# Large Global Volatility Matrix Analysis Based on Structural Information

Sung Hoon Choi
Department of Economics, University of Connecticut, Storrs, CT 06269, USA.
E-mail<EMAIL_ADDRESS>Donggyu Kim
College of Business, Korea Advanced Institute of Science and Technology
(KAIST), Seoul, Republic of Korea. Email<EMAIL_ADDRESS>
###### Abstract

In this paper, we develop a novel large volatility matrix estimation procedure
for analyzing global financial markets. Practitioners often use lower-
frequency data, such as weekly or monthly returns, to address the issue of
different trading hours in the international financial market. However, this
approach can lead to inefficiency due to information loss. To mitigate this
problem, our proposed method, called Structured Principal Orthogonal
complEment Thresholding (Structured-POET), incorporates structural information
for both global and national factor models. We establish the asymptotic
properties of the Structured-POET estimator, and also demonstrate the
drawbacks of conventional covariance matrix estimation procedures when using
lower-frequency data. Finally, we apply the Structured-POET estimator to an
out-of-sample portfolio allocation study using international stock market
data.

Key words: High-dimensionality, international financial market, low-rank
matrix, multi-level factor model, POET, sparsity.

##  1 Introduction

Factor analysis and principal component analysis (PCA) are commonly used for
various applications, including macroeconomic variable forecasting and
portfolio allocation optimization (Bai, 2003; Bernanke et al., 2005; Fan et
al., 2016; Stock and Watson, 2002). Recent research has highlighted the
importance of considering local factors in addition to global factors. These
local factors have an impact on individuals in each local group and can be
defined by regional, country, or industry level (Bekaert et al., 2009; Fama
and French, 2012; Kose et al., 2003; Moench et al., 2013). To account for
different level factors, a multi-level factor structure has been developed.
See Ando and Bai (2015, 2016); Bai and Wang (2015); Choi et al. (2018); Han
(2021) for more information.

Several large volatility matrix estimation procedures have been developed
based on latent factor models to account for the strong cross-sectional
correlation in the stock market (Ait-Sahalia and Xiu, 2017; Fan et al., 2016,
2018a; Fan and Kim, 2019; Jung et al., 2022; Wang and Fan, 2017). For
instance, under the single-level factor model, Fan et al. (2013) proposed a
principal orthogonal component thresholding (POET) covariance matrix
estimation procedure when the factors are unobservable. This method can
consistently estimate unobservable factors using PCA and cross-sectionally
correlated errors via thresholding with a large number of assets. Recently, to
account for the latent local factor structure, Choi and Kim (2023) developed a
Double-POET covariance matrix estimation procedure based on the multi-level
factor structure. Specifically, Double-POET is a two-step estimation procedure
that estimates global and national factors separately by applying PCA at each
factor level based on the block local factor structure.

When analyzing global financial market data, it is common practice to use
lower frequencies, such as two-day average, weekly, monthly, or quarterly
data, instead of daily returns (Ando and Bai, 2017; Bekaert et al., 2009; Chib
et al., 2006; Choi and Kim, 2023; Fama and French, 2012; Hou et al., 2011).
This is because international stock markets operate at different times, which
results in returns being based on different information sets when measured on
any given date over a short period, such as daily. Hence, practitioners often
opt for using weekly or monthly returns to mitigate the impact of non-
synchronized trading hours in international markets. Choi and Kim (2023) also
used weekly returns to analyze the large global volatility matrix. However,
using lower-frequency data can lead to a loss of information and less
efficient estimators. Thus, it is important to study the non-synchronized
trading hours in international markets and develop an efficient large global
volatility matrix estimation procedure.

This paper proposes a novel large volatility matrix estimation procedure that
incorporates structural information with an entire set of observations in the
global and national factor model. Specifically, we consider the international
financial market and impose a latent multi-level factor structure to account
for both global and national risk factors. To handle the non-synchronized
trading hours in the international market, we assume that the correlation is
stationary with respect to the relative trading hour difference. For example,
if the proportion of the overlapped time goes to one, its corresponding
correlation converges to the correlation of the synchronized time, which is
the parameter of interest. This structure helps theoretically understand the
non-synchronized trading hour problem in the international market, and we
study the large global volatility matrix estimation problem under the proposed
stationary global and national factor model. For example, national factor
membership is assumed to be naturally known, and we further assume that
countries within the same continent have the same information set. To estimate
the global volatility matrix, we first apply the Double-POET procedure in each
continental group using daily returns, which have the finest information in
our model setup. To capture the spillover effect between continents, we
conduct a low-rank approximation to each continental pair using a lower-
frequency, which helps mitigate the effect of the non-synchronized trading
hours. Finally, to accommodate the latent global factors, we employ PCA on the
structurally fitted global factor components from previous procedures, which
we call Structured Principal Orthogonal complEment Thresholding (Structured-
POET). We derive the rates of convergence for the Structured-POET estimator
under different matrix norms and discuss its benefits compared to the Double-
POET procedure. For example, the Double-POET estimator may not be consistent
using daily returns, while the Double-POET estimator with lower-frequency
returns has a slower convergence rate than the Structured-POET estimator. An
empirical study on portfolio allocation also supports the theoretical findings
and shows the benefit of the proposed Structured-POET estimator.

The remainder of the paper is organized as follows. In Section 2, we introduce
the model and propose the Structured-POET estimation procedure. Section 3
provides an asymptotic analysis of the Structured-POET estimator. In Section
4, we conduct a simulation study to evaluate the finite sample performance of
the proposed method. Section 5 applies the proposed method to a real data
problem of portfolio allocation using global stock market data. Finally, the
conclusion is provided in Section 6. All proofs are presented in Appendix A.

##  2 Model Setup and Estimation Procedure

Throughout this paper, let $\lambda_{\min}(\mbox{\bf A})$ and
$\lambda_{\max}(\mbox{\bf A})$ denote the minimum and maximum eigenvalues of
matrix A, respectively. In addition, we denote by $\|\mbox{\bf A}\|_{F}$,
$\|\mbox{\bf A}\|_{2}$ (or $\|\mbox{\bf A}\|$ for short), $\|\mbox{\bf
A}\|_{\infty}$, and $\|\mbox{\bf A}\|_{\max}$ the Frobenius norm, operator
norm, $l_{\infty}$-norm and elementwise norm, which are defined, respectively,
as $\|\mbox{\bf A}\|_{F}=\mathrm{tr}^{1/2}(\mbox{\bf A}^{\prime}\mbox{\bf
A})$, $\|\mbox{\bf A}\|_{2}=\lambda_{\max}^{1/2}(\mbox{\bf
A}^{\prime}\mbox{\bf A})$, $\|\mbox{\bf
A}\|_{\infty}=\max_{i}\sum_{j}|a_{ij}|$, and $\|\mbox{\bf
A}\|_{\max}=\max_{i,j}|a_{ij}|$. When A is a vector, the maximum norm is
denoted as $\|\mbox{\bf A}\|_{\infty}=\max_{i}|a_{i}|$, and both $\|\mbox{\bf
A}\|$ and $\|\mbox{\bf A}\|_{F}$ are equal to the Euclidean norm. We denote
$\mathrm{diag}(\mbox{\bf A}_{1},\ldots,\mbox{\bf A}_{n})$ with the diagonal
block entries as $\mbox{\bf A}_{1},\ldots,\mbox{\bf A}_{n}$.

###  2.1 Multi-Level Factor Model

We consider a global and national factor model (Choi and Kim, 2023):

$y_{it}=b_{i}^{\prime}G_{t}+{\lambda_{i}^{l}}^{\prime}f_{t}^{l}+u_{it},\text{
for }i=1,\dots,p,t=1,\dots,T,\text{ and }l=1,\dots,L,$ (2.1)

where $y_{it}$ is the observed data for the $i$th individual belonging to
country $l$ at time $t$; $G_{t}$ is a $k\times 1$ vector of unobserved latent
global factors, and $b_{i}$ is the global factor loadings; $f_{t}^{l}$ is an
$r_{l}\times 1$ vector of unobserved latent national factors that affect
individuals belonging to country $l$, and $\lambda_{i}^{l}$ is the
corresponding national factor loadings; and $u_{it}$ is an idiosyncratic error
term, which is uncorrelated with $G_{t}$ and $f_{t}^{l}$. Throughout the
paper, global and national factors are uncorrelated, while their factor
loadings may not be orthogonal to each other. In addition, we assume that the
group membership of the national factors is known and the numbers of factors,
$k$ and $r_{l}$, are fixed.

Due to the known national factor membership, we can stack the observations and
write the model (2.1) in a vector form as follows:

$y_{t}=\mbox{\bf B}G_{t}+\mbox{\boldmath$\Lambda$}F_{t}+u_{t},$ (2.2)

where $y_{t}=({y_{t}^{1}}^{\prime},\dots,{y_{t}^{L}}^{\prime})^{\prime}$,
where
$y_{t}^{l}=(y_{p_{1}+\cdots+p_{l-1}+1t},\dots,y_{p_{1}+\cdots+p_{l}t})^{\prime}$
and the number of individuals $p_{l}$ within country $l$; the $p\times k$
matrix $\mbox{\bf B}=(b_{1},\dots,b_{p})^{\prime}$; the $p\times r$ block
diagonal matrix
$\mbox{\boldmath$\Lambda$}=\mathrm{diag}(\Lambda^{1},\dots,\Lambda^{L})$,
where $\Lambda^{l}=(\lambda_{1}^{l},\dots,\lambda_{p_{l}}^{l})^{\prime}$ is a
$p_{l}\times r_{l}$ matrix of local factor loadings for each $l$ such that
$r=\sum_{l=1}^{L}r_{l}$; the $r\times 1$ vector
$F_{t}={(f_{t}^{1}}^{\prime},\dots,{f_{t}^{l}}^{\prime})^{\prime}$, where
$f_{t}^{l}$ is an $r_{l}\times 1$ vector of local factors; and
$u_{t}=(u_{1t},\dots,u_{pt})^{\prime}$.

In this paper, we are interested in the $p\times p$ covariance matrix of
$y_{t}$ and its inverse matrix:

$\mbox{\boldmath$\Sigma$}=\mbox{\bf B}\mathrm{cov}(G_{t})\mbox{\bf
B}^{\prime}+\mbox{\boldmath$\Lambda$}\mathrm{cov}(F_{t})\mbox{\boldmath$\Lambda$}^{\prime}+\mbox{\boldmath$\Sigma$}_{u}:=\mbox{\boldmath$\Sigma$}_{g}+\mbox{\boldmath$\Sigma$}_{l}+\mbox{\boldmath$\Sigma$}_{u},$
(2.3)

where $\mbox{\boldmath$\Sigma$}_{u}$ is a sparse idiosyncratic covariance
matrix of $u_{t}$. In particular, we measure the sparsity level of
$\mbox{\boldmath$\Sigma$}_{u}=(\sigma_{u,ij})_{p\times p}$ as follows (Bickel
and Levina, 2008; Cai and Liu, 2011; Rothman et al., 2009): for some
$q\in[0,1)$,

$\displaystyle m_{p}=\max_{i\leq p}\sum_{j\leq p}|\sigma_{u,ij}|^{q},$ (2.4)

which diverges slowly, such as $\log p$. This implies that most pairs are
weakly cross-sectionally correlated in the idiosyncratic error component.
Decomposition (2.3) is a multi-level factor-based covariance matrix. Under the
pervasive assumption, there are distinguished eigenvalues among the global
factor components, the local factor components, and the idiosyncratic error
components. Hence, we can analyze the model by the presence of distinguished
eigenvalues at different levels (see Section 2.2). We note that the
correlation matrix of $y_{t}$ can be obtained by

$\mbox{\bf R}_{0}=(\rho_{0,ij})_{p\times
p}=D_{0}^{-\frac{1}{2}}\mbox{\boldmath$\Sigma$}D_{0}^{-\frac{1}{2}},$ (2.5)

where $D_{0}$ is the diagonal matrix consisting of the diagonal elements of
$\Sigma$.

When we consider the international stock market, each stock exchange has its
own trading hours. Hence, stocks in different exchanges have different
observation time points, so we cannot estimate $\rho_{0,ij}$ if stocks $i$ and
$j$ are not in the same region. To demonstrate this, we represent the $i$th
observations in region $s$ as $\\{y_{i,t+\delta_{s}}\\}_{t=1}^{T}$, and
$\delta_{s}\in[0,1)$ is the market close time for $s\in\\{1,\dots,S\\}$. In
this paper, we assume that
$\\{(y_{1,t+\delta_{1}},\dots,y_{p,t+\delta_{S}})^{\prime}\\}_{t\geq 1}$ is
stationary. We then denote the “estimable” correlation by $\rho_{h,ij}$ for
assets $i$ and $j$ that are located in regions $s,q\in\\{1,\dots,S\\}$,
respectively, where the relative time difference
$h=\frac{|\delta_{s}-\delta_{q}|}{d}$ and the window size of frequency
$d=T^{1-\alpha}$ for $\alpha\in(0,1]$. Finally, we impose the following
Lipschitz condition for $\rho_{h,ij}$:

$|\rho_{h,ij}-\rho_{0,ij}|\leq Ch^{\beta},$ (2.6)

for some $\beta>0$ and a positive constant $C$. This model setup provides a
mathematical framework to understand a fraction of the global market. For
example, from the proposed model setup perspective, when we use data with
$d=1$, $\rho_{h,ij}$ does not converge to the synchronized correlation
$\rho_{0,ij}$. In contrast, when using lower-frequency data, $\rho_{h,ij}$
converges to $\rho_{0,ij}$. Thus, in practice, researchers often use weekly or
monthly returns (i.e., $d=5\text{ or }21$) instead of daily returns (i.e.,
$d=1$) to mitigate the effect of different trading hours based on daily
transaction prices (Ando and Bai, 2017; Bekaert et al., 2009; Chib et al.,
2006; Fama and French, 2012; Hou et al., 2011). However, this causes
inefficiency. We discuss this inefficiency theoretically in Section 3.

In this paper, for simplicity, we assume that stocks in the same continent
have the same observation time points; hence, regional membership is the
continent. Naturally, regional membership is known. In addition, we assume
that the number of regions, $S$, is fixed. Given the regional membership, we
can stack the observations by country within each continent. Then, we define
the “estimable” correlation matrix as follows:

$\displaystyle\mbox{\bf
R}_{h}=\begin{bmatrix}R_{0,11}&R_{h,12}&\cdots&R_{h,1S}\\\
R_{h,21}&R_{0,22}&\cdots&R_{h,2S}\\\ \vdots&\vdots&\ddots&\vdots\\\
R_{h,S1}&R_{h,S2}&\cdots&R_{0,SS}\end{bmatrix},$ (2.7)

where $R_{0,ss}=(\rho_{0,ij})_{p_{s}\times p_{s}}$ and
$R_{h,sq}=(\rho_{h,ij})_{p_{s}\times p_{q}}$ for $s,q\in\\{1,\dots,S\\}$. We
note that, for $s\neq q$, $R_{h,sq}$ represents the spillover effect between
continents $s$ and $q$, and we assume that its rank is $k^{*}_{sq}$. Moreover,
without loss of generality, each rank is at most equal to the number of global
factors (i.e., $k^{*}_{sq}\leq k$). We denote the corresponding covariance
matrix by $\mbox{\boldmath$\Sigma$}_{h}=D_{0}^{\frac{1}{2}}\mbox{\bf
R}_{h}D_{0}^{\frac{1}{2}}$.

Let $\mbox{\boldmath$\Sigma$}^{s}$ be the covariance matrix for continent $s$,
which is a $p_{s}\times p_{s}$ diagonal block of $\Sigma$. We then decompose
$\mbox{\boldmath$\Sigma$}^{s}$ as follows:

$\mbox{\boldmath$\Sigma$}^{s}=\mbox{\boldmath$\Sigma$}_{g}^{s}+\mbox{\boldmath$\Sigma$}_{l}^{s}+\mbox{\boldmath$\Sigma$}_{u}^{s},\;\;\;\;\text{
for }s=1,\dots,S,$ (2.8)

where $\mbox{\boldmath$\Sigma$}_{g}^{s}$ is the global factor component,
$\mbox{\boldmath$\Sigma$}_{l}^{s}$ is the national factor component, and
$\mbox{\boldmath$\Sigma$}_{l}^{s}$ is the idiosyncratic component. The
equation (2.8) represents the multi-level factor-based covariance matrix.
Thus, when we consider markets that have the same observation time, we can
directly apply the Double-POET procedure proposed by Choi and Kim (2023) with
all possible observations (i.e., $d=1$) to estimate the covariance matrix
$\mbox{\boldmath$\Sigma$}^{s}$. Specifically, we first employ PCA to estimate
$\mbox{\boldmath$\Sigma$}_{g}^{s}$ using the leading principal components
based on the sample covariance matrix. Then, we apply PCA on each diagonal
block of the remainder terms after removing the latent global factor
components. Finally, we apply an adaptive thresholding method to the remaining
idiosyncratic components. The specific algorithm is described in Appendix A.1.
However, due to the problem of different trading hours, when analyzing the
global stock market, lower-frequency data (e.g., $d=5$) is often used, which
causes inefficiency. To handle this issue, we propose a novel procedure for
estimating large global covariance matrix $\Sigma$ in the following
subsection, which incorporates the structural information with an entire set
of observations.

###  2.2 Structured-POET Procedure

Following Choi and Kim (2023), we assume the canonical conditions that
$\mathrm{cov}(G_{t})=\mbox{\bf I}_{k}$, $\mathrm{cov}(f_{t}^{l})=\mbox{\bf
I}_{r_{l}}$, and $\mbox{\bf B}^{\prime}\mbox{\bf B}$ and
${\Lambda^{l}}^{\prime}\Lambda^{l}$ are diagonal matrices for
$l\in\\{1,\dots,L\\}$. We note that $G_{t}$ and $f_{t}^{l}$ are uncorrelated
with each other. Let $a_{1},a_{2}\in(0,1]$ be the strengths of global and
local factors, respectively. We then impose the following pervasiveness
conditions: for each $l$, the eigenvalues of $p^{-a_{1}}\mbox{\bf
B}^{\prime}\mbox{\bf B}$ and $p_{l}^{-a_{2}}{\Lambda^{l}}^{\prime}\Lambda^{l}$
are distinct and bounded away from zero. This condition implies that the first
$k$ eigenvalues of $\mbox{\bf B}\mathrm{cov}(G_{t})\mbox{\bf B}^{\prime}$
diverge at rate $O(p^{a_{1}})$, while the first $r$ eigenvalues of
$\mbox{\boldmath$\Lambda$}\mathrm{cov}(F_{t})\mbox{\boldmath$\Lambda$}^{\prime}$
diverge at rate $O(p^{ca_{2}})$, where $a_{1}>ca_{2}$. For $c\in(0,1]$, we
note that $p^{c}\asymp p_{l}$ for each country $l$. Also, all eigenvalues of
$\mbox{\boldmath$\Sigma$}_{u}$ are bounded.

To incorporate the structure of the global financial market discussed in
Section 2.1, we propose a Structured-POET procedure to estimate $\Sigma$ as
follows:

  1. 1.

For each continent $s$, we compute the Double-POET estimator (Choi and Kim,
2023) using $T$ observations and denote it as
$\widehat{\mbox{\boldmath$\Sigma$}}^{s,\mathcal{D}}\equiv\widehat{\mbox{\boldmath$\Sigma$}}_{g}^{s,\mathcal{D}}+\widehat{\mbox{\boldmath$\Sigma$}}_{l}^{s,\mathcal{D}}+\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{s,\mathcal{D}}$.
The specific procedure is described in Appendix A.1. Let
$\widetilde{\mbox{\boldmath$\Sigma$}}_{g}^{\mathcal{D}}=\mathrm{diag}(\widehat{\mbox{\boldmath$\Sigma$}}_{g}^{1,\mathcal{D}},\dots,\widehat{\mbox{\boldmath$\Sigma$}}_{g}^{S,\mathcal{D}})$,
$\widetilde{\mbox{\boldmath$\Sigma$}}_{l}^{\mathcal{D}}=\mathrm{diag}(\widehat{\mbox{\boldmath$\Sigma$}}_{l}^{1,\mathcal{D}},\dots,\widehat{\mbox{\boldmath$\Sigma$}}_{l}^{S,\mathcal{D}})$,
and
$\widetilde{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}}=\mathrm{diag}(\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{1,\mathcal{D}},\dots,\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{S,\mathcal{D}})$.
Then, we construct a block diagonal matrix

$\widetilde{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}=\mathrm{diag}(\widehat{\mbox{\boldmath$\Sigma$}}^{1,\mathcal{D}},\dots,\widehat{\mbox{\boldmath$\Sigma$}}^{S,\mathcal{D}})\equiv\widetilde{\mbox{\boldmath$\Sigma$}}_{g}^{\mathcal{D}}+\widetilde{\mbox{\boldmath$\Sigma$}}_{l}^{\mathcal{D}}+\widetilde{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}}.$
(2.9)

  2. 2.

