# Factor Strength Estimation in Vector and Matrix Time Series Factor Models

Weilin Chen Weilin Chen is PhD student, Department of Statistics, London
School of Economics. Email<EMAIL_ADDRESS>Department of Statistics,
London School of Economics and Political Science  Clifford Lam Clifford Lam is
Professor, Department of Statistics, London School of Economics. Email:
<EMAIL_ADDRESS>Department of Statistics, London School of Economics and
Political Science

Most factor modelling research in vector or matrix-valued time series assume
all factors are pervasive/strong and leave weaker factors and their
corresponding series to the noise. Weaker factors can in fact be important to
a group of observed variables, for instance a sector factor in a large
portfolio of stocks may only affect particular sectors, but can be important
both in interpretations and predictions for those stocks. While more recent
factor modelling researches do consider “local” factors which are weak factors
with sparse corresponding factor loadings, there are real data examples in the
literature where factors are weak because of weak influence on most/all
observed variables, so that the corresponding factor loadings are not sparse
(non-local). As a first in the literature, we propose estimators of factor
strengths for both local and non-local weak factors, and prove their
consistency with rates of convergence spelt out for both vector and matrix-
valued time series factor models. Factor strength has an important indication
in what estimation procedure of factor models to follow, as well as the
estimation accuracy of various estimators (Chen and Lam,, 2024). Simulation
results show that our estimators have good performance in recovering the true
factor strengths, and an analysis on the NYC taxi traffic data indicates the
existence of weak factors in the data which may not be localized.

Key words and phrases: Factor strength identification; non-localized weak
factors; matrix-valued time series; realized factor strengths; factor strength
assignment.

##  1 Introduction

Factor modelling has become an increasingly important tool for analyzing high
dimensional data across various academic fields, including finance, economics,
psychology, and biology. In high dimensional time series, it is generally
assumed that a small number of factors drive the dynamics of all variables,
leading to significant dimension reduction. Traditional factor models
primarily focus on vector time series, exploring various assumptions regarding
cross-correlation and serial dependence structures (Bai and Ng,, 2002, Bai,,
2003, Forni et al.,, 2000, Chamberlain and Rothschild,, 1983, Stock and
Watson,, 2002, Fan et al.,, 2013, Stock and Watson,, 2005, Bai and Ng,, 2007,
2023, Fan et al.,, 2019, Lam et al.,, 2011, Lam and Yao,, 2012, Pan and Yao,,
2008). More recently, studies have extended their scope to include matrix
factor models (Wang et al.,, 2019, Chen and Fan,, 2021, He et al.,, 2022, Yu
et al.,, 2022) and tensor factor models (Chen et al.,, 2022, Han et al.,,
2020, 2022, Barigozzi et al., 2023b, , Barigozzi et al., 2023a, , Chen and
Lam,, 2024), incorporating emerging data in more complex matrix or tensor
formats.

In factor modelling, a crucial assumption pertains to the strengths of
factors. In the early studies of standard vector factor models (Bai and Ng,,
2002, Bai,, 2003, Stock and Watson,, 2002), it is typically assumed that all
$r$ factors are strong, commonly referred to as pervasive. Specifically, in
the model

$\displaystyle\mathbf{x}_{t}=\mathbf{A}\mathbf{f}_{t}+\boldsymbol{\epsilon}_{t},\quad
t=1,\cdots,T,$

where $\mathbf{x}_{t}\in\mathbb{R}^{d}$, $\mathbf{A}\in\mathbb{R}^{d\times r}$
and $\mathbf{f}_{t}\in\mathbb{R}^{r}$, the pervasive factor assumption implies
that all $r$ eigenvalues of $\mathbf{A}^{\mathrm{\scriptscriptstyle
T}}\mathbf{A}$ diverge proportionally to $d$, i.e.,
$\lambda_{j}(\mathbf{A}^{\mathrm{\scriptscriptstyle T}}\mathbf{A})\asymp d$
for $j=1,\cdots,r$. This results in a clear partition of the eigenvalues of
the observed covariance matrix into two sets: large eigenvalues representing
factor-related variation and small eigenvalues representing idiosyncratic
variation. Such a clear partition is also crucial for validating the procedure
to estimate the number of factors by analyzing the empirical behaviors of
eigenvalues (Bai,, 2003, Ahn and Horenstein,, 2013, Onatski,, 2010).

However, a clear separation of the eigenvalues into one set of large
eigenvalues and a second set of small eigenvalues is typically not found in
practice. Empirical studies in economics and finance indicate that eigenvalues
often diverge at varying rates (Ross,, 1976, Trzcinka,, 1986, Freyaldenhoven,,
2022). In response, models introducing weak factors have been proposed for
analyzing vector time series (Lam et al.,, 2011, Lam and Yao,, 2012, Bai and
Ng,, 2023, Freyaldenhoven,, 2022, Onatski,, 2012, Uematsu and Yamagata,,
2022). For the $j$-th column $\mathbf{a}_{j}$ of $\mathbf{A}$, its factor
strength $\alpha_{j}$, ranging between 0 and 1, is defined such that

$\displaystyle\|\mathbf{a}_{j}\|^{2}\asymp d^{\alpha_{j}},\ \ j=1,\cdots,r,$

ensuring that

$\displaystyle\lambda_{j}(\mathbf{A}^{\mathrm{\scriptscriptstyle
T}}\mathbf{A})\asymp d^{\alpha_{j}},\ \ j=1,\cdots,r.$

Thus, a strong (or pervasive) factor has $\alpha_{j}=1$, while a weak factor
has $\alpha_{j}<1$. Theoretically, a weak factor can result from two
scenarios: (i) the factor has a weak effect on all observables, or (ii) it
affects only a subset of observables, referred to as a “local” factor by
Freyaldenhoven, (2022).

Building on assumptions about weak factors for vector time series, the
literature has developed studies focusing on the estimation of the factor
loading space and the number of factors when weak factors are present in the
model (Lam et al.,, 2011, Lam and Yao,, 2012, Bai and Ng,, 2023,
Freyaldenhoven,, 2022, Uematsu and Yamagata,, 2022). Despite these efforts,
there is limited research on directly estimating factor strengths themselves.
Uematsu and Yamagata, (2022) assume sparsity in the factor loading matrix
$\mathbf{A}$ and employ techniques akin to adaptive LASSO for factor
selection. They calculate the estimated factor strengths by counting the
number of nonzero elements in the estimated factor loading matrix. Another
study with similar sparsity assumptions, Bailey et al., (2021), proposes
estimating factor strengths based on the proportion of statistically
significant factor loadings, but it concentrates on cases where factors are
observed, while our primary emphasis is on latent factor models.

The sparsity assumptions in the above mentioned works specifically address
scenarios akin to case (ii) mentioned earlier, i.e., when factors are weak due
to being “local”. This framework does not cover situations where a factor is
weak because of its weak impact on all observed units. Connor and Korajczyk,
(2022) considers such a scenario when factors are observed, and provides test-
statistics for differentiating strong from weak factors. They demonstrate in
an analysis of US equity returns how weak factors can have effects on some or
all variables (thus no sparsity assumption since these weak factors are not
“local”). While their test-statistics can differentiate strong from weak
factors, factor strengths are not estimated in the paper. Factor strength
provides important indication on how well a factor loading matrix can be
estimated (see Lam and Yao, (2012) and Chen and Lam, (2024) for rates of
convergence for factor loading matrices in the presence of weak factors). In
recent years, matrix and tensor factor models with assumptions on weak factors
have also emerged (Chen et al.,, 2022, Han et al.,, 2020, 2022, Chen and Lam,,
2024). However, none of these papers provide a method to estimate factor
strengths. Consequently, factor strength estimation remains an important yet
challenging issue, especially when we are not relying on sparsity assumptions
on the factor loading matrices.

In this paper, we propose a novel method to estimate factor strengths in
factor models for vector and matrix time series. Our method does not assume
the factor loading matrix is sparse. Instead, we make use of covariance
information and the estimated factor loading matrices to extract factor
strengths directly. To the best of our knowledge, this is the first method to
estimate factor strengths that can be applied in general settings when the
factor loading matrices are not necessarily sparse. Moreover, it represents
the first method to estimate factor strengths in matrix factor models, i.e.,
tensor factor models with order $K=2$. For matrix factor models, the factor
strengths on the row loading matrices and column loading matrices are
estimated with specific identifiability conditions provided. Numerical
experiments show that our method performs well in various settings, shedding
light on future research directions in this field.

The rest of this paper is organized as follows. Section 2 introduces our
method for estimating factor strengths for vector factor models, accompanied
by theoretical results. In Section 3, we extend the approach to matrix factor
models and provide an identifiability condition that enables the simultaneous
estimation of factor strengths on both modes, with theoretical guarantees.
Section 4 presents our simulation studies, showcasing the performance of our
method in various settings, with a matrix-valued NYC taxi data set analyzed in
Section 4.3. All proofs are presented in Section 5.

##  2 Estimation Method in Vector Factor Models

The models we consider are time series factor models in vector or matrix
formats. We start with a vector factor model, which takes the form

$\displaystyle\mathbf{x}_{t}=\mathbf{A}\mathbf{f}_{t}+\boldsymbol{\epsilon}_{t},\quad
t\in[T],$ (2.1)

where $\mathbf{x}_{t}\in\mathbb{R}^{d}$,
$\boldsymbol{\epsilon}_{t}\in\mathbb{R}^{d}$,
$\mathbf{A}\in\mathbb{R}^{d\times r}$ is the factor loading matrix, and
$\mathbf{f}_{t}\in\mathbb{R}^{r}$ are the latent factors. We also define, for
any positive integer $m$, $[m]:={1,...,m}$. We assume $d\gg r$ and $r$ is
finite, and present the following assumptions to identify the factor loadings
and factor strengths.

  * (V1)

(Factor strengths) $\mathbf{A}$ is of full rank, and
$\mathbf{A}^{\mathrm{\scriptscriptstyle T}}\mathbf{A}=\mathbf{D}$, where
$\mathbf{D}$ is a diagonal matrix. Define the diagonal entries of $\mathbf{D}$
as $d_{jj}:=(\mathbf{D})_{jj}$, then $d_{jj}\asymp d^{\alpha_{j}}$ for
$j\in[r]$, and $0<\alpha_{r}\leq\dots\leq\alpha_{1}\leq 1$.

  * (V2)

(Latent factors) There is $\mathbf{z}_{f,t}$ the same dimension as
$\mathbf{f}_{t}$, such that $\mathbf{f}_{t}=\sum_{q\geq
0}a_{f,q}\mathbf{z}_{f,t-q}$. The time series $\\{\mathbf{z}_{f,q}\\}$ has
i.i.d. elements with mean 0, variance 1 and uniformly bounded fourth order
moments. The coefficients $a_{f,q}$ are so that $\sum_{q\geq 0}a_{f,q}^{2}=1$.

Assumption (V1) defines factor strengths in the model.
$\mathbf{A}^{\mathrm{\scriptscriptstyle T}}\mathbf{A}$ being diagonal is
necessary to identify and estimate a spectrum of different factor strengths.
Otherwise, different rotations could mix weak factors with stronger ones,
making the factor strengths unidentifiable. This assumption is not as non-
general as it appears. As an illustrative example, consider a $d\times 2$
factor loading matrix $\mathbf{A}=(\mathbf{a}_{1},\mathbf{a}_{2})$, with
$\|\mathbf{a}_{1}\|^{2}\asymp d^{\alpha_{1}}$ and
$\|\mathbf{a}_{2}\|^{2}\asymp d^{\alpha_{2}}$, $\alpha_{1}>\alpha_{2}$. Also,
write $\mathbf{a}_{1}^{\mathrm{\scriptscriptstyle
T}}\mathbf{a}_{2}=\|\mathbf{a}_{1}\|\cdot\|\mathbf{a}_{2}\|\cos(\theta)$. Then
by the QR decomposition,

$\displaystyle\mathbf{A}=\mathbf{Q}\mathbf{R},\;\text{ where
}\;\mathbf{R}:=\left(\begin{array}[]{cc}\|\mathbf{a}_{1}\|&\|\mathbf{a}_{2}\|\cos(\theta)\\\
0&\|\mathbf{a}_{2}\|\\\ \end{array}\right),$

and $\mathbf{Q}$ is a $d\times 2$ matrix with orthogonal columns. The matrix
$\mathbf{R}$ can in fact be written as

$\displaystyle\mathbf{R}=\mathbf{D}^{1/2}\widetilde{\mathbf{R}},\;\text{ where
}\;\widetilde{\mathbf{R}}:=\left(\begin{array}[]{cc}1&\|\mathbf{a}_{2}\|\cos(\theta)/\|\mathbf{a}_{1}\|\\\
0&1\\\ \end{array}\right)\;\text{ and
}\;\mathbf{D}:=\left(\begin{array}[]{cc}\|\mathbf{a}_{1}\|^{2}&0\\\
0&\|\mathbf{a}_{2}\|^{2}\\\ \end{array}\right),$

with the entry
$\|\mathbf{a}_{2}\|\cos(\theta)/\|\mathbf{a}_{1}\|=O(d^{(\alpha_{2}-\alpha_{1})/2})=o(1)$.
Hence the common component in model (2.1) can now be written as
$\mathbf{A}\mathbf{f}_{t}=\widetilde{\mathbf{A}}\widetilde{\mathbf{f}}_{t}$,
where $\widetilde{\mathbf{A}}:=\mathbf{Q}\mathbf{D}^{1/2}$ and
$\widetilde{\mathbf{f}}_{t}:=\widetilde{\mathbf{R}}\mathbf{f}_{t}$. Now the
new factor loading matrix indeed has
$\widetilde{\mathbf{A}}^{\mathrm{\scriptscriptstyle
T}}\widetilde{\mathbf{A}}=\mathbf{D}$, a diagonal matrix with
$d_{jj}=\|\mathbf{a}_{j}\|^{2}\asymp d^{\alpha_{j}}$ as in Assumption (V1).
The new factor series $\widetilde{\mathbf{f}}_{t}$ is asymptotically the same
as $\mathbf{f}_{t}$ since $\widetilde{\mathbf{R}}$ is asymptotically the
identity matrix. If $\alpha_{1}=\alpha_{2}$, $\widetilde{\mathbf{R}}$ is not
asymptotically the identity matrix, but we only need to trivially modify all
proofs in the paper (omitted).

It’s important to note that we do not impose any sparsity assumptions on the
factor loading matrix $\mathbf{A}$, in contrast to other recent literature
dealing with weak factors (Uematsu and Yamagata,, 2022, Freyaldenhoven,, 2022,
Bailey et al.,, 2021). Consequently, a factor in our model can be weak if
either (i) the factor has a weak effect on all observables, or (ii) it affects
only a subset of observables. Such relaxed assumptions provide more
flexibility for our approach to be used in practice. Assumption (V2) states
that $\mathbf{f}_{t}$ has uncorrelated elements. Define
$\mathbf{F}=\left[\mathbf{f}_{1},\dots,\mathbf{f}_{T}\right]^{\mathrm{\scriptscriptstyle
T}}$, then Assumption (V2) implies
$\mathbf{E}[\frac{\mathbf{F}^{\mathrm{\scriptscriptstyle
T}}\mathbf{F}}{T}]=\mathbf{I}_{r}$ and
$\big{\|}\frac{\mathbf{F}^{\mathrm{\scriptscriptstyle
T}}\mathbf{F}}{T}\big{\|}=O_{\mathbb{P}}(1)$, facilitating the rationality of
our method as described later.

