}...B_{i_{d}},|\varphi^{*}_{2}/\sigma_{NT}|<\epsilon\right)+N^{-c}$
$\displaystyle\leq\mathbb{P}\left(|\varphi^{*}_{1}/\sigma_{NT}|<\epsilon+x,B_{i_{1}}...B_{i_{d}}\right)+N^{-c}$
$\displaystyle=\mathbb{P}\left(|\varphi^{*}_{1}/\sigma_{NT}|<\epsilon+x\right)\mathbb{P}\left(B_{i_{1}}...B_{i_{d}}\right)+N^{-c}$
$\displaystyle\leq\left\\{\mathbb{P}\left(|\varphi^{*}_{1}/\sigma_{NT}|<\epsilon+x,|\varphi^{*}_{2}/\sigma_{NT}|<\epsilon\right)+N^{-c}\right\\}\mathbb{P}\left(B_{i_{1}}...B_{i_{d}}\right)+N^{-c}$
$\displaystyle\leq\mathbb{P}\left(A_{N}(x+2\epsilon)\right)\mathbb{P}\left(B_{i_{1}}...B_{i_{d}}\right)+2N^{-c}.$

Likewise,

$\mathbb{P}\left(A_{N}(x)B_{i_{1}}...B_{i_{d}}\right)\geq\mathbb{P}\left(A_{N}(x-2\epsilon)\right)\mathbb{P}\left(B_{i_{1}}...B_{i_{d}}\right)-2N^{-c}.$

Hence,

$\left|\mathbb{P}\left(A_{N}(x)B_{i_{1}}...B_{i_{d}}\right)-\mathbb{P}\left(A_{N}(x)\right)\mathbb{P}\left(B_{i_{1}}...B_{i_{d}}\right)\right|\leq\Delta_{N,\epsilon}\cdot\mathbb{P}\left(B_{i_{1}}...B_{i_{d}}\right)+2N^{-c},$

where

$\displaystyle\Delta_{N,\epsilon}$
$\displaystyle=\left|\mathbb{P}\left(A_{N}(x)\right)-\mathbb{P}\left(A_{N}(x+2\epsilon)\right)\right|+\left|\mathbb{P}\left(A_{N}(x)\right)-\mathbb{P}\left(A_{N}(x-2\epsilon)\right)\right|$
$\displaystyle=\mathbb{P}\left(A_{N}(x+2\epsilon)\right)-\mathbb{P}\left(A_{N}(x-2\epsilon)\right)$

since $\mathbb{P}(A_{N}(x))$ is increasing in $x$. By running over all
possible combinations of $1\leq i_{1}<...<i_{d}\leq N$, we have

$\zeta(N,d)\leq\Delta_{N,\epsilon}\cdot H(N,d)+2\binom{N}{d}\cdot N^{-c}.$

Since $\mathbb{P}(A_{N}(x))\to\Phi(x)$, we have $\lim_{\epsilon\downarrow
0}\limsup_{N\to\infty}\Delta_{N,\epsilon}=\lim_{\epsilon\downarrow
0}\\{\Phi(x+2\epsilon)-\Phi(x-2\epsilon)\\}=0$. Since for each $d\geq 1$,
$H(N,d)\to\pi^{-1/2}\exp(-dx/2)/d!$ as $N\to\infty$, we have
$\limsup_{N\to\infty}H(N,d)<\infty$. Due to the fact that $\binom{N}{d}\leq
N^{d}$ for fixed $d\geq 1$, first sending $N\to\infty$ and then sending
$\epsilon\downarrow 0$, we get (C.1).

It remain to prove that the claim (C.2) indeed holds.

By (B.7) in Ma et al. (2020), $\exists$ constants $0<c_{M}<C_{M}<\infty$ s.t.

$2c_{M}^{2}N^{-2}\text{Tr}(\bm{\Sigma}^{2})\leq\sigma_{NT}^{2}\leq
2C_{M}^{2}N^{-2}\text{Tr}(\bm{\Sigma}^{2})\\{1+o(1)\\}.$ (C.3)

Define $\sigma_{d}^{2}=2c^{2}N^{-2}\text{Tr}(\bm{\Sigma}_{11}^{2})$ with
$c_{M}\leq c\leq C_{M}$, then