Given a sample covariance matrix using $d$ frequency data,
$\widehat{\mbox{\boldmath$\Sigma$}}_{h}=T^{-\alpha}\sum_{t=1}^{T^{\alpha}}(y_{t}-\bar{y})(y_{t}-\bar{y})^{\prime}$,
we compute the sample correlation matrix $\widehat{\mbox{\bf
R}}_{h}=\hat{D}_{h}^{-\frac{1}{2}}\widehat{\mbox{\boldmath$\Sigma$}}_{h}\hat{D}_{h}^{-\frac{1}{2}}$,
where $\hat{D}_{h}$ is the diagonal matrix consisting of the diagonal elements
of $\widehat{\mbox{\boldmath$\Sigma$}}_{h}$. We denote the sample correlation
matrix $\widehat{\mbox{\bf R}}_{h}$ as the following block matrix form:

$\widehat{\mbox{\bf R}}_{h}=(\hat{\rho}_{h,ij})_{p\times
p}=\begin{bmatrix}\widehat{R}_{h,11}&\widehat{R}_{h,12}&\cdots&\widehat{R}_{h,1S}\\\
\widehat{R}_{h,21}&\widehat{R}_{h,22}&\cdots&\widehat{R}_{h,2S}\\\
\vdots&\vdots&\ddots&\vdots\\\
\widehat{R}_{h,S1}&\widehat{R}_{h,S2}&\cdots&\widehat{R}_{h,SS}\end{bmatrix}.$

For each $(s,q)$th off-diagonal partitioned block, we conduct the best
rank-$k^{\ast}_{sq}$ matrix approximation to $\widehat{R}_{h,sq}$ such that
$\widehat{\Theta}_{sq}=\sum_{i=1}^{k^{*}_{sq}}\widehat{\xi}_{i}\widehat{u}_{i}\widehat{w}_{i}^{\prime}$,
where
$\\{\widehat{\xi}_{i},\widehat{u}_{i},\widehat{w}_{i}\\}_{i=1}^{p_{s}\wedge
p_{q}}$ are the ordered singular values, left-singular and right-singular
vectors of $\widehat{R}_{sq}$ in decreasing order. Then, we define

$\widehat{\mbox{\boldmath$\Theta$}}=\begin{bmatrix}\mathbf{0}&\widehat{\Theta}_{12}&\cdots&\widehat{\Theta}_{1S}\\\
\widehat{\Theta}_{21}&\mathbf{0}&\cdots&\widehat{\Theta}_{2S}\\\
\vdots&\vdots&\ddots&\vdots\\\
\widehat{\Theta}_{S1}&\widehat{\Theta}_{S2}&\cdots&\mathbf{0}\end{bmatrix}.$

  3. 3.

Let
$\widetilde{\delta}_{1}\geq\widetilde{\delta}_{2}\geq\dots\geq\widetilde{\delta}_{k}$
be the $k$ largest eigenvalues of
$\widetilde{\mbox{\boldmath$\Sigma$}}_{g}=(\widetilde{\mbox{\boldmath$\Sigma$}}_{g}^{\mathcal{D}}+\hat{D}^{\frac{1}{2}}\widehat{\mbox{\boldmath$\Theta$}}\hat{D}^{\frac{1}{2}})$
and $\\{\widetilde{v}_{i}\\}_{i=1}^{k}$ be their corresponding eigenvectors,
where $\hat{D}$ is the diagonal matrix consisting of the diagonal elements of
(2.9). The final estimator of $\Sigma$ is then defined as

$\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{S}}=\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\widetilde{\mbox{\bf
V}}_{g}^{\prime}+\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{S}},$ (2.10)

where
$\widetilde{\mbox{\boldmath$\Gamma$}}_{g}=\mathrm{diag}(\widetilde{\delta}_{1},\dots,\widetilde{\delta}_{k})$,
$\widetilde{\mbox{\bf V}}_{g}=(\widetilde{v}_{1},\dots,\widetilde{v}_{k})$,
$\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{S}}=\widetilde{\mbox{\boldmath$\Sigma$}}_{l}^{\mathcal{D}}+\widetilde{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}}$,
and $\widetilde{\mbox{\boldmath$\Sigma$}}_{l}^{\mathcal{D}}$ and
$\widetilde{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}}$ are defined in (2.9).

######  Remark 2.1.

To implement Structured-POET, we need to determine the rank $k^{*}_{sq}$ and
the number of factors, which are unknown in practice. We note that each
$(s,q)$th off-diagonal partitioned block $R_{h,sq}$ in (2.7) is a low-rank
matrix, and each rank is less than or equal to the number of global factors
(i.e., $k^{*}_{sq}\leq k$). Thus, to determine the rank and number of global
factors, we can use the data-driven methods proposed by Ahn and Horenstein
(2013); Bai and Ng (2002); Onatski (2010). For example, the rank $k^{*}_{sq}$
can be determined by finding the largest singular value gap such that
$\max_{i\leq\bar{k}_{sq}}(\widehat{\xi}_{i}-\widehat{\xi}_{i+1})$, where
$\bar{k}_{sq}=\min\\{p_{s},p_{q}\\}$. On the other hand, to estimate the
number of global factors based on $\widehat{\mbox{\boldmath$\Sigma$}}_{h}$, we
use a modified version of the eigenvalue ratio method introduced by Choi and
Kim (2023) in the numerical study.

In summary, for given continent and country memberships, we apply Double-POET
to each continental block, which incorporates all observations to estimate the
national factor and idiosyncratic components. We then conduct a low-rank
approximation to each off-diagonal block using lower-frequency observations.
Finally, we perform PCA on the structured global factor components obtained
from the previous procedures. This procedure is called Structured-POET.
Structured-POET efficiently estimates global and local factor components by
utilizing all observations and considering the block structure of the local
factors. In contrast, the Double-POET or POET estimator may not be efficient
due to the loss of the structural information and observations. In Section
3.1, we discuss the theoretical inefficiency of Double-POET. Also, the
numerical study in Sections 4 and 5 shows that Structured-POET outperforms
Double-POET and POET.

##  3 Asymptotic Properties

This section establishes the asymptotic properties of the proposed Structured-
POET estimator. To investigate asymptotic behaviors, we require the following
technical assumption.

######  Assumption 3.1.

  * (i)

For some constants $c\in(0,1]$, $a_{1}\in(\frac{3+2c}{5},1]$, and
$a_{2}\in(\frac{3}{5},1]$, all eigenvalues of $\mbox{\bf B}^{\prime}\mbox{\bf
B}/p^{a_{1}}$ and $\Lambda^{l\prime}\Lambda^{l}/p_{l}^{a_{2}}$ are strictly
bigger than zero as $p,p_{l}\rightarrow\infty$, for $l\in\\{1,\dots,L\\}$. In
addition, $p_{l}\asymp p^{c}$, for each country $l$, and $a_{1}\geq ca_{2}$.
There is a constant $C>0$ such that $\|\mbox{\bf B}\|_{\max}\leq C$ and
$\|\mbox{\boldmath$\Lambda$}\|_{\max}\leq C$.

  * (ii)

There exist constants $C_{1},C_{2}>0$ such that
$\lambda_{\min}(\mbox{\boldmath$\Sigma$}_{u})>C_{1}$ and
$\|\mbox{\boldmath$\Sigma$}_{u}\|_{1}\leq C_{2}m_{p}$.

  * (iii)

Let $d=T^{1-\alpha}$ for $\alpha\in(0,1)$. The sample correlation matrix using
$d$ frequency data, $\widehat{\mbox{\bf
R}}_{h}=\hat{D}_{h}^{-\frac{1}{2}}\widehat{\mbox{\boldmath$\Sigma$}}_{h}\hat{D}_{h}^{-\frac{1}{2}}$,
where $\hat{D}_{h}$ is the diagonal matrix consisting of the diagonal elements
of
$\widehat{\mbox{\boldmath$\Sigma$}}_{h}=T^{-\alpha}\sum_{t=1}^{T^{\alpha}}(y_{t}-\bar{y})(y_{t}-\bar{y})^{\prime}$,
satisfies

$\|\widehat{\mbox{\bf R}}_{h}-\mbox{\bf R}_{h}\|_{\max}=O_{P}(\sqrt{\log
p/T^{\alpha}}).$

  * (iv)

Denote $\mbox{\boldmath$\Sigma$}=(\Sigma_{ij})_{p\times p}$. The sample
covariance matrix using $T$ observation,
$\widehat{\mbox{\boldmath$\Sigma$}}=T^{-1}\sum_{t=1}^{T}(y_{t}-\bar{y})(y_{t}-\bar{y})^{\prime}=(\widehat{\Sigma}_{ij})_{p\times
p}$, satisfies that, for $s\in\\{1,\dots,S\\}$,

$\max_{\\{i,j\\}\in s}|\widehat{\Sigma}_{ij}-\Sigma_{ij}|=O_{P}(\sqrt{\log
p/T}).$

######  Remark 3.1.

Assumption 3.1(i) is known as the factor pervasiveness assumption, which is
closely related to the incoherence structure (Fan et al., 2018b). This
assumption can hold in macroeconomic and financial applications and is used
for analyzing low-rank matrices (Bai, 2003; Chamberlain and Rothschild, 1983;
Fan et al., 2013, 2016; Lam and Yao, 2012; Stock and Watson, 2002).
Specifically, in the context of a multi-level factor model, global factors
have a broad impact on most individuals, while national factors only affect
individuals within each national group. This illustrates the pervasive
condition at different levels in an intuitive manner. Assumptions
3.1(iii)-(iv) provide a high-level sufficient condition for analyzing large
matrices. The sample correlation matrix with $d$ frequency data serves as the
initial estimator for $\mbox{\bf R}_{h}$ in Assumption 3.1(iii). This
condition is required to account for the spillover effect between continents.
Here, the correlation matrix is considered to overcome the amplified scale
issue of using the sample covariance matrix based on lower-frequency data (see
Remark 3.3). On the other hand, we can impose the element-wise convergence
condition for each continent using the sample covariance matrix based on all
observations (Assumption 3.1(iv)). These element-wise convergence rate
conditions are easily satisfied under the sub-Gaussian condition and mixing
time dependency (Fan et al., 2018a, b; Vershynin, 2010; Wang and Fan, 2017).
It can also be satisfied under heavy-tailed observations with bounded fourth
moments (Fan et al., 2017, 2018a, 2018b, 2021).

In addition to the max norm and spectral norm, the relative Frobenius norm
(Stein and James, 1961) is considered as a measure of large matrix estimation
errors:

$\|\widehat{\mbox{\boldmath$\Sigma$}}-\mbox{\boldmath$\Sigma$}\|_{\Sigma}=p^{-1/2}\|\mbox{\boldmath$\Sigma$}^{-1/2}\widehat{\mbox{\boldmath$\Sigma$}}\mbox{\boldmath$\Sigma$}^{-1/2}-\mbox{\bf
I}_{p}\|_{F}.$

Note that the factor $p^{-1/2}$ plays the role of normalization, i.e.,
$\|\mbox{\boldmath$\Sigma$}\|_{\Sigma}=1$. Under this relative Frobenius norm,
in the approximate single-level factor model, the conventional POET estimator
is consistent if $p=o(T^{2})$, while the sample covariance does not converge
if $p>T$ (Fan et al., 2013). In the multi-level factor model, the Double-POET
estimator is consistent as long as $p=o(T^{2})$ and
$\frac{1}{4}<c<\frac{3}{4}$, while the POET estimator does not converge if
$c>\frac{1}{2}$ or $p^{\kappa}>T$ with $\kappa=\min\\{\frac{1}{2},2c\\}$ (Choi
and Kim, 2023). In Section 3.1, we compare the convergence rates of Double-
POET and Structured-POET.

We obtain the following convergence rates for Structured-POET and its inverse
under various norms.

######  Theorem 3.1.

Suppose that $m_{p}=o(p^{c(5a_{2}-3)/2})$ and Assumption 3.1 holds. Let
$\omega_{T}=p^{\frac{5}{2}(1-a_{1})+\frac{5}{2}c(1-a_{2})}\sqrt{\log
p/T}+1/p^{\frac{5}{2}a_{1}-\frac{3}{2}+c(\frac{5}{2}a_{2}-\frac{7}{2})}+m_{p}/\sqrt{p^{c(5a_{2}-3)}}$
and $\tau\asymp\omega_{T}$. If $m_{p}\omega_{T}^{1-q}=o(1)$, we have

$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{S}}-\mbox{\boldmath$\Sigma$}\|_{\max}=O_{P}\left(\omega_{T}+p^{5(1-a_{1})}\Big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\Big{)}+\frac{1}{p^{5a_{1}-4-c}}\right),$
(3.1)
$\displaystyle\|(\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{S}})^{-1}-\mbox{\boldmath$\Sigma$}^{-1}\|=O_{P}\left(m_{p}\omega_{T}^{1-q}+p^{\frac{c}{2}(1-a_{2})}\omega_{T}+p^{\frac{11}{2}(1-a_{1})}\Big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\Big{)}+\frac{1}{p^{\frac{11}{2}a_{1}-\frac{9}{2}-c}}\right).$
(3.2)

In addition, if $a_{1}>\frac{3}{4}$ and $a_{2}>\frac{3}{4}$, we have

$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{S}}-\mbox{\boldmath$\Sigma$}\|_{\Sigma}=$
$\displaystyle
O_{P}\Big{(}m_{p}\omega_{T}^{1-q}+p^{\frac{7}{2}(1-a_{1})}\big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\big{)}+\frac{1}{p^{\frac{7}{2}a_{1}-\frac{5}{2}-c}}$
$\displaystyle\qquad\qquad+p^{\frac{21}{2}-10a_{1}}\big{(}\frac{\log
p}{T^{\alpha}}+\frac{1}{T^{2(1-\alpha)\beta}}\big{)}+\frac{1}{p^{10a_{1}-\frac{17}{2}-2c}}+\frac{m_{p}^{2}}{p^{5ca_{2}-3c-\frac{1}{2}}}\Big{)}.$
(3.3)

######  Remark 3.2.

$\omega_{T}$ is related to the estimation of latent local factors and
idiosyncratic components using $T$ observations. The additional terms
$\sqrt{\log p/T^{\alpha}}$ and $1/T^{(1-\alpha)\beta}$ are the cost to handle
the non-synchronized trading hours, when estimating latent global factors. In
particular, the first term is coming from sub-sampled observations,
$T^{\alpha}=T/d$, while the second term is the cost to estimate the
synchronized correlation matrix $\mbox{\bf R}_{0}$. The optimal choice of
$\alpha$ is $\alpha^{\ast}=\frac{2\beta}{1+2\beta}$, which simultaneously
minimizes the convergence rates. This implies that the choice of frequency
window $d$ is important in practice. In the numerical study, we use the weekly
data, that is, $d=5$.

For simplicity, consider the case of strong global and local factors (i.e.,
$a_{1}=1$ and $a_{2}=1$), $q=0$, and $m_{p}=O(1)$. Then, the proposed
Structured-POET method yields

$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{S}}-\mbox{\boldmath$\Sigma$}\|_{\Sigma}=$
$\displaystyle O_{P}\left(\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}+\frac{1}{p^{1-c}+p^{c}}+\sqrt{p}\Big{(}\frac{\log
p}{T^{\alpha}}+\frac{1}{T^{2(1-\alpha)\beta}}\Big{)}+\frac{1}{p^{\frac{3}{2}-2c}}+\frac{1}{p^{2c-\frac{1}{2}}}\right),$

which can be convergent as long as $p=o(T^{\alpha})$ and
$\frac{1}{4}<c<\frac{3}{4}$. Similar to the Double-POET estimator (Choi and
Kim, 2023), the upper and lower bounds of $c$ are required to estimate both
global and national factor components.

###  3.1 Double-POET Using Lower-Frequency Data

In this subsection, we compare the Double-POET method with the proposed
Structured-POET method.

To capture the global factor, local factor, and idiosyncratic components, we
can apply the Double-POET method. However, when considering international
stocks, practitioners commonly use lower-frequency data to minimize the impact
of different observation time points. Let
$\widehat{\mbox{\boldmath$\Sigma$}}_{h}=T^{-\alpha}\sum_{t=1}^{T^{\alpha}}(y_{t}-\bar{y})(y_{t}-\bar{y})^{\prime}$
be the sample covariance matrix using $d$ frequency return data. Then,
$d^{-1}\widehat{\mbox{\boldmath$\Sigma$}}_{h}$ is used for the initial pilot
estimator for covariance matrix $\Sigma$, since
$\widehat{\mbox{\boldmath$\Sigma$}}_{h}$ is the amplified estimator by $d$,
which slowly grows (see Remark 3.3). Let
$\widehat{\mbox{\boldmath$\Gamma$}}=\mathrm{diag}(\widehat{\delta}_{1},\dots,\widehat{\delta}_{k})$
and $\widehat{\mbox{\bf V}}=(\widehat{v}_{1},\dots,\widehat{v}_{k})$ be the
leading eigenvalues and their corresponding eigenvectors of
$d^{-1}\widehat{\mbox{\boldmath$\Sigma$}}_{h}$. Next, let
$\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{l}$ be the $l$th $p_{l}\times p_{l}$
diagonal block of
$\widehat{\mbox{\boldmath$\Sigma$}}_{E}=d^{-1}\widehat{\mbox{\boldmath$\Sigma$}}_{h}-\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}$. Let
$\widehat{\Psi}^{l}=\mathrm{diag}(\widehat{\kappa}_{1}^{l},\dots,\widehat{\kappa}_{r_{l}}^{l})$
and
$\widehat{\Phi}^{l}=(\widehat{\eta}_{1}^{l},\dots,\widehat{\eta}_{r_{l}}^{l})$
be the leading eigenvalues and their corresponding eigenvectors of
$\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{l}$. Let
$\widehat{\mbox{\boldmath$\Psi$}}=\mathrm{diag}(\widehat{\Psi}^{1},\dots,\widehat{\Psi}^{L})$,
$\widehat{\mbox{\boldmath$\Phi$}}=\mathrm{diag}(\widehat{\Phi}^{1},\dots,\widehat{\Phi}^{L})$,
and
$\widehat{\mbox{\boldmath$\Sigma$}}_{u}=d^{-1}\widehat{\mbox{\boldmath$\Sigma$}}_{h}-\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf
V}}^{\prime}-\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}$.
Then, the Double-POET estimator is defined as follows:

$\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}=\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf
V}}^{\prime}+\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}+\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}},$

where $\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}}$ is the
thresholded error covariance matrix estimator based on
$\widehat{\mbox{\boldmath$\Sigma$}}_{u}=(\widehat{\sigma}_{u,ij})_{p\times p}$
(Bickel and Levina, 2008; Fan et al., 2013):

$\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}}=(\widehat{\sigma}_{u,ij}^{\mathcal{D}})_{p\times
p},\;\;\;\;\;\widehat{\sigma}_{u,ij}^{\mathcal{D}}=\left\\{\begin{array}[]{ll}\widehat{\sigma}_{u,ij},&i=j\\\
s_{ij}(\widehat{\sigma}_{u,ij})I(|\widehat{\sigma}_{u,ij}|\geq\tau_{ij}),&i\neq
j\end{array}\right.,$

where $s_{ij}(\cdot)$ is a generalized shrinkage function (Cai and Liu, 2011;
Rothman et al., 2009) and an entry-dependent threshold
$\tau_{ij}=\tau(\widehat{\sigma}_{u,ii}\widehat{\sigma}_{u,jj})^{1/2}$. The
thresholding constant $\tau$ is determined in Theorem 3.2.