To estimate the factor strengths $\alpha_{j}$, $j\in[r]$, note that from
Assumption (V1), the factor loading matrix $\mathbf{A}$ can be written as
$\mathbf{A}=\mathbf{Q}\mathbf{D}^{1/2}$, where
$\mathbf{Q}\in\mathbb{R}^{d\times r}$ has orthogonal columns such that
$\mathbf{Q}^{\mathrm{\scriptscriptstyle T}}\mathbf{Q}=\mathbf{I}_{r}$, and
$\mathbf{D}\in\mathbb{R}^{r\times r}$ is a diagonal matrix defined in
Assumption (V1). Since $\mathbf{Q}$ is orthogonal, the information about
factor strengths in $\mathbf{A}$ is fully encapsulated in $\mathbf{D}$, given
that $d_{jj}\asymp d^{\alpha_{j}}$. Consequently, we can estimate factor
strengths by estimating the diagonal elements of $\mathbf{D}$. To achieve
this, we define
$\widehat{\mathbf{S}}=\widehat{\mathbf{Q}}^{\mathrm{\scriptscriptstyle
T}}\widehat{\boldsymbol{\Sigma}}_{x}\widehat{\mathbf{Q}}$, where
$\widehat{\mathbf{Q}}$ is an estimator of $\mathbf{Q}$, and
$\widehat{\boldsymbol{\Sigma}}_{x}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{x}_{t}\mathbf{x}_{t}^{\mathrm{\scriptscriptstyle
T}}$. Then

$\displaystyle\widehat{\mathbf{S}}=$
$\displaystyle\widehat{\mathbf{Q}}^{\mathrm{\scriptscriptstyle
T}}\mathbf{Q}\mathbf{D}^{1/2}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{f}_{t}\mathbf{f}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\mathbf{D}^{1/2}\mathbf{Q}^{\mathrm{\scriptscriptstyle
T}}\widehat{\mathbf{Q}}$
$\displaystyle+\widehat{\mathbf{Q}}^{\mathrm{\scriptscriptstyle
T}}\mathbf{Q}\mathbf{D}^{1/2}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{f}_{t}\boldsymbol{\epsilon}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\widehat{\mathbf{Q}}+\widehat{\mathbf{Q}}^{\mathrm{\scriptscriptstyle
T}}\left(\frac{1}{T}\sum_{t=1}^{T}\boldsymbol{\epsilon}_{t}\mathbf{f}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\mathbf{D}^{1/2}\mathbf{Q}^{\mathrm{\scriptscriptstyle
T}}\widehat{\mathbf{Q}}+\widehat{\mathbf{Q}}^{\mathrm{\scriptscriptstyle
T}}\left(\frac{1}{T}\sum_{t=1}^{T}\boldsymbol{\epsilon}_{t}\boldsymbol{\epsilon}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\widehat{\mathbf{Q}}.$ (2.2)

If the error terms $\boldsymbol{\epsilon}_{t}$ are appropriately bounded, with
proper assumptions on its cross-correlation and serial dependence, the last
three terms in (2.2) become small in comparison to the first term. Moreover,
considering that
$\mathbf{E}[\mathbf{f}_{t}\mathbf{f}_{t}^{\mathrm{\scriptscriptstyle
T}}]=\mathbf{I}_{r}$ by Assumption (V2), and assuming we have an estimator
$\widehat{\mathbf{Q}}$ that is close to $\mathbf{Q}$, we can make the
following approximation:

$\displaystyle\widehat{\mathbf{S}}\approx\widehat{\mathbf{Q}}^{\mathrm{\scriptscriptstyle
T}}\mathbf{Q}\mathbf{D}^{1/2}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{f}_{t}\mathbf{f}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\mathbf{D}^{1/2}\mathbf{Q}^{\mathrm{\scriptscriptstyle
T}}\widehat{\mathbf{Q}}\approx\mathbf{D}.$

In practice, the estimated $\widehat{\mathbf{Q}}$ can be obtained through
various approaches, depending on the model assumptions (see Bai and Ng,
(2002), Lam et al., (2011), Lam and Yao, (2012), Bai and Li, (2012), Bai and
Liao, (2016) for examples). Now, given that $\mathbf{D}$ is diagonal, we can
directly derive the estimator for $d_{jj}$, $j\in[r]$, by using the diagonal
entries of $\widehat{\mathbf{S}}$, such that
$\widehat{d}_{jj}:=\widehat{s}_{jj}$, where $\widehat{d}_{jj}$ and
$\widehat{s}_{jj}$ represent the $j$-th diagonal entries of
$\widehat{\mathbf{D}}$ and $\widehat{\mathbf{S}}$, respectively. Thus, the
factor strengths on $\mathbf{A}$ can be estimated as

$\displaystyle\widehat{\alpha}_{j}=\frac{\textnormal{log}\left(\widehat{d}_{jj}\right)}{\textnormal{log}(d)},\
\ j\in[r],$ (2.3)

and we can further obtain $\widehat{\mathbf{A}}$ as
$\widehat{\mathbf{A}}=\widehat{\mathbf{Q}}\widehat{\mathbf{D}}^{1/2}$, where
$\widehat{\mathbf{D}}$ is a diagonal matrix with diagonal entries given by
$\widehat{d}_{jj}$.

To assess the estimator $\widehat{\alpha}_{j}$ obtained by (2.3), note that
from Assumption (V1), the true factor strength $\alpha_{j}$ is defined as

$\displaystyle\big{\|}\mathbf{a}_{j}\big{\|}^{2}=Cd^{\alpha_{j}},\ \ j\in[r],$
(2.4)

where $C$ is a constant that may vary across different $j$. Additionally,
introduce the realized factor strength $\widetilde{\alpha}_{j}$ as

$\displaystyle\widetilde{\alpha}_{j}:=\frac{\textnormal{log}(\big{\|}\mathbf{a}_{j}\big{\|}^{2})}{\textnormal{log}(d)}=\alpha_{j}+\frac{\textnormal{log}(C)}{\textnormal{log}(d)}.$
(2.5)

It is important to note that our estimator $\widehat{\alpha}_{j}$ is, in fact,
an estimator for $\widetilde{\alpha}_{j}$ rather than the true $\alpha_{j}$,
as $C$ and $\alpha_{j}$ are not identifiable. However, given that $C$ is a
constant, when the dimension $d$ grows large, we expect
$\frac{\textnormal{log}(C)}{\textnormal{log}(d)}\rightarrow 0$, leading to a
negligible difference between $\widetilde{\alpha}_{j}$ and $\alpha_{j}$. In
the special case where $C=1$, we always have
$\widetilde{\alpha}_{j}=\alpha_{j}$. In practical situations with finite
samples and a moderately sized $d$, it is desirable for $C$ to be close to 1,
ensuring that $\widetilde{\alpha}_{j}$ does not significantly differ from
$\alpha_{j}$. In such cases, $\widehat{\alpha}_{j}$ serves as a reliable
approximation to the true $\alpha_{j}$.

To introduce the theory for the consistency of $\widehat{\alpha}_{j}$, we need
the following additional assumptions:

  * (V3)

(Noise series) Define
$\mathbf{E}=\left[\boldsymbol{\epsilon}_{1},\dots,\boldsymbol{\epsilon}_{T}\right]^{\mathrm{\scriptscriptstyle
T}}$, then $\left\|\frac{\mathbf{E}^{\mathrm{\scriptscriptstyle
T}}\mathbf{E}}{T}\right\|=O_{\mathbb{P}}\left(1+\frac{d}{T}\right)$.

  * (V4)

(Accuracy of the estimated $\widehat{\mathbf{Q}}$) The estimated factor
loading $\widehat{\mathbf{Q}}$ satisfies

$\displaystyle\|\widehat{\mathbf{Q}}-\mathbf{Q}\|=O_{\mathbb{P}}(d^{\alpha_{r}-\alpha_{1}}).$
(2.6)

  * (V5)

(Model parameters) We assume $\alpha_{r}\geq\frac{\alpha_{1}}{2}$ and

$\displaystyle\frac{d^{1+\alpha_{1}-2\alpha_{r}}}{T}=O(1).$ (2.7)

Assumption (V3) is standard in the literature that addresses the possibly
correlated noise (Bai,, 2003, Bai and Ng,, 2007). It asserts that (weak)
cross-correlations and serial dependence can be allowed in the noise series,
which can be inferred from more primitive conditions (Moon and Weidner,, 2015,
Onatski,, 2015). Assumption (V4) states that the estimated
$\widehat{\mathbf{Q}}$ should be close to the true $\mathbf{Q}$, with the
specified rate of convergence required. In the special case when all factors
have the same strength, (2.6) reduces to
$\|\widehat{\mathbf{Q}}-\mathbf{Q}\|=O_{\mathbb{P}}(1)$, which is naturally
satisfied by any consistent estimator of $\mathbf{Q}$. It is important to note
that depending on the method used to obtain $\widehat{\mathbf{Q}}$, additional
technical assumptions may be necessary to ensure the error bound of
$\widehat{\mathbf{Q}}$ is satisfied, although these details are not provided
here. Assumption (V5) outlines the necessary relationships between the weakest
and the strongest factor, as well as between $d$ and $T$. To consistently
estimate $\alpha_{j}$, it is crucial that the weakest factor is not
excessively weak compared to the strongest ones. This relationship also
influences the required magnitude of $T$. Consider, for example, the scenario
where the strongest factor is pervasive (i.e., $\alpha_{1}=1$). In this case,
we will need $\alpha_{r}\geq 0.5$, and (2.7) will be automatically satisfied
if the weakest factor is also pervasive (i.e., $\alpha_{r}=1$). However, if
$\alpha_{r}=0.5$, then we will require $\frac{d}{T}=O(1)$ to fulfill the rate
condition (2.7).

The following theorem shows the consistency of estimated factor strengths
$\widehat{\alpha}_{j}$ obtained by (2.3).

######  Theorem 1.

Under Assumptions (V1)-(V5), if the constant $C$ defined in (2.4) is unknown,
we have

$\displaystyle|\widehat{\alpha}_{j}-\alpha_{j}|=O_{\mathbb{P}}\left(\frac{1}{\textnormal{log}(d)}\right),\
\ j\in[r].$

Theorem 1 asserts that $\widehat{\alpha}_{j}$ converges to the true factor
strength $\alpha_{j}$ with a rate of $1/\textnormal{log}(d)$ when we do not
know the constant $C$ defined in (2.4). Indeed, this rate is optimal, as we
have demonstrated that $\widehat{\alpha}_{j}$ is an estimator for the realized
factor strength $\widetilde{\alpha}_{j}$ defined in (2.5) when $C\neq 1$.
Consequently, $\widetilde{\alpha}_{j}$ converges to the true $\alpha_{j}$ with
a rate of $1/\textnormal{log}(d)$, and the rate of
$|\widehat{\alpha}_{j}-\alpha_{j}|$ cannot surpass this bound. Theorem 1
highlights that, with proper assumptions, we can achieve this optimal rate.

Nevertheless, if we assume $C=1$, then $\widetilde{\alpha}_{j}=\alpha_{j}$,
making the factor strength $\alpha_{j}$ exactly identifiable. In such case, we
can achieve a better rate of convergence for
$|\widehat{\alpha}_{j}-\alpha_{j}|$. To accomplish this, we need the following
Assumption (V4’) and (V5’).

  * (V4’)

The estimated factor loading $\widehat{\mathbf{Q}}$ satisfies
$\|\widehat{\mathbf{Q}}-\mathbf{Q}\|=o_{\mathbb{P}}(d^{\alpha_{r}-\alpha_{1}}).$

  * (V5’)

We assume $\alpha_{r}>\frac{\alpha_{1}}{2}$ and
$\frac{d^{1+\alpha_{1}-2\alpha_{r}}}{T}=o(1).$

Assumptions (V4’) and (V5’) are parallel to Assumptions (V4) and (V5),
respectively. The slightly more restrictive rate conditions are necessary for
the proof of the following theorem.

######  Theorem 2.

Under Assumption (V1), (V2), (V3), (V4) and (V5’), if the constant $C$ defined
in (2.4) equals 1, then we have, for any $j\in[r-1]$ satisfying
$\alpha_{j}>\alpha_{r}$,

$\displaystyle|\widehat{\alpha}_{j}-\alpha_{j}|=O_{\mathbb{P}}\left(\frac{c_{j}+d^{\alpha_{r}-\alpha_{j}}}{\textnormal{log}(d)}\right),$
(2.8)

where

$\displaystyle
c_{j}:=(d^{\frac{\alpha_{1}}{2}-\alpha_{j}})(1+d^{1/2}T^{-\frac{1}{2}})+d^{-\alpha_{j}}(1+dT^{-1})=o(1).$

Furthermore, if Assumption (V4’) is satisfied, then for any $j\in[r]$,

$\displaystyle|\widehat{\alpha}_{j}-\alpha_{j}|=O_{\mathbb{P}}\left(\frac{c_{j}}{\textnormal{log}(d)}\right)+o_{\mathbb{P}}\left(\frac{d^{\alpha_{r}-\alpha_{j}}}{\textnormal{log}(d)}\right).$
(2.9)

Theorem 2 presents the improved rate of convergence for the estimated factor
strengths when they are exactly identifiable. Equation (2.8) indicates that
when $\alpha_{r}$ is not too small and $T$ is large enough to satisfy (V5’),
all factor strengths except for the weakest ones can be estimated at a rate
faster than $1/\textnormal{log}(d)$, assuming some factors are stronger than
others. Moreover, stronger factors can achieve faster rates, and the rate
increases as $T$ or $d$ grows larger.

Furthermore, (2.9) states that if $\mathbf{Q}$ is accurately estimated such
that (V4’) is also satisfied, then all factor strengths, including the weakest
ones, can be estimated at an even faster rate. Note that if all factors have
the same strengths, then (V4’) is automatically satisfied for any consistent
estimator of $\mathbf{Q}$, and the rate (2.9) directly applies.