$\displaystyle\mathbb{P}(|\Theta_{1}|\geq\epsilon\sigma_{NT})$
$\displaystyle=\mathbb{P}\left(\left|2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{e}_{(1),t}^{\prime}\bm{e}_{(1),s}h_{t}h_{s}\right|\geq\epsilon\sigma_{NT}\right)$
$\displaystyle=\mathbb{P}\left(\left|\sigma_{d}^{-1}\cdot
2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{e}_{(1),t}^{\prime}\bm{e}_{(1),s}h_{t}h_{s}\right|\geq\epsilon^{\prime}\sqrt{\frac{\text{Tr}(\bm{\Sigma}^{2})}{\text{Tr}(\bm{\Sigma}_{11}^{2})}}\right)$
$\displaystyle\leq\exp\left\\{-\epsilon^{\prime}\sqrt{\frac{\text{Tr}(\bm{\Sigma}^{2})}{\text{Tr}(\bm{\Sigma}_{11}^{2})}}\right\\}\cdot\mathbb{E}\left(\exp\left|\sigma_{d}^{-1}\cdot
2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{e}_{(1),t}^{\prime}\bm{e}_{(1),s}h_{t}h_{s}\right|\right)$
$\displaystyle\leq\exp\left\\{-\epsilon^{\prime}\sqrt{\frac{\text{Tr}(\bm{\Sigma}^{2})}{\text{Tr}(\bm{\Sigma}_{11}^{2})}}\right\\}\cdot\log(T),$

where the last inequality follows since

$\sigma_{d}^{-1}\cdot
2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{e}_{(1),t}^{\prime}\bm{e}_{(1),s}h_{t}h_{s}/\log\\{\log(T)\\}\to
0,\text{ a.s. }$

by the law of the iterated logarithm of zero-mean square integrable martingale
(see Theorem 4.8 in Hall et al. (2014)). Similarly,

$\mathbb{P}(|\Theta_{2}|\geq\epsilon\sigma_{NT})\leq\exp\left\\{-\frac{\epsilon^{\prime}}{2}\sqrt{\frac{\text{Tr}(\bm{\Sigma}^{2})}{\text{Tr}(\bm{\Sigma}_{22\cdot
1}\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}\bm{\Sigma}_{12})}}\right\\}\cdot\log(T),$

where $\bm{\Sigma}_{22\cdot
1}=\bm{\Sigma}_{22}-\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}\bm{\Sigma}_{12}$,
and

$\mathbb{P}(|\Theta_{3}|\geq\epsilon\sigma_{NT})\leq\exp\left\\{-\frac{\epsilon^{\prime}}{2}\sqrt{\frac{\text{Tr}(\bm{\Sigma}^{2})}{\text{Tr}[(\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}\bm{\Sigma}_{12})^{2}]}}\right\\}\cdot\log(T).$

It is then easy to see that (C.2) holds.

Step 2. Investigate the asymptotic independence of $M_{NT}$ and
$(S_{NT}-\widehat{\mu}_{NT})/\widehat{\sigma}_{NT}$ when $\bm{e}_{t}$’s are
sub-Gaussian.

According to (C.3), define

$W(\bm{e}_{1},...,\bm{e}_{T})=\sigma_{s}^{-1}\varphi_{NT}=\sigma_{s}^{-1}\left(2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{e}_{t}^{\prime}\bm{e}_{s}h_{t}h_{s}\right),$

where $\sigma_{s}^{2}=2c^{2}N^{-2}\text{Tr}(\bm{\Sigma}^{2})$ with $c_{M}\leq
c\leq C_{M}$. For $\bm{X}=(x_{1},...,x_{q})^{\prime}\in\mathbb{R}^{q}$, we
consider a smooth approximation of the maximum function $\bm{X}\to\max_{1\leq
i\leq q}x_{i}$, namely,

$F_{\beta}(\bm{X})=\beta^{-1}\log\left\\{\sum_{i=1}^{q}\exp(\beta
x_{i})\right\\},$

where $\beta>0$ is the smoothing parameter that controls the level of
approximation. An elementary calculation shows that
$\forall\bm{X}\in\mathbb{R}^{q}$,

$0\leq F_{\beta}(\bm{X})-\max_{1\leq i\leq q}x_{i}\leq\beta^{-1}\log(q),$

see Chernozhukov et al. (2019). W.L.O.G. assume that $\sigma_{ii}=1$ for
$i=1,...,N$. Define

$V(\bm{e}_{1},...,\bm{e}_{T})=F_{\beta}(\sqrt{\varphi_{1}},...,\sqrt{\varphi_{N}})=\beta^{-1}\log\left\\{\sum_{i=1}^{N}\exp\left(\beta
T^{-1/2}\sum_{t=1}^{T}e_{it}h_{t}\right)\right\\}.$

By Lemma 7.5 and setting $\beta=T^{1/8}\log(N)$, it suffices to show that

$\mathbb{P}\left(W(\bm{e}_{1},...,\bm{e}_{T})\leq
x,V(\bm{e}_{1},...,\bm{e}_{T})\leq\sqrt{2\log(N)-\log\\{\log(N)\\}+y}\right)\to\Phi(x)F(y).$