######  Assumption 3.2.

Let $d=T^{1-\alpha}$ for $\alpha\in(0,1)$. The sample covariance matrix using
$d$ frequency data,
$\widehat{\mbox{\boldmath$\Sigma$}}_{h}=T^{-\alpha}\sum_{t=1}^{T^{\alpha}}(y_{t}-\bar{y})(y_{t}-\bar{y})^{\prime}$,
satisfies

$\|d^{-1}\widehat{\mbox{\boldmath$\Sigma$}}_{h}-\mbox{\boldmath$\Sigma$}_{h}\|_{\max}=O_{P}(\sqrt{\log
p/T^{\alpha}}).$

######  Remark 3.3.

Assumption 3.2 is a necessary condition for the analysis of large covariance
matrix inference. This element-wise convergence condition is similar to
Assumption 3.1(iii) in Choi and Kim (2023). However, to match the scale of
$\mbox{\boldmath$\Sigma$}_{h}$, the sample covariance matrix using $d$
frequency return data, $\widehat{\mbox{\boldmath$\Sigma$}}_{h}$, needs to be
divided by $d$. To illustrate this point, consider the case when $p=1$ and a
collection of $T$ i.i.d. random variables, $\\{y_{1},\dots,y_{T}\\}$, where
$y_{t}$ is a log-return defined as $y_{t}=\log x_{t}-\log x_{t-1}$ and $x_{t}$
is the asset price at time $t$. Assume that $y_{t}$ has a mean of zero and a
variance of $\sigma^{2}$. We can obtain lower-frequency data by summing daily
log-returns for each $d$ frequency window, and this is equivalent to sub-
sampling based on the price data. The variance of the resulting $d$ frequency
data is $d\times\sigma^{2}$. Therefore, we can compare the estimator
$\hat{\sigma}_{h}/d$ with the true variance $\sigma^{2}$, where
$\hat{\sigma}_{h}=T^{-\alpha}\sum_{t=1}^{T^{\alpha}}(y_{t}-\bar{y})^{2}$ using
$d$ frequency log-returns. Using this fact and Assumption 3.1(iii), we can
impose the above element-wise convergence condition. However, Structured-POET
does not require this assumption because it can remove the scale issue by
using the correlation matrix structure and recovering with daily-based
variance estimator $\hat{D}$ in Section 2.2. In the simulation study, we used
$d^{-1}\widehat{\mbox{\boldmath$\Sigma$}}_{h}$ for the initial sample
covariance matrix.

Similar to the proofs of Choi and Kim (2023), we can show that Double-POET
yields the following convergence rates.

######  Theorem 3.2.

Suppose that $m_{p}=o(p^{c(5a_{2}-3)/2})$ and Assumptions 3.1–3.2 hold. Let
$\omega_{T^{\alpha}}=p^{\frac{5}{2}(1-a_{1})+\frac{5}{2}c(1-a_{2})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{5}{2}a_{1}-\frac{3}{2}+c(\frac{5}{2}a_{2}-\frac{7}{2})}+m_{p}/\sqrt{p^{c(5a_{2}-3)}}$
and $\tau\asymp\omega_{T^{\alpha}}$. If $m_{p}\omega_{T^{\alpha}}^{1-q}=o(1)$,
we have

$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}-\mbox{\boldmath$\Sigma$}\|_{\max}=O_{P}\left(\omega_{T^{\alpha}}\right),$
(3.4)
$\displaystyle\|(\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}})^{-1}-\mbox{\boldmath$\Sigma$}^{-1}\|_{2}=O_{P}\left(m_{p}\omega_{T^{\alpha}}^{1-q}+p^{\frac{c}{2}(1-a_{2})}\omega_{T^{\alpha}}+p^{3(1-a_{1})}\Big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\Big{)}+\frac{1}{p^{3a_{1}-2-c}}\right).$
(3.5)

In addition, if $a_{1}>\frac{3}{4}$ and $a_{2}>\frac{3}{4}$, we have

$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}-\mbox{\boldmath$\Sigma$}\|_{\Sigma}=$
$\displaystyle
O_{P}\Big{(}m_{p}\omega_{T^{\alpha}}^{1-q}+p^{\frac{11}{2}-5a_{1}+5c(1-a_{2})}\big{(}\frac{\log
p}{T^{\alpha}}+\frac{1}{T^{2(1-\alpha)\beta}}\big{)}$
$\displaystyle\qquad\qquad\qquad\qquad+\frac{1}{p^{5a_{1}-\frac{7}{2}-c(7-5a_{2})}}+\frac{m_{p}^{2}}{p^{5ca_{2}-3c-\frac{1}{2}}}\Big{)}.$
(3.6)

We compare the rates of convergence between Structured-POET and Double-POET as
follows. For simplicity, consider $m_{p}=O(1)$, $a_{1}=1$, and $a_{2}=1$, and
ignore the log order terms. Define the optimal
$\alpha^{\ast}=\frac{2\beta}{1+2\beta}$. Under the relative Frobenius norm
with $\alpha=\alpha^{\ast}$, we have

$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}-\mbox{\boldmath$\Sigma$}\|_{\Sigma}$
$\displaystyle=O_{P}\left(\left(\frac{1}{T^{\frac{\beta}{1+2\beta}}}+\frac{1}{p^{1-c}}+\frac{1}{p^{c}}\right)^{1-q}+\frac{\sqrt{p}}{T^{\frac{2\beta}{1+2\beta}}}+\frac{1}{p^{\frac{3}{2}-2c}}+\frac{1}{p^{2c-\frac{1}{2}}}\right),$
$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{S}}-\mbox{\boldmath$\Sigma$}\|_{\Sigma}$
$\displaystyle=O_{P}\left(\left(\frac{1}{\sqrt{T}}+\frac{1}{p^{1-c}}+\frac{1}{p^{c}}\right)^{1-q}+\frac{1}{T^{\frac{\beta}{1+2\beta}}}+\frac{\sqrt{p}}{T^{\frac{2\beta}{1+2\beta}}}+\frac{1}{p^{\frac{3}{2}-2c}}+\frac{1}{p^{2c-\frac{1}{2}}}\right).$

In particular, when $q\neq 0$, Structured-POET can obtain a faster convergence
rate under the relative Frobenius norm. In contrast, if $q=0$, the convergence
rates of both estimators are the same. This is because the estimation error of
the correlations between continents dominates the convergence rate. However,
this does not mean that their estimation errors are exactly the same. In fact,
from the simulation study, we can conjecture that Structured-POET has smaller
convergence rates than Double-POET for $q=0$. That is, the relative ratio of
the convergence rate of Structured-POET with respect to that of Double-POET
may be less than 1. Unfortunately, due to the complex upper bound calculations
used to handle high-dimensional matrices, we cannot theoretically show this
statement for $q=0$. We leave this for a future study. Similarly, under the
spectral norm for the inverse matrix, the convergence rate of Structure-POET
can be faster than that of Double-POET when $q\neq 0$.

##  4 Simulation Study

In this section, simulations are carried out to examine the finite sample
performance of the proposed Structured-POET method. We considered the true
covariance as $\Sigma=BB^{\prime}+\Lambda\Lambda^{\prime}+\Sigma_{u}$, where
each row of $B$ was drawn from $\mathcal{N}(\mu_{B},I_{k})$, where each
element of $\mu_{B}$ is i.i.d. Uniform$(-0.5,0.5)$; for
$\Lambda=\mathrm{diag}(\Lambda^{1},\dots,\Lambda^{L})$, each row of
$\Lambda^{l}$ for each $l$ was drawn from
$\mathcal{N}(\mu_{\Lambda^{l}},I_{r_{l}})$, where each element of
$\mu_{\Lambda^{l}}$ is i.i.d. Uniform$(-0.3,0.3)$. We generated $\Sigma_{u}$
as follows. Let $D_{u}=\mathrm{diag}(d_{1},\dots,d_{p})$, where each
$\\{d_{i}\\}$ was generated independently from Uniform$(0.5,1.5)$. Let
$\pi=(\pi_{1},\dots,\pi_{p})^{\prime}$ be a sparse vector, where each
$\pi_{i}$ was drawn from $\mathcal{N}(0,1)$ with probability
$\frac{0.5}{\sqrt{p}\log{p}}$, and $\pi_{i}=0$ otherwise. Then, we set
$\Sigma_{u}=D_{u}+\pi\pi^{\prime}-\mathrm{diag}\\{\pi_{1}^{2},\dots,\pi_{p}^{2}\\}$.
In the simulation, we generated $\Sigma_{u}$ until it was positive definite.

Let $D$ be the diagonal matrix consisting of the diagonal elements of
$\Sigma$. We then obtained the true correlation matrix
$R=D^{-\frac{1}{2}}\Sigma D^{-\frac{1}{2}}=(\rho_{0,ij})_{p\times p}$. Next,
we set $R_{h}=(\rho_{h,ij})_{p\times p}$, where
$\rho_{h,ij}=\mathrm{sgn}(\rho_{0,ij})(|\rho_{0,ij}|+0.5h^{\beta})$ if $i$ and
$j$ belong to different continent groups, for $h=\frac{0.5}{d}$ and
$\beta=0.75$, and $\rho_{h,ij}=\rho_{0,ij}$ if $i$ and $j$ are in the same
continent group. Let $\\{\gamma_{i},v_{i}\\}_{i=1}^{k}$ be the leading
eigenvalues and eigenvectors of
$\widetilde{\Sigma}_{g}=D^{\frac{1}{2}}R_{h}D^{\frac{1}{2}}-\Lambda\Lambda^{\prime}-\Sigma_{u}$.
Then, we obtained $B_{h}=V\Gamma^{\frac{1}{2}}$, where
$\Gamma=\mathrm{diag}(\gamma_{1},\dots,\gamma_{k})$ and
$V=(v_{1},\dots,v_{k})$. Finally, we generated observations by

$y_{t}=B_{h}G_{t}+\Lambda F_{t}+u_{t},$

where $G_{t}$, $F_{t}$, and $u_{t}$ were drawn from $\mathcal{N}(0,I_{k})$,
$\mathcal{N}(0,I_{r})$, and $\mathcal{N}(0,\Sigma_{u})$, respectively.

In this simulation study, we fixed the number of individuals $p=500$. We set
the number of continents $S=2$ and the number of local groups $L=20$ such that
each continent group included $10$ local groups (i.e., $p_{l}=25$). Also, we
chose the numbers of factors as $k=3$ and $r=L\times r_{l}$, where $r_{l}=2$
for each local group $l$. Then, we considered two cases: (i) increasing $T$
from 100 to 600 in increments of 50 with the size of frequency $d\in\\{1,5\\}$
(i.e., in-sample size is $T/d$) and (ii) increasing $d$ from 1 to 10 with a
fixed $T=600$. For each case, 200 simulations were conducted.

For comparison, the sample covariance matrix (SamCov), POET, Double-POET
(D-POET), and Structured-POET (S-POET) methods were employed to estimate
$\Sigma$. The average estimation errors were measured under the following
norms:
$\|\widehat{\mbox{\boldmath$\Sigma$}}-\mbox{\boldmath$\Sigma$}\|_{\Sigma}$,
$\|\widehat{\mbox{\boldmath$\Sigma$}}-\mbox{\boldmath$\Sigma$}\|_{\max}$, and
$\|(\widehat{\mbox{\boldmath$\Sigma$}})^{-1}-\mbox{\boldmath$\Sigma$}^{-1}\|$,
where $\widehat{\mbox{\boldmath$\Sigma$}}$ is one of the covariance matrix
estimators. We note that the lower-frequency data is obtained by summing the
observations for each $d$ frequency window. Therefore, for the SamCov
estimator and the initial pilot estimator of POET and D-POET, we used
$d^{-1}\widehat{\mbox{\boldmath$\Sigma$}}_{h}$ (see Section 3.1). For each
estimation, we determined the number of factors for D-POET and POET using the
eigenvalue ratio methods suggested by Choi and Kim (2023) and Ahn and
Horenstein (2013), with $k_{\max}=10$ and $r_{l,\max}=10$, respectively. For
the S-POET estimation, we chose the number of ranks for each off-diagonal
block by the largest singular value ratio. In addition, we employed the soft
thresholding scheme for the idiosyncratic covariance matrix estimation.

Figure 1: Averages of
$\|\widehat{\mbox{\boldmath$\Sigma$}}-\mbox{\boldmath$\Sigma$}\|_{\Sigma}$,
$\|(\widehat{\mbox{\boldmath$\Sigma$}})^{-1}-\mbox{\boldmath$\Sigma$}^{-1}\|$,
and $\|\widehat{\mbox{\boldmath$\Sigma$}}-\mbox{\boldmath$\Sigma$}\|_{\max}$
for the sample covariance matrix, POET, Double-POET, and Structured-POET
against $T$ with fixed $p=500$ and $L=20$. Figure 2: Averages of
$\|\widehat{\mbox{\boldmath$\Sigma$}}-\mbox{\boldmath$\Sigma$}\|_{\Sigma}$,
$\|(\widehat{\mbox{\boldmath$\Sigma$}})^{-1}-\mbox{\boldmath$\Sigma$}^{-1}\|$,
and $\|\widehat{\mbox{\boldmath$\Sigma$}}-\mbox{\boldmath$\Sigma$}\|_{\max}$
for the sample covariance matrix, POET, Double-POET, and Structured-POET
against $d$ with fixed $p=500$, $T=600$, and $L=20$.

Figures 1 and 2 depict the averages of estimation errors under different norms
against $T$ and $d$, respectively. From Figures 1 and 2, we find that S-POET
has smaller estimation errors than the other methods under different norms.
Specifically, in Figure 1, as $T$ increases, the estimation errors of S-POET
and estimators with $d=5$ decrease, while the estimation errors of estimators
with $d=1$ do not decrease. This is because the estimators with $d=1$ actually
estimate $\mbox{\boldmath$\Sigma$}_{h}$ not $\mbox{\boldmath$\Sigma$}_{0}$,
and when $d=1$, $\mbox{\boldmath$\Sigma$}_{h}$ is not close to
$\mbox{\boldmath$\Sigma$}_{0}$. When comparing the estimation procedures with
$d=5$, S-POET shows the best performance. This is because S-POET can
accurately estimate the local factors and idiosyncratic components by
utilizing whole observations, while other estimators utilize lower-frequency
observations, which causes inefficiency. Figure 2 indicates that the
estimation errors of all methods dramatically drop from $d=1$ to $d=2$, which
is consistent with the results shown in Figure 1. S-POET shows stable results
and has the minimum estimation errors when $d=4$. In contrast, as the
frequency size $d$ increases, the estimation errors of SamCov and D-POET tend
to increase again due to the smaller sample sizes. That is, the loss of
information is severe only when using lower-frequency observations. From this
result, we can conjecture that for a fixed $T$, the estimation error resulting
from a small sample size with larger $d$ is greater than the error resulting
from the effect of observation time gaps. However, the proposed S-POET
incorporates all available data to estimate the same regional covariance
matrices, which helps enjoy the efficiency. It is worth noting that the
estimation errors of POET seem constant as $d$ increases. This is because POET
does not estimate the local covariance matrix, which may dominate other
estimation errors. The above results support the theoretical findings
established in Section 3.

##  5 Empirical Study

We conducted a minimum variance portfolio allocation study using the proposed
Structured-POET method with global financial data. We obtained the daily
transaction prices of international stock markets over 15 countries by the
total market capitalization. The whole sample period is from January 3, 2017,
to December 30, 2022. After excluding stocks with missing returns and no
variation, we picked $1500$ stocks for this period based on the market cap for
each country. In particular, we selected $500$ firms for each continent and
calculated both daily and weekly log-returns. The distribution of our sample
is presented in Table 1.

Table 1: Distributions of the number of firms America | Asia | Europe
---|---|---
United States (US) | 221 | China (CN) | 100 | United Kingdom (GB) | 100
Canada (CA) | 100 | Japan (JP) | 100 | France (FR) | 100
Brazil (BR) | 100 | Hong Kong (HK) | 100 | Germany (DE) | 100
Mexico (MX) | 48 | India (IN) | 100 | Switzerland (CH) | 100
Chile (CL) | 31 | Korea (KR) | 100 | Sweden (SE) | 100
| Total | 1500

We computed several estimators, including Structured-POET (S-POET), Double-
POET (D-POET), POET, and the sample covariance matrix (SamCov) estimators, for
each week. In the case of the S-POET procedure, we used weekly returns to
estimate the global factor component and daily returns to estimate the local
factor and idiosyncratic components. We employed both daily and weekly returns
for the other procedures. For all POET-type procedures, we estimated the
idiosyncratic volatility matrix using information of the 11 Global Industrial
Classification Standard (GICS) sectors (Ait-Sahalia and Xiu, 2017; Fan et al.,
2016). Specifically, we set the idiosyncratic components to zero for the
different sectors, while maintaining them for the same sector. We used
different numbers of global factors, $k$, ranging from 1 to 5 for both D-POET
and POET. For D-POET, we chose the number of local factors using the
eigenvalue ratio method proposed by Ahn and Horenstein (2013) with
$r_{l,\max}=5$. For S-POET, we utilized the best rank-one approximation for
each off-diagonal continental block.

We considered the following constrained minimum variance portfolio allocation
problem (Fan et al., 2012; Jagannathan and Ma, 2003) to analyze the out-of-
sample portfolio allocation performance:

$\min_{\omega}\omega^{T}\widehat{\mbox{\boldmath$\Sigma$}}\omega,\text{
subject to }\omega^{\top}\mathbf{1}=1,\;\|\omega\|_{1}\leq c,$

where $\mathbf{1}=(1,\dots,1)^{\top}\in\mathbb{R}^{p}$, the gross exposure
constraint $c$ varies from 1 to 4, and $\widehat{\mbox{\boldmath$\Sigma$}}$ is
one of the volatility matrix estimators obtained from S-POET, D-POET, POET,
and SamCov. At the beginning of each week, we obtained optimal portfolios
based on each estimator using the past 12 months’ returns and held these
portfolios for one week. We then computed the square root of the realized
volatility using the weekly log-returns. Their averages were recorded for the
out-of-sample risk. We examined six out-of-sample periods: 2018, 2019, 2020,
2021, 2022, and the full period from 2018 to 2022.

Figure 3: Out-of-sample risks of the optimal portfolios constructed by the
SamCov, POET, Double-POET, and Structured-POET estimators for the global stock
market.

Figure 3 illustrates the out-of-sample risks of the portfolios constructed by
SamCov, POET, D-POET, and S-POET under varying exposure constraints. To draw
readable plots, we presented the best performing results among $k=1,\ldots,5$
for each estimator type and each period. We used subscripts to explicitly
denote the frequency of the data used, with W (blue lines) and D (red lines)
representing weekly and daily data, respectively. As shown in Figure 3, S-POET
consistently outperforms the other estimators. Specifically, for all periods
except 2021, S-POET reduces the minimum risks by 4.7%–10.2% compared to the
best estimator among the other methods. When comparing the estimation
procedures with the same frequency observations, D-POET exhibits lower risks
than POET and SamCov. Furthermore, D-POET, POET, and SamCov estimators using
daily data tend to have lower risks than those using weekly data. This may be
because, in practice, the impact of estimation inefficiency resulting from the
smaller sample size could be greater than that resulting from the different
observation time points. In summary, for portfolio allocation in the global
stock market, S-POET, which incorporates both daily and weekly returns under
the specific structure, outperforms POET and Double-POET, which use only daily
or weekly returns. From this result, we can conjecture that the estimation
accuracy for the national factor and idiosyncratic components can be improved
using more frequent data (i.e., daily returns) for each continent group. In
addition, for global factor estimation, using less frequent data (i.e., weekly
returns) can manage the different trading hour problem.

##  6 Conclusion

In this paper, we introduce a novel large global volatility matrix inference
procedure. The proposed Structured-POET method leverages structural
information from global financial markets based on latent global and national
factor models. We establish the asymptotic properties of Structured-POET and
demonstrate its efficiency in estimating a large global covariance matrix
compared to the Double-POET procedure.

In our empirical study, we demonstrate that the proposed Structured-POET
estimator outperforms the other existing methods in minimum variance portfolio
allocation problems. This is because the Structured-POET procedure accurately
estimates the latent national factors and idiosyncratic components using daily
returns. Additionally, using weekly returns to estimate the latent global
factors can mitigate the effect of different trading hours across markets.
Overall, our findings support the effectiveness of the Structured-POET method
in estimating large global volatility matrices.