##  3 Extension to matrix factor models

In Section 2, we discuss our method to estimate factor strengths in a vector
factor model. The similar approach can be extended to a matrix factor model,
which is developed for analyzing time series observations recorded in matrix
form (Wang et al.,, 2019, Chen and Fan,, 2021, He et al.,, 2023, Yu et al.,,
2022). Consider the matrix factor model:

$\displaystyle\mathbf{X}_{t}=\mathbf{A}_{1}\mathbf{F}_{t}\mathbf{A}_{2}^{\mathrm{\scriptscriptstyle
T}}+\mathbf{E}_{t},\quad t\in[T],$ (3.10)

where $\mathbf{X}_{t}\in\mathbf{R}^{d_{1}\times d_{2}}$,
$\mathbf{E}_{t}\in\mathbf{R}^{d_{1}\times d_{2}}$,
$\mathbf{F}_{t}\in\mathbf{R}^{r_{1}\times r_{2}}$, and
$\mathbf{A}_{k}\in\mathbf{R}^{d_{k}\times r_{k}}$ for $k=1,2$. The following
assumptions for matrix factor models are direct extensions of Assumptions (V1)
and (V2) for vector factor models:

  * (M1)

(Factor strengths) For $k=1,2$, $\mathbf{A}_{k}$ is of full rank, and
$\mathbf{A}_{k}^{\mathrm{\scriptscriptstyle T}}\mathbf{A}_{k}=\mathbf{D}_{k}$,
where $\mathbf{D}_{k}$ is a diagonal matrix. Define the diagonal entries of
$\mathbf{D}_{k}$ as $d_{k,jj}:=(\mathbf{D}_{k})_{jj}$, then $d_{k,jj}\asymp
d^{\alpha_{k,j}}$ for $j\in[r_{k}]$, and
$0<\alpha_{k,r_{k}}\leq\dots\leq\alpha_{k,1}\leq 1$.

  * (M2)

(Latent factors) There is $\mathbf{Z}_{f,t}$ the same dimension as
$\mathbf{F}_{t}$, such that $\mathbf{F}_{t}=\sum_{q\geq
0}a_{f,q}\mathbf{Z}_{f,t-q}$. The time series $\\{\mathbf{Z}_{f,q}\\}$ has
i.i.d. elements with mean 0, variance 1 and uniformly bounded fourth order
moments. The coefficients $a_{f,q}$ are so that $\sum_{q\geq 0}a_{f,q}^{2}=1$.

Assumption (M2) is parallel to the assumptions made for the factor series in
Chen and Lam, (2024) when $K=2$. Assumption (M1) fixes the concept of factor
strength similar to Assumption (V1) in Section 2. With Assumption (M1), we can
write $\mathbf{A}_{1}=\mathbf{Q}_{1}\mathbf{D}_{1}^{1/2}$ and
$\mathbf{A}_{2}=\mathbf{Q}_{2}\mathbf{D}_{2}^{1/2}$, where
$\mathbf{Q}_{k}\in\mathbf{R}^{d_{k}\times r_{k}}$ has orthogonal columns for
$k=1,2$. Then (3.10) can be written as

$\displaystyle\mathbf{X}_{t}=\mathbf{Q}_{1}\mathbf{D}_{1}^{1/2}\mathbf{F}_{t}\mathbf{D}_{2}^{1/2}\mathbf{Q}_{2}^{\mathrm{\scriptscriptstyle
T}}+\mathbf{E}_{t},\ \ t\in[T].$

To estimate the factor strengths on $\mathbf{A}_{1}$, similar to the vector
case, we can create
$\widehat{\mathbf{S}}_{1}=\widehat{\mathbf{Q}}_{1}^{\mathrm{\scriptscriptstyle
T}}\widehat{\boldsymbol{\Sigma}}_{1x}\widehat{\mathbf{Q}}_{1}$, where
$\widehat{\mathbf{Q}}_{1}$ is an estimator of $\mathbf{Q}_{1}$, and
$\widehat{\boldsymbol{\Sigma}}_{1x}=\frac{1}{T}\sum_{t=1}^{T}\mathbf{X}_{t}\mathbf{X}_{t}^{\mathrm{\scriptscriptstyle
T}}$. Then

$\displaystyle\widehat{\mathbf{S}}_{1}=$
$\displaystyle\widehat{\mathbf{Q}}_{1}^{\mathrm{\scriptscriptstyle
T}}\mathbf{Q}_{1}\mathbf{D}_{1}^{1/2}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\mathbf{D}_{1}^{1/2}\mathbf{Q}_{1}^{\mathrm{\scriptscriptstyle
T}}\widehat{\mathbf{Q}}_{1}$
$\displaystyle+\widehat{\mathbf{Q}}_{1}^{\mathrm{\scriptscriptstyle
T}}\mathbf{Q}_{1}\mathbf{D}_{1}^{1/2}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}^{1/2}\mathbf{Q}_{2}^{\mathrm{\scriptscriptstyle
T}}\mathbf{E}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\widehat{\mathbf{Q}}_{1}+\widehat{\mathbf{Q}}_{1}^{\mathrm{\scriptscriptstyle
T}}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{E}_{t}\mathbf{Q}_{2}\mathbf{D}_{2}^{1/2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\mathbf{D}_{1}^{1/2}\mathbf{Q}_{1}^{\mathrm{\scriptscriptstyle
T}}\widehat{\mathbf{Q}}+\widehat{\mathbf{Q}}_{1}^{\mathrm{\scriptscriptstyle
T}}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{E}_{t}\mathbf{E}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\widehat{\mathbf{Q}}_{1}.$ (3.11)

The last three terms in (3.11) will become small compared to the first term if
the error terms $\mathbf{E}_{t}$ are small with proper assumptions on its
cross-correlation and serial dependence. For matrix factor models, literature
has been developed to obtain $\widehat{\mathbf{Q}}_{1}$ using different
approaches under various model assumptions (see Wang et al., (2019), Chen et
al., (2022), Chen and Fan, (2021), Chen and Lam, (2024), He et al., (2023),
Barigozzi et al., 2023b  for examples). If $\widehat{\mathbf{Q}}_{1}$ is close
to $\mathbf{Q}_{1}$, then

$\displaystyle\widehat{\mathbf{S}}_{1}$
$\displaystyle\approx\widehat{\mathbf{Q}}_{1}^{\mathrm{\scriptscriptstyle
T}}\mathbf{Q}_{1}\mathbf{D}_{1}^{1/2}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\mathbf{D}_{1}^{1/2}\mathbf{Q}_{1}^{\mathrm{\scriptscriptstyle
T}}\widehat{\mathbf{Q}}_{1}$
$\displaystyle\approx\mathbf{D}_{1}^{1/2}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\mathbf{D}_{1}^{1/2}$
$\displaystyle\approx\mathbf{D}_{1}^{1/2}\textnormal{tr}(\mathbf{D}_{2})\mathbf{D}_{1}^{1/2}$
$\displaystyle=\textnormal{tr}(\mathbf{D}_{2})\mathbf{D}_{1}.$ (3.12)

If $\mathbf{D}_{2}$ is known, or we have an estimate for it, we can then
estimate the diagonal entries of $\mathbf{D}_{1}$ by using the diagonal
entries of $\widehat{\mathbf{S}}_{1}/\textnormal{tr}(\mathbf{D}_{2})$. With
$\widehat{\mathbf{Q}}_{2}$ an estimator of $\mathbf{Q}_{2}$ and if they are
close, parallel arguments (by swapping the index 1 with 2 and vice versa) show

$\displaystyle\widehat{\mathbf{S}}_{2}$
$\displaystyle\approx\textnormal{tr}(\mathbf{D}_{1})\mathbf{D}_{2}.$ (3.13)

Thus, if $\mathbf{D}_{1}$ is known or if we have an estimate of it, then we
can estimate the diagonal entries of $\mathbf{D}_{2}$ by using the diagonal
entries of $\widehat{\mathbf{S}}_{2}/\textnormal{tr}(\mathbf{D}_{1})$.

However, in most practical scenarios, neither $\mathbf{D}_{1}$ nor
$\mathbf{D}_{2}$ is known. Consequently, we aim to estimate both
$\mathbf{D}_{1}$ and $\mathbf{D}_{2}$ simultaneously from (3.12) and (3.13).
In such situations, it is crucial to note that due to matrix multiplication,
the factor strengths on $\mathbf{A}_{1}$ and $\mathbf{A}_{2}$ are not
identifiable in matrix factor models. This lack of identifiability is
reflected in the relationships derived from (3.12) and (3.13):

$\displaystyle\textnormal{tr}(\widehat{\mathbf{S}}_{1})\approx\textnormal{tr}(\mathbf{D}_{1})\textnormal{tr}(\mathbf{D}_{2})\approx\textnormal{tr}(\widehat{\mathbf{S}}_{2}).$
(3.14)

Therefore, to estimate the factor strengths on $\mathbf{D}_{1}$ and
$\mathbf{D}_{2}$ simultaneously, it is necessary to define the identifiability
condition as:

  * (IC)

(Factor strength identifiability)

$\displaystyle\frac{\textnormal{tr}(\mathbf{D}_{1})}{r_{1}d_{1}}=\frac{\textnormal{tr}(\mathbf{D}_{2})}{r_{2}d_{2}}.$
(3.15)

Note that the identifiability condition is not unique. However, we choose
(3.15) as it is convenient for interpretations. The intuition behind (3.15) is
that, in general, larger factor strengths will be “assigned” to larger
dimensions. For instance, consider $r_{1}=r_{2}=1$ and
$\alpha_{1,1}=\alpha_{2,1}=1$. In this case,
$\textnormal{tr}(\mathbf{A}_{1}^{\mathrm{\scriptscriptstyle
T}}\mathbf{A}_{1})=\textnormal{tr}(\mathbf{D}_{1})\approx r_{1}d_{1}$ and
$\textnormal{tr}(\mathbf{A}_{2}^{\mathrm{\scriptscriptstyle
T}}\mathbf{A}_{2})=\textnormal{tr}(\mathbf{D}_{2})\approx r_{2}d_{2}$ (each up
to multiplication of an unknown constant). Consequently, by (3.15), the
estimated factor strengths can recover the true ones if we know one of it,
i.e., $\widehat{\alpha}_{1,1}\approx\widehat{\alpha}_{2,1}\approx 1$. On the
other hand, if $\mathbf{A}_{1}$ and $\mathbf{A}_{2}$ have the exact same
dimensions ($r_{1}=r_{2}$ and $d_{1}=d_{2}$), they will be “assigned” the same
factor strengths under (IC), as the factor strengths on them are completely
symmetric and indistinguishable from each other.

With identifiability condition (3.15), together with (3.14), we can allocate
the proper factor strengths on $\mathbf{D}_{1}$ and $\mathbf{D}_{2}$
accordingly. For more accuracy and consistency in calculations, we can use the
average of $\textnormal{tr}(\widehat{\mathbf{S}}_{1})$ and
$\textnormal{tr}(\widehat{\mathbf{S}}_{2})$ as an estimate of
$\textnormal{tr}(\mathbf{D}_{1})\textnormal{tr}(\mathbf{D}_{2})$ and solve for
$\textnormal{tr}(\mathbf{D}_{1})$ and $\textnormal{tr}(\mathbf{D}_{2})$,
respectively. This leads to the following approximations:

$\displaystyle\textnormal{tr}(\mathbf{D}_{1})\approx\left(\frac{\textnormal{tr}(\widehat{\mathbf{S}}_{1}+\widehat{\mathbf{S}}_{2})}{2}\cdot\frac{r_{1}d_{1}}{r_{2}d_{2}}\right)^{1/2},$
(3.16)

and

$\displaystyle\textnormal{tr}(\mathbf{D}_{2})\approx\left(\frac{\textnormal{tr}(\widehat{\mathbf{S}}_{1}+\widehat{\mathbf{S}}_{2})}{2}\cdot\frac{r_{2}d_{2}}{r_{1}d_{1}}\right)^{1/2}.$
(3.17)

By substituting (3.17) and (3.16) back into (3.12) and (3.13), we can estimate
the diagonal entries of $\mathbf{D}_{1}$ and $\mathbf{D}_{2}$ by taking the
corresponding diagonal entries in $\widehat{\mathbf{S}}_{1}$ and
$\widehat{\mathbf{S}}_{2}$, respectively, normalized to specific magnitudes.
This leads to:

$\displaystyle\widehat{d}_{1,jj}:=\frac{\widehat{s}_{1,{jj}}}{\left(\frac{\textnormal{tr}(\widehat{\mathbf{S}}_{1}+\widehat{\mathbf{S}}_{2})}{2}\cdot\frac{r_{2}d_{2}}{r_{1}d_{1}}\right)^{1/2}},\
\ j\in[r_{1}],$

where $\widehat{d}_{1,jj}$ and $\widehat{s}_{1,{jj}}$ are the $j$-th diagonal
entries of $\widehat{\mathbf{D}}_{1}$ and $\widehat{\mathbf{S}}_{1}$,
respectively, and

$\displaystyle\widehat{d}_{2,jj}:=\frac{\widehat{s}_{2,{jj}}}{\left(\frac{\textnormal{tr}(\widehat{\mathbf{S}}_{1}+\widehat{\mathbf{S}}_{2})}{2}\cdot\frac{r_{1}d_{1}}{r_{2}d_{2}}\right)^{1/2}},\
\ j\in[r_{2}],$

where $\widehat{d}_{2,jj}$ and $\widehat{s}_{2,{jj}}$ are the $j$-th diagonal
entries of $\widehat{\mathbf{D}}_{2}$ and $\widehat{\mathbf{S}}_{2}$,
respectively. Finally, the factor strengths on $\mathbf{A}_{1}$ and
$\mathbf{A}_{2}$ can be estimated as:

$\displaystyle\widehat{\alpha}_{1,j}=\frac{\textnormal{log}\left(\widehat{d}_{1,jj}\right)}{\textnormal{log}(d_{1})},\
\ j\in[r_{1}],$

and

$\displaystyle\widehat{\alpha}_{2,j}=\frac{\textnormal{log}\left(\widehat{d}_{2,jj}\right)}{\textnormal{log}(d_{2})},\
\ j\in[r_{2}],$

and we can further obtain
$\widehat{\mathbf{A}}_{k}=\widehat{\mathbf{Q}}_{k}\widehat{\mathbf{D}}_{k}^{1/2}$,
where $\widehat{\mathbf{D}}_{k}$ is a diagonal matrix with diagonal entries
given by $\widehat{d}_{k,jj}$, for $k=1,2$.

To introduce the theoretical guarantee for $\widehat{\alpha}_{1,j}$ and
$\widehat{\alpha}_{2,j}$, we similarly define the following assumptions for
the matrix factor models, as an extension of Assumptions (V3) to (V5) for
vector factor models:

  * (M3)

(Noise series)
$\left\|\frac{1}{d_{2}T}\sum_{t=1}^{T}\frac{\mathbf{E}_{t}\mathbf{E}_{t}^{\mathrm{\scriptscriptstyle
T}}}{T}\right\|=O_{\mathbb{P}}\left(1+\frac{d_{1}}{d_{2}T}\right)$ and
$\left\|\frac{1}{d_{1}T}\sum_{t=1}^{T}\frac{\mathbf{E}_{t}^{\mathrm{\scriptscriptstyle
T}}\mathbf{E}_{t}}{T}\right\|=O_{\mathbb{P}}\left(1+\frac{d_{2}}{d_{1}T}\right)$.