Suppose $\\{\bm{z}_{1},...,\bm{z}_{T}\\}$ are i.i.d. from
$N(\bm{0},\bm{\Sigma})$, and are independent of
$\\{\bm{e}_{1},...,\bm{e}_{T}\\}$. According to the results of step 1, it
remains to show that
$(W(\bm{e}_{1},...,\bm{e}_{T}),V(\bm{e}_{1},...,\bm{e}_{T}))$ has the same
limiting distribution as
$(W(\bm{z}_{1},...,\bm{z}_{T}),V(\bm{z}_{1},...,\bm{z}_{T}))$.

Let $\mathcal{C}_{b}^{3}(\mathbb{R})$ denote the class of bounded functions
with bounded and continuous derivatives up to order 3. It is known that a
sequence of random variables $\\{Z_{n}\\}_{n=1}^{\infty}$ converges weakly to
a random variable $Z$ if and only if for every
$f\in\mathcal{C}_{b}^{3}(\mathbb{R})$,
$\mathbb{E}(f(Z_{n}))\to\mathbb{E}(f(Z))$, see, e.g. Pollard (1984). It
suffices to show that

$\mathbb{E}\\{f(W(\bm{e}_{1},...,\bm{e}_{T}),V(\bm{e}_{1},...,\bm{e}_{T}))\\}-\mathbb{E}\\{f(W(\bm{z}_{1},...,\bm{z}_{T}),V(\bm{z}_{1},...,\bm{z}_{T}))\\}\to
0,$

for every $f\in\mathcal{C}_{b}^{3}(\mathbb{R})$ as $(N,T)\to\infty$. We
introduce $W_{d}=W(\bm{e}_{1},...,\bm{e}_{d-1},\bm{z}_{d},...,\bm{z}_{T})$ and
$V_{d}=V(\bm{e}_{1},...,\bm{e}_{d-1},\bm{z}_{d},...,\bm{z}_{T})$ for
$d=1,...,T+1$. Then

$\displaystyle|\mathbb{E}\\{f(W(\bm{e}_{1},...,\bm{e}_{T}),V(\bm{e}_{1},...,\bm{e}_{T}))\\}-\mathbb{E}\\{f(W(\bm{z}_{1},...,\bm{z}_{T}),V(\bm{z}_{1},...,\bm{z}_{T}))\\}|$
$\displaystyle\leq$
$\displaystyle\sum_{d=1}^{T}|\mathbb{E}\\{f(W_{d},V_{d})\\}-\mathbb{E}\\{f(W_{d+1},V_{d+1})\\}|.$

Let

$\displaystyle
W_{d,0}=2N^{-1}T^{-1}\sigma_{s}^{-1}\left(\sum_{t=2}^{d-1}\sum_{s=1}^{t-1}\bm{e}_{t}^{\prime}\bm{e}_{s}h_{t}h_{s}+\sum_{t=d+2}^{T}\sum_{s=d+1}^{t-1}\bm{z}_{t}^{\prime}\bm{z}_{s}h_{t}h_{s}+\sum_{t=d+1}^{T}\sum_{s=1}^{d-1}\bm{z}_{t}^{\prime}\bm{e}_{s}h_{t}h_{s}\right),$
$\displaystyle V_{d,0}=\beta^{-1}\log\left[\sum_{i=1}^{N}\exp\left\\{\beta
T^{-1/2}\left(\sum_{t=1}^{d-1}e_{it}h_{t}+\sum_{t=d+1}^{T}z_{it}h_{t}\right)\right\\}\right],$

which only rely on
$\mathcal{F}_{d}=\sigma\\{\bm{e}_{1},...,\bm{e}_{d-1},\bm{z}_{d+1},...,\bm{z}_{T}\\}$.
By Taylor’s expansion, we have

$\displaystyle f(W_{d},V_{d})-f(W_{d,0},V_{d,0})$ $\displaystyle=$
$\displaystyle
f_{1}(W_{d,0},V_{d,0})(W_{d}-W_{d,0})+f_{2}(W_{d,0},V_{d,0})(V_{d}-V_{d,0})$
$\displaystyle+\frac{1}{2}f_{11}(W_{d,0},V_{d,0})(W_{d}-W_{d,0})^{2}+\frac{1}{2}f_{22}(W_{d,0},V_{d,0})(V_{d}-V_{d,0})^{2}$
$\displaystyle+\frac{1}{2}f_{12}(W_{d,0},V_{d,0})(W_{d}-W_{d,0})(V_{d}-V_{d,0})$
$\displaystyle+O(|W_{d}-W_{d,0}|^{3})+O(|V_{d}-V_{d,0}|^{3}),$