## References

  * Ahn and Horenstein (2013) Ahn, S. C. and A. R. Horenstein (2013): “Eigenvalue ratio test for the number of factors,” _Econometrica_ , 81, 1203–1227.
  * Ait-Sahalia and Xiu (2017) Ait-Sahalia, Y. and D. Xiu (2017): “Using principal component analysis to estimate a high dimensional factor model with high-frequency data,” _Journal of Econometrics_ , 201, 384–399.
  * Ando and Bai (2015) Ando, T. and J. Bai (2015): “Asset pricing with a general multifactor structure,” _Journal of Financial Econometrics_ , 13, 556–604.
  * Ando and Bai (2016) ——— (2016): “Panel data models with grouped factor structure under unknown group membership,” _Journal of Applied Econometrics_ , 31, 163–191.
  * Ando and Bai (2017) ——— (2017): “Clustering huge number of financial time series: A panel data approach with high-dimensional predictors and factor structures,” _Journal of the American Statistical Association_ , 112, 1182–1198.
  * Bai (2003) Bai, J. (2003): “Inferential theory for factor models of large dimensions,” _Econometrica_ , 71, 135–171.
  * Bai and Ng (2002) Bai, J. and S. Ng (2002): “Determining the number of factors in approximate factor models,” _Econometrica_ , 70, 191–221.
  * Bai and Wang (2015) Bai, J. and P. Wang (2015): “Identification and Bayesian estimation of dynamic factor models,” _Journal of Business & Economic Statistics_, 33, 221–240.
  * Bekaert et al. (2009) Bekaert, G., R. J. Hodrick, and X. Zhang (2009): “International stock return comovements,” _The Journal of Finance_ , 64, 2591–2626.
  * Bernanke et al. (2005) Bernanke, B. S., J. Boivin, and P. Eliasz (2005): “Measuring the effects of monetary policy: a factor-augmented vector autoregressive (FAVAR) approach,” _The Quarterly journal of economics_ , 120, 387–422.
  * Bickel and Levina (2008) Bickel, P. J. and E. Levina (2008): “Covariance regularization by thresholding,” _The Annals of Statistics_ , 36, 2577–2604.
  * Cai and Liu (2011) Cai, T. and W. Liu (2011): “Adaptive thresholding for sparse covariance matrix estimation,” _Journal of the American Statistical Association_ , 106, 672–684.
  * Chamberlain and Rothschild (1983) Chamberlain, G. and M. Rothschild (1983): “Arbitrage, Factor Structure, and Mean-Variance Analysis on Large Asset Markets,” _Econometrica_ , 51, 1281–1304.
  * Chib et al. (2006) Chib, S., F. Nardari, and N. Shephard (2006): “Analysis of high dimensional multivariate stochastic volatility models,” _Journal of Econometrics_ , 134, 341–371.
  * Choi et al. (2018) Choi, I., D. Kim, Y. J. Kim, and N.-S. Kwark (2018): “A multilevel factor model: Identification, asymptotic theory and applications,” _Journal of Applied Econometrics_ , 33, 355–377.
  * Choi and Kim (2023) Choi, S. H. and D. Kim (2023): “Large volatility matrix analysis using global and national factor models,” _Journal of Econometrics_.
  * Fama and French (2012) Fama, E. F. and K. R. French (2012): “Size, value, and momentum in international stock returns,” _Journal of financial economics_ , 105, 457–472.
  * Fan et al. (2016) Fan, J., A. Furger, and D. Xiu (2016): “Incorporating global industrial classification standard into portfolio allocation: A simple factor-based large covariance matrix estimator with high-frequency data,” _Journal of Business & Economic Statistics_, 34, 489–503.
  * Fan and Kim (2019) Fan, J. and D. Kim (2019): “Structured volatility matrix estimation for non-synchronized high-frequency financial data,” _Journal of Econometrics_ , 209, 61–78.
  * Fan et al. (2017) Fan, J., Q. Li, and Y. Wang (2017): “Estimation of high dimensional mean regression in the absence of symmetry and light tail assumptions,” _Journal of the Royal Statistical Society: Series B (Statistical Methodology)_ , 79, 247–265.
  * Fan et al. (2011) Fan, J., Y. Liao, and M. Mincheva (2011): “High dimensional covariance matrix estimation in approximate factor models,” _Annals of statistics_ , 39, 3320.
  * Fan et al. (2013) ——— (2013): “Large covariance estimation by thresholding principal orthogonal complements,” _Journal of the Royal Statistical Society. Series B, Statistical methodology_ , 75.
  * Fan et al. (2018a) Fan, J., H. Liu, and W. Wang (2018a): “Large covariance estimation through elliptical factor models,” _Annals of statistics_ , 46, 1383.
  * Fan et al. (2018b) Fan, J., W. Wang, and Y. Zhong (2018b): “An $\infty$ eigenvector perturbation bound and its application to robust covariance estimation,” _Journal of Machine Learning Research_ , 18, 1–42.
  * Fan et al. (2021) Fan, J., W. Wang, and Z. Zhu (2021): “A shrinkage principle for heavy-tailed data: High-dimensional robust low-rank matrix recovery,” _Annals of statistics_ , 49, 1239.
  * Fan et al. (2012) Fan, J., J. Zhang, and K. Yu (2012): “Vast portfolio selection with gross-exposure constraints,” _Journal of the American Statistical Association_ , 107, 592–606.
  * Han (2021) Han, X. (2021): “Shrinkage estimation of factor models with global and group-specific factors,” _Journal of Business & Economic Statistics_, 39, 1–17.
  * Hou et al. (2011) Hou, K., G. A. Karolyi, and B.-C. Kho (2011): “What factors drive global stock returns?” _The Review of Financial Studies_ , 24, 2527–2574.
  * Jagannathan and Ma (2003) Jagannathan, R. and T. Ma (2003): “Risk reduction in large portfolios: Why imposing the wrong constraints helps,” _The Journal of Finance_ , 58, 1651–1683.
  * Jung et al. (2022) Jung, K., D. Kim, and S. Yu (2022): “Next generation models for portfolio risk management: An approach using financial big data,” _Journal of Risk and Insurance_ , 89, 765–787.
  * Kose et al. (2003) Kose, M. A., C. Otrok, and C. H. Whiteman (2003): “International business cycles: World, region, and country-specific factors,” _american economic review_ , 93, 1216–1239.
  * Lam and Yao (2012) Lam, C. and Q. Yao (2012): “Factor modeling for high-dimensional time series: inference for the number of factors,” _The Annals of Statistics_ , 694–726.
  * Moench et al. (2013) Moench, E., S. Ng, and S. Potter (2013): “Dynamic hierarchical factor models,” _Review of Economics and Statistics_ , 95, 1811–1817.
  * Onatski (2010) Onatski, A. (2010): “Determining the number of factors from empirical distribution of eigenvalues,” _The Review of Economics and Statistics_ , 92, 1004–1016.
  * Rothman et al. (2009) Rothman, A. J., E. Levina, and J. Zhu (2009): “Generalized thresholding of large covariance matrices,” _Journal of the American Statistical Association_ , 104, 177–186.
  * Stein and James (1961) Stein, C. and W. James (1961): “Estimation with quadratic loss,” in _Proc. 4th Berkeley Symp. Mathematical Statistics Probability_ , vol. 1, 361–379.
  * Stock and Watson (2002) Stock, J. H. and M. W. Watson (2002): “Forecasting using principal components from a large number of predictors,” _Journal of the American statistical association_ , 97, 1167–1179.
  * Vershynin (2010) Vershynin, R. (2010): “Introduction to the non-asymptotic analysis of random matrices,” _arXiv preprint arXiv:1011.3027_.
  * Wang and Fan (2017) Wang, W. and J. Fan (2017): “Asymptotics of empirical eigenstructure for high dimensional spiked covariance,” _Annals of statistics_ , 45, 1342.

##  Appendix A Appendix

###  A.1 Double-POET procedure

We decompose the covariance matrix of the $s$th continent as follows:

$\mbox{\boldmath$\Sigma$}^{s}=\mbox{\boldmath$\Sigma$}_{g}^{s}+\mbox{\boldmath$\Sigma$}_{l}^{s}+\mbox{\boldmath$\Sigma$}_{u}^{s}.$

Then, each component as well as $\mbox{\boldmath$\Sigma$}^{s}$ can be
estimated by the Double-POET procedure (Choi and Kim, 2023) as follows:

  1. 1.

Given a sample covariance matrix, $\widehat{\mbox{\boldmath$\Sigma$}}^{s}$,
using $T$ observations, let
$\\{\widehat{\delta}_{i}^{s},\widehat{v}_{i}^{s}\\}_{i=1}^{p}$ be the
eigenvalues and eigenvectors of $\widehat{\mbox{\boldmath$\Sigma$}}^{s}$ in
decreasing order. We compute

$\widehat{\mbox{\boldmath$\Sigma$}}_{g}^{s,\mathcal{D}}=\widehat{\mbox{\bf
V}}^{s}\widehat{\mbox{\boldmath$\Gamma$}}^{s}\widehat{\mbox{\bf
V}}^{s\prime},$

where
$\widehat{\mbox{\boldmath$\Gamma$}}^{s}=\mathrm{diag}(\widehat{\delta}_{1}^{s},\dots,\widehat{\delta}_{k}^{s})$
and $\widehat{\mbox{\bf
V}}^{s}=(\widehat{v}_{1}^{s},\dots,\widehat{v}_{k}^{s})$.

  2. 2.

Define $\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{l,s}$ as each $p_{l}\times
p_{l}$ diagonal block of
$\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{s}=\widehat{\mbox{\boldmath$\Sigma$}}^{s}-\widehat{\mbox{\boldmath$\Sigma$}}_{g}^{s,\mathcal{D}}$.
For the $l$th block, let
$\\{\widehat{\kappa}_{i}^{l,s},\widehat{\eta}_{i}^{l,s}\\}_{i=1}^{p_{l}}$ be
the eigenvalues and eigenvectors of
$\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{l,s}$ in decreasing order. Then, we
compute

$\widehat{\mbox{\boldmath$\Sigma$}}_{l}^{s,\mathcal{D}}=\widehat{\mbox{\boldmath$\Phi$}}^{s}\widehat{\mbox{\boldmath$\Psi$}}^{s}\widehat{\mbox{\boldmath$\Phi$}}^{s\prime},$

where
$\widehat{\mbox{\boldmath$\Psi$}}^{s}=\mathrm{diag}(\widehat{\Psi}^{1},\dots,\widehat{\Psi}^{L_{s}})$
for
$\widehat{\Psi}^{l}=\mathrm{diag}(\widehat{\kappa}_{1}^{l,s},\dots,\widehat{\kappa}_{r_{l}}^{l,s})$,
and the block diagonal matrix
$\widehat{\mbox{\boldmath$\Phi$}}^{s}=\mathrm{diag}(\widehat{\Phi}^{1},\dots,\widehat{\Phi}^{L_{s}})$
for
$\widehat{\Phi}^{l}=(\widehat{\eta}_{1}^{l,s},\dots,\widehat{\eta}_{r_{l}}^{l,s})$
for $l=1,2,\dots,L_{s}$, where $L_{s}$ is the number of countries in continent
$s$.

  3. 3.

Let
$\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{s}=\widehat{\mbox{\boldmath$\Sigma$}}^{s}-\widehat{\mbox{\boldmath$\Sigma$}}_{g}^{s,\mathcal{D}}-\widehat{\mbox{\boldmath$\Sigma$}}_{l}^{s,\mathcal{D}}$
be the principal orthogonal complement. We apply the adaptive thresholding
method on
$\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{s}=(\widehat{\sigma}_{u,ij})_{p\times
p}$ following Bickel and Levina (2008) and Fan et al. (2013). Specifically,
define $\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{s,\mathcal{D}}$ as the
thresholded error covariance matrix estimator:

$\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{s,\mathcal{D}}=(\widehat{\sigma}_{u,ij}^{s,\mathcal{D}})_{p_{s}\times
p_{s}},\;\;\;\;\;\widehat{\sigma}_{u,ij}^{s,\mathcal{D}}=\left\\{\begin{array}[]{ll}\widehat{\sigma}_{u,ij},&i=j\\\
s_{ij}(\widehat{\sigma}_{u,ij})I(|\widehat{\sigma}_{u,ij}|\geq\tau_{ij}),&i\neq
j\end{array}\right.,$

where an entry-dependent threshold
$\tau_{ij}=\tau(\widehat{\sigma}_{u,ii}\widehat{\sigma}_{u,jj})^{1/2}$ and
$s_{ij}(\cdot)$ is a generalized shrinkage function (Cai and Liu, 2011;
Rothman et al., 2009). The thresholding constant $\tau$ is determined in
Theorem 3.1.

  4. 4.

The final estimator of $\mbox{\boldmath$\Sigma$}^{s}$ is then defined as

$\widehat{\mbox{\boldmath$\Sigma$}}^{s,\mathcal{D}}=\widehat{\mbox{\boldmath$\Sigma$}}_{g}^{s,\mathcal{D}}+\widehat{\mbox{\boldmath$\Sigma$}}_{l}^{s,\mathcal{D}}+\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{s,\mathcal{D}}.$

By using the proof of Theorem 3.1 of Choi and Kim (2023) and Assumption 3.1,
we can obtain the following results: for each continent $s\in\\{1,\dots,S\\}$,

$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}_{g}^{s,\mathcal{D}}-\mbox{\boldmath$\Sigma$}_{g}^{s}\|_{\max}$
$\displaystyle=O_{P}(p^{\frac{5}{2}(1-a_{1})}\sqrt{\log
p/T}+1/p^{\frac{5a_{1}}{2}-\frac{3}{2}-c}),$ (A.1)
$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}_{l}^{s,\mathcal{D}}-\mbox{\boldmath$\Sigma$}_{l}^{s}\|_{\max}$
$\displaystyle=O_{P}(\omega_{T}),$ (A.2)
$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{s,\mathcal{D}}-\mbox{\boldmath$\Sigma$}_{u}^{s}\|_{\max}$
$\displaystyle=O_{P}(\omega_{T}),$ (A.3)

where $\omega_{T}=p^{\frac{5}{2}(1-a_{1})+\frac{5}{2}c(1-a_{2})}\sqrt{\log
p/T}+1/p^{\frac{5}{2}a_{1}-\frac{3}{2}+c(\frac{5}{2}a_{2}-\frac{7}{2})}+m_{p}/\sqrt{p^{c(5a_{2}-3)}}$.

###  A.2 Proof of Theorem 3.1

We first provide useful lemmas below. Let $\\{\delta_{i},v_{i}\\}_{i=1}^{p}$
be the eigenvalues and their corresponding eigenvectors of $\Sigma$ in
decreasing order. Let $\\{\bar{\delta}_{i},\bar{v}_{i}\\}_{i=1}^{k}$ and
$\\{\widetilde{\delta}_{i},\widetilde{v}_{i}\\}_{i=1}^{k}$ be the leading
eigenvalues and eigenvectors of $\mbox{\bf B}\mbox{\bf B}^{\prime}$ and
$\widetilde{\mbox{\boldmath$\Sigma$}}_{g}$, respectively, where
$\widetilde{\mbox{\boldmath$\Sigma$}}_{g}=(\widetilde{\mbox{\boldmath$\Sigma$}}_{g}^{\mathcal{D}}+\hat{D}^{\frac{1}{2}}\widehat{\mbox{\boldmath$\Theta$}}\hat{D}^{\frac{1}{2}})$.
Define
$\mbox{\boldmath$\Sigma$}_{E}=\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime}+\mbox{\boldmath$\Sigma$}_{u}$
and let
$\mbox{\boldmath$\Sigma$}_{E}^{l}=\Lambda^{l}{\Lambda^{l}}^{\prime}+\mbox{\boldmath$\Sigma$}_{u}^{l}$
be the $l$th diagonal block of $\mbox{\boldmath$\Sigma$}_{E}$. For each
country $l$, let $\\{\kappa_{i}^{l},\eta_{i}^{l}\\}_{i=1}^{p_{l}}$ be the
eigenvalues and eigenvectors of $\mbox{\boldmath$\Sigma$}_{E}^{l}$ in
decreasing order, and
$\\{\bar{\kappa}_{i}^{l},\bar{\eta}_{i}^{l}\\}_{i=1}^{r_{l}}$ for
$\Lambda^{l}{\Lambda^{l}}^{\prime}$.

By Weyl’s theorem, we have the following lemma under the pervasive conditions.

######  Lemma A.1.

Under Assumption 3.1(i), we have

$|\delta_{i}-\bar{\delta}_{i}|\leq\|\mbox{\boldmath$\Sigma$}_{E}\|\text{ for
}i\leq k,\;\;\;\;\;|\delta_{i}|\leq\|\mbox{\boldmath$\Sigma$}_{E}\|\text{ for
}i>k,$

and, for $i\leq k$, $\bar{\delta}_{i}/p^{a_{1}}$ is strictly bigger than zero
for all $p$. In addition, for each national group $l$, we have

$|\kappa_{i}^{l}-\bar{\kappa}_{i}^{l}|\leq\|\mbox{\boldmath$\Sigma$}_{u}^{l}\|\text{
for }i\leq
r_{l},\;\;\;\;\;|\kappa_{i}^{l}|\leq\|\mbox{\boldmath$\Sigma$}_{u}^{l}\|\text{
for }i>r_{l},$

and, for $i\leq r_{l}$, $\bar{\kappa}_{i}^{l}/p_{l}^{a_{2}}$ is strictly
bigger than zero for all $p_{l}$.

The following lemma presents the individual convergence rate of leading
eigenvectors using Lemma A.1 and the $l_{\infty}$ norm perturbation bound
theorem of Fan et al. (2018b).

######  Lemma A.2.

Under Assumption 3.1(i), we have the following results.