  * (M4)

(Accuracy of the estimated $\widehat{\mathbf{Q}}_{k}$) For $k=1,2$, the
estimated factor loading $\widehat{\mathbf{Q}}_{k}$ satisfies

$\displaystyle\|\widehat{\mathbf{Q}}_{k}-\mathbf{Q}_{k}\|=o_{\mathbb{P}}(d^{\alpha_{k,r_{k}}-\alpha_{k,1}}).$

  * (M5)

(Model parameters) For $k=1,2$, we assume
$\alpha_{k,r_{k}}\geq\frac{\alpha_{k,1}}{2}$, $\alpha_{k,1}\geq 0.5$.
Furthermore

$\displaystyle\frac{d_{1}}{d_{2}^{\alpha_{2,1}}T}=o(1),\ \ \
\frac{d_{2}}{d_{1}^{\alpha_{1,1}}T}=o(1),\ \ \
\frac{d_{1}^{1-\alpha_{1,1}}}{d_{2}^{\alpha_{2,r_{2}}}}=o(1),\ \ \
\frac{d_{2}^{1-\alpha_{2,1}}}{d_{1}^{\alpha_{1,r_{1}}}}=o(1).$

Assumption (M3) extends Assumption (V3) from the vector model to the matrix
model, allowing for (weak) cross-correlations among fibers and serial
dependence in the noise series. Specifically, we can express
$\frac{1}{d_{2}T}\sum_{t=1}^{T}\frac{\mathbf{E}_{t}\mathbf{E}_{t}^{\mathrm{\scriptscriptstyle
T}}}{T}$ as
$\frac{1}{d_{2}T}\sum_{t=1}^{T}\sum_{i=1}^{d_{2}}\mathbf{e}_{t,i}\mathbf{e}_{t,i}^{T}$,
where the $\mathbf{e}_{t,i}$’s represent the columns of $\mathbf{E}_{t}$. Then
Assumption (M3) will be satisfied by applying random matrix theory if the
correlations among columns of $\mathbf{E}_{t}$, rows of $\mathbf{E}_{t}$, and
serial dependence of $\mathbf{E}_{t}$ are not too strong (Ahn and Horenstein,,
2013, Bai and Yin,, 1993). Assumption (M4) states that the estimated
$\widehat{\mathbf{Q}}_{k}$ should be close to the true $\mathbf{Q}_{k}$.
Consider a common scenario that $d_{1}\asymp d_{2}\asymp T$, and the strongest
factors for both modes are pervasive, i.e., $\alpha_{1,1}=\alpha_{2,2}=1$.
Then the rate requirement for Assumption (M4) can be satisfied by the
projection estimator of Chen and Lam, (2024) when $\alpha_{k,r_{k}}\geq 0.5$
(representing a very weak factor). In the special case when all factors have
the same strength, Assumption (M4) is naturally satisfied by any consistent
estimator of $\mathbf{Q}_{k}$. Assumption (M5) delineates the requisite
relationships between the weakest and strongest factors of each mode, as well
as among $d_{1}$, $d_{2}$, and $T$. These conditions are relatively mild, as
$\alpha_{k,1}\geq 0.5$ denotes a very weak strongest factor for each mode. In
a typical scenario where $d_{1}\asymp d_{2}\asymp T$ and
$\alpha_{1,1}=\alpha_{2,2}=1$, Assumption (M5) will be satisfied as long as
$\alpha_{k,r_{k}}\geq 0.5$.

Similar to the vector factor model, the true factor strength $\alpha_{k,j}$
for a factor under a matrix factor model is defined as

$\displaystyle\big{\|}\mathbf{a}_{k,j}\big{\|}^{2}=Cd_{k}^{\alpha_{k,j}},\ \
j\in[r_{k}],\ \ k\in[1,2],$ (3.18)

where $C$ is a constant that may vary across different $k,j$. Additionally,
define the realized factor strength $\widetilde{\alpha}_{k,j}$ as

$\displaystyle\widetilde{\alpha}_{k,j}:=\frac{\textnormal{log}(\big{\|}\mathbf{a}_{k,j}\big{\|}^{2})}{\textnormal{log}(d_{k})}=\alpha_{k,j}+\frac{\textnormal{log}(C)}{\textnormal{log}(d_{k})},\
\ j\in[r_{k}],\ \ k\in[1,2].$

Similar to the vector case, our estimator $\widehat{\alpha}_{k,j}$ is an
estimator for $\widetilde{\alpha}_{k,j}$ rather than the true $\alpha_{k,j}$.
When the constant $C\neq 1$, the convergence rate of $\widehat{\alpha}_{k,j}$
towards the true $\alpha_{k,j}$ cannot be faster than
$1/\textnormal{log}(d_{k})$. However, if $C=1$, then the rate of convergence
can be much faster. The following theorem shows the consistency of estimated
factor strengths $\widehat{\alpha}_{k,j}$ for matrix factor models by
specifying the rates under different scenarios.

######  Theorem 3.

Under Assumptions (M1)-(M5), and assuming the identifiability condition (3.15)
holds. For each $j\in[r_{k}]$, $k=1,2$, if the constant $C$ defined in (3.18)
is unknown, then we have

$\displaystyle|\widehat{\alpha}_{k,j}-\alpha_{k,j}|=O_{\mathbb{P}}\left(\frac{1}{\textnormal{log}(d_{k})}\right).$

Furthermore, if $\alpha_{k,r_{k}}>\frac{\alpha_{k,1}}{2}$ and the constant $C$
defined in (3.18) equals 1, then for $j\in[r_{k}]$, $k=1,2$,

$\displaystyle|\widehat{\alpha}_{k,j}-\alpha_{k,j}|=O_{\mathbb{P}}\left(\frac{c_{k,j}}{\textnormal{log}(d_{k})}\right)+o_{\mathbb{P}}\left(\frac{d^{\alpha_{k,r_{k}}-\alpha_{k,j}}}{\textnormal{log}(d_{k})}\right),$
(3.19)

where

$\displaystyle
c_{1,j}:=d_{1}^{\frac{\alpha_{1,1}}{2}-\alpha_{1,j}}d_{2}^{\frac{1-\alpha_{2,1}}{2}}\left(1+\frac{d_{1}}{d_{2}T}\right)^{1/2}+d_{1}^{-\alpha_{1,j}}d_{2}^{1-\alpha_{2,1}}\left(1+\frac{d_{1}}{d_{2}T}\right)=o(1),$
$\displaystyle
c_{2,j}:=d_{2}^{\frac{\alpha_{2,1}}{2}-\alpha_{2,j}}d_{1}^{\frac{1-\alpha_{1,1}}{2}}\left(1+\frac{d_{2}}{d_{1}T}\right)^{1/2}+d_{2}^{-\alpha_{2,j}}d_{1}^{1--\alpha_{1,1}}\left(1+\frac{d_{2}}{d_{1}T}\right)=o(1).$

Theorem 3 extends the results of Theorem 1 and Theorem 2 from vector time
series to matrix time series. Specifically, in general scenarios when $C\neq
1$, the optimal rate of $1/\textnormal{log}(d_{k})$ is achieved by our
estimated factor strengths. In the special case when $C=1$ such that the
factor strengths are exactly identifiable, then we can obtain an improved rate
of convergence as outlined by (3.19). The improved rate (3.19) for matrix
factor models can be compared to the rate (2.9) for vector factor models in
Theorem 2. If the strongest factor for mode-2 is pervasive, i.e.,
$\alpha_{2,1}=1$, then for estimating the factor strengths in
$\mathbf{A}_{1}$, we have

$\displaystyle
c_{1,j}=d_{1}^{\frac{\alpha_{1,1}}{2}-\alpha_{1,j}}\left(1+\frac{d_{1}}{d_{2}T}\right)^{1/2}+d_{1}^{-\alpha_{1,j}}\left(1+\frac{d_{1}}{d_{2}T}\right),$

which is faster than the rate $c_{j}$ in Theorem 2 when $d_{1}\succ d_{2}T$,
and at the same rate as $c_{j}$ when $d_{1}\preceq d_{2}T$. Thus, with the
matrix factor model, we can potentially obtain a more accurate estimator of
the factor strengths.

##  4 Simulation Experiments

In this section, we conduct simulation experiments to test the performances of
our proposed method to estimate factor strengths in vector and matrix factor
models.

###  4.1 Simulation settings

For generating our data, we use model (2.1) for vector time series, and (3.10)
for matrix time series. For vector time series, the factor loading matrix
$\mathbf{A}$ is generated with $\mathbf{A}=\mathbf{B}\mathbf{R}$, where the
elements in $\mathbf{B}\in\mathbb{R}^{d\times r}$ are i.i.d.
$U(-\sqrt{3},\sqrt{3})$, and $\mathbf{R}\in\mathbb{R}^{r\times r}$ is diagonal
with the $j$-th diagonal element being $d^{-\zeta_{j}}$, $0\leq\zeta_{j}\leq
0.5$. Pervasive (strong) factors have $\zeta_{j}=0$, while weak factors have
$0<\zeta_{j}\leq 0.5$. In this way, the constant $C$ in (2.4) will be close to
1, so that $\widetilde{\alpha}_{j}\approx\alpha_{j}$ and
$\|\mathbf{a}_{j}\|\approx d_{k}^{\alpha_{j}}$ for $j\in[r]$. For matrix time
series, we independently generate $\mathbf{A}_{1}$ and $\mathbf{A}_{2}$ using
the same procedure described above. The factor loading matrix $\mathbf{A}_{k}$
for $k=1,2$ is generated independently with
$\mathbf{A}_{k}=\mathbf{B}_{k}\mathbf{R}_{k}$, where the elements in
$\mathbf{B}_{k}\in\mathbb{R}^{d_{k}\times r_{k}}$ are i.i.d.
$U(-\sqrt{3},\sqrt{3})$, and $\mathbf{R}_{k}\in\mathbb{R}^{r_{k}\times r_{k}}$
is diagonal with the $j$-th diagonal element being $d_{k}^{-\zeta_{k,j}}$,
$0\leq\zeta_{k,j}\leq 0.5$. Pervasive (strong) factors have $\zeta_{k,j}=0$,
while weak factors have $0<\zeta_{k,j}\leq 0.5$.

The elements in $\mathbf{f}_{t}$ for vector time series (or $\mathbf{F}_{t}$
for matrix time series) are independent standardized AR(1) with AR
coefficients 0.8. The elements in $\boldsymbol{\epsilon}_{t}$ (or
$\mathbf{E}_{t}$) are generated based on Assumption (E1) and (E2) in Chen and
Lam, (2024), where they decompose the noise series into a common component
part and an independent noise part, facilitating weak cross-correlations and
serial dependence in the noise series. We use the same parameters to generate
$\boldsymbol{\epsilon}_{t}$ (or $\mathbf{E}_{t}$) as in Chen and Lam, (2024),
except that we add an additional step to normalize the magnitude of the noise
based on the signal-to-noise ratio $\delta$. This ratio is defined as the
average ratio of standard errors of $\mathbf{f}_{t}$ and
$\boldsymbol{\epsilon}_{t}$ (or $\mathbf{F}_{t}$ and $\mathbf{E}_{t}$), which
ensures that
$\frac{1}{d}\sum_{j=1}^{d}\text{var}(\mathbf{e}_{t,j})=\frac{1}{\delta^{2}}$
(or
$\frac{1}{d_{1}d_{2}}\sum_{i=1}^{d_{1}}\sum_{j=1}^{d_{2}}\text{var}(\mathbf{e}_{t,i,j})=\frac{1}{\delta^{2}}$).
We assume $\delta=2$ for all simulation experiments in this section.

We set $r=2$ for vector time series, and $r_{1}=r_{2}=2$ for matrix time
series. We consider two settings of factor strengths for vector time series:

  * (I)

One strong factor and one weak factor with $\zeta_{1}=0$ and $\zeta_{2}=0.2$,
so that $\alpha_{1}=1$, $\alpha_{2}=0.6$.

  * (II)

Two weak factors with $\zeta_{1}=0.1$ and $\zeta_{2}=0.2$, so that
$\alpha_{1}=0.8$, $\alpha_{2}=0.6$.

Similarly, two settings of factor strengths are considered for matrix time
series:

  * (I)

One strong factor and one weak factor with $\zeta_{k,1}=0$ and
$\zeta_{k,2}=0.2$ for $k=1,2$, so that $\alpha_{k,1}=1$, $\alpha_{k,2}=0.6$.

  * (II)

Two weak factors with $\zeta_{k,1}=0.1$ and $\zeta_{k,2}=0.2$ for $k=1,2$, so
that $\alpha_{k,1}=0.8$, $\alpha_{k,2}=0.6$.

Each experiment is repeated for 500 times.

###  4.2 Results

For vector factor models, we consider all combinations of dimensions
$d=50,100,200,400,800$ and $T=50,100,200,400,800$ for each of the two settings
of factor strengths outlined in Section 4.1. We estimate
$\widehat{\alpha}_{1}$ and $\widehat{\alpha}_{2}$ following the process
described in Section 2, where $\widehat{\mathbf{Q}}$ is estimated using PCA of
the sample covariance matrix (Bai and Ng,, 2002). Tables 1 and 2 record the
mean and standard deviation over 500 repetitions of factor strengths
estimations under different settings and dimensions.

Based on the results presented in Table 1 and 2, our factor strengths
estimators demonstrate good performance across all settings in vector factor
models. Both $\widehat{\alpha}_{1}$ and $\widehat{\alpha}_{2}$ converge to the
true factor strengths $\alpha_{1}$ and $\alpha_{2}$, with a particularly
notable improvement as $T$ increases. Furthermore, the standard deviation of
the estimators decreases with the increase in $T$ or $d$. It’s essential to
note that the standard deviation of estimation is influenced not only by
errors in the estimation procedure but also by the fact that
$\widetilde{\alpha}_{j}$ is not generated to be exactly $\alpha_{j}$ but with
some small variance (i.e., the constant $C$ in (2.4) is not exactly 1).
Nevertheless, given that $\widetilde{\alpha}_{j}\approx\alpha_{j}$, the
estimated $\widehat{\alpha}_{j}$ still serves as a good approximation of
$\alpha_{j}$.

| $d$ | $T=50$ | $T=100$ | $T=200$ | $T=400$ | $T=800$
---|---|---|---|---|---|---
$\widehat{\alpha}_{1}$ | $50$ | 1.00(0.10) | 0.99(0.07) | 1.00(0.06) | 1.00(0.05) | 1.00(0.04)
$100$ | 0.99(0.09) | 0.98(0.06) | 0.98(0.05) | 1.00(0.04) | 1.00(0.03)
$200$ | 0.99(0.08) | 0.98(0.06) | 0.99(0.04) | 1.00(0.03) | 1.00(0.02)
$400$ | 0.98(0.07) | 0.99(0.04) | 1.00(0.04) | 1.00(0.03) | 1.00(0.02)
$800$ | 0.98(0.07) | 1.00(0.04) | 0.99(0.03) | 1.00(0.02) | 1.00(0.02)
|  |  |  |  |  |
$\widehat{\alpha}_{2}$ | $50$ | 0.56(0.08) | 0.59(0.08) | 0.59(0.06) | 0.59(0.05) | 0.59(0.05)
$100$ | 0.58(0.07) | 0.59(0.07) | 0.59(0.05) | 0.59(0.04) | 0.60(0.03)
$200$ | 0.59(0.06) | 0.60(0.05) | 0.60(0.04) | 0.60(0.03) | 0.60(0.02)
$400$ | 0.62(0.05) | 0.59(0.05) | 0.60(0.04) | 0.60(0.03) | 0.60(0.02)
$800$ | 0.64(0.04) | 0.61(0.04) | 0.60(0.03) | 0.60(0.02) | 0.60(0.02)
Table 1: The mean and standard deviation (in brackets) of the estimated factor strengths for Setting (I) under vector factor models. The true factor strengths are $\alpha_{1}=1$, $\alpha_{2}=0.6$. | $d$ | $T=50$ | $T=100$ | $T=200$ | $T=400$ | $T=800$
---|---|---|---|---|---|---
$\widehat{\alpha}_{1}$ | $50$ | 0.82(0.09) | 0.81(0.08) | 0.82(0.05) | 0.81(0.05) | 0.81(0.04)
$100$ | 0.80(0.09) | 0.80(0.07) | 0.80(0.05) | 0.80(0.03) | 0.80(0.03)
$200$ | 0.79(0.07) | 0.79(0.05) | 0.80(0.04) | 0.81(0.03) | 0.80(0.03)
$400$ | 0.80(0.07) | 0.80(0.05) | 0.80(0.04) | 0.80(0.02) | 0.80(0.02)
$800$ | 0.81(0.05) | 0.80(0.05) | 0.80(0.03) | 0.80(0.03) | 0.80(0.02)
|  |  |  |  |  |
$\widehat{\alpha}_{2}$ | $50$ | 0.56(0.09) | 0.56(0.08) | 0.59(0.06) | 0.60(0.05) | 0.59(0.05)
$100$ | 0.57(0.07) | 0.59(0.05) | 0.59(0.05) | 0.59(0.03) | 0.59(0.03)
$200$ | 0.59(0.05) | 0.59(0.05) | 0.60(0.04) | 0.59(0.03) | 0.60(0.02)
$400$ | 0.61(0.05) | 0.60(0.05) | 0.60(0.03) | 0.60(0.03) | 0.60(0.02)
$800$ | 0.63(0.03) | 0.61(0.04) | 0.60(0.03) | 0.60(0.02) | 0.60(0.02)
Table 2: The mean and standard deviation (in brackets) of the estimated factor
strengths for Setting (II) under vector factor models. The true factor
strengths are $\alpha_{1}=0.8$, $\alpha_{2}=0.6$.