and

$\displaystyle f(W_{d+1},V_{d+1})-f(W_{d,0},V_{d,0})$ $\displaystyle=$
$\displaystyle
f_{1}(W_{d,0},V_{d,0})(W_{d+1}-W_{d,0})+f_{2}(W_{d,0},V_{d,0})(V_{d+1}-V_{d,0})$
$\displaystyle+\frac{1}{2}f_{11}(W_{d,0},V_{d,0})(W_{d+1}-W_{d,0})^{2}+\frac{1}{2}f_{22}(W_{d,0},V_{d,0})(V_{d+1}-V_{d,0})^{2}$
$\displaystyle+\frac{1}{2}f_{12}(W_{d,0},V_{d,0})(W_{d+1}-W_{d,0})(V_{d+1}-V_{d,0})$
$\displaystyle+O(|W_{d+1}-W_{d,0}|^{3})+O(|V_{d+1}-V_{d,0}|^{3}),$

where $f=f(x,y)$, $f_{1}=\partial f/\partial x$, $f_{2}=\partial f/\partial
y$, $f_{11}=\partial^{2}f/\partial x^{2}$, $f_{22}=\partial^{2}f/\partial
y^{2}$ and $f_{12}=\partial^{2}f/\partial x\partial y$. Notice that

$\displaystyle
W_{d}-W_{d,0}=2N^{-1}T^{-1}\sigma_{s}^{-1}\left(\sum_{s=1}^{d-1}\bm{z}_{d}^{\prime}\bm{e}_{s}h_{d}h_{s}+\sum_{t=d+1}^{T}\bm{z}_{t}^{\prime}\bm{z}_{d}h_{t}h_{d}\right),\text{
and}$ $\displaystyle
W_{d+1}-W_{d,0}=2N^{-1}T^{-1}\sigma_{s}^{-1}\left(\sum_{s=1}^{d-1}\bm{e}_{d}^{\prime}\bm{e}_{s}h_{d}h_{s}+\sum_{t=d+1}^{T}\bm{z}_{t}^{\prime}\bm{e}_{d}h_{t}h_{d}\right).$
(C.4)

Due to $\mathbb{E}(\bm{e}_{t})=\mathbb{E}(\bm{z}_{t})=0$ and
$\mathbb{E}(\bm{e}_{t}\bm{e}_{t}^{\prime})=\mathbb{E}(\bm{z}_{t}\bm{z}_{t}^{\prime})$,
it can be verified that

$\displaystyle\mathbb{E}(W_{d}-W_{d,0}|\mathcal{F}_{d})=\mathbb{E}(W_{d+1}-W_{d,0}|\mathcal{F}_{d}),\text{
and}$
$\displaystyle\mathbb{E}\\{(W_{d}-W_{d,0})^{2}|\mathcal{F}_{d}\\}=\mathbb{E}\\{(W_{d+1}-W_{d,0})^{2}|\mathcal{F}_{d}\\}.$

Hence,

$\displaystyle\mathbb{E}\\{f_{1}(W_{d},V_{d,0})(W_{d+1}-W_{d,0})\\}=\mathbb{E}\\{f_{1}(W_{d,0},V_{d,0})(W_{d+1}-W_{d,0})\\}\text{
and}$
$\displaystyle\mathbb{E}\\{f_{11}(W_{d,0},V_{d,0})(W_{d}-W_{d,0})^{2}\\}=\mathbb{E}\\{f_{11}(W_{d,0},V_{d,0})(W_{d+1}-W_{d,0})^{2}\\}.$

Next consider $V_{d}-V_{d,0}$. Let
$v_{d,0,i}=T^{-1/2}\sum_{t=1}^{d-1}e_{it}h_{t}+T^{-1/2}\sum_{t=d+1}^{T}z_{it}h_{t}$,
$v_{d,i}=v_{d,0,i}+T^{-1/2}z_{id}h_{d}$,
$v_{d+1,i}=v_{d,0,i}+T^{-1/2}e_{id}h_{d}$,
$\bm{v}_{d,0}=(v_{d,0,1},...,v_{d,0,N})^{\prime}$ and
$\bm{v}_{d}=(v_{d,1},...,v_{d,N})^{\prime}$. By Taylor’s expansion, we have