  * (i)

We have

$\max_{i\leq k}\|\bar{v}_{i}-v_{i}\|_{\infty}\leq
C\frac{\|\mbox{\boldmath$\Sigma$}_{E}\|_{\infty}}{p^{3(a_{1}-\frac{1}{2})}}.$

  * (ii)

For each national group $l$, we have

$\max_{i\leq r_{l}}\|\bar{\eta}_{i}^{l}-\eta_{i}^{l}\|_{\infty}\leq
C\frac{\|\mbox{\boldmath$\Sigma$}_{u}^{l}\|_{\infty}}{p_{l}^{3(a_{2}-\frac{1}{2})}}.$

###### Proof.

(i) Let $\mbox{\bf B}=(\tilde{b}_{1},\dots,\tilde{b}_{k})$. Then, for $i\leq
k$, $\bar{\delta}_{i}=\|\tilde{b}_{i}\|^{2}\asymp p^{a_{1}}$ from Lemma A.1
and $\bar{v}_{i}=\tilde{b}_{i}/\|\tilde{b}_{i}\|$. Hence,
$\|\bar{v}_{i}\|_{\infty}\leq\|\mbox{\bf B}\|_{\max}/\|\tilde{b}_{i}\|\leq
C/\sqrt{p^{a_{1}}}$. In addition, for $\widetilde{\mbox{\bf
V}}=(\bar{v}_{1},\dots,\bar{v}_{k})$, the coherence $\mu(\widetilde{\mbox{\bf
V}})=p\max_{i}\sum_{j=1}^{k}\widetilde{\mbox{\bf V}}_{ij}^{2}/k\leq
Cp^{1-a_{1}}$, where $\widetilde{\mbox{\bf V}}_{ij}$ is the $(i,j)$ entry of
$\widetilde{\mbox{\bf V}}$. Thus, by Theorem 1 of Fan et al. (2018b), we have

$\max_{i\leq k}\|\bar{v}_{i}-v_{i}\|_{\infty}\leq
Cp^{2(1-a_{1})}\frac{\|\mbox{\boldmath$\Sigma$}_{E}\|_{\infty}}{\bar{\gamma}\sqrt{p}},$

where the eigengap
$\bar{\gamma}=\min\\{\bar{\delta}_{i}-\bar{\delta}_{i+1}:1\leq i\leq k\\}$ and
$\delta_{k+1}=0$. By the similar argument, we can show the result (ii). ∎

######  Lemma A.3.

Let $\mbox{\bf R}_{0}=(R_{0,sq})_{S\times S}$, where $R_{0,sq}$ is the
$(s,q)$th off-diagonal partitioned block matrix for $s,q\in\\{1,\dots,S\\}$.
Under Assumption 3.1, for $s\neq q$, we have

$\displaystyle\|\widehat{\Theta}_{sq}-R_{0,sq}\|_{\max}=O_{P}\left(p^{\frac{5}{2}(1-a_{1})}\big{(}\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta}\big{)}\right).$

###### Proof.

For $s\neq q$, let the singular value decomposition be $R_{0,sq}=\mbox{\bf
U}\Xi\mbox{\bf
W}^{\prime}=\sum_{i=1}^{k_{sq}^{\ast}}\xi_{i}u_{i}w_{i}^{\prime}$, where
$k_{sq}^{\ast}$ is the rank of $R_{0,sq}$, the singular values are
$\xi_{1}\geq\xi_{2}\geq\dots\geq\xi_{k_{sq}^{\ast}}>0$, and the matrices
$\mbox{\bf U}=(u_{1},\dots,u_{k_{sq}^{\ast}})$, $\mbox{\bf
W}=(w_{1},\dots,w_{k_{sq}^{\ast}})$ consist of the singular vectors. By
Lipschitz condition, $\|\mbox{\bf R}_{h}-\mbox{\bf
R}_{0}\|_{\max}=O(1/T^{(1-\alpha)\beta})$, and Assumption 3.1 (iii), we have

$\displaystyle\|\widehat{\mbox{\bf R}}_{h}-\mbox{\bf
R}_{0}\|_{\max}=O_{P}\left(\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\right).$ (A.4)

Note that $R_{0,sq}$ is $k_{sq}^{\ast}$-rank matrix for $s\neq
q\in\\{1,\dots,S\\}$. By Weyl’s inequality, we have

$\displaystyle|\widehat{\xi}_{i}-\xi_{i}|$
$\displaystyle\leq\|\widehat{R}_{sq}-R_{0,sq}\|_{F}=O_{P}\left(p\big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\big{)}\right).$ (A.5)

By Theorem 1 of Fan et al. (2018b), we have

$\displaystyle\|\widehat{u}_{i}-u_{i}\|_{\infty}$ $\displaystyle\leq
Cp^{2(1-a_{1})}\frac{\|\widehat{R}_{sq}-R_{0,sq}\|_{\infty}}{p^{a_{1}}\sqrt{p}}\leq
Cp^{2(1-a_{1})}\frac{\|\widehat{R}_{sq}-R_{0,sq}\|_{\max}}{p^{a_{1}-1}\sqrt{p}}$
$\displaystyle=O_{P}\left(p^{\frac{5}{2}-3a_{1}}\big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\big{)}\right).$ (A.6)

Similarly, we can obtain the same rate for
$\|\widehat{w}_{i}-w_{i}\|_{\infty}$. Note that $\|\mbox{\bf
U}\Xi^{\frac{1}{2}}\|_{\max}=O_{P}(1)$. By (A.5) and (A.6), we have

$\displaystyle\|\widehat{\mbox{\bf U}}\widehat{\Xi}^{\frac{1}{2}}-\mbox{\bf
U}\Xi^{\frac{1}{2}}\|_{\max}$ $\displaystyle\leq\|\widehat{\mbox{\bf
U}}(\widehat{\Xi}^{\frac{1}{2}}-\Xi^{\frac{1}{2}})\|_{\max}+\|(\widehat{\mbox{\bf
U}}-\mbox{\bf U})\Xi^{\frac{1}{2}}\|_{\max}$
$\displaystyle=O_{P}\left(p^{\frac{5}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})\right)=o\left(1\right).$

Then, we have $\|\widehat{\mbox{\bf U}}\|_{\max}=O_{P}(1/\sqrt{p^{a_{1}}})$.
Similarly, we can obtain $\|\widehat{\mbox{\bf
W}}\|_{\max}=O_{P}(1/\sqrt{p^{a_{1}}})$. Therefore, we have

$\displaystyle\|\widehat{\Theta}_{sq}-R_{0,sq}\|_{\max}\leq\|\widehat{\mbox{\bf
U}}(\widehat{\Xi}-\Xi)\widehat{\mbox{\bf
W}}^{\prime}\|_{\max}+\|(\widehat{\mbox{\bf U}}-\mbox{\bf
U})\Xi(\widehat{\mbox{\bf W}}-\mbox{\bf
W})^{\prime}\|_{\max}+2\|(\widehat{\mbox{\bf U}}-\mbox{\bf U})\Xi\mbox{\bf
W}^{\prime}\|_{\max}$
$\displaystyle\qquad=O_{P}(p^{-a_{1}}\|\widehat{\Xi}-\Xi\|_{\max}+\sqrt{p^{a_{1}}}\|\widehat{\mbox{\bf
U}}-\mbox{\bf U}\|_{\max})=O_{P}\big{(}p^{\frac{5}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})\big{)}.$

∎

######  Lemma A.4.

Under Assumption 3.1, for $i\leq k$, we have

$\displaystyle|\widetilde{\delta}_{i}-\bar{\delta}_{i}|=O_{P}\left(p^{\frac{7}{2}-\frac{5}{2}a_{1}}\Big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\Big{)}+\frac{1}{p^{\frac{5}{2}a_{1}-\frac{5}{2}-c}}\right),$
$\displaystyle\|\widetilde{v}_{i}-\bar{v}_{i}\|_{\infty}=O_{P}\left(p^{5-\frac{11}{2}a_{1}}\Big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\Big{)}+\frac{1}{p^{\frac{11}{2}a_{1}-4-c}}\right).$

###### Proof.

Let $\widetilde{\mbox{\bf R}}_{0}=(\tilde{\rho}_{0,ij})_{p\times p}$, where
$\tilde{\rho}_{0,ij}=0$ if $\\{i,j\\}\in s$, and
$\tilde{\rho}_{0,ij}=\rho_{0,ij}$ if $i\in s$, $j\in q$, and $s\neq q$. By
Assumption 3.1 (iv), we have
$\max_{i}|\widehat{\Sigma}_{ii}-\Sigma_{ii}|=O_{P}(\sqrt{\log p/T})$. Then, by
Lemma A.3, we can easily obtain that

$\displaystyle\|\hat{D}^{\frac{1}{2}}\widehat{\mbox{\boldmath$\Theta$}}\hat{D}^{\frac{1}{2}}-D^{\frac{1}{2}}\widetilde{\mbox{\bf
R}}_{0}D^{\frac{1}{2}}\|_{\max}=O_{P}\big{(}p^{\frac{5}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})\big{)}.$ (A.7)

By using the fact that $\widetilde{\mbox{\boldmath$\Sigma$}}_{g}$ and
$\mbox{\boldmath$\Sigma$}_{g}$ are low-rank matrices, (A.1) and (A.7), we have

$\displaystyle|\widetilde{\delta}_{i}-\bar{\delta}_{i}|$
$\displaystyle\leq\|\widetilde{\mbox{\boldmath$\Sigma$}}_{g}-\mbox{\boldmath$\Sigma$}_{g}\|_{F}$
$\displaystyle=O_{P}\left(\sqrt{\frac{p^{2}}{S}\big{(}p^{5(1-a_{1})}\frac{\log
p}{T}+\frac{1}{p^{5a_{1}-3-2c}}\big{)}+\frac{p^{2}(S-1)}{S}\big{(}p^{5(1-a_{1})}(\frac{\log
p}{T^{\alpha}}+\frac{1}{T^{2(1-\alpha)\beta}})\big{)}}\right)$
$\displaystyle=O_{P}\left(p\Big{(}p^{\frac{5}{2}(1-a_{1})}(\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}})+\frac{1}{p^{\frac{5}{2}a_{1}-\frac{3}{2}-c}}\Big{)}\right).$

By Theorem 1 of Fan et al. (2018b), (A.1) and (A.7), we have

$\displaystyle\|\widetilde{v}_{i}-\bar{v}_{i}\|_{\infty}$ $\displaystyle\leq
Cp^{2(1-a_{1})}\frac{\|\widetilde{\mbox{\boldmath$\Sigma$}}_{g}-\mbox{\boldmath$\Sigma$}_{g}\|_{\infty}}{p^{a_{1}}\sqrt{p}}=O_{P}\left(p^{5-\frac{11}{2}a_{1}}\big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\big{)}+\frac{1}{p^{\frac{11}{2}a_{1}-4-c}}\right).$

∎

Proof of Theorem 3.1. Consider (3.1). Let $\mbox{\bf B}\mbox{\bf
B}^{\prime}=\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}\widetilde{\mbox{\bf V}}^{\prime}$,
where
$\widetilde{\mbox{\boldmath$\Gamma$}}=\mathrm{diag}(\bar{\delta}_{1},\dots,\bar{\delta}_{k})$
and their corresponding leading $k$ eigenvectors $\widetilde{\mbox{\bf
V}}=(\bar{v}_{1},\dots,\bar{v}_{k})$. By Lemma A.4, we have

$\displaystyle\|\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\widetilde{\mbox{\bf
V}}_{g}^{\prime}-\mbox{\bf B}\mbox{\bf B}^{\prime}\|_{\max}$
$\displaystyle\leq\|\widetilde{\mbox{\bf
V}}_{g}(\widetilde{\mbox{\boldmath$\Gamma$}}_{g}-\widetilde{\mbox{\boldmath$\Gamma$}})\widetilde{\mbox{\bf
V}}_{g}^{\prime}\|_{\max}+\|(\widetilde{\mbox{\bf V}}_{g}-\widetilde{\mbox{\bf
V}})\widetilde{\mbox{\boldmath$\Gamma$}}(\widetilde{\mbox{\bf
V}}_{g}-\widetilde{\mbox{\bf V}})^{\prime}\|_{\max}+2\|\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}(\widetilde{\mbox{\bf
V}}_{g}-\widetilde{\mbox{\bf V}})^{\prime}\|_{\max}$
$\displaystyle=O(p^{-a_{1}}\|\widetilde{\mbox{\boldmath$\Gamma$}}_{g}-\widetilde{\mbox{\boldmath$\Gamma$}}\|_{\max}+\sqrt{p^{a_{1}}}\|\widetilde{\mbox{\bf
V}}_{g}-\widetilde{\mbox{\bf V}}\|_{\max})$
$\displaystyle=O_{P}\left(p^{5(1-a_{1})}\Big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\Big{)}+\frac{1}{p^{5a_{1}-4-c}}\right).$

In addition, by (A.2) and (A.3), we have
$\|\widetilde{\mbox{\boldmath$\Sigma$}}_{l}^{\mathcal{D}}-\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime}\|_{\max}=O_{P}(\omega_{T})$
and
$\|\widetilde{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}}-\mbox{\boldmath$\Sigma$}_{u}\|_{\max}=O_{P}(\omega_{T})$.
Therefore, we have

$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{S}}-\mbox{\boldmath$\Sigma$}\|_{\max}$
$\displaystyle\leq\|\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\widetilde{\mbox{\bf
V}}_{g}^{\prime}-\mbox{\bf B}\mbox{\bf
B}^{\prime}\|_{\max}+\|\widetilde{\mbox{\boldmath$\Sigma$}}_{l}^{\mathcal{D}}-\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime}\|_{\max}+\|\widetilde{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}_{u}-\mbox{\boldmath$\Sigma$}_{u}\|_{\max}$
$\displaystyle=O_{P}\left(p^{5(1-a_{1})}\Big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\Big{)}+\frac{1}{p^{5a_{1}-4-c}}+\omega_{T}\right).$

Consider (3.2). Similar to the proofs of (A.23), we can show
$\|(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{S}})^{-1}-\mbox{\boldmath$\Sigma$}_{E}^{-1}\|=O_{P}(m_{p}\omega_{T}^{1-q}+p^{\frac{c}{2}(1-a_{2})}\omega_{T})$.
Let $\widehat{\mbox{\bf
H}}=\widetilde{\mbox{\boldmath$\Gamma$}}_{g}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}_{g}^{\prime}(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{S}})^{-1}\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}^{\frac{1}{2}}$ and
$\widetilde{\mbox{\bf
H}}=\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}^{\prime}\mbox{\boldmath$\Sigma$}_{E}^{-1}\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}$. Using the Sherman-
Morrison-Woodbury formula, we have

$\|(\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{S}})^{-1}-\mbox{\boldmath$\Sigma$}^{-1}\|\leq\|(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{S}})^{-1}-\mbox{\boldmath$\Sigma$}_{E}^{-1}\|+\Delta,$

where
$\Delta=\|(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{S}})^{-1}\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}^{\frac{1}{2}}(\mbox{\bf
I}_{k}+\widehat{\mbox{\bf
H}})^{-1}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}_{g}^{\prime}(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{S}})^{-1}-\mbox{\boldmath$\Sigma$}_{E}^{-1}\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}(\mbox{\bf
I}_{k}+\widetilde{\mbox{\bf
H}})^{-1}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}^{\prime}\mbox{\boldmath$\Sigma$}_{E}^{-1}\|$. Then, the right hand side
can be bounded by following terms:

$\displaystyle
L_{1}=\|((\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{S}})^{-1}-\mbox{\boldmath$\Sigma$}_{E}^{-1})\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}(\mbox{\bf
I}_{k}+\widetilde{\mbox{\bf
H}})^{-1}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}^{\prime}\mbox{\boldmath$\Sigma$}_{E}^{-1}\|,$ $\displaystyle
L_{2}=\|\mbox{\boldmath$\Sigma$}_{E}^{-1}(\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}^{{\frac{1}{2}}}-\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{{\frac{1}{2}}})(\mbox{\bf
I}_{k}+\widetilde{\mbox{\bf
H}})^{-1}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}^{\prime}\mbox{\boldmath$\Sigma$}_{E}^{-1}\|,$ $\displaystyle
L_{3}=\|\mbox{\boldmath$\Sigma$}_{E}^{-1}\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{{\frac{1}{2}}}((\mbox{\bf
I}_{k}+\widehat{\mbox{\bf H}})^{-1}-(\mbox{\bf I}_{k}+\widetilde{\mbox{\bf
H}})^{-1})\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}^{\prime}\mbox{\boldmath$\Sigma$}_{E}^{-1}\|.$

By Weyl’s inequality, we have $\lambda_{\min}(\mbox{\boldmath$\Sigma$}_{E})>c$
since $\lambda_{\min}(\mbox{\boldmath$\Sigma$}_{u})>c$ and
$\lambda_{\min}(\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime})=0$.
Hence, $\|\mbox{\boldmath$\Sigma$}_{E}^{-1}\|=O_{P}(1)$. Note that
$\|\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{{\frac{1}{2}}}\|=O_{P}(p^{\frac{a_{1}}{2}})$.
By Lemma A.4, we have $\|\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}^{{\frac{1}{2}}}-\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{{\frac{1}{2}}}\|_{\max}=O_{P}(p^{5(1-a_{1})}\big{(}\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta}\big{)}+1/p^{5a_{1}-4-c})$. Then, we have

$\displaystyle\|\widehat{\mbox{\bf H}}-\widetilde{\mbox{\bf H}}\|$
$\displaystyle\leq\|(\widetilde{\mbox{\boldmath$\Gamma$}}_{g}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}_{g}^{\prime}-\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}^{\prime})(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{S}})^{-1}(\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}^{\frac{1}{2}}-\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}})\|$
$\displaystyle\;\;\;+\|(\widetilde{\mbox{\boldmath$\Gamma$}}_{g}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}_{g}^{\prime}-\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}^{\prime})(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{S}})^{-1}\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\|+\|\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}^{\prime}((\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{S}})^{-1}-\mbox{\boldmath$\Sigma$}_{E}^{-1})\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\|$
$\displaystyle=O_{P}\left(p^{a_{1}}m_{p}\omega_{T}^{1-q}+p^{\frac{11}{2}-\frac{9}{2}a_{1}}\Big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\Big{)}+\frac{1}{p^{\frac{9}{2}(a_{1}-1)-c}}\right).$

Since $\lambda_{\min}(\mbox{\bf I}_{k}+\widetilde{\mbox{\bf
H}})\geq\lambda_{\min}(\widetilde{\mbox{\bf
H}})\geq\lambda_{\min}(\mbox{\boldmath$\Sigma$}_{E}^{-1})\lambda_{\min}^{2}(\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}})\geq Cp^{a_{1}}$, we
have $\|(\mbox{\bf I}_{k}+\widetilde{\mbox{\bf
H}})^{-1}\|=O_{P}(1/p^{a_{1}})$. Then, $L_{1}=O_{P}(m_{p}\omega_{T}^{1-q})$.
In addition, $L_{2}=O_{P}(p^{\frac{-a_{1}}{2}}\|\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}^{\frac{1}{2}}-\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\|)=O_{P}(p^{\frac{11}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{11}{2}a_{1}-\frac{9}{2}-c})$
and $L_{3}=O_{P}(p^{a_{1}}\|(\mbox{\bf I}_{k}+\widehat{\mbox{\bf
H}})^{-1}-(\mbox{\bf I}_{k}+\widetilde{\mbox{\bf
H}})^{-1}\|)=O_{P}(p^{-a_{1}}\|\widehat{\mbox{\bf H}}-\widetilde{\mbox{\bf
H}}\|)=O_{P}(m_{p}\omega_{T}^{1-q}+p^{\frac{11}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{11}{2}a_{1}-\frac{9}{2}-c})$.
Thus, we have

$\displaystyle\Delta=O_{P}(m_{p}\omega_{T}^{1-q}+p^{\frac{11}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{11}{2}a_{1}-\frac{9}{2}-c}).$
(A.8)

Therefore, we have

$\|(\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{S}})^{-1}-\mbox{\boldmath$\Sigma$}^{-1}\|=O_{P}\left(m_{p}\omega_{T}^{1-q}+p^{\frac{c}{2}(1-a_{2})}\omega_{T}+p^{\frac{11}{2}(1-a_{1})}\Big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\Big{)}+\frac{1}{p^{\frac{11}{2}a_{1}-\frac{9}{2}-c}}\right).$
(A.9)

Consider (3.3). We derive the rate of convergence for
$\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{S}}-\mbox{\boldmath$\Sigma$}\|_{\Sigma}$.
The SVD decomposition of $\Sigma$ is

$\mbox{\boldmath$\Sigma$}=(\mbox{\bf V}_{p\times
k}\;\;\mbox{\boldmath$\Phi$}_{p\times
r}\;\;\mbox{\boldmath$\Omega$}_{p\times(p-k-r)})\begin{pmatrix}\mbox{\boldmath$\Gamma$}_{k\times
k}&&\\\ &\mbox{\boldmath$\Psi$}_{r\times r}&\\\
&&\mbox{\boldmath$\Theta$}_{(p-k-r)\times(p-k-r)}\end{pmatrix}\begin{pmatrix}\mbox{\bf
V}^{\prime}\\\ \mbox{\boldmath$\Phi$}^{\prime}\\\
\mbox{\boldmath$\Omega$}^{\prime}\end{pmatrix}.$

Note that $\Omega$ is used to denote the precision matrix in Section 2.2.
Moreover, since all the eigenvalues of $\Sigma$ are strictly bigger than 0,
for any maxtrix A, we have $\|\mbox{\bf
A}\|_{\Sigma}^{2}=O_{P}(p^{-1})\|\mbox{\bf A}\|_{F}^{2}$. Then, we have

$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{S}}-\mbox{\boldmath$\Sigma$}\|_{\Sigma}$
$\displaystyle\leq
p^{-1/2}\Big{(}\|\mbox{\boldmath$\Sigma$}^{-1/2}(\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\widetilde{\mbox{\bf
V}}_{g}^{\prime}-\mbox{\bf B}\mbox{\bf
B}^{\prime})\mbox{\boldmath$\Sigma$}^{-1/2}\|_{F}$
$\displaystyle\;\;\;\;\;+\|\mbox{\boldmath$\Sigma$}^{-1/2}(\widetilde{\mbox{\boldmath$\Sigma$}}_{l}^{\mathcal{D}}-\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime})\mbox{\boldmath$\Sigma$}^{-1/2}\|_{F}+\|\mbox{\boldmath$\Sigma$}^{-1/2}(\widetilde{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}_{u}-\mbox{\boldmath$\Sigma$}_{u})\mbox{\boldmath$\Sigma$}^{-1/2}\|_{F}\Big{)}$
$\displaystyle=:\Delta_{G}+\Delta_{L}+\Delta_{S}.$

By using the fact that $S$ is fixed and proofs of (3.5) in Choi and Kim
(2023), we can obtain