For matrix factor models, we consider the following five settings of different
dimensions for $d_{1}$ and $d_{2}$:

  * i.

$d_{1}=d_{2}=25$;

  * ii.

$d_{1}=d_{2}=50$;

  * iii.

$d_{1}=d_{2}=100$;

  * iv.

$d_{1}=25$, $d_{2}=50$;

  * v.

$d_{1}=50$, $d_{2}=100$.

We consider all combinations of $T=50,100,200,400,800$, and the above five
settings of dimensions for each of the two settings of factor strengths
outlined in Section 4.1. We estimate $\widehat{\alpha}_{1,1}$,
$\widehat{\alpha}_{1,2}$, $\widehat{\alpha}_{2,1}$, and
$\widehat{\alpha}_{2,2}$ following the process described in Section 3, where
$\widehat{\mathbf{Q}}_{1}$ and $\widehat{\mathbf{Q}}_{2}$ are estimated using
the pre-averaging and iterative projection algorithm developed in Chen and
Lam, (2024). Table 3 and 4 record the mean and standard deviation over 100
repetitions of factor strengths estimations under different settings and
dimensions.

From Table 3 and 4, our estimation procedure performs effectively across all
settings in matrix factor models. The identifiability condition (3.15)
efficiently allocates factor strengths between $\mathbf{A}_{1}$ and
$\mathbf{A}_{2}$. When $d_{1}=d_{2}$, we estimate relatively similar factor
strengths for $\mathbf{A}_{1}$ and $\mathbf{A}_{2}$ since they are
indistinguishable. Moreover, all estimated factor strengths converge to the
true values as $T$ and $d$ increase. In cases where $d_{1}\neq d_{2}$, the
estimated factor strengths on $\mathbf{A}_{1}$ and $\mathbf{A}_{2}$ are
allocated based on the relative magnitudes of $d_{1}$ and $d_{2}$,
contributing to the recovery of true factor strengths. This tendency is
particularly pronounced in Setting (I), where the strongest factors on
$\mathbf{A}_{1}$ and $\mathbf{A}_{2}$ are pervasive.

| $(d_{1},d_{2})$ | $T=50$ | $T=100$ | $T=200$ | $T=400$ | $T=800$
---|---|---|---|---|---|---
$\widehat{\alpha}_{1,1}$ | $(25,25)$ | 1.00(0.07) | 1.00(0.06) | 0.99(0.05) | 0.99(0.05) | 1.00(0.04)
$(50,50)$ | 0.99(0.06) | 1.00(0.05) | 1.00(0.04) | 1.00(0.03) | 1.00(0.03)
$(100,100)$ | 0.99(0.04) | 0.99(0.03) | 1.00(0.02) | 1.00(0.02) | 1.00(0.02)
$(25,50)$ | 1.01(0.08) | 0.99(0.06) | 0.99(0.05) | 0.99(0.04) | 0.99(0.04)
| $(50,100)$ | 0.98(0.06) | 1.00(0.04) | 0.99(0.04) | 0.99(0.03) | 0.99(0.02)
|  |  |  |  |  |
$\widehat{\alpha}_{1,2}$ | $(25,25)$ | 0.55(0.11) | 0.57(0.09) | 0.58(0.09) | 0.58(0.06) | 0.59(0.06)
$(50,50)$ | 0.57(0.08) | 0.58(0.07) | 0.59(0.07) | 0.59(0.05) | 0.60(0.04)
$(100,100)$ | 0.57(0.07) | 0.59(0.06) | 0.59(0.04) | 0.59(0.04) | 0.59(0.03)
$(25,50)$ | 0.54(0.11) | 0.55(0.10) | 0.57(0.08) | 0.57(0.06) | 0.57(0.06)
| $(50,100)$ | 0.57(0.09) | 0.56(0.07) | 0.59(0.06) | 0.59(0.05) | 0.59(0.04)
|  |  |  |  |  |
$\widehat{\alpha}_{2,1}$ | $(25,25)$ | 1.00(0.07) | 1.00(0.05) | 0.99(0.05) | 0.99(0.05) | 1.00(0.04)
$(50,50)$ | 0.99(0.06) | 0.99(0.05) | 1.00(0.04) | 1.00(0.03) | 1.00(0.03)
$(100,100)$ | 0.99(0.04) | 0.99(0.03) | 1.00(0.02) | 1.00(0.02) | 1.00(0.02)
$(25,50)$ | 1.01(0.06) | 1.00(0.04) | 1.00(0.04) | 1.00(0.03) | 1.00(0.03)
| $(50,100)$ | 0.99(0.05) | 1.01(0.03) | 1.00(0.03) | 1.00(0.02) | 1.00(0.02)
|  |  |  |  |  |
$\widehat{\alpha}_{2,2}$ | $(25,25)$ | 0.55(0.12) | 0.56(0.09) | 0.58(0.08) | 0.58(0.06) | 0.58(0.06)
$(50,50)$ | 0.59(0.09) | 0.59(0.07) | 0.58(0.06) | 0.59(0.04) | 0.59(0.04)
$(100,100)$ | 0.58(0.08) | 0.58(0.06) | 0.59(0.04) | 0.59(0.04) | 0.59(0.03)
$(25,50)$ | 0.58(0.10) | 0.59(0.07) | 0.59(0.06) | 0.59(0.04) | 0.60(0.05)
| $(50,100)$ | 0.59(0.08) | 0.58(0.06) | 0.59(0.05) | 0.60(0.04) | 0.60(0.03)
Table 3: The mean and standard deviation (in brackets) of the estimated factor strengths for Setting (I) under matrix factor models. The true factor strengths are $\alpha_{1,1}=\alpha_{2,1}=1$, $\alpha_{1,2}=\alpha_{2,2}=0.6$. | $(d_{1},d_{2})$ | $T=50$ | $T=100$ | $T=200$ | $T=400$ | $T=800$
---|---|---|---|---|---|---
$\widehat{\alpha}_{1,1}$ | $(25,25)$ | 0.82(0.07) | 0.81(0.05) | 0.80(0.05) | 0.81(0.05) | 0.81(0.05)
$(50,50)$ | 0.80(0.06) | 0.81(0.04) | 0.80(0.03) | 0.80(0.03) | 0.81(0.03)
$(100,100)$ | 0.79(0.05) | 0.80(0.03) | 0.80(0.02) | 0.80(0.02) | 0.80(0.02)
$(25,50)$ | 0.79(0.07) | 0.79(0.05) | 0.78(0.05) | 0.78(0.05) | 0.78(0.04)
| $(50,100)$ | 0.78(0.06) | 0.78(0.05) | 0.78(0.03) | 0.78(0.03) | 0.78(0.02)
|  |  |  |  |  |
$\widehat{\alpha}_{1,2}$ | $(25,25)$ | 0.56(0.10) | 0.55(0.08) | 0.58(0.06) | 0.58(0.06) | 0.58(0.06)
$(50,50)$ | 0.57(0.06) | 0.58(0.06) | 0.58(0.05) | 0.59(0.04) | 0.59(0.04)
$(100,100)$ | 0.58(0.08) | 0.60(0.05) | 0.59(0.03) | 0.60(0.03) | 0.59(0.02)
$(25,50)$ | 0.53(0.10) | 0.54(0.08) | 0.55(0.07) | 0.55(0.06) | 0.55(0.06)
| $(50,100)$ | 0.56(0.08) | 0.56(0.06) | 0.57(0.04) | 0.57(0.04) | 0.57(0.04)
|  |  |  |  |  |
$\widehat{\alpha}_{2,1}$ | $(25,25)$ | 0.83(0.07) | 0.80(0.06) | 0.81(0.05) | 0.81(0.04) | 0.81(0.05)
$(50,50)$ | 0.81(0.05) | 0.81(0.04) | 0.80(0.04) | 0.81(0.03) | 0.81(0.03)
$(100,100)$ | 0.79(0.04) | 0.80(0.03) | 0.80(0.03) | 0.80(0.02) | 0.80(0.02)
$(25,50)$ | 0.83(0.06) | 0.83(0.04) | 0.81(0.04) | 0.82(0.03) | 0.82(0.03)
| $(50,100)$ | 0.82(0.04) | 0.82(0.04) | 0.82(0.03) | 0.82(0.02) | 0.82(0.02)
|  |  |  |  |  |
$\widehat{\alpha}_{2,2}$ | $(25,25)$ | 0.55(0.09) | 0.57(0.08) | 0.57(0.06) | 0.57(0.06) | 0.58(0.06)
$(50,50)$ | 0.57(0.06) | 0.57(0.06) | 0.59(0.05) | 0.59(0.04) | 0.59(0.04)
$(100,100)$ | 0.60(0.06) | 0.59(0.05) | 0.60(0.04) | 0.60(0.03) | 0.59(0.02)
$(25,50)$ | 0.58(0.08) | 0.60(0.06) | 0.62(0.05) | 0.61(0.04) | 0.61(0.04)
| $(50,100)$ | 0.60(0.06) | 0.61(0.05) | 0.61(0.03) | 0.62(0.03) | 0.62(0.02)
Table 4: The mean and standard deviation (in brackets) of the estimated factor
strengths for Setting (II) under matrix factor models. The true factor
strengths are $\alpha_{1,1}=\alpha_{2,1}=0.8$,
$\alpha_{1,2}=\alpha_{2,2}=0.6$.

###  4.3 NYC taxi traffic analysis

We analyze taxi traffic pattern in New York city. The data includes all
individual taxi rides operated by Yellow Taxi within New York City, published
at

https://www1.nyc.gov/site/tlc/about/tlc-trip-record-data.page.

To simplify the discussion, we only consider rides within Manhattan Island.
The dataset contains 1.1 billion trip records within the period of January 1,
2011 to December 31, 2021\. Each trip record includes fields capturing pick-up
and drop-off dates/times, pick-up and drop-off locations, trip distances,
itemized fares, rate types, payment types, and driver-reported passenger
counts. Our study focuses on the pick-up and drop-off dates/times, and pick-up
and drop-off locations of each ride.

The pick-up and drop-off locations in Manhattan are coded according to 69
predefined zones and we will use them to classify the pick-up and drop-off
locations. While Chen and Lam, (2024) and Chen et al., (2018) further divide
each day into 24 hourly periods and analyze the
$\mathcal{X}_{t}\in\mathbb{R}^{69\times 69\times 24}$ tensor time series to
estimate factor loadings and the number of factors, our focus in this paper is
on estimating factor strengths in matrix time series. To achieve this, we
record the total number of rides moving among the zones within specific hours
on each day. As an example, we utilize data from 10pm to 12am on each non-
business day, representing taxi traffic patterns during nighttime (Chen and
Lam,, 2024, Chen et al.,, 2022). We analyze the non-business-day series within
the period from January 1, 2011, to December 31, 2021, encompassing 1248 days.
Thus, $\mathbf{X}_{t}\in\mathbb{R}^{69\times 69}$ for each day, where
$x_{i_{1},i_{2},t}$ represents the number of trips from zone $i_{1}$ (the
pick-up zone) to zone $i_{2}$ (the drop-off zone) between 10pm and 12am on day
$t$. Similar analysis can be conducted to examine traffic patterns by
aggregating data from other hours as well.

Chen and Lam, (2024) utilizes their bootstrapped correlation thresholding
method to estimate the number of factors for both pick-up and drop-off
locations, resulting in $\widehat{r}_{1}=\widehat{r}_{2}=3$. This observation
suggests the potential existence of weak factors, as opposed to other rank
estimators designed to analyze only pervasive factors, all of which yield
$\widehat{r}_{1}=\widehat{r}_{2}=1$ (Yu et al.,, 2022, Barigozzi et al.,
2023b, , He et al.,, 2022, 2023). To assess the factor strengths for these
potential weak factors, we incorporate $\widehat{r}_{1}=\widehat{r}_{2}=3$ in
our analysis.

To estimate factor strengths, we first obtain estimators of
$\widehat{\mathbf{Q}}_{1}$ and $\widehat{\mathbf{Q}}_{2}$ using the pre-
averaging and iterative projection method proposed by Chen and Lam, (2024).
This method is specifically designed to provide more accurate estimators of
the factor loadings particularly in the presence of weak factors.
Subsequently, using the estimated $\widehat{\mathbf{Q}}_{1}$ and
$\widehat{\mathbf{Q}}_{2}$, we calculate the estimated factor strengths using
the method outlined in Section 3. The heatmaps of the loading matrices
$\mathbf{A}_{1}$ (for pick-up locations) and $\mathbf{A}_{2}$ (for drop-off
locations) are shown in Figure 1 and 2, respectively. The estimated factor
strengths $\widehat{\alpha}_{k,j}$ are summarized in Table 5.

Pick-up Factors | $\widehat{\alpha}_{1,1}$ | $\widehat{\alpha}_{1,2}$ | $\widehat{\alpha}_{1,3}$
---|---|---|---
1.39 | 0.65 | 0.50
Drop-off Factors | $\widehat{\alpha}_{2,1}$ | $\widehat{\alpha}_{2,2}$ | $\widehat{\alpha}_{2,3}$
1.40 | 0.82 | 0.63
Table 5: Factor strengths estimators for NYC Taxi Traffic Data.