$\displaystyle V_{d}-V_{d,0}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\partial_{i}F_{\beta}(\bm{v}_{d,0})(v_{d,i}-v_{d,0,i})+\frac{1}{2}\sum_{i=1}^{N}\sum_{j=1}^{N}\partial_{i}\partial_{j}F_{\beta}(\bm{v}_{d,0})(v_{d,i}-v_{d,0,i})(v_{d,j}-v_{d,0,j})$
$\displaystyle+\frac{1}{6}\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{N}\partial_{i}\partial_{j}\partial_{k}F_{\beta}(\bm{v}_{d,0}+\delta(v_{d}-v_{d,0}))(v_{d,i}-v_{d,0,i})(v_{d,j}-v_{d,0,j})(v_{d,k}-v_{d,0,k}),$
(C.5)

for some $\delta\in(0,1)$. Again, due to
$\mathbb{E}(\bm{e}_{t})=\mathbb{E}(\bm{z}_{t})=0$ and
$\mathbb{E}(\bm{e}_{t}\bm{e}_{t}^{\prime})=\mathbb{E}(\bm{z}_{t}\bm{z}_{t}^{\prime})$,
it can be verified that

$\mathbb{E}(v_{d,i}-v_{d,0,i}|\mathcal{F}_{d})=\mathbb{E}(v_{d+1,i}-v_{d,0,i}|\mathcal{F}_{d}),\text{
and
}\mathbb{E}\\{(v_{d,i}-v_{d,0,i})^{2}|\mathcal{F}_{d}\\}=\mathbb{E}\\{(v_{d+1,i}-v_{d,0,i})^{2}|\mathcal{F}_{d}\\}.$

By Lemma A.2 in Chernozhukov et al. (2019), we have

$\left|\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{N}\partial_{i}\partial_{j}\partial_{k}F_{\beta}(\bm{v}_{d,0}+\delta(\bm{v}_{d}-\bm{v}_{d,0}))\right|\leq
C\beta^{2}$

for some positive constant $C$. By Assumption 2.3 (ii), we have
$\mathbb{P}(\max_{1\leq i\leq N,1\leq t\leq T}|e_{it}|>C\log(NT))\to 0$, and
since $z_{it}\sim N(0,1)$, $\mathbb{P}(\max_{1\leq i\leq N,1\leq t\leq
T}|z_{it}|>C\log(NT))\to 0$. Hence,

$\displaystyle\left|\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{k=1}^{N}\partial_{i}\partial_{j}\partial_{k}F_{\beta}(\bm{v}_{d,0}+\delta(\bm{v}_{d}-\bm{v}_{d,0}))(v_{d,i}-v_{d,0,i})(v_{d,j}-v_{d,0,j})(v_{d,k}-v_{d,0,k})\right|$
$\displaystyle\leq$ $\displaystyle C\beta^{2}T^{-3/2}\log^{3}(NT)$

holds with probability approaching one. Consequently, we have with probability
approaching one,

$\left|\mathbb{E}\\{f_{2}(W_{d,0},V_{d,0})(V_{d}-V_{d,0})\\}-\mathbb{E}\\{f_{2}(W_{d,0},V_{d,0})(V_{d+1}-V_{d,0})\\}\right|\leq
C\beta^{2}T^{-3/2}\log^{3}(NT).$

Similarly, it can be verified that

$\left|\mathbb{E}\\{f_{22}(W_{d,0},V_{d,0})(V_{d}-V_{d,0})^{2}\\}-\mathbb{E}\\{f_{2}(W_{d,0},V_{d,0})(V_{d+1}-V_{d,0})^{2}\\}\right|\leq
C\beta^{2}T^{-3/2}\log^{3}(NT),$

and

$\displaystyle\left|\mathbb{E}\\{f_{12}(W_{d,0},V_{d,0})(W_{d}-W_{d,0})(V_{d}-V_{d,0})\\}-\mathbb{E}\\{f_{12}(W_{d,0},V_{d,0})(W_{d+1}-W_{d,0})(V_{d+1}-V_{d,0})\\}\right|$
$\displaystyle\leq$ $\displaystyle C\beta^{2}T^{-3/2}\log^{3}(NT).$

Again, Lemma A.2 in Chernozhukov et al. (2019), together with (7.3), implies
that $\mathbb{E}(|V_{d}-V_{d,0}|^{3})=O(T^{-3/2}\log^{3}(NT))$. According to
(7.3) and the proof of Lemma A.5 in Ma et al. (2020), we have
$\mathbb{E}(|W_{d}-W_{d,0}|^{4})=O(T^{-2})$, thus

$\sum_{d=1}^{T}\mathbb{E}(|W_{d}-W_{d,0}|^{3})\leq\sum_{d=1}^{T}\\{\mathbb{E}(|W_{d}-W_{d,0}|^{4})\\}^{3/4}=O(T^{-1/2}).$