$\displaystyle\Delta_{L}$
$\displaystyle=O_{P}\Big{(}p^{\frac{5}{2}(1-a_{1})+c(1-a_{2})}\sqrt{\frac{\log
p}{T}}+\frac{1}{p^{\frac{5}{2}a_{1}-\frac{3}{2}-2c+ca_{2}}}+\frac{m_{p}}{p^{ca_{2}}}$
$\displaystyle\qquad\qquad+p^{\frac{11}{2}-5a_{1}+5c(1-a_{2})}\frac{\log
p}{T}+\frac{1}{p^{5a_{1}-\frac{7}{2}-c(7-5a_{2})}}+\frac{m_{p}^{2}}{p^{5ca_{2}-3c-\frac{1}{2}}}\Big{)}$
(A.10)

and

$\displaystyle\Delta_{S}=O_{P}(p^{-1/2}\|\widetilde{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}_{u}-\mbox{\boldmath$\Sigma$}_{u}\|_{F})=O_{P}(\|\widetilde{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}_{u}-\mbox{\boldmath$\Sigma$}_{u}\|_{2})=O_{P}(m_{p}\omega_{T}^{1-q}).$
(A.11)

We have

$\displaystyle\Delta_{G}$
$\displaystyle=p^{-1/2}\left\|\begin{pmatrix}\mbox{\boldmath$\Gamma$}^{-\frac{1}{2}}\mbox{\bf
V}^{\prime}\\\
\mbox{\boldmath$\Psi$}^{-\frac{1}{2}}\mbox{\boldmath$\Phi$}^{\prime}\\\
\mbox{\boldmath$\Theta$}^{-\frac{1}{2}}\mbox{\boldmath$\Omega$}^{\prime}\end{pmatrix}(\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\widetilde{\mbox{\bf
V}}_{g}^{\prime}-\mbox{\bf B}\mbox{\bf B}^{\prime})\left(\mbox{\bf
V}\mbox{\boldmath$\Gamma$}^{-\frac{1}{2}}\;\;\mbox{\boldmath$\Phi$}\mbox{\boldmath$\Psi$}^{-\frac{1}{2}}\;\;\mbox{\boldmath$\Omega$}\mbox{\boldmath$\Theta$}^{-\frac{1}{2}}\right)\right\|_{F}$
$\displaystyle\leq p^{-1/2}\big{(}\|\mbox{\boldmath$\Gamma$}^{-1/2}\mbox{\bf
V}^{\prime}(\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\widetilde{\mbox{\bf
V}}_{g}^{\prime}-\mbox{\bf B}\mbox{\bf B}^{\prime})\mbox{\bf
V}\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}+\|\mbox{\boldmath$\Psi$}^{-1/2}\mbox{\boldmath$\Phi$}^{\prime}(\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\widetilde{\mbox{\bf
V}}_{g}^{\prime}-\mbox{\bf B}\mbox{\bf
B}^{\prime})\mbox{\boldmath$\Phi$}\mbox{\boldmath$\Psi$}^{-1/2}\|_{F}$
$\displaystyle\;\;\;\;+\|\mbox{\boldmath$\Theta$}^{-1/2}\mbox{\boldmath$\Omega$}^{\prime}(\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\widetilde{\mbox{\bf
V}}_{g}^{\prime}-\mbox{\bf B}\mbox{\bf
B}^{\prime})\mbox{\boldmath$\Omega$}\mbox{\boldmath$\Theta$}^{-1/2}\|_{F}+2\|\mbox{\boldmath$\Gamma$}^{-1/2}\mbox{\bf
V}^{\prime}(\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\widetilde{\mbox{\bf
V}}_{g}^{\prime}-\mbox{\bf B}\mbox{\bf
B}^{\prime})\mbox{\boldmath$\Phi$}\mbox{\boldmath$\Psi$}^{-1/2}\|_{F}$
$\displaystyle\;\;\;\;+2\|\mbox{\boldmath$\Gamma$}^{-1/2}\mbox{\bf
V}^{\prime}(\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\widetilde{\mbox{\bf
V}}_{g}^{\prime}-\mbox{\bf B}\mbox{\bf
B}^{\prime})\mbox{\boldmath$\Omega$}\mbox{\boldmath$\Theta$}^{-1/2}\|_{F}+2\|\mbox{\boldmath$\Psi$}^{-1/2}\mbox{\boldmath$\Phi$}^{\prime}(\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\widetilde{\mbox{\bf
V}}_{g}^{\prime}-\mbox{\bf B}\mbox{\bf
B}^{\prime})\mbox{\boldmath$\Omega$}\mbox{\boldmath$\Theta$}^{-1/2}\|_{F}\big{)}$
$\displaystyle=:\Delta_{G1}+\Delta_{G2}+\Delta_{G3}+2\Delta_{G4}+2\Delta_{G5}+2\Delta_{G6}.$

In order to find the convergence rate of relative Frobenius norm, we consider
the above terms separately. Note that
$\mbox{\boldmath$\Gamma$}=\mathrm{diag}(\delta_{1},\dots,\delta_{k})$ and
$\mbox{\bf V}=(v_{1},\dots,v_{k})$. For $\Delta_{G1}$, we have

$\displaystyle\Delta_{G1}$ $\displaystyle\leq
p^{-1/2}\left(\|\mbox{\boldmath$\Gamma$}^{-1/2}\mbox{\bf
V}^{\prime}(\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\widetilde{\mbox{\bf
V}}_{g}^{\prime}-\mbox{\bf V}\mbox{\boldmath$\Gamma$}\mbox{\bf
V}^{\prime})\mbox{\bf
V}\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}+\|\mbox{\boldmath$\Gamma$}^{-1/2}\mbox{\bf
V}^{\prime}(\mbox{\bf V}\mbox{\boldmath$\Gamma$}\mbox{\bf
V}^{\prime}-\mbox{\bf B}\mbox{\bf B}^{\prime})\mbox{\bf
V}\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}\right)$
$\displaystyle=:\Delta_{G1}^{(a)}+\Delta_{G1}^{(b)}.$

We bound the two terms separately. We have

$\displaystyle\Delta_{G1}^{(a)}$ $\displaystyle\leq
p^{-1/2}\big{(}\|\mbox{\boldmath$\Gamma$}^{-1/2}(\mbox{\bf
V}^{\prime}\widetilde{\mbox{\bf V}}_{g}-\mbox{\bf
I})\widetilde{\mbox{\boldmath$\Gamma$}}_{g}(\widetilde{\mbox{\bf
V}}_{g}^{\prime}\mbox{\bf V}-\mbox{\bf
I})\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}+2\|\mbox{\boldmath$\Gamma$}^{-1/2}(\mbox{\bf
V}^{\prime}\widetilde{\mbox{\bf V}}_{g}-\mbox{\bf
I})\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}$
$\displaystyle\;\;\;+\|(\mbox{\boldmath$\Gamma$}^{-1/2}(\widetilde{\mbox{\boldmath$\Gamma$}}_{g}-\mbox{\boldmath$\Gamma$})\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}\big{)}=:I+II+III.$

By Lemmas A.2 and A.4, we obtain $\|\mbox{\bf V}^{\prime}\widetilde{\mbox{\bf
V}}_{g}-\mbox{\bf I}\|_{F}=\|\mbox{\bf V}^{\prime}(\widetilde{\mbox{\bf
V}}_{g}-\mbox{\bf V})\|_{F}\leq\|\widetilde{\mbox{\bf V}}_{g}-\mbox{\bf
V}\|_{F}=O_{P}(p^{\frac{11}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{11}{2}a_{1}-\frac{9}{2}-c})$.
Then, $II=O_{P}(p^{5-\frac{11}{2}a_{1}}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{11}{2}a_{1}-4-c})$ and $I$ is
of smaller order. In addition, we have
$III\leq\|\mbox{\boldmath$\Gamma$}^{-1/2}(\widetilde{\mbox{\boldmath$\Gamma$}}_{g}-\mbox{\boldmath$\Gamma$})\mbox{\boldmath$\Gamma$}^{-1/2}\|=O_{P}(p^{\frac{7}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{7}{2}a_{1}-\frac{5}{2}-c}+1/p^{a_{1}-ca_{2}})$
by Lemma A.4. Thus,
$\Delta_{G1}^{(a)}=O_{P}(p^{\frac{7}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{7}{2}a_{1}-\frac{5}{2}-c}+1/p^{a_{1}-ca_{2}})$.
Similarly, we have

$\displaystyle\Delta_{G1}^{(b)}$ $\displaystyle\leq
p^{-1/2}\big{(}\|\mbox{\boldmath$\Gamma$}^{-1/2}(\mbox{\bf
V}^{\prime}\widetilde{\mbox{\bf V}}-\mbox{\bf
I})\widetilde{\mbox{\boldmath$\Gamma$}}(\widetilde{\mbox{\bf
V}}^{\prime}\mbox{\bf V}-\mbox{\bf
I})\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}+2\|\mbox{\boldmath$\Gamma$}^{-1/2}(\mbox{\bf
V}^{\prime}\widetilde{\mbox{\bf V}}-\mbox{\bf
I})\widetilde{\mbox{\boldmath$\Gamma$}}\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}$
$\displaystyle\;\;\;+\|(\mbox{\boldmath$\Gamma$}^{-1/2}(\widetilde{\mbox{\boldmath$\Gamma$}}-\mbox{\boldmath$\Gamma$})\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}\big{)}=:I^{\prime}+II^{\prime}+III^{\prime}.$

By $\sin\theta$ theorem, $\|\mbox{\bf V}^{\prime}\widetilde{\mbox{\bf
V}}-\mbox{\bf I}\|=\|\mbox{\bf V}^{\prime}(\widetilde{\mbox{\bf V}}-\mbox{\bf
V})\|\leq\|\widetilde{\mbox{\bf V}}-\mbox{\bf
V}\|=O(\|\mbox{\boldmath$\Sigma$}_{E}\|/p^{a_{1}})$. Then, we have
$II^{\prime}=O(1/p^{a_{1}-ca_{2}})$ and $I^{\prime}$ is of smaller order. By
Lemma A.1, we have $III^{\prime}=O(1/p^{a_{1}-ca_{2}})$. Thus,
$\Delta_{G1}^{(b)}=O(1/p^{a_{1}-ca_{2}})$. Then, we obtain

$\Delta_{G1}=O_{P}\left(p^{\frac{7}{2}(1-a_{1})}\Big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\Big{)}+\frac{1}{p^{\frac{7}{2}a_{1}-\frac{5}{2}-c}}+\frac{1}{p^{a_{1}-ca_{2}}}\right).$
(A.12)

For $\Delta_{G3}$, we have

$\Delta_{G3}\leq
p^{-1/2}\|\mbox{\boldmath$\Theta$}^{-1/2}\mbox{\boldmath$\Omega$}^{\prime}\widetilde{\mbox{\bf
V}}_{g}\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\widetilde{\mbox{\bf
V}}_{g}^{\prime}\mbox{\boldmath$\Omega$}\mbox{\boldmath$\Theta$}^{-1/2}\|_{F}+p^{-1/2}\|\mbox{\boldmath$\Theta$}^{-1/2}\mbox{\boldmath$\Omega$}^{\prime}\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}\widetilde{\mbox{\bf
V}}^{\prime}\mbox{\boldmath$\Omega$}\mbox{\boldmath$\Theta$}^{-1/2}\|_{F}=:\Delta_{G3}^{(a)}+\Delta_{G3}^{(b)}.$

By Lemmas A.2 and A.4, we have

$\displaystyle\|\mbox{\boldmath$\Omega$}^{\prime}\widetilde{\mbox{\bf
V}}_{g}\|_{F}=$
$\displaystyle\|\mbox{\boldmath$\Omega$}^{\prime}(\widetilde{\mbox{\bf
V}}_{g}-\mbox{\bf V})\|_{F}=O(\sqrt{p}\|\widetilde{\mbox{\bf V}}_{g}-\mbox{\bf
V}\|_{\max})$ $\displaystyle=O_{P}(p^{\frac{11}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{11}{2}a_{1}-\frac{9}{2}-c}).$

Since $\|\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\|=O_{P}(p^{a_{1}})$, we have

$\Delta_{G3}^{(a)}\leq
p^{-1/2}\|\mbox{\boldmath$\Theta$}^{-1}\|\|\mbox{\boldmath$\Omega$}^{\prime}\widetilde{\mbox{\bf
V}}_{g}\|_{F}^{2}\|\widetilde{\mbox{\boldmath$\Gamma$}}_{g}\|=O_{P}(p^{\frac{21}{2}-10a_{1}}(\log
p/T^{\alpha}+1/T^{2(1-\alpha)\beta})+1/p^{10a_{1}-\frac{17}{2}-2c}).$

Similarly, $\Delta_{G3}^{(b)}=O_{P}(1/p^{5a_{1}-\frac{7}{2}-2c})$ because
$\|\mbox{\boldmath$\Omega$}^{\prime}\widetilde{\mbox{\bf
V}}\|_{F}=O(\sqrt{p}\|\widetilde{\mbox{\bf V}}-\mbox{\bf
V}\|_{\max})=O_{P}(1/p^{3a_{1}-2-c})$ by Lemma A.2. Then, we obtain

$\Delta_{G3}=O_{P}\left(p^{\frac{21}{2}-10a_{1}}\Big{(}\frac{\log
p}{T^{\alpha}}+\frac{1}{T^{2(1-\alpha)\beta}}\Big{)}+\frac{1}{p^{10a_{1}-\frac{17}{2}-2c}}\right).$

Similarly, we can show that the terms $\Delta_{G2}$, $\Delta_{G4}$,
$\Delta_{G5}$ and $\Delta_{G6}$ are dominated by $\Delta_{G1}$ and
$\Delta_{G3}$. Therefore, we have

$\displaystyle\Delta_{G}$
$\displaystyle=O_{P}\Big{(}p^{\frac{7}{2}(1-a_{1})}\big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\big{)}+\frac{1}{p^{\frac{7}{2}a_{1}-\frac{5}{2}-c}}+\frac{1}{p^{a_{1}-ca_{2}}}$
$\displaystyle\qquad\qquad+p^{\frac{21}{2}-10a_{1}}\big{(}\frac{\log
p}{T^{\alpha}}+\frac{1}{T^{2(1-\alpha)\beta}}\big{)}+\frac{1}{p^{10a_{1}-\frac{17}{2}-2c}}\Big{)}.$
(A.13)

Combining the terms $\Delta_{G}$, $\Delta_{L}$ and $\Delta_{S}$ together, we
complete the proof of (3.3). $\square$

###  A.3 Proof of Theorem 3.2

We provide useful technical lemmas below.

######  Lemma A.5.

Under Assumptions 3.1–3.2, for $i\leq k$, we have

$\displaystyle|\widehat{\delta}_{i}/\delta_{i}-1|=O_{P}\left(p^{1-a_{1}}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})\right),$
$\displaystyle\|\widehat{v}_{i}-v_{i}\|_{\infty}=O_{P}\left(\cfrac{1}{p^{3(a_{1}-\frac{1}{2})-c}}+p^{\frac{5}{2}-3a_{1}}\Big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\Big{)}\right).$

###### Proof.

By Lipschitz condition and Assumption 3.2, we have

$\|d^{-1}\widehat{\mbox{\boldmath$\Sigma$}}_{h}-\mbox{\boldmath$\Sigma$}\|_{\max}=O_{P}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta}).$ (A.14)

Then, we can obtain the first statement by Weyl’s theorem. We have

$d^{-1}\widehat{\mbox{\boldmath$\Sigma$}}_{h}=\mbox{\bf B}\mbox{\bf
B}^{\prime}+\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime}+\mbox{\boldmath$\Sigma$}_{u}+(d^{-1}\widehat{\mbox{\boldmath$\Sigma$}}_{h}-\mbox{\boldmath$\Sigma$})=\mbox{\bf
B}\mbox{\bf
B}^{\prime}+\mbox{\boldmath$\Sigma$}_{E}+(d^{-1}\widehat{\mbox{\boldmath$\Sigma$}}_{h}-\mbox{\boldmath$\Sigma$}).$

We can treat $\mbox{\bf B}\mbox{\bf B}^{\prime}$ as a low rank matrix and the
remaining terms as a perturbation matrix. Note that
$\|\mbox{\boldmath$\Sigma$}_{E}\|_{\infty}=O(p^{c})$. By Theorem 1 of Fan et
al. (2018b), Lemma A.2, Assumption 3.1 and (A.14), we have

$\displaystyle\|\widehat{v}_{i}-v_{i}\|_{\infty}$ $\displaystyle\leq
Cp^{2(1-a_{1})}\frac{\|\mbox{\boldmath$\Sigma$}_{E}\|_{\infty}}{p^{a_{1}}\sqrt{p}}+Cp^{2(1-a_{1})}\frac{\|d^{-1}\widehat{\mbox{\boldmath$\Sigma$}}_{h}-\mbox{\boldmath$\Sigma$}\|_{\max}}{p^{a_{1}-1}\sqrt{p}}$
$\displaystyle=O_{P}\left(\cfrac{1}{p^{3(a_{1}-\frac{1}{2})-c}}+p^{\frac{5}{2}-3a_{1}}\Big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\Big{)}\right).$

∎

######  Lemma A.6.

Under Assumptions 3.1–3.2, for $i\leq r_{l}$, we have

$\displaystyle|\widehat{\kappa}_{i}^{l}/\kappa_{i}^{l}-1|=O_{P}\left(p^{\frac{5}{2}(1-a_{1})+c(1-a_{2})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{5a_{1}}{2}-\frac{3}{2}-2c+ca_{2}}\right),$
$\displaystyle\|\widehat{\eta}_{i}^{l}-\eta_{i}^{l}\|_{\infty}=O_{P}\left(p^{\frac{5}{2}(1-a_{1})+c(\frac{5}{2}-3a_{2})}\big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\big{)}+\cfrac{1}{p^{\frac{5a_{1}}{2}-\frac{3}{2}+c(3a_{2}-\frac{7}{2})}}+\cfrac{m_{p}}{p^{3c(a_{2}-\frac{1}{2})}}\right).$

###### Proof.

We have

$\displaystyle\|\mbox{\boldmath$\Sigma$}_{E}\|\leq\|\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime}\|+\|\mbox{\boldmath$\Sigma$}_{u}\|\leq\|\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime}\|+O(m_{p})=O(p^{ca_{2}}).$

Let $\mbox{\bf B}\mbox{\bf B}^{\prime}=\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}\widetilde{\mbox{\bf V}}^{\prime}$,
where
$\widetilde{\mbox{\boldmath$\Gamma$}}=\mathrm{diag}(\bar{\delta}_{1},\dots,\bar{\delta}_{k})$
and their corresponding leading $k$ eigenvectors $\widetilde{\mbox{\bf
V}}=(\bar{v}_{1},\dots,\bar{v}_{k})$. Also, we let
$\mbox{\boldmath$\Gamma$}=\mathrm{diag}(\delta_{1},\dots,\delta_{k})$ and the
corresponding eigenvectors $\mbox{\bf V}=(v_{1},\dots,v_{k})$ of covariance
matrix $\Sigma$. Note that $\|\mbox{\bf B}\|_{\max}=\|\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{1/2}\|_{\max}=O(1)$. By Lemmas
A.1-A.2, we have

$\displaystyle\|\mbox{\bf
V}\mbox{\boldmath$\Gamma$}^{\frac{1}{2}}-\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{{\frac{1}{2}}}\|_{\max}$
$\displaystyle\leq\|\mbox{\bf
B}\widetilde{\mbox{\boldmath$\Gamma$}}^{-{\frac{1}{2}}}(\mbox{\boldmath$\Gamma$}^{{\frac{1}{2}}}-\widetilde{\mbox{\boldmath$\Gamma$}}^{{\frac{1}{2}}})\|_{\max}+\|(\mbox{\bf
V}-\widetilde{\mbox{\bf V}})\mbox{\boldmath$\Gamma$}^{{\frac{1}{2}}}\|_{\max}$
$\displaystyle\leq
C\cfrac{\|\mbox{\boldmath$\Sigma$}_{E}\|}{p^{a_{1}}}+C\cfrac{\|\mbox{\boldmath$\Sigma$}_{E}\|_{\infty}}{\sqrt{p^{5a_{1}-3}}}=o\left(1\right).$

Hence, we have $\|\mbox{\bf
V}\mbox{\boldmath$\Gamma$}^{\frac{1}{2}}\|_{\max}=O(1)$ and $\|\mbox{\bf
V}\|_{\max}=O(1/\sqrt{p^{a_{1}}})$. By this fact and the results from Lemmas
A.1-A.5, we have