From Table 5, both $\widehat{\alpha}_{1,1}$ and $\widehat{\alpha}_{2,1}$ have
values greater than 1, indicating that the strongest factor for both the pick-
up and drop-off loadings should be pervasive with $C>1$. This observation is
further supported by Figure 1 and Figure 2, where both figures show that the
first factor has a large number of elements that are not close to zero.
Specifically, both factors load most heavily on East Village, where a
significant number of arts, music venues, and restaurants are located.
Moreover, the values of $\widehat{\alpha}_{1,1}$ and $\widehat{\alpha}_{2,1}$
are very close because $d_{1}=d_{2}$ and $r_{1}=r_{2}$, and the
identifiability condition (3.15) assigns similar strengths for the strongest
factors to both modes. This alignment is reasonable in practice.

Furthermore, it can be observed from Table 5 that the estimated strengths of
the second and third factors for both pick-up and drop-off loadings are much
less than 1. This suggests that these factors are likely to be weak (non-
pervasive), a conclusion supported by Figures 1 and 2 as well. In these
figures, the second and third factors exhibit certain localized behavior, with
many entries near zero. Specifically, Factor 2 in the pick-up loadings loads
heavily on Penn Station, a major transportation hub, while Factor 2 in the
drop-off loadings loads heavily on the Lower East Side, known for its
nightlife and entertainment venues. Furthermore, both Factor 3 in the pick-up
and drop-off loadings highlight Times Square/Theatre District, another popular
tourist destination and hub of nightlife activity. It is also important to
note that sparsity alone may not fully account for the presence of weak
factors. For example, Factor 2 in the pick-up loadings demonstrates greater
sparsity compared to Factor 3, yet it exhibits stronger estimated strengths.
This suggests the potential weak influence of certain factors on some or all
observed variables.

Figure 1: Loadings on three pickup factors.
Figure 2: Loadings on three dropoff factors.

## References

  * Ahn and Horenstein, (2013) Ahn, S. C. and Horenstein, A. R. (2013).  Eigenvalue ratio test for the number of factors.  Econometrica, 81(3):1203–1227.
  * Bai, (2003) Bai, J. (2003).  Inferential theory for factor models of large dimensions.  Econometrica, 71(1):135–171.
  * Bai and Li, (2012) Bai, J. and Li, K. (2012).  Statistical analysis of factor models of high dimension.  The Annals of Statistics, 40(1).
  * Bai and Liao, (2016) Bai, J. and Liao, Y. (2016).  Efficient estimation of approximate factor models via penalized maximum likelihood.  Journal of Econometrics, 191(1):1–18.
  * Bai and Ng, (2002) Bai, J. and Ng, S. (2002).  Determining the number of factors in approximate factor models.  Econometrica, 70(1):191–221.
  * Bai and Ng, (2007) Bai, J. and Ng, S. (2007).  Determining the number of primitive shocks in factor models.  Journal of Business and Economic Statistics, 25(1):52–60.
  * Bai and Ng, (2023) Bai, J. and Ng, S. (2023).  Approximate factor models with weaker loadings.  Journal of Econometrics, 235(2):1893–1916.
  * Bai and Yin, (1993) Bai, Z. D. and Yin, Y. Q. (1993).  Limit of the smallest eigenvalue of a large dimensional sample covariance matrix.  The Annals of Probability, 21(3):1275–1294.
  * Bailey et al., (2021) Bailey, N., Kapetanios, G., and Pesaran, M. H. (2021).  Measurement of factor strength: Theory and practice.  Journal of Applied Econometrics, 36(5):587–613.
  * (10) Barigozzi, M., He, Y., Li, L., and Trapani, L. (2023a).  Robust tensor factor analysis.  arXiv:2303.18163.
  * (11) Barigozzi, M., He, Y., Li, L., and Trapani, L. (2023b).  Statistical inference for large-dimensional tensor factor models by iterative projections.  arXiv:2206.09800.
  * Chamberlain and Rothschild, (1983) Chamberlain, G. and Rothschild, M. (1983).  Arbitrage, factor structure, and mean-variance analysis on large asset markets.  Econometrica, 51(5):1281–1304.
  * Chen and Fan, (2021) Chen, E. Y. and Fan, J. (2021).  Statistical inference for high-dimensional matrix-variate factor models.  Journal of the American Statistical Association, 118(542):1038–1055.
  * Chen et al., (2018) Chen, M., Gao, C., and Ren, Z. (2018).  Robust covariance and scatter matrix estimation under Huber’s contamination model.  Ann. Statist., 46(5):1932–1960.
  * Chen et al., (2022) Chen, R., Yang, D., and Zhang, C.-H. (2022).  Factor models for high-dimensional tensor time series.  Journal of the American Statistical Association, 117(537):94–116.
  * Chen and Lam, (2024) Chen, W. and Lam, C. (2024).  Rank and factor loadings estimation in time series tensor factor model by pre-averaging.  The Annals of Statistics, 52(1):364 – 391.
  * Connor and Korajczyk, (2022) Connor, G. and Korajczyk, R. A. (2022).  Semi-Strong Factors in Asset Returns*.  Journal of Financial Econometrics, 22(1):70–93.
  * Fan et al., (2013) Fan, J., Liao, Y., and Mincheva, M. (2013).  Large covariance estimation by thresholding principal orthogonal complements.  Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(4):603–680.
  * Fan et al., (2019) Fan, J., Wang, W., and Zhong, Y. (2019).  Robust covariance estimation for approximate factor models.  Journal of Econometrics, 208(1):5 – 22.  Special Issue on Financial Engineering and Risk Management.
  * Forni et al., (2000) Forni, M., Hallin, M., Lippi, M., and Reichlin, L. (2000).  The generalized dynamic-factor model: Identification and estimation.  Review of Economics and Statistics, 82(4):540–554.
  * Freyaldenhoven, (2022) Freyaldenhoven, S. (2022).  Factor models with local factors — determining the number of relevant factors.  Journal of Econometrics, 229(1):80–102.
  * Han et al., (2020) Han, Y., Chen, R., Yang, D., and Zhang, C.-H. (2020).  Tensor factor model estimation by iterative projection.  arXiv: 2006.02611.
  * Han et al., (2022) Han, Y., Zhang, C. H., and Chen, R. (2022).  Rank determination in tensor factor model.  Electronic Journal of Statistic, 16:1726–1803.
  * He et al., (2023) He, Y., Kong, X., Yu, L., Zhang, X., and Zhao, C. (2023).  Matrix factor analysis: From least squares to iterative projection.  Journal of Business and Economic Statistics, 0(0):1 – 13.
  * He et al., (2022) He, Y., Wang, Y., Yu, L., Zhou, W., and Zhou, W.-X. (2022).  Matrix kendall’s tau in high-dimensions: A robust statistic for matrix factor model.  arXiv:2207.09633.
  * Lam and Yao, (2012) Lam, C. and Yao, Q. (2012).  Factor modeling for high-dimensional time series: Inference for the number of factors.  The Annals of Statistics, 40(2):694–726.
  * Lam et al., (2011) Lam, C., Yao, Q., and Bathia, N. (2011).  Estimation of latent factors for high-dimensional time series.  Biometrika, 98(4):901–918.
  * Moon and Weidner, (2015) Moon, H. R. and Weidner, M. (2015).  Dynamic linear panel regression models with interactive fixed effects.  Econometric Theory, 33(1):158–195.
  * Onatski, (2010) Onatski, A. (2010).  Determining the number of factors from empirical distribution of eigenvalues.  Review of Economics and Statistics, 92(4):1004–1016.
  * Onatski, (2012) Onatski, A. (2012).  Asymptotics of the principal components estimator of large factor models with weakly influential factors.  Journal of Econometrics, 168(2):244–258.
  * Onatski, (2015) Onatski, A. (2015).  Asymptotic analysis of the squared estimation error in misspecified factor models.  Journal of Econometrics, 186(2):388–406.
  * Pan and Yao, (2008) Pan, J. and Yao, Q. (2008).  Modelling multiple time series via common factors.  Biometrika, 95(2):365–379.
  * Ross, (1976) Ross, S. A. (1976).  The arbitrage theory of capital asset pricing.  Journal of Economic Theory, 13(3):341 – 360.
  * Stock and Watson, (2005) Stock, J. and Watson, M. (2005).  Implications of dynamic factor models for var analysis.  NBER Working Papers.  No. 11467.
  * Stock and Watson, (2002) Stock, J. H. and Watson, M. W. (2002).  Forecasting using principal components from a large number of predictors.  Journal of the American Statistical Association, 97(460):1167–1179.
  * Trzcinka, (1986) Trzcinka, C. (1986).  On the number of factors in the arbitrage pricing model.  The Journal of Finance, 41(2):347–368.
  * Uematsu and Yamagata, (2022) Uematsu, Y. and Yamagata, T. (2022).  Estimation of sparsity-induced weak factor models.  Journal of Business Economic Statistics, 41(1):213–227.
  * Wang et al., (2019) Wang, D., Liu, X., and Chen, R. (2019).  Factor models for matrix-valued high-dimensional time series.  Journal of Econometrics, 208(1):231–248.  Special Issue on Financial Engineering and Risk Management.
  * Yu et al., (2022) Yu, L., He, Y., Kong, X., and Zhang, X. (2022).  Projected estimation for large-dimensional matrix factor models.  Journal of Econometrics, 229(1):201–217.

##  5 Appendix

Proof of Theorem 1. To start with, note that (2.2) can be written as

$\displaystyle\widehat{\mathbf{S}}=\mathbf{S}_{11}+\mathbf{S}_{12}+\mathbf{S}_{21}+\mathbf{S}_{22},$

where

$\displaystyle\mathbf{S}_{11}:$
$\displaystyle=\widehat{\mathbf{Q}}^{\mathrm{\scriptscriptstyle
T}}\mathbf{Q}\mathbf{D}^{1/2}\left(\frac{\mathbf{F}^{\mathrm{\scriptscriptstyle
T}}\mathbf{F}}{T}\right)\mathbf{D}^{1/2}\mathbf{Q}^{\mathrm{\scriptscriptstyle
T}}\widehat{\mathbf{Q}},$ $\displaystyle\mathbf{S}_{12}:$
$\displaystyle=\widehat{\mathbf{Q}}^{\mathrm{\scriptscriptstyle
T}}\mathbf{Q}\mathbf{D}^{1/2}\left(\frac{\mathbf{F}^{\mathrm{\scriptscriptstyle
T}}\mathbf{E}}{T}\right)\widehat{\mathbf{Q}},$ $\displaystyle\mathbf{S}_{21}:$
$\displaystyle=\widehat{\mathbf{Q}}^{\mathrm{\scriptscriptstyle
T}}\left(\frac{\mathbf{E}^{\mathrm{\scriptscriptstyle
T}}\mathbf{F}}{T}\right)\mathbf{D}^{1/2}\mathbf{Q}^{\mathrm{\scriptscriptstyle
T}}\widehat{\mathbf{Q}},$ $\displaystyle\mathbf{S}_{22}:$
$\displaystyle=\widehat{\mathbf{Q}}^{\mathrm{\scriptscriptstyle
T}}\left(\frac{\mathbf{E}^{\mathrm{\scriptscriptstyle
T}}\mathbf{E}}{T}\right)\widehat{\mathbf{Q}}.$

For each $j=1,\cdots,r$, we have
$\widehat{\alpha}_{j}=\frac{\textnormal{log}(\widehat{d}_{jj})}{\textnormal{log}(d)}=\frac{\textnormal{log}(\widehat{s}_{jj})}{\textnormal{log}(d)}$.
Thus, we need to bound the distance between $\widehat{s}_{jj}$ and
$d^{\alpha}_{j}$. We start with $\mathbf{S}_{11}$ which contains the true
signal part. Denote
$\mathbf{M}:=\widehat{\mathbf{Q}}^{\mathrm{\scriptscriptstyle T}}\mathbf{Q}$,
then we can further decompose $\mathbf{S}_{11}$ as

$\displaystyle\mathbf{S}_{11}=\mathbf{S}_{11,1}+\mathbf{S}_{11,2}+\mathbf{S}_{11,3}+\mathbf{S}_{11,4},$

where

$\displaystyle\mathbf{S}_{11,1}:$
$\displaystyle=\mathbf{D}^{1/2}\left(\frac{\mathbf{F}^{\mathrm{\scriptscriptstyle
T}}\mathbf{F}}{T}\right)\mathbf{D}^{1/2},$ $\displaystyle\mathbf{S}_{11,2}:$
$\displaystyle=\mathbf{M}\mathbf{D}^{1/2}\left(\frac{\mathbf{F}^{\mathrm{\scriptscriptstyle
T}}\mathbf{F}}{T}\right)\mathbf{D}^{1/2}(\mathbf{M}-\mathbf{I}_{r})^{\mathrm{\scriptscriptstyle
T}},$ $\displaystyle\mathbf{S}_{11,3}:$
$\displaystyle=(\mathbf{M}-\mathbf{I}_{r})\mathbf{D}^{1/2}\left(\frac{\mathbf{F}^{\mathrm{\scriptscriptstyle
T}}\mathbf{F}}{T}\right)\mathbf{D}^{1/2}\mathbf{M}^{\mathrm{\scriptscriptstyle
T}},$ $\displaystyle\mathbf{S}_{11,4}:$
$\displaystyle=(\mathbf{M}-\mathbf{I}_{r})\mathbf{D}^{1/2}\left(\frac{\mathbf{F}^{\mathrm{\scriptscriptstyle
T}}\mathbf{F}}{T}\right)\mathbf{D}^{1/2}(\mathbf{I}_{r}-\mathbf{M})^{\mathrm{\scriptscriptstyle
T}}.$

By Assumption (V2), the diagonal entries of
$\frac{\mathbf{F}^{\mathrm{\scriptscriptstyle T}}\mathbf{F}}{T}$ are all
bounded with constant magnitude $O(1)$. Thus, $(\mathbf{S}_{11,1})_{jj}\asymp
d^{\alpha}_{j}$, which incorporates the true factor strengths. The estimation
errors between $\widehat{s}_{jj}$ and $d^{\alpha}_{j}$ comes from the diagonal
entries of all remaining terms
$\mathbf{S}_{11,2},\mathbf{S}_{11,3},\mathbf{S}_{11,4},\mathbf{S}_{12},\mathbf{S}_{21},\mathbf{S}_{22}$,
and we bound each of them accordingly.