Combining all facts together, we conclude that

$\sum_{d=1}^{T}|\mathbb{E}\\{f(W_{d},V_{d})\\}-\mathbb{E}\\{f(W_{d+1},V_{d+1})\\}|=O(\beta^{2}T^{-3/2}\log^{3}(NT))+O(T^{-1/2})\to
0,$

as $(N,T)\to\infty$, due to $\log(N)=o(T^{1/4})$. Then the proof of Theorem
3.1 is complete. ∎

###  7.4 Proof of Theorem 3.2

###### Proof.

It suffices to show that the conclusion holds for Gaussian $\bm{e}_{t}$’s.
Using the notation given in (7.1), we have

$\displaystyle M_{NT}=\max_{1\leq i\leq
N}\widehat{\sigma}_{ii}^{-1}(\varphi_{i}+\zeta_{i1}+\zeta_{i2}+\zeta_{i3}+\zeta_{i4}+\zeta_{i5})\text{
and }$ $\displaystyle
S_{NT}=N^{-1}\sum_{i=1}^{N}(\varphi_{i}+\zeta_{i1}+\zeta_{i2}+\zeta_{i3}+\zeta_{i4}+\zeta_{i5}).$

Under the alternative hypothesis given in (10), according to the proof of
Theorem 2.1, we have

$\displaystyle M_{NT}$ $\displaystyle=\max_{1\leq i\leq
N}\sigma_{ii}^{-1}\left\\{\varphi_{i}+T^{-1}(\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{1}_{T})^{2}(\delta_{i}^{0})^{2}\right\\}+o_{p}(1)$
$\displaystyle=\max_{i\in\mathcal{A}}\sigma_{ii}^{-1}\left\\{\varphi_{i}+T^{-1}(\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{1}_{T})^{2}(\delta_{i}^{0})^{2}\right\\}+\max_{i\in\mathcal{A}^{c}}\sigma_{ii}^{-1}\varphi_{i}+o_{p}(1),$

where $\varphi_{i}=T^{-1}(\sum_{t=1}^{T}e_{it}h_{t})^{2}$. In addition, by
Lemma A.7 in Ma et al. (2020), we have

$(S_{NT}-\mu_{NT})/\sigma_{NT}=\sigma_{NT}^{-1}\varphi_{NT}+\sigma_{NT}^{-1}N^{-1}T^{-1}(\bm{1}_{T}^{\prime}M_{\bm{Z}}\bm{1}_{T})^{2}\sum_{i=1}^{N}(\delta_{i}^{0})^{2}+o_{p}(1),$

where
$\varphi_{NT}=2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{e}_{t}^{\prime}\bm{e}_{s}h_{t}h_{s}$.
Define $\bm{e}_{\mathcal{A},t}=(e_{it},i\in\mathcal{A})^{\prime}$ and
$\bm{e}_{\mathcal{A}^{c},t}=(e_{it},i\in\mathcal{A}^{c})^{\prime}$. Then, we
rewrite

$\displaystyle\varphi_{NT}$
$\displaystyle=2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{e}_{\mathcal{A},t}^{\prime}\bm{e}_{\mathcal{A},s}h_{t}h_{s}+2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{e}_{\mathcal{A}^{c},t}^{\prime}\bm{e}_{\mathcal{A}^{c},s}h_{t}h_{s}$
$\displaystyle=:\varphi_{NT,\mathcal{A}}+\varphi_{NT,\mathcal{A}^{c}}.$

According to the proof of Theorem 3.1, we have known that
$\sigma_{NT}^{-1}\varphi_{NT,\mathcal{A}^{c}}$ and
$\max_{i\in\mathcal{A}^{c}}\sigma_{ii}^{-1}\varphi_{i}$ are asymptotically
independent. Hence, it suffices to show that
$\sigma_{NT}^{-1}\varphi_{NT,\mathcal{A}^{c}}$ is asymptotically independent
of $\varphi_{i},i\in\mathcal{A}$.