$\displaystyle\|\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}\widetilde{\mbox{\bf
V}}^{\prime}-\mbox{\bf V}\mbox{\boldmath$\Gamma$}\mbox{\bf
V}^{\prime}\|_{\max}$ $\displaystyle\leq\|\widetilde{\mbox{\bf
V}}(\widetilde{\mbox{\boldmath$\Gamma$}}-\mbox{\boldmath$\Gamma$})\widetilde{\mbox{\bf
V}}^{\prime}\|_{\max}+\|(\widetilde{\mbox{\bf V}}-\mbox{\bf
V})\mbox{\boldmath$\Gamma$}(\widetilde{\mbox{\bf V}}-\mbox{\bf
V})^{\prime}\|_{\max}+2\|\mbox{\bf
V}\mbox{\boldmath$\Gamma$}(\widetilde{\mbox{\bf V}}-\mbox{\bf
V})^{\prime}\|_{\max}$
$\displaystyle=O(p^{-a_{1}}\|\widetilde{\mbox{\boldmath$\Gamma$}}-\mbox{\boldmath$\Gamma$}\|_{\max}+\sqrt{p^{a_{1}}}\|\widetilde{\mbox{\bf
V}}-\mbox{\bf V}\|_{\max})=O(1/p^{\frac{5a_{1}}{2}-\frac{3}{2}-c}),$
$\displaystyle\|\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}-\mbox{\bf
V}\mbox{\boldmath$\Gamma$}\mbox{\bf V}^{\prime}\|_{\max}$
$\displaystyle\leq\|\widehat{\mbox{\bf
V}}(\widehat{\mbox{\boldmath$\Gamma$}}-\mbox{\boldmath$\Gamma$})\widehat{\mbox{\bf
V}}^{\prime}\|_{\max}+\|(\widehat{\mbox{\bf V}}-\mbox{\bf
V})\mbox{\boldmath$\Gamma$}(\widehat{\mbox{\bf V}}-\mbox{\bf
V})^{\prime}\|_{\max}+2\|\mbox{\bf
V}\mbox{\boldmath$\Gamma$}(\widehat{\mbox{\bf V}}-\mbox{\bf
V})^{\prime}\|_{\max}$
$\displaystyle=O_{P}(p^{-a_{1}}\|\widehat{\mbox{\boldmath$\Gamma$}}-\mbox{\boldmath$\Gamma$}\|_{\max}+\sqrt{p^{a_{1}}}\|\widehat{\mbox{\bf
V}}-\mbox{\bf V}\|_{\max})$
$\displaystyle=O_{P}(p^{\frac{5}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{5a_{1}}{2}-\frac{3}{2}-c}).$

Thus, we have

$\|\widehat{\mbox{\bf V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf
V}}^{\prime}-\mbox{\bf B}\mbox{\bf
B}^{\prime}\|_{\max}=O_{P}(p^{\frac{5}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{5a_{1}}{2}-\frac{3}{2}-c}).$
(A.15)

Then, we have

$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}_{E}-\mbox{\boldmath$\Sigma$}_{E}\|_{\max}$
$\displaystyle\leq$
$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}_{h}-\mbox{\boldmath$\Sigma$}\|_{\max}+\|\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}-\mbox{\bf
B}\mbox{\bf B}^{\prime}\|_{\max}$ (A.16) $\displaystyle=$ $\displaystyle
O_{P}(p^{\frac{5}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{5a_{1}}{2}-\frac{3}{2}-c}).$
(A.17)

Therefore, the first statement is followed by (A.17) and the Weyl’s theorem.

We decompose the sample covariance matrix
$\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{l}$ for each group $l$ as follows:

$\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{l}=\Lambda^{l}\Lambda^{l\prime}+\mbox{\boldmath$\Sigma$}_{u}^{l}+(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{l}-\mbox{\boldmath$\Sigma$}_{E}^{l}).$

Then, by Theorem 1 of Fan et al. (2018b), Lemma A.2 and (A.17), we have

$\displaystyle\|\widehat{\eta}_{i}^{l}-\eta_{i}^{l}\|_{\infty}$
$\displaystyle\leq
Cp_{l}^{2(1-a_{2})}\frac{\|\mbox{\boldmath$\Sigma$}_{u}^{l}+(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{l}-\mbox{\boldmath$\Sigma$}_{E}^{l})\|_{\infty}}{p_{l}^{a_{2}}\sqrt{p_{l}}}$
$\displaystyle\leq
Cp_{l}^{2(1-a_{2})}\frac{\|\mbox{\boldmath$\Sigma$}_{u}^{l}\|_{\infty}}{p_{l}^{a_{2}}\sqrt{p_{l}}}+Cp_{l}^{2(1-a_{2})}\frac{\|\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{l}-\mbox{\boldmath$\Sigma$}_{E}^{l}\|_{\max}}{p_{l}^{a_{2}-1}\sqrt{p_{l}}}$
$\displaystyle=O_{P}\left(p^{\frac{5}{2}(1-a_{1})+c(\frac{5}{2}-3a_{2})}\big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\big{)}+\cfrac{1}{p^{\frac{5a_{1}}{2}-\frac{3}{2}+c(3a_{2}-\frac{7}{2})}}+\cfrac{m_{p}}{p^{3c(a_{2}-\frac{1}{2})}}\right).$

∎

Proof of Theorem 3.2. We first consider (3.4). We have

$\|\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}-\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime}\|_{\max}=\max_{l}\|\widehat{\Phi}^{l}\widehat{\Psi}^{l}\widehat{\Phi}^{j\prime}-\Lambda^{l}{\Lambda^{l}}^{\prime}\|_{\max}.$

For each group $l$, let
$\Lambda^{l}{\Lambda^{l}}^{\prime}=\widetilde{\Phi}^{l}\widetilde{\Psi}^{l}\widetilde{\Phi}^{l\prime}$,
where
$\widetilde{\Psi}^{l}=\mathrm{diag}(\bar{\kappa}_{1}^{l},\dots,\bar{\kappa}_{r_{l}}^{l})$
and the corresponding eigenvectors
$\widetilde{\Phi}^{l}=(\bar{\eta}_{1},\dots,\bar{\eta}_{r_{l}})$. In addition,
let $\Psi^{l}=\mathrm{diag}(\kappa_{1}^{l},\dots,\kappa_{r_{l}}^{l})$ and
$\Phi^{l}=(\eta_{1},\dots,\eta_{r_{l}})$ to be the leading eigenvalues and the
corresponding eigenvectors of $\mbox{\boldmath$\Sigma$}_{E}^{l}$,
respectively. Then, we have

$\displaystyle\|\Phi^{l}{\Psi^{l}}^{\frac{1}{2}}-\widetilde{\Phi}^{l}{{}\widetilde{\Psi}^{l}}^{\frac{1}{2}}\|_{\max}$
$\displaystyle\leq\|\Lambda^{l}{{}\widetilde{\Psi}^{l}}^{-\frac{1}{2}}({\Psi^{l}}^{\frac{1}{2}}-{{}\widetilde{\Psi}^{l}}^{\frac{1}{2}})\|_{\max}+\|(\Phi^{l}-\widetilde{\Phi}^{l}){\Psi^{l}}^{\frac{1}{2}}\|_{\max}$
$\displaystyle\leq\frac{\|\mbox{\boldmath$\Sigma$}_{u}^{l}\|}{p_{l}^{a_{2}}}+\frac{\|\mbox{\boldmath$\Sigma$}_{u}^{l}\|_{\infty}}{\sqrt{p_{l}^{5a_{2}-3}}}=o(1).$
(A.18)

Since
$\|\Lambda^{l}\|_{\max}=\|\widetilde{\Phi}^{l}{{}\widetilde{\Psi}^{l}}^{\frac{1}{2}}\|_{\max}=O(1)$,
$\|\Phi^{l}{\Psi^{l}}^{\frac{1}{2}}\|_{\max}=O(1)$ and
$\|\Phi^{l}\|_{\max}=O(1/\sqrt{p_{l}^{a_{2}}})$. Using this fact and results
from Lemmas A.1, A.2 and A.6, we can show

$\displaystyle\|\widetilde{\Phi}^{l}\widetilde{\Psi}^{l}\widetilde{\Phi}^{l\prime}-\Phi^{l}\Psi^{l}\Phi^{l\prime}\|_{\max}\leq
O(p_{l}^{-a_{2}}\|\widetilde{\Psi}^{l}-\Psi^{l}\|_{\max}+\sqrt{p_{l}^{a_{2}}}\|\widetilde{\Phi}^{l}-\Phi^{l}\|_{\max})=O(m_{p}/\sqrt{p^{c(5a_{2}-3)}}),$
$\displaystyle\|\widehat{\Phi}^{l}\widehat{\Psi}^{l}\widehat{\Phi}^{l\prime}-\Phi^{l}\Psi^{l}\Phi^{l\prime}\|_{\max}\leq
O_{P}(p_{l}^{-a_{2}}\|\widehat{\Psi}^{l}-\Psi^{l}\|_{\max}+\sqrt{p_{l}^{a_{2}}}\|\widehat{\Phi}^{l}-\Phi^{l}\|_{\max})$
$\displaystyle\qquad\qquad=O_{P}\left(p^{\frac{5}{2}(1-a_{1})+\frac{5}{2}c(1-a_{2})}\big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\big{)}+\frac{1}{p^{\frac{5}{2}a_{1}-\frac{3}{2}+c(\frac{5}{2}a_{2}-\frac{7}{2})}}+\frac{m_{p}}{\sqrt{p^{c(5a_{2}-3)}}}\right).$

By using these rates, we obtain

$\displaystyle\|\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}-\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime}\|_{\max}$
$\displaystyle\qquad=O_{P}\left(p^{\frac{5}{2}(1-a_{1})+\frac{5}{2}c(1-a_{2})}\big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\big{)}+\frac{1}{p^{\frac{5}{2}a_{1}-\frac{3}{2}+c(\frac{5}{2}a_{2}-\frac{7}{2})}}+\frac{m_{p}}{\sqrt{p^{c(5a_{2}-3)}}}\right).$
(A.19)

By (A.14), (A.15) and (A.19), we then have

$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}_{u}-\mbox{\boldmath$\Sigma$}_{u}\|_{\max}$
$\displaystyle\leq\|d^{-1}\widehat{\mbox{\boldmath$\Sigma$}}_{h}-\mbox{\boldmath$\Sigma$}\|_{\max}+\|\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}-\mbox{\bf
B}\mbox{\bf
B}^{\prime}\|_{\max}+\|\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}-\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime}\|_{\max}$
$\displaystyle=O_{P}\left(p^{\frac{5}{2}(1-a_{1})+\frac{5}{2}c(1-a_{2})}\big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\big{)}+\frac{1}{p^{\frac{5}{2}a_{1}-\frac{3}{2}+c(\frac{5}{2}a_{2}-\frac{7}{2})}}+\frac{m_{p}}{\sqrt{p^{c(5a_{2}-3)}}}\right).$
(A.20)

By definition,
$\|\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}}-\widehat{\mbox{\boldmath$\Sigma$}}_{u}\|_{\max}=\max_{ij}|s_{ij}(\widehat{\sigma}_{ij})-\widehat{\sigma}_{ij}|\leq\max_{ij}\tau_{ij}=O_{P}(\tau)$.
Then, we have

$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}_{u}-\mbox{\boldmath$\Sigma$}_{u}\|_{\max}=O_{P}(\tau+\omega_{T^{\alpha}})=O_{P}(\omega_{T^{\alpha}}),$
(A.21)

when $\tau$ is chosen as the same order of
$\omega_{T^{\alpha}}=p^{\frac{5}{2}(1-a_{1})+\frac{5}{2}c(1-a_{2})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{5}{2}a_{1}-\frac{3}{2}+c(\frac{5}{2}a_{2}-\frac{7}{2})}+m_{p}/\sqrt{p^{c(5a_{2}-3)}}$.
Therefore, by the results of (A.15), (A.19) and (A.21), we have

$\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}-\mbox{\boldmath$\Sigma$}\|_{\max}\leq\|\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}-\mbox{\bf
B}\mbox{\bf
B}^{\prime}\|_{\max}+\|\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}-\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime}\|_{\max}+\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}_{u}-\mbox{\boldmath$\Sigma$}_{u}\|_{\max}=O_{P}(\omega_{T^{\alpha}}).$

Consider (3.5). Similar to the proofs of Theorem 2.1 in Fan et al. (2011), we
can show
$\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}_{u}-\mbox{\boldmath$\Sigma$}_{u}\|_{2}=O_{P}(m_{p}\omega_{T^{\alpha}}^{1-q})$.
In addition, since $\lambda_{\min}(\mbox{\boldmath$\Sigma$}_{u})>c_{1}$ and
$m_{p}\omega_{T^{\alpha}}^{1-q}=o(1)$, the minimum eigenvalue of
$\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}}$ is strictly bigger than
0 with probability approaching 1. Then, we have

$\|(\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}})^{-1}-\mbox{\boldmath$\Sigma$}_{u}^{-1}\|_{2}\leq\lambda_{\min}(\mbox{\boldmath$\Sigma$}_{u})^{-1}\|\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}}-\mbox{\boldmath$\Sigma$}_{u}\|_{2}\lambda_{\min}(\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}})^{-1}=O_{P}(m_{p}\omega_{T^{\alpha}}^{1-q}).$
(A.22)

Define
$\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{D}}=\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}+\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}}$.
We first show that
$\|(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{D}})^{-1}-\mbox{\boldmath$\Sigma$}_{E}^{-1}\|=O_{P}(p^{\frac{c}{2}(1-a_{2})}\omega_{T^{\alpha}}+m_{p}\omega_{T^{\alpha}}^{1-q})$.
Let $\widehat{\mbox{\bf
J}}=\widehat{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}(\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}})^{-1}\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}$
and $\widetilde{\mbox{\bf
J}}=\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\widetilde{\mbox{\boldmath$\Phi$}}^{\prime}\mbox{\boldmath$\Sigma$}_{u}^{-1}\widetilde{\mbox{\boldmath$\Phi$}}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}$.
Using the Sherman-Morrison-Woodbury formula, we have

$\|(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{D}})^{-1}-\mbox{\boldmath$\Sigma$}_{E}^{-1}\|\leq\|(\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}})^{-1}-\mbox{\boldmath$\Sigma$}_{u}^{-1}\|+\Delta_{1^{\prime}},$

where
$\Delta_{1^{\prime}}=\|(\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}})^{-1}\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}(\mbox{\bf
I}_{r}+\widehat{\mbox{\bf
J}})^{-1}\widehat{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}(\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}})^{-1}-\mbox{\boldmath$\Sigma$}_{u}^{-1}\widetilde{\mbox{\boldmath$\Phi$}}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}(\mbox{\bf
I}_{r}+\widetilde{\mbox{\bf
J}})^{-1}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\widetilde{\mbox{\boldmath$\Phi$}}^{\prime}\mbox{\boldmath$\Sigma$}_{u}^{-1}\|$.
Then, the right hand side can be bounded by following terms:

$\displaystyle
L_{1^{\prime}}=\|((\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}})^{-1}-\mbox{\boldmath$\Sigma$}_{u}^{-1})\widetilde{\mbox{\boldmath$\Phi$}}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}(\mbox{\bf
I}_{r}+\widetilde{\mbox{\bf
J}})^{-1}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\widetilde{\mbox{\boldmath$\Phi$}}^{\prime}\mbox{\boldmath$\Sigma$}_{u}^{-1}\|,$
$\displaystyle
L_{2^{\prime}}=\|\mbox{\boldmath$\Sigma$}_{u}^{-1}(\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}-\widetilde{\mbox{\boldmath$\Phi$}}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}})(\mbox{\bf
I}_{r}+\widetilde{\mbox{\bf
J}})^{-1}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\widetilde{\mbox{\boldmath$\Phi$}}^{\prime}\mbox{\boldmath$\Sigma$}_{u}^{-1}\|,$
$\displaystyle
L_{3^{\prime}}=\|\mbox{\boldmath$\Sigma$}_{u}^{-1}\widetilde{\mbox{\boldmath$\Phi$}}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}((\mbox{\bf
I}_{r}+\widehat{\mbox{\bf J}})^{-1}-(\mbox{\bf I}_{r}+\widetilde{\mbox{\bf
J}})^{-1})\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\widetilde{\mbox{\boldmath$\Phi$}}^{\prime}\mbox{\boldmath$\Sigma$}_{u}^{-1}\|.$

By Lemma A.6,
$\|\Phi^{l}{\Psi^{l}}^{\frac{1}{2}}-\widehat{\Phi}^{l}{{}\widehat{\Psi}^{l}}^{\frac{1}{2}}\|_{\max}\leq\|\Lambda^{l}{{}\widehat{\Psi}^{l}}^{-\frac{1}{2}}({\Psi^{l}}^{\frac{1}{2}}-{{}\widehat{\Psi}^{l}}^{\frac{1}{2}})\|_{\max}+\|(\Phi^{l}-\widehat{\Phi}^{l}){\Psi^{l}}^{\frac{1}{2}}\|_{\max}=O_{P}(\omega_{T^{\alpha}})$,
and by (A.18) and (A.22), we then have

$\displaystyle\|\widetilde{\mbox{\boldmath$\Phi$}}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\|\leq\max_{l}\|\widetilde{\Phi}^{l}{{}\widetilde{\Psi}^{l}}^{\frac{1}{2}}\|=O_{P}(\sqrt{p^{ca_{2}}}),$
$\displaystyle\|\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}-\widetilde{\mbox{\boldmath$\Phi$}}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\|\leq\max_{l}\sqrt{p^{c}}\|\widehat{\Phi}^{l}{{}\widehat{\Psi}^{l}}^{\frac{1}{2}}-\widetilde{\Phi}^{l}{{}\widetilde{\Psi}^{l}}^{\frac{1}{2}}\|_{\max}=O_{P}\left(\sqrt{p^{c}}\omega_{T^{\alpha}}\right),$

and

$\displaystyle\|\widehat{\mbox{\bf J}}-\widetilde{\mbox{\bf J}}\|$
$\displaystyle\leq\|(\widehat{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}-\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\widetilde{\mbox{\boldmath$\Phi$}}^{\prime})(\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}})^{-1}(\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}-\widetilde{\mbox{\boldmath$\Phi$}}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}})\|$
$\displaystyle\;\;\;+\|(\widehat{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}-\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\widetilde{\mbox{\boldmath$\Phi$}}^{\prime})(\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}})^{-1}\widetilde{\mbox{\boldmath$\Phi$}}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\|+\|\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\widetilde{\mbox{\boldmath$\Phi$}}^{\prime}((\widehat{\mbox{\boldmath$\Sigma$}}_{u}^{\mathcal{D}})^{-1}-\mbox{\boldmath$\Sigma$}_{u}^{-1})\widetilde{\mbox{\boldmath$\Phi$}}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\|$
$\displaystyle=O_{P}(p^{\frac{c}{2}(1+a_{2})}\omega_{T^{\alpha}}+p^{ca_{2}}m_{p}\omega_{T^{\alpha}}^{1-q}).$

Since $\lambda_{\min}(\mbox{\bf I}_{r}+\widetilde{\mbox{\bf
J}})\geq\lambda_{\min}(\widetilde{\mbox{\bf
J}})\geq\lambda_{\min}(\mbox{\boldmath$\Sigma$}_{u}^{-1})\lambda_{\min}^{2}(\widetilde{\mbox{\boldmath$\Phi$}}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}})\geq
Cp^{ca_{2}}$, we have $\|(\mbox{\bf I}_{r}+\widetilde{\mbox{\bf
J}})^{-1}\|=O_{P}(1/p^{ca_{2}})$. Then,
$L_{1^{\prime}}=O_{P}(m_{p}\omega_{T^{\alpha}}^{1-q})$ by (A.22). In addition,
$L_{2^{\prime}}=O_{P}(p^{-ca_{2}/2}\|\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}-\widetilde{\mbox{\boldmath$\Phi$}}\widetilde{\mbox{\boldmath$\Psi$}}^{\frac{1}{2}}\|)=O_{P}(p^{\frac{c}{2}(1-a_{2})}\omega_{T^{\alpha}})$
and $L_{3^{\prime}}=O_{P}(p^{ca_{2}}\|(\mbox{\bf I}_{r}+\widehat{\mbox{\bf
J}})^{-1}-(\mbox{\bf I}_{r}+\widetilde{\mbox{\bf
J}})^{-1}\|)=O_{P}(p^{-ca_{2}}\|\widehat{\mbox{\bf J}}-\widetilde{\mbox{\bf
J}}\|)=O_{P}(p^{\frac{c}{2}(1-a_{2})}\omega_{T^{\alpha}}+m_{p}\omega_{T^{\alpha}}^{1-q})$.
Thus, we have

$\Delta_{1^{\prime}}=O_{P}(p^{\frac{c}{2}(1-a_{2})}\omega_{T^{\alpha}}+m_{p}\omega_{T^{\alpha}}^{1-q}),$
(A.23)

which yields
$\|(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{D}})^{-1}-\mbox{\boldmath$\Sigma$}_{E}^{-1}\|=O_{P}(p^{\frac{c}{2}(1-a_{2})}\omega_{T^{\alpha}}+m_{p}\omega_{T^{\alpha}}^{1-q})$.