Note that $\mathbf{S}_{11,2}+\mathbf{S}_{11,3}$ is a symmetric matrix, so by
the Schur’s majorization theorem,
$\max_{j}|(\mathbf{S}_{11,2}+\mathbf{S}_{11,3})_{jj}|\leq|\lambda_{1}(\mathbf{S}_{11,2}+\mathbf{S}_{11,3})|\leq\big{\|}\mathbf{S}_{11,2}+\mathbf{S}_{11,3}\big{\|}$,
and

$\displaystyle\big{\|}\mathbf{S}_{11,2}+\mathbf{S}_{11,3}\big{\|}$
$\displaystyle\leq
2\big{\|}\mathbf{M}\big{\|}\big{\|}\mathbf{D}\big{\|}\left\|\frac{\mathbf{F}^{\mathrm{\scriptscriptstyle
T}}\mathbf{F}}{T}\right\|\big{\|}\mathbf{M}-\mathbf{I}_{r}\big{\|}$
$\displaystyle\preceq d^{\alpha_{1}}\big{\|}\mathbf{M}-\mathbf{I}_{r}\big{\|}$
$\displaystyle=O_{\mathbb{P}}(d^{\alpha_{r}}),$

where the last step follows as
$\big{\|}\mathbf{M}-\mathbf{I}_{r}\big{\|}=\big{\|}\widehat{\mathbf{Q}}^{\mathrm{\scriptscriptstyle
T}}(\mathbf{Q}-\widehat{\mathbf{Q}})\big{\|}\leq\big{\|}\widehat{\mathbf{Q}}-\mathbf{Q}\big{\|}=O_{\mathbb{P}}(d^{\alpha_{r}-\alpha_{1}})$.
Thus,
$\max_{j}|(\mathbf{S}_{11,2}+\mathbf{S}_{11,3})_{jj}|=O_{\mathbb{P}}(d^{\alpha_{r}})$.
Next,

$\displaystyle\big{\|}\mathbf{S}_{11,4}\big{\|}$
$\displaystyle\leq\big{\|}\mathbf{D}\big{\|}\left\|\frac{\mathbf{F}^{\mathrm{\scriptscriptstyle
T}}\mathbf{F}}{T}\right\|\big{\|}\mathbf{M}-\mathbf{I}_{r}\big{\|}^{2}$
$\displaystyle\preceq
d^{\alpha_{1}}\big{\|}\mathbf{M}-\mathbf{I}_{r}\big{\|}^{2}$
$\displaystyle=O_{\mathbb{P}}(d^{2\alpha_{r}-\alpha_{1}})$
$\displaystyle\preceq O_{\mathbb{P}}(d^{\alpha_{r}}),$

and
$\max_{j}|(\mathbf{S}_{11,4})_{jj}|\leq|\lambda_{1}(\mathbf{S}_{11,4})|\leq\big{\|}\mathbf{S}_{11,4}\big{\|}=O_{\mathbb{P}}(d^{\alpha_{r}})$
since $\mathbf{S}_{11,4}$ is also symmetric. Similar steps can be applied to
deal with $\mathbf{S}_{12}+\mathbf{S}_{21}$ and $\mathbf{S}_{22}$,
respectively, since they are both symmetric matrices. We have

$\displaystyle\big{\|}\mathbf{S}_{12}+\mathbf{S}_{21}\big{\|}$
$\displaystyle\leq
2\big{\|}\mathbf{D}^{1/2}\big{\|}\left\|\frac{\mathbf{F}^{\mathrm{\scriptscriptstyle
T}}\mathbf{E}}{T}\right\|$ $\displaystyle\leq
2\big{\|}\mathbf{D}^{1/2}\big{\|}\sqrt{\left\|\frac{\mathbf{F}^{\mathrm{\scriptscriptstyle
T}}\mathbf{F}}{T}\right\|}\sqrt{\left\|\frac{\mathbf{E}^{\mathrm{\scriptscriptstyle
T}}\mathbf{E}}{T}\right\|}$ $\displaystyle\preceq
O_{\mathbb{P}}\left(d^{\frac{\alpha_{1}}{2}}\left(1+\sqrt{\frac{d}{T}}\right)\right),$

and

$\displaystyle\big{\|}\mathbf{S}_{22}\big{\|}\leq\left\|\frac{\mathbf{E}^{\mathrm{\scriptscriptstyle
T}}\mathbf{E}}{T}\right\|=O_{\mathbb{P}}\left(1+\frac{d}{T}\right).$

Thus,
$\max_{j}|(\mathbf{S}_{12}+\mathbf{S}_{21})_{jj}|=O_{\mathbb{P}}\left(d^{\frac{\alpha_{1}}{2}}\left(1+\sqrt{\frac{d}{T}}\right)\right)$
and
$\max_{j}|(\mathbf{S}_{22})_{jj}|=O_{\mathbb{P}}\left(1+\frac{d}{T}\right)$.
Finally, we have

$\displaystyle|\widehat{\alpha}_{j}-\alpha_{j}|$
$\displaystyle=\left|\frac{\textnormal{log}(\widehat{s}_{jj})-\textnormal{log}(d^{\alpha_{j}})}{\textnormal{log}(d)}\right|$
$\displaystyle=\left|\textnormal{log}\left(\frac{(\mathbf{S}_{11,1})_{jj}+(\mathbf{S}_{11,2}+\mathbf{S}_{11,3})_{jj}+(\mathbf{S}_{11,4})_{jj}+(\mathbf{S}_{12}+\mathbf{S}_{21})_{jj}+(\mathbf{S}_{22})_{jj}}{d^{\alpha_{j}}}\right)\right|\frac{1}{\textnormal{log}(d)}$
(5.20)
$\displaystyle=\left|\textnormal{log}\left(\frac{Cd^{\alpha_{j}}\left[1+(\mathbf{S}_{11,2}+\mathbf{S}_{11,3})_{jj}/d^{\alpha_{j}}+(\mathbf{S}_{11,4})_{jj}/d^{\alpha_{j}}+(\mathbf{S}_{12}+\mathbf{S}_{21})_{jj}/d^{\alpha_{j}}+(\mathbf{S}_{22})_{jj}/d^{\alpha_{j}}\right]}{d^{\alpha_{j}}}\right)\right|\frac{1}{\textnormal{log}(d)}$
$\displaystyle=\frac{\textnormal{log}(C)+\textnormal{log}(R_{s})}{\textnormal{log}(d)},$

where $C$ is a constant, and

$\displaystyle
R_{s}:=1+\frac{(\mathbf{S}_{11,2}+\mathbf{S}_{11,3})_{jj}}{d^{\alpha_{j}}}+\frac{(\mathbf{S}_{11,4})_{jj}}{d^{\alpha_{j}}}+\frac{(\mathbf{S}_{12}+\mathbf{S}_{21})_{jj}}{d^{\alpha_{j}}}+\frac{(\mathbf{S}_{22})_{jj}}{d^{\alpha_{j}}}.$

We have obtained the upper bound for each term in $\mathbf{R}_{s}$, such that

$\displaystyle\left|\frac{(\mathbf{S}_{11,2}+\mathbf{S}_{11,3})_{jj}}{d^{\alpha_{j}}}\right|$
$\displaystyle=O_{\mathbb{P}}\left(d^{\alpha_{r}-\alpha_{j}}\right)\preceq
O_{\mathbb{P}}(1),$ (5.21)
$\displaystyle\left|\frac{(\mathbf{S}_{11,4})_{jj}}{d^{\alpha_{j}}}\right|$
$\displaystyle=O_{\mathbb{P}}\left(d^{\alpha_{r}-\alpha_{j}}\right)\preceq
O_{\mathbb{P}}(1),$ (5.22)
$\displaystyle\left|\frac{(\mathbf{S}_{12}+\mathbf{S}_{21})_{jj}}{d^{\alpha_{j}}}\right|$
$\displaystyle=O_{\mathbb{P}}\left((d^{\frac{\alpha_{1}}{2}-\alpha_{j}})(1+d^{1/2}T^{-\frac{1}{2}})\right)\preceq
O_{\mathbb{P}}(1),$ (5.23)
$\displaystyle\left|\frac{(\mathbf{S}_{22})_{jj}}{d^{\alpha_{j}}}\right|$
$\displaystyle=O_{\mathbb{P}}\left(d^{-\alpha_{j}}(1+dT^{-1})\right)\preceq
O_{\mathbb{P}}(1),$ (5.24)

where the last two lines follow from Assumption (V5). Therefore,
$R_{s}=O_{\mathbb{P}}(1)$ and
$|\widehat{\alpha}_{j}-\alpha_{j}|=O_{\mathbb{P}}(1/\textnormal{log}(d))$.
This completes the proof of Theorem 1. $\square$

Proof of Theorem 2. We use the same definitions of
$\mathbf{S}_{11,1},\mathbf{S}_{11,2},\mathbf{S}_{11,3},\mathbf{S}_{11,4},\mathbf{S}_{12},\mathbf{S}_{21},\mathbf{S}_{22}$
as in the proof of Theorem 1. Define
$\mathbf{f}_{j}=[f_{1j},\cdots,f_{Tj}]^{\mathrm{\scriptscriptstyle
T}}\in\mathbb{R}^{T}$. If $C=1$, then

$\displaystyle(\mathbf{S}_{11,1})_{jj}$
$\displaystyle=d^{\alpha_{j}}\left(\mathbb{E}\left[\frac{\mathbf{f}_{j}^{\mathrm{\scriptscriptstyle
T}}\mathbf{f}_{j}}{T}\right]+O_{\mathbb{P}}\left(\sqrt{Var\left[\frac{\mathbf{f}_{j}^{\mathrm{\scriptscriptstyle
T}}\mathbf{f}_{j}}{T}\right]}\right)\right)$
$\displaystyle=d^{\alpha_{j}}+d^{\alpha_{j}}O_{\mathbb{P}}\left(\sqrt{\frac{\mathbb{E}\left[(\mathbf{f}_{j}^{\mathrm{\scriptscriptstyle
T}}\mathbf{f}_{j})^{2}\right]}{T^{2}}-1}\right),$

since $\mathbb{E}[f_{tj}]^{2}=1$ for any $t\in[T],j\in[r]$ by Assumption (V2).
Next, we have

$\displaystyle\mathbb{E}\left[(\mathbf{f}_{j}^{\mathrm{\scriptscriptstyle
T}}\mathbf{f}_{j})^{2}\right]$
$\displaystyle=\mathbb{E}\left[\sum_{t=1}^{T}f_{tj}^{2}\right]^{2}$
$\displaystyle=\sum_{t=1}^{T}\sum_{s=1}^{T}\sum_{q\geq
0}a_{q}^{2}a_{q-|t-s|}^{2}\mathbb{E}(z^{4})+\sum_{t=1}^{T}\sum_{s=1}^{T}\left(\sum_{q\geq
0}\sum_{p\geq 0}^{p\neq
q-|t-s|}a_{q}^{2}a_{p}^{2}\right)\mathbb{E}(z^{2})^{2},$

where $z$ is a random variable with mean 0, variance 1, and uniformly bounded
fourth moment. Let $\mathbf{B}_{f,T}\in\mathbb{R}^{T\times T}$ to be a matrix
such that $(\mathbf{B}_{f,T})_{ts}=\sum_{q\geq 0}a_{q}^{2}a_{q-|t-s|}^{2}$.
Then

$\displaystyle\sum_{t=1}^{T}\sum_{s=1}^{T}\sum_{q\geq
0}a_{q}^{2}a_{q-|t-s|}^{2}=\sum_{t=1}^{T}\sum_{s=1}^{T}(\mathbf{B}_{f,T})_{ts}$
$\displaystyle\leq
T\|\mathbf{B}_{f,T}\|_{1}=T\max_{t}\sum_{s=1}^{T}|(\mathbf{B}_{f,T})_{ts}|$
$\displaystyle\leq 2T\sum_{v=0}^{T}|\sum_{q\geq 0}a_{q}^{2}a_{q+v}^{2}|\leq
2T\left(\sum_{q\geq 0}a_{q}^{2}\right)^{2}\leq 2T.$

Thus, $\sum_{t=1}^{T}\sum_{s=1}^{T}\sum_{q\geq
0}a_{q}^{2}a_{q-|t-s|}^{2}\mathbb{E}(z^{4})=O(T)$. Next, since
$\mathbb{E}(z^{2})=1$,

$\displaystyle\sum_{t=1}^{T}\sum_{s=1}^{T}\left(\sum_{q\geq 0}\sum_{p\geq
0}^{p\neq
q-|t-s|}a_{q}^{2}a_{p}^{2}\right)\mathbb{E}(z^{2})^{2}\leq\sum_{t=1}^{T}\sum_{s=1}^{T}\left(\sum_{q\geq
0}\sum_{p\geq
0}a_{q}^{2}a_{p}^{2}\right)\leq\sum_{t=1}^{T}\sum_{s=1}^{T}\left(\sum_{q\geq
0}a_{q}^{2}\right)^{2}\leq T^{2}.$

Moreover, since $\mathbb{E}\left[(\mathbf{f}_{j}^{\mathrm{\scriptscriptstyle
T}}\mathbf{f}_{j})^{2}\right]\geq T^{2}$ by definition, we will also have
$\sum_{t=1}^{T}\sum_{s=1}^{T}\left(\sum_{q\geq 0}\sum_{p\geq 0}^{p\neq
q-|t-s|}a_{q}^{2}a_{p}^{2}\right)\mathbb{E}(z^{2})^{2}\geq T^{2}-O(T)$. Thus,
we finally have that
$\mathbb{E}\left[(\mathbf{f}_{j}^{\mathrm{\scriptscriptstyle
T}}\mathbf{f}_{j})^{2}\right]=T^{2}+O(T),$ so

$\displaystyle(\mathbf{S}_{11,1})_{jj}=d^{\alpha_{j}}+d^{\alpha_{j}}O_{\mathbb{P}}\left(T^{-\frac{1}{2}}\right),$

and we can define
$(\widetilde{\mathbf{S}}_{11,1})_{jj}:=(\mathbf{S}_{11,1})_{jj}-d^{\alpha_{j}}=O_{\mathbb{P}}\left(d^{\alpha_{j}}T^{-\frac{1}{2}}\right)$.
From (5.20), using the same notations, we have

$\displaystyle|\widehat{\alpha}_{j}-\alpha_{j}|$
$\displaystyle=\left|\textnormal{log}\left(\frac{(\mathbf{S}_{11,1})_{jj}+(\mathbf{S}_{11,2}+\mathbf{S}_{11,3})_{jj}+(\mathbf{S}_{11,4})_{jj}+(\mathbf{S}_{12}+\mathbf{S}_{21})_{jj}+(\mathbf{S}_{22})_{jj}}{d^{\alpha_{j}}}\right)\right|\frac{1}{\textnormal{log}(d)}$
$\displaystyle=\left|\textnormal{log}\left(\frac{d^{\alpha_{j}}\left[1+r_{s}\right]}{d^{\alpha_{j}}}\right)\right|\frac{1}{\textnormal{log}(d)}$
$\displaystyle=\frac{\textnormal{log}(1+r_{s})}{\textnormal{log}(d)},$

where

$\displaystyle
r_{s}:=(\widetilde{\mathbf{S}}_{11,1})_{jj}/d^{\alpha_{j}}+(\mathbf{S}_{11,2}+\mathbf{S}_{11,3})_{jj}/d^{\alpha_{j}}+(\mathbf{S}_{11,4})_{jj}/d^{\alpha_{j}}+(\mathbf{S}_{12}+\mathbf{S}_{21})_{jj}/d^{\alpha_{j}}+(\mathbf{S}_{22})_{jj}/d^{\alpha_{j}}.$
(5.25)

If $r_{s}$ is small such that $r_{s}=o_{\mathbb{P}}(1)$, then
$\textnormal{log}(1+r_{s})=r_{s}+O(r_{s}^{2})$, so we can achieve a
convergence rate such that