Define
$\bm{\Sigma}_{\mathcal{A},\mathcal{A}^{c}}=\text{Cov}(\bm{e}_{\mathcal{A},t},\bm{e}_{\mathcal{A}^{c},t})$.
By Lemma 7.6, $\bm{e}_{\mathcal{A}^{c},t}$ can be decomposed as
$\bm{e}_{\mathcal{A}^{c},t}=\bm{U}_{t}+\bm{V}_{t}$, where
$\bm{U}_{t}=\bm{e}_{\mathcal{A}^{c},t}-\bm{\Sigma}_{\mathcal{A}^{c},\mathcal{A}}\bm{\Sigma}_{\mathcal{A},\mathcal{A}}^{-1}\bm{e}_{\mathcal{A},t}$
and
$\bm{V}_{t}=\bm{\Sigma}_{\mathcal{A}^{c},\mathcal{A}}\bm{\Sigma}_{\mathcal{A},\mathcal{A}}^{-1}\bm{e}_{\mathcal{A},t}$
satisfying that $\bm{U}_{t}\sim
N(\bm{0},\bm{\Sigma}_{\mathcal{A}^{c},\mathcal{A}^{c}}-\bm{\Sigma}_{\mathcal{A}^{c},\mathcal{A}}\bm{\Sigma}_{\mathcal{A},\mathcal{A}}^{-1}\bm{\Sigma}_{\mathcal{A},\mathcal{A}^{c}})$,
$\bm{V}_{t}\sim
N(\bm{0},\bm{\Sigma}_{\mathcal{A}^{c},\mathcal{A}}\bm{\Sigma}_{\mathcal{A},\mathcal{A}}^{-1}\bm{\Sigma}_{\mathcal{A},\mathcal{A}^{c}})$
and

$\bm{U}_{t}\text{ and }\bm{e}_{\mathcal{A},t}\text{ are independent}.$ (C.6)

Then, we have

$\varphi_{NT,\mathcal{A}^{c}}=2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}(\bm{U}_{t}^{\prime}\bm{U}_{s}+2\bm{U}_{t}^{\prime}\bm{V}_{s}+\bm{V}_{t}^{\prime}\bm{V}_{s})h_{t}h_{s}.$

By using the arguments similar to those in the proof of (C.2), we have

$\displaystyle\mathbb{P}\left(2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}2\bm{U}_{t}^{\prime}\bm{V}_{s}h_{t}h_{s}\geq\epsilon\sigma_{NT}\right)\leq\log(T)\exp(-c_{\epsilon}N^{1/2}|\mathcal{A}|^{-1/2})\to
0\text{ and}$
$\displaystyle\mathbb{P}\left(2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{V}_{t}^{\prime}\bm{V}_{s}h_{t}h_{s}\geq\epsilon\sigma_{NT}\right)\leq\log(T)\exp(-c_{\epsilon}N^{1/2}|\mathcal{A}|^{-1/2})\to
0,$

due to $|\mathcal{A}|=o(N/[\log\\{\log(N)\\}]^{2})$ and $\log(N)=o(T^{1/4})$.
Consequently, we conclude that

$\sigma_{NT}^{-1}\varphi_{NT,\mathcal{A}^{c}}=\sigma_{NT}^{-1}\cdot
2N^{-1}T^{-1}\sum_{t=2}^{T}\sum_{s=1}^{t-1}\bm{U}_{t}^{\prime}\bm{U}_{s}h_{t}h_{s}+o_{p}(1),$

which, together with Lemma 7.5 and (C.6), implies that
$\sigma_{NT}^{-1}\varphi_{NT,\mathcal{A}^{c}}$ is asymptotically independent
of $e_{it},i\in\mathcal{A}$. Hence, Theorem 3.2 follows. ∎

###  7.5 Some useful facts

######  Lemma 7.1.

(Bonferroni inequality) Let $A=\cup_{t=1}^{n}A_{t}$. For any integer $k$,
where $1\leq k\leq n/2$, we have

$\sum\nolimits_{t=1}^{2k}(-1)^{t-1}E_{t}\leq\mathbb{P}(A)\leq\sum\nolimits_{t=1}^{2k-1}(-1)^{t-1}E_{t},$

where $E_{t}=\sum_{1\leq i_{1}<...<i_{t}\leq n}\mathbb{P}(A_{i_{1}}\cap...\cap
A_{i_{t}})$.

###### Proof.

See Lemma 1 in Cai et al. (2014). ∎

######  Lemma 7.2.

Let $(Z_{1},...,Z_{N})^{\prime}$ be a zero mean multivariate normal random
vector with covariance matrix $\bm{\Sigma}=(\sigma_{ij})_{N\times N}$ and
diagonal $\sigma_{ii}=1$ for $1\leq i\leq N$. Suppose that
$\max\nolimits_{1\leq i<j\leq N}|\sigma_{ij}|\leq r<1$ and
$\max\nolimits_{1\leq j\leq N}\sum\nolimits_{i=1}^{N}\sigma_{ij}^{2}\leq c$
for some $r$ and $c$. Then foe any $x\in\mathbb{R}$ as $N\to\infty$,

$\mathbb{P}\left[\max\nolimits_{1\leq i\leq
N}Z_{i}^{2}-2\log(N)+\log\\{\log(N)\\}\leq
x\right]\to\exp\left\\{-\pi^{-1/2}\exp(-x/2)\right\\}.$