Let $\widehat{\mbox{\bf
H}}=\widehat{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\widehat{\mbox{\bf
V}}^{\prime}(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{D}})^{-1}\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}$ and $\widetilde{\mbox{\bf
H}}=\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}^{\prime}\mbox{\boldmath$\Sigma$}_{E}^{-1}\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}$. Using the Sherman-
Morrison-Woodbury formula again, we have

$\|(\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}})^{-1}-\mbox{\boldmath$\Sigma$}^{-1}\|\leq\|(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{D}})^{-1}-\mbox{\boldmath$\Sigma$}_{E}^{-1}\|+\Delta_{2^{\prime}},$

where
$\Delta_{2^{\prime}}=\|(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{D}})^{-1}\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}(\mbox{\bf
I}_{k}+\widehat{\mbox{\bf
H}})^{-1}\widehat{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\widehat{\mbox{\bf
V}}^{\prime}(\widehat{\mbox{\boldmath$\Sigma$}}_{E}^{\mathcal{D}})^{-1}-\mbox{\boldmath$\Sigma$}_{E}^{-1}\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}(\mbox{\bf
I}_{k}+\widetilde{\mbox{\bf
H}})^{-1}\widetilde{\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}\widetilde{\mbox{\bf
V}}^{\prime}\mbox{\boldmath$\Sigma$}_{E}^{-1}\|$. By Weyl’s inequality, we
have $\lambda_{\min}(\mbox{\boldmath$\Sigma$}_{E})>c$ since
$\lambda_{\min}(\mbox{\boldmath$\Sigma$}_{u})>c$ and
$\lambda_{\min}(\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime})=0$.
Hence, $\|\mbox{\boldmath$\Sigma$}_{E}^{-1}\|=O_{P}(1)$. By Lemmas A.1-A.5, we
have $\|\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}^{{\frac{1}{2}}}-\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}^{{\frac{1}{2}}}\|_{\max}=O_{P}(p^{\frac{5}{2}(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{\frac{5}{2}a_{1}-\frac{3}{2}-c})$.
Similar to the proof of (A.8), we can show
$\Delta_{2^{\prime}}=O_{P}(m_{p}\omega_{T^{\alpha}}^{1-q}+p^{3(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{3a_{1}-2-c})$. Therefore, we have
$\|(\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}})^{-1}-\mbox{\boldmath$\Sigma$}^{-1}\|=O_{P}(m_{p}\omega_{T^{\alpha}}^{1-q}+p^{\frac{c}{2}(1-a_{2})}\omega_{T^{\alpha}}+p^{3(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{3a_{1}-2-c})$.

Consider (3.6). We derive the rate of convergence for
$\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}-\mbox{\boldmath$\Sigma$}\|_{\Sigma}$.
The SVD decomposition of $\Sigma$ is

$\mbox{\boldmath$\Sigma$}=(\mbox{\bf V}_{p\times
k}\;\;\mbox{\boldmath$\Phi$}_{p\times
r}\;\;\mbox{\boldmath$\Omega$}_{p\times(p-k-r)})\begin{pmatrix}\mbox{\boldmath$\Gamma$}_{k\times
k}&&\\\ &\mbox{\boldmath$\Psi$}_{r\times r}&\\\
&&\mbox{\boldmath$\Theta$}_{(p-k-r)\times(p-k-r)}\end{pmatrix}\begin{pmatrix}\mbox{\bf
V}^{\prime}\\\ \mbox{\boldmath$\Phi$}^{\prime}\\\
\mbox{\boldmath$\Omega$}^{\prime}\end{pmatrix}.$

Note that $\Omega$ is used to denote the precision matrix in Section 2.2.
Moreover, since all the eigenvalues of $\Sigma$ are strictly bigger than 0,
for any maxtrix A, we have $\|\mbox{\bf
A}\|_{\Sigma}^{2}=O_{P}(p^{-1})\|\mbox{\bf A}\|_{F}^{2}$. Then, we have

$\displaystyle\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}-\mbox{\boldmath$\Sigma$}\|_{\Sigma}$
$\displaystyle\leq
p^{-1/2}\Big{(}\|\mbox{\boldmath$\Sigma$}^{-1/2}(\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}-\mbox{\bf
B}\mbox{\bf B}^{\prime})\mbox{\boldmath$\Sigma$}^{-1/2}\|_{F}$
$\displaystyle\;\;\;\;\;+\|\mbox{\boldmath$\Sigma$}^{-1/2}(\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}-\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime})\mbox{\boldmath$\Sigma$}^{-1/2}\|_{F}+\|\mbox{\boldmath$\Sigma$}^{-1/2}(\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}_{u}-\mbox{\boldmath$\Sigma$}_{u})\mbox{\boldmath$\Sigma$}^{-1/2}\|_{F}\Big{)}$
$\displaystyle=:\Delta_{G^{\prime}}+\Delta_{L^{\prime}}+\Delta_{S^{\prime}}$

and

$\Delta_{S^{\prime}}=O_{P}(p^{-1/2}\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}_{u}-\mbox{\boldmath$\Sigma$}_{u}\|_{F})=O_{P}(\|\widehat{\mbox{\boldmath$\Sigma$}}^{\mathcal{D}}_{u}-\mbox{\boldmath$\Sigma$}_{u}\|_{2})=O_{P}(m_{p}\omega_{T^{\alpha}}^{1-q}).$

We have

$\displaystyle\Delta_{G^{\prime}}$
$\displaystyle=p^{-1/2}\left\|\begin{pmatrix}\mbox{\boldmath$\Gamma$}^{-\frac{1}{2}}\mbox{\bf
V}^{\prime}\\\
\mbox{\boldmath$\Psi$}^{-\frac{1}{2}}\mbox{\boldmath$\Phi$}^{\prime}\\\
\mbox{\boldmath$\Theta$}^{-\frac{1}{2}}\mbox{\boldmath$\Omega$}^{\prime}\end{pmatrix}(\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}-\mbox{\bf
B}\mbox{\bf B}^{\prime})\left(\mbox{\bf
V}\mbox{\boldmath$\Gamma$}^{-\frac{1}{2}}\;\;\mbox{\boldmath$\Phi$}\mbox{\boldmath$\Psi$}^{-\frac{1}{2}}\;\;\mbox{\boldmath$\Omega$}\mbox{\boldmath$\Theta$}^{-\frac{1}{2}}\right)\right\|_{F}$
$\displaystyle\leq p^{-1/2}\big{(}\|\mbox{\boldmath$\Gamma$}^{-1/2}\mbox{\bf
V}^{\prime}(\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}-\mbox{\bf
B}\mbox{\bf B}^{\prime})\mbox{\bf
V}\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}+\|\mbox{\boldmath$\Psi$}^{-1/2}\mbox{\boldmath$\Phi$}^{\prime}(\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}-\mbox{\bf
B}\mbox{\bf
B}^{\prime})\mbox{\boldmath$\Phi$}\mbox{\boldmath$\Psi$}^{-1/2}\|_{F}$
$\displaystyle\;\;\;\;+\|\mbox{\boldmath$\Theta$}^{-1/2}\mbox{\boldmath$\Omega$}^{\prime}(\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}-\mbox{\bf
B}\mbox{\bf
B}^{\prime})\mbox{\boldmath$\Omega$}\mbox{\boldmath$\Theta$}^{-1/2}\|_{F}+2\|\mbox{\boldmath$\Gamma$}^{-1/2}\mbox{\bf
V}^{\prime}(\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}-\mbox{\bf
B}\mbox{\bf
B}^{\prime})\mbox{\boldmath$\Phi$}\mbox{\boldmath$\Psi$}^{-1/2}\|_{F}$
$\displaystyle\;\;\;\;+2\|\mbox{\boldmath$\Gamma$}^{-1/2}\mbox{\bf
V}^{\prime}(\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}-\mbox{\bf
B}\mbox{\bf
B}^{\prime})\mbox{\boldmath$\Omega$}\mbox{\boldmath$\Theta$}^{-1/2}\|_{F}+2\|\mbox{\boldmath$\Psi$}^{-1/2}\mbox{\boldmath$\Phi$}^{\prime}(\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}-\mbox{\bf
B}\mbox{\bf
B}^{\prime})\mbox{\boldmath$\Omega$}\mbox{\boldmath$\Theta$}^{-1/2}\|_{F}\big{)}$
$\displaystyle=:\Delta_{G1^{\prime}}+\Delta_{G2^{\prime}}+\Delta_{G3^{\prime}}+2\Delta_{G4^{\prime}}+2\Delta_{G5^{\prime}}+2\Delta_{G6^{\prime}}.$

In order to find the convergence rate of relative Frobenius norm, we consider
the above terms separately. For $\Delta_{G1^{\prime}}$, we have

$\displaystyle\Delta_{G1^{\prime}}$ $\displaystyle\leq
p^{-1/2}\left(\|\mbox{\boldmath$\Gamma$}^{-1/2}\mbox{\bf
V}^{\prime}(\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf V}}^{\prime}-\mbox{\bf
V}\mbox{\boldmath$\Gamma$}\mbox{\bf V}^{\prime})\mbox{\bf
V}\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}+\|\mbox{\boldmath$\Gamma$}^{-1/2}\mbox{\bf
V}^{\prime}(\mbox{\bf V}\mbox{\boldmath$\Gamma$}\mbox{\bf
V}^{\prime}-\mbox{\bf B}\mbox{\bf B}^{\prime})\mbox{\bf
V}\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}\right)$
$\displaystyle=:\Delta_{G1^{\prime}}^{(a)}+\Delta_{G1^{\prime}}^{(b)}.$

We bound the two terms separately. We have

$\displaystyle\Delta_{G1^{\prime}}^{(a)}$ $\displaystyle\leq
p^{-1/2}\big{(}\|\mbox{\boldmath$\Gamma$}^{-1/2}(\mbox{\bf
V}^{\prime}\widehat{\mbox{\bf V}}-\mbox{\bf
I})\widehat{\mbox{\boldmath$\Gamma$}}(\widehat{\mbox{\bf V}}^{\prime}\mbox{\bf
V}-\mbox{\bf
I})\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}+2\|\mbox{\boldmath$\Gamma$}^{-1/2}(\mbox{\bf
V}^{\prime}\widehat{\mbox{\bf V}}-\mbox{\bf
I})\widehat{\mbox{\boldmath$\Gamma$}}\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}$
$\displaystyle\;\;\;+\|(\mbox{\boldmath$\Gamma$}^{-1/2}(\widehat{\mbox{\boldmath$\Gamma$}}-\mbox{\boldmath$\Gamma$})\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}\big{)}=:I+II+III.$

By Lemma A.5, $\|\mbox{\bf V}^{\prime}\widehat{\mbox{\bf V}}-\mbox{\bf
I}\|_{F}=\|\mbox{\bf V}^{\prime}(\widehat{\mbox{\bf V}}-\mbox{\bf
V})\|_{F}\leq\|\widehat{\mbox{\bf V}}-\mbox{\bf
V}\|_{F}=O_{P}(p^{3(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{3a_{1}-2-c})$. Then, $II$ is of
order $O_{P}(p^{3(1-a_{1})-\frac{1}{2}}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{3(a_{1}-\frac{1}{2})-c})$ and $I$ is
of smaller order. In addition, we have
$III\leq\|\mbox{\boldmath$\Gamma$}^{-1/2}(\widehat{\mbox{\boldmath$\Gamma$}}-\mbox{\boldmath$\Gamma$})\mbox{\boldmath$\Gamma$}^{-1/2}\|=O_{P}(p^{1-a_{1}}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta}))$ by Lemma A.5. Thus,
$\Delta_{G1^{\prime}}^{(a)}=O_{P}(p^{1-a_{1}}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{3(a_{1}-\frac{1}{2})-c})$.
Similarly, we have

$\displaystyle\Delta_{G1^{\prime}}^{(b)}$ $\displaystyle\leq
p^{-1/2}\big{(}\|\mbox{\boldmath$\Gamma$}^{-1/2}(\mbox{\bf
V}^{\prime}\widetilde{\mbox{\bf V}}-\mbox{\bf
I})\widetilde{\mbox{\boldmath$\Gamma$}}(\widetilde{\mbox{\bf
V}}^{\prime}\mbox{\bf V}-\mbox{\bf
I})\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}+2\|\mbox{\boldmath$\Gamma$}^{-1/2}(\mbox{\bf
V}^{\prime}\widetilde{\mbox{\bf V}}-\mbox{\bf
I})\widetilde{\mbox{\boldmath$\Gamma$}}\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}$
$\displaystyle\;\;\;+\|(\mbox{\boldmath$\Gamma$}^{-1/2}(\widetilde{\mbox{\boldmath$\Gamma$}}-\mbox{\boldmath$\Gamma$})\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}\big{)}=:I^{\prime}+II^{\prime}+III^{\prime}.$

By $\sin\theta$ theorem, $\|\mbox{\bf V}^{\prime}\widetilde{\mbox{\bf
V}}-\mbox{\bf I}\|=\|\mbox{\bf V}^{\prime}(\widetilde{\mbox{\bf V}}-\mbox{\bf
V})\|\leq\|\widetilde{\mbox{\bf V}}-\mbox{\bf
V}\|=O(\|\mbox{\boldmath$\Sigma$}_{E}\|/p^{a_{1}})$. Then, we have
$II^{\prime}=O(1/p^{a_{1}-ca_{2}})$ and $I^{\prime}$ is of smaller order. By
Lemma A.1, we have $III^{\prime}=O(1/p^{a_{1}-ca_{2}})$. Thus,
$\Delta_{G1^{\prime}}^{(b)}=O(1/p^{a_{1}-ca_{2}})$. Then, we obtain

$\Delta_{G1^{\prime}}=O_{P}\left(p^{1-a_{1}}\big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\big{)}+\frac{1}{p^{3(a_{1}-\frac{1}{2})-c}}+\frac{1}{p^{a_{1}-ca_{2}}}\right).$
(A.24)

For $\Delta_{G3^{\prime}}$, we have

$\Delta_{G3^{\prime}}\leq
p^{-1/2}\|\mbox{\boldmath$\Theta$}^{-1/2}\mbox{\boldmath$\Omega$}^{\prime}\widehat{\mbox{\bf
V}}\widehat{\mbox{\boldmath$\Gamma$}}\widehat{\mbox{\bf
V}}^{\prime}\mbox{\boldmath$\Omega$}\mbox{\boldmath$\Theta$}^{-1/2}\|_{F}+p^{-1/2}\|\mbox{\boldmath$\Theta$}^{-1/2}\mbox{\boldmath$\Omega$}^{\prime}\widetilde{\mbox{\bf
V}}\widetilde{\mbox{\boldmath$\Gamma$}}\widetilde{\mbox{\bf
V}}^{\prime}\mbox{\boldmath$\Omega$}\mbox{\boldmath$\Theta$}^{-1/2}\|_{F}=:\Delta_{G3^{\prime}}^{(a)}+\Delta_{G3^{\prime}}^{(b)}.$

By Lemma A.5, we have

$\|\mbox{\boldmath$\Omega$}^{\prime}\widehat{\mbox{\bf
V}}\|_{F}=\|\mbox{\boldmath$\Omega$}^{\prime}(\widehat{\mbox{\bf V}}-\mbox{\bf
V})\|_{F}=O(\sqrt{p}\|\widehat{\mbox{\bf V}}-\mbox{\bf
V}\|_{\max})=O_{P}(p^{3(1-a_{1})}(\sqrt{\log
p/T^{\alpha}}+1/T^{(1-\alpha)\beta})+1/p^{3a_{1}-2-c}).$

Since $\|\widehat{\mbox{\boldmath$\Gamma$}}\|=O_{P}(p^{a_{1}})$, we have

$\Delta_{G3^{\prime}}^{(a)}\leq
p^{-1/2}\|\mbox{\boldmath$\Theta$}^{-1}\|\|\mbox{\boldmath$\Omega$}^{\prime}\widehat{\mbox{\bf
V}}\|_{F}^{2}\|\widehat{\mbox{\boldmath$\Gamma$}}\|=O_{P}(p^{11/2-5a_{1}}(\log
p/T^{\alpha}+1/T^{2(1-\alpha)\beta})+1/p^{5a_{1}-7/2-2c}).$

Similarly, $\Delta_{G3^{\prime}}^{(b)}=O_{P}(1/p^{5a_{1}-7/2-2c})$ because
$\|\mbox{\boldmath$\Omega$}^{\prime}\widetilde{\mbox{\bf
V}}\|_{F}=O(\sqrt{p}\|\widetilde{\mbox{\bf V}}-\mbox{\bf
V}\|_{\max})=O_{P}(1/p^{3a_{1}-2-c})$ by Lemma A.2. Then, we obtain

$\Delta_{G3^{\prime}}=O_{P}\left(p^{\frac{11}{2}-5a_{1}}\big{(}\frac{\log
p}{T^{\alpha}}+\frac{1}{T^{2(1-\alpha)\beta}}\big{)}+\frac{1}{p^{5a_{1}-\frac{7}{2}-2c}}\right).$

Similarly, we can show that the terms $\Delta_{G2^{\prime}}$,
$\Delta_{G4^{\prime}}$, $\Delta_{G5^{\prime}}$ and $\Delta_{G6^{\prime}}$ are
dominated by $\Delta_{G1^{\prime}}$ and $\Delta_{G3^{\prime}}$. Therefore, we
have

$\Delta_{G^{\prime}}=O_{P}\left(p^{1-a_{1}}\big{(}\sqrt{\frac{\log
p}{T^{\alpha}}}+\frac{1}{T^{(1-\alpha)\beta}}\big{)}+\frac{1}{p^{a_{1}-ca_{2}}}+p^{\frac{11}{2}-5a_{1}}\big{(}\frac{\log
p}{T^{\alpha}}+\frac{1}{T^{2(1-\alpha)\beta}}\big{)}+\frac{1}{p^{5a_{1}-\frac{7}{2}-2c}}\right).$
(A.25)

Similarly, we consider

$\displaystyle\Delta_{L^{\prime}}$
$\displaystyle=p^{-1/2}\left\|\begin{pmatrix}\mbox{\boldmath$\Gamma$}^{-\frac{1}{2}}\mbox{\bf
V}^{\prime}\\\
\mbox{\boldmath$\Psi$}^{-\frac{1}{2}}\mbox{\boldmath$\Phi$}^{\prime}\\\
\mbox{\boldmath$\Theta$}^{-\frac{1}{2}}\mbox{\boldmath$\Omega$}^{\prime}\end{pmatrix}(\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}-\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime})\left(\mbox{\bf
V}\mbox{\boldmath$\Gamma$}^{-\frac{1}{2}}\;\;\mbox{\boldmath$\Phi$}\mbox{\boldmath$\Psi$}^{-\frac{1}{2}}\;\;\mbox{\boldmath$\Omega$}\mbox{\boldmath$\Theta$}^{-\frac{1}{2}}\right)\right\|_{F}$
$\displaystyle\leq p^{-1/2}\big{(}\|\mbox{\boldmath$\Gamma$}^{-1/2}\mbox{\bf
V}^{\prime}(\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}-\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime})\mbox{\bf
V}\mbox{\boldmath$\Gamma$}^{-1/2}\|_{F}+\|\mbox{\boldmath$\Psi$}^{-1/2}\mbox{\boldmath$\Phi$}^{\prime}(\widehat{\mbox{\boldmath$\Phi$}}\widehat{\mbox{\boldmath$\Psi$}}\widehat{\mbox{\boldmath$\Phi$}}^{\prime}-\mbox{\boldmath$\Lambda$}\mbox{\boldmath$\Lambda$}^{\prime})\mbox{\boldmath$\Phi$}\mbox{\boldmath$\Psi$}^{-1/2}\|_{F}$