###### Proof.

See Lemma 6 in Cai et al. (2014). ∎

######  Lemma 7.3.

Define
$\rho_{i0t}=\delta_{i}(t/T)-\bm{\lambda}_{0,i0}^{\prime}\widetilde{\bm{B}}(t/T)$
and $\rho_{ijt}=\beta_{ij}(t/T)-\bm{\lambda}_{0,ij}^{\prime}\bm{B}(t/T)$ for
$1\leq j\leq d$ and $1\leq i\leq N$. Then, under Assumption 2.1, there exist
$\bm{\lambda}_{0,i0}\in\mathbb{R}^{L}$ and
$\bm{\lambda}_{0,ij}\in\mathbb{R}^{L}$ such that

$\sup\nolimits_{1\leq t\leq T}|\rho_{i0t}|=O(L^{-r})\text{ and
}\sup\nolimits_{1\leq t\leq T}|\rho_{ijt}|=O(L^{-r})\text{ as }T\to\infty.$

###### Proof.

See Lemma A.1 in Ma et al. (2020). ∎

######  Lemma 7.4.

Under Assumption 2.2, $\exists$ constants $0<c_{1}\leq C_{1}<\infty$ and
$0>C_{2}<\infty$, with probability 1,

$c_{1}L^{-1}\leq\lambda_{\min}(\bm{Z}^{\prime}\bm{Z}/T)\leq\lambda_{\max}(\bm{Z}^{\prime}\bm{Z}/T)\leq
C_{1}L^{-1},$
$C_{1}L\leq\lambda_{\min}\\{(\bm{Z}^{\prime}\bm{Z}/T)^{-1}\\}\leq\lambda_{\max}\\{(\bm{Z}^{\prime}\bm{Z}/T)^{-1}\\}\leq
c_{1}L,$

as $T\to\infty$, and for any onozero vector $\bm{a}\in\mathbb{R}^{T}$ with
$||\bm{a}||=1$, $\bm{a}^{\prime}(\bm{Z}^{\prime}\bm{Z}/T)\bm{a}\leq
C_{2}L^{-1}$.

###### Proof.

See Lemma A.2 in Ma et al. (2020). ∎

######  Lemma 7.5.

Let $\\{(U,U_{N},\widetilde{U}_{N})\in\mathbb{R}^{3};N\geq 1\\}$ and
$\\{(V,V_{N},\widetilde{V}_{N})\in\mathbb{R}^{3};N\geq 1\\}$ be two sequences
of random variables with $U_{N}\stackrel{{\scriptstyle d}}{{\to}}U$ and
$V_{N}\stackrel{{\scriptstyle d}}{{\to}}V$ as $N\to\infty$. Assume $U$ and $V$
are continuous random variables. We assume that

$\widetilde{U}_{N}=U_{N}+o_{p}(1)\text{ and
}\widetilde{V}_{N}=V_{N}+o_{p}(1).$

If $U_{N}$ and $V_{N}$ are asymptotically independent, then
$\widetilde{U}_{N}$ and $\widetilde{V}_{N}$ are also asymptotically
independent.

###### Proof.

See Lemma 7.10 in Feng et al. (2023). ∎

######  Lemma 7.6.

Let $\bm{X}\sim N(\bm{\mu},\bm{\Sigma})$ with invertible $\bm{\Sigma}$, and
partition $\bm{X}$, $\bm{\mu}$ and $\bm{\Sigma}$ as

$\bm{X}=\left(\begin{array}[]{l}\bm{X}_{1}\\\
\bm{X}_{2}\end{array}\right),\quad\bm{\mu}=\left(\begin{array}[]{l}\bm{\mu}_{1}\\\
\bm{\mu}_{2}\end{array}\right),\quad\bm{\Sigma}=\left(\begin{array}[]{ll}\bm{\Sigma}_{11}&\bm{\Sigma}_{12}\\\
\bm{\Sigma}_{21}&\bm{\Sigma}_{22}\end{array}\right).$

Then $\bm{X}_{2}-\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}\bm{X}_{1}\sim
N(\bm{\mu}_{2}-\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}\bm{\mu}_{1},\bm{\Sigma}_{22\cdot
1})$ and is independent of $\bm{X}_{1}$, where $\bm{\Sigma}_{22\cdot
1}=\bm{\Sigma}_{22}-\bm{\Sigma}_{21}\bm{\Sigma}_{11}^{-1}\bm{\Sigma}_{12}$.

###### Proof.

See Theorem 1.2.11 in Muirhead (2009). ∎

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