$\displaystyle|\widehat{\alpha}_{j}-\alpha_{j}|=O(r_{s}/\textnormal{log}(d)).$
(5.26)

Note that
$(\widetilde{\mathbf{S}}_{11,1})_{jj}/d^{\alpha_{j}}=O_{\mathbb{P}}\left(T^{-1/2}\right)=o_{\mathbb{P}}(1)$
as $T\rightarrow\infty$. Also, from (5.21) to (5.24), if Assumption (V5’) is
satisfied, then for any $j$ such that $\alpha_{j}>\alpha_{r}$,

$\displaystyle\left|\frac{(\mathbf{S}_{11,2}+\mathbf{S}_{11,3})_{jj}}{d^{\alpha_{j}}}\right|$
$\displaystyle=O_{\mathbb{P}}\left(d^{\alpha_{r}-\alpha_{j}}\right)\preceq
o_{\mathbb{P}}(1),$ (5.27)
$\displaystyle\left|\frac{(\mathbf{S}_{11,4})_{jj}}{d^{\alpha_{j}}}\right|$
$\displaystyle=O_{\mathbb{P}}\left(d^{\alpha_{r}-\alpha_{j}}\right)\preceq
o_{\mathbb{P}}(1),$ (5.28)
$\displaystyle\left|\frac{(\mathbf{S}_{12}+\mathbf{S}_{21})_{jj}}{d^{\alpha_{j}}}\right|$
$\displaystyle=O_{\mathbb{P}}\left((d^{\frac{\alpha_{1}}{2}-\alpha_{j}})(1+d^{1/2}T^{-\frac{1}{2}})\right)\preceq
o_{\mathbb{P}}(1),$ (5.29)
$\displaystyle\left|\frac{(\mathbf{S}_{22})_{jj}}{d^{\alpha_{j}}}\right|$
$\displaystyle=O_{\mathbb{P}}\left(d^{-\alpha_{j}}(1+dT^{-1})\right)\preceq
o_{\mathbb{P}}(1),$ (5.30)

so we can show (2.8) by substituting (5.27)(5.28)(5.29)(5.30) into (5.25) and
(5.26). Furthermore, if Assumption (V4’) is also satisfied, then for all
$j\in[r]$,

$\displaystyle\left|\frac{(\mathbf{S}_{11,2}+\mathbf{S}_{11,3})_{jj}}{d^{\alpha_{j}}}\right|$
$\displaystyle=o_{\mathbb{P}}\left(d^{\alpha_{r}-\alpha_{j}}\right)\preceq
o_{\mathbb{P}}(1),$ (5.31)
$\displaystyle\left|\frac{(\mathbf{S}_{11,4})_{jj}}{d^{\alpha_{j}}}\right|$
$\displaystyle=o_{\mathbb{P}}\left(d^{\alpha_{r}-\alpha_{j}}\right)\preceq
o_{\mathbb{P}}(1),$ (5.32)

and we can show (2.9) by substituting (5.31), (5.32), (5.29), (5.30) into
(5.25) and (5.26). This completes the proof of Theorem 2. $\square$

Before presenting the proof of Theorem 3, we first state the following Lemma.

######  Lemma 1.

Denote $g_{1}:=\textnormal{tr}(\mathbf{D}_{1})$,
$g_{2}:=\textnormal{tr}(\mathbf{D}_{2})$. From (3.16) and (3.17), denote
$\widehat{g}_{1}:=\left(\frac{\textnormal{tr}(\widehat{\mathbf{S}}_{1}+\widehat{\mathbf{S}}_{2})}{2}\cdot\frac{r_{1}d_{1}}{r_{2}d_{2}}\right)^{1/2}$
and
$\widehat{g}_{2}:=\left(\frac{\textnormal{tr}(\widehat{\mathbf{S}}_{1}+\widehat{\mathbf{S}}_{2})}{2}\cdot\frac{r_{2}d_{2}}{r_{1}d_{1}}\right)^{1/2}$.
Then, under Assumptions (M1) - (M5), if the identifiability condition (3.15)
holds, we have

$\displaystyle\frac{\textnormal{tr}(\widehat{\mathbf{S}}_{1})}{g_{1}g_{2}}=1+o_{\mathbb{P}}(1),\
\ \
\frac{\textnormal{tr}(\widehat{\mathbf{S}}_{2})}{g_{1}g_{2}}=1+o_{\mathbb{P}}(1),$

and thus,

$\displaystyle\frac{\widehat{g}_{1}}{g_{1}}=1+o_{\mathbb{P}}(1),\ \ \
\frac{\widehat{g}_{2}}{g_{2}}=1+o_{\mathbb{P}}(1),$

which means $\widehat{g}_{1}$ and $\widehat{g}_{2}$ are ratio-consistent
estimates of $g_{1}$ and $g_{2}$, respectively.

Proof of Lemma 1 We prove the results for $k=1$ WLOG, and the results for
$k=2$ will similarly follows. Similar to the proof of Theorem 1, we start by
writing (3.11) as

$\displaystyle\widehat{\mathbf{S}}_{1}=\mathbf{S}_{11}^{(1)}+\mathbf{S}_{12}^{(1)}+\mathbf{S}_{21}^{(1)}+\mathbf{S}_{22}^{(1)},$

where

$\displaystyle\mathbf{S}_{11}^{(1)}:$
$\displaystyle=\widehat{\mathbf{Q}}_{1}^{\mathrm{\scriptscriptstyle
T}}\mathbf{Q}_{1}\mathbf{D}_{1}^{1/2}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\mathbf{D}_{1}^{1/2}\mathbf{Q}_{1}^{\mathrm{\scriptscriptstyle
T}}\widehat{\mathbf{Q}}_{1},$ $\displaystyle\mathbf{S}_{12}^{(1)}:$
$\displaystyle=\widehat{\mathbf{Q}}_{1}^{\mathrm{\scriptscriptstyle
T}}\mathbf{Q}_{1}\mathbf{D}_{1}^{1/2}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}^{1/2}\mathbf{Q}_{2}^{\mathrm{\scriptscriptstyle
T}}\mathbf{E}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\widehat{\mathbf{Q}}_{1},$ $\displaystyle\mathbf{S}_{21}^{(1)}:$
$\displaystyle=\widehat{\mathbf{Q}}_{1}^{\mathrm{\scriptscriptstyle
T}}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{E}_{t}\mathbf{Q}_{2}\mathbf{D}_{2}^{1/2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\mathbf{D}_{1}^{1/2}\mathbf{Q}_{1}^{\mathrm{\scriptscriptstyle
T}}\widehat{\mathbf{Q}}_{1},$ $\displaystyle\mathbf{S}_{22}^{(1)}:$
$\displaystyle=\widehat{\mathbf{Q}}_{1}^{\mathrm{\scriptscriptstyle
T}}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{E}_{t}\mathbf{E}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\widehat{\mathbf{Q}}_{1}.$

Let $\mathbf{M}_{1}:=\widehat{\mathbf{Q}}_{1}^{\mathrm{\scriptscriptstyle
T}}\mathbf{Q}_{1}$, we can further decompose $\mathbf{S}_{11}^{(1)}$ as

$\displaystyle\mathbf{S}_{11}^{(1)}=\mathbf{S}_{11,1}^{(1)}+\mathbf{S}_{11,2}^{(1)}+\mathbf{S}_{11,3}^{(1)}+\mathbf{S}_{11,4}^{(1)},$

where

$\displaystyle\mathbf{S}_{11,1}^{(1)}:$
$\displaystyle=\mathbf{D}_{1}^{1/2}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\mathbf{D}_{1}^{1/2},$ $\displaystyle\mathbf{S}_{11,2}^{(1)}:$
$\displaystyle=\mathbf{M}_{1}\mathbf{D}_{1}^{1/2}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\mathbf{D}_{1}^{1/2}(\mathbf{M}_{1}-\mathbf{I}_{r_{1}})^{\mathrm{\scriptscriptstyle
T}},$ $\displaystyle\mathbf{S}_{11,3}^{(1)}:$
$\displaystyle=(\mathbf{M}_{1}-\mathbf{I}_{r_{1}})\mathbf{D}_{1}^{1/2}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\mathbf{D}_{1}^{1/2}\mathbf{M}_{1}^{\mathrm{\scriptscriptstyle T}},$
$\displaystyle\mathbf{S}_{11,4}^{(1)}:$
$\displaystyle=(\mathbf{M}_{1}-\mathbf{I}_{r_{1}})\mathbf{D}_{1}^{1/2}\left(\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right)\mathbf{D}_{1}^{1/2}(\mathbf{I}_{r_{1}}-\mathbf{M}_{1})^{\mathrm{\scriptscriptstyle
T}}.$

Thus,

$\displaystyle\textnormal{tr}(\widehat{\mathbf{S}}_{1})=\textnormal{tr}(\mathbf{S}_{11,1}^{(1)})+\textnormal{tr}(\mathbf{S}_{11,2}^{(1)})+\textnormal{tr}(\mathbf{S}_{11,3}^{(1)})+\textnormal{tr}(\mathbf{S}_{11,4}^{(1)})+\textnormal{tr}(\mathbf{S}_{12}^{(1)})+\textnormal{tr}(\mathbf{S}_{21}^{(1)})+\textnormal{tr}(\mathbf{S}_{22}^{(1)}).$
(5.33)

Next, we want to show that
$\frac{\textnormal{tr}(\widehat{\mathbf{S}}_{1})}{\textnormal{tr}(\mathbf{D}_{1})\textnormal{tr}(\mathbf{D}_{2})}=1+o_{\mathbb{P}}(1)$.
We start by $\textnormal{tr}(\mathbf{S}_{11,1}^{(1)})$. For each
$j=1,\cdots,r_{1}$, we have

$\displaystyle(\mathbf{S}_{11,1}^{(1)})_{jj}$
$\displaystyle=d_{1}^{\alpha_{1,j}}\frac{1}{T}\sum_{t=1}^{T}\left(\sum_{i=1}^{r_{1}}d_{2}^{\alpha_{2,i}}f_{tji}^{2}\right)$
$\displaystyle=d_{1}^{\alpha_{1,j}}\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}\left(\sum_{i=1}^{r_{1}}d_{2}^{\alpha_{2,i}}f_{tji}^{2}\right)+d_{1}^{\alpha_{1,j}}O_{\mathbb{P}}\left(\sqrt{Var\left[\frac{1}{T}\sum_{t=1}^{T}\left(\sum_{i=1}^{r_{2}}d_{2}^{\alpha_{2,i}}f_{tji}^{2}\right)\right]}\right).$
(5.34)

For the first term in (5.34),

$\displaystyle
d_{1}^{\alpha_{1,j}}\frac{1}{T}\sum_{t=1}^{T}\mathbb{E}\left(\sum_{i=1}^{r_{1}}d_{2}^{\alpha_{2,i}}f_{tji}^{2}\right)=Cd_{1}^{\alpha_{1,j}}\textnormal{tr}(\mathbf{D}_{2}),$

where $C$ is the constant such that
$\|\mathbf{a}_{1j}\|^{2}=Cd_{1}^{\alpha_{1,j}}$. For the second term in
(5.34), denote
$\mathbf{f}_{ji}=[f_{1ji},\cdots,f_{Tji}]^{\mathrm{\scriptscriptstyle
T}}\in\mathbb{R}^{T}$, then

$\displaystyle
Var\left[\frac{1}{T}\sum_{t=1}^{T}\left(\sum_{i=1}^{r_{2}}d_{2}^{\alpha_{2,i}}f_{tji}^{2}\right)\right]$
$\displaystyle=\sum_{i=1}^{r_{2}}d_{2}^{\alpha_{2,i}}Var\left(\frac{\sum_{t=1}^{T}f_{tji}^{2}}{T}\right)$
$\displaystyle=\sum_{i=1}^{r_{2}}d_{2}^{\alpha_{2,i}}Var\left(\frac{\sum_{t=1}^{T}\mathbf{f}_{ji}^{\mathrm{\scriptscriptstyle
T}}\mathbf{f}_{ji}}{T}\right)$
$\displaystyle=\sum_{i=1}^{r_{2}}d_{2}^{\alpha_{2,i}}O_{\mathbb{P}}(T^{-1})$
$\displaystyle=\textnormal{tr}(\mathbf{D}_{2})O_{\mathbb{P}}(T^{-1}),$

where the second last step follows from the proof of Theorem 2 by simply
replacing $\mathbf{f}_{j}$ in the proof of Theorem 2 with $\mathbf{f}_{ji}$
here. Thus,

$\displaystyle(\mathbf{S}_{11,1}^{(1)})_{jj}=\textnormal{tr}(\mathbf{D}_{2})Cd_{1}^{\alpha_{1,j}}+[\textnormal{tr}(\mathbf{D}_{2})]^{1/2}d_{1}^{\alpha_{1,j}}O_{\mathbb{P}}(T^{-1/2}).$
(5.35)

Therefore,

$\displaystyle\textnormal{tr}(\mathbf{S}_{11,1}^{(1)})=\textnormal{tr}(\mathbf{D}_{1})\textnormal{tr}(\mathbf{D}_{2})+\textnormal{tr}(\mathbf{D}_{1})[\textnormal{tr}(\mathbf{D}_{2})]^{1/2}O_{\mathbb{P}}(T^{-1/2}),$

and the dominating term in $\textnormal{tr}(\mathbf{S}_{11,1}^{(1)})$ is
$\textnormal{tr}(\mathbf{D}_{1})\textnormal{tr}(\mathbf{D}_{2})$. In other
words,

$\displaystyle\frac{\textnormal{tr}(\mathbf{S}_{11,1}^{(1)})}{\textnormal{tr}(\mathbf{D}_{1})\textnormal{tr}(\mathbf{D}_{2})}=1+o_{\mathbb{P}}(1).$

Next, we show that $\textnormal{tr}(\mathbf{S}_{11,1}^{(1)})$ is dominating
all remaining terms in (5.33), such that all remaining terms are dominated by
the rate $\textnormal{tr}(\mathbf{D}_{1})\textnormal{tr}(\mathbf{D}_{2})\asymp
d_{1}^{\alpha_{1,1}}d_{2}^{\alpha_{2,1}}$. We have

$\displaystyle\textnormal{tr}(\mathbf{S}_{11,2}^{(1)})+\textnormal{tr}(\mathbf{S}_{11,3}^{(1)})$
$\displaystyle\leq
r_{1}d_{1}^{\alpha_{1,1}}\left\|\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right\|\big{\|}\mathbf{M}_{1}-\mathbf{I}_{r_{1}}\big{\|}\preceq
r_{1}d_{1}^{\alpha_{1,1}}d_{2}^{\alpha_{2,1}}o_{\mathbb{P}}(1)\prec
d_{1}^{\alpha_{1,1}}d_{2}^{\alpha_{2,1}},$
$\displaystyle\textnormal{tr}(\mathbf{S}_{11,4}^{(1)})$ $\displaystyle\leq
r_{1}d_{1}^{\alpha_{1,1}}\left\|\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right\|\big{\|}\mathbf{M}_{1}-\mathbf{I}_{r_{1}}\big{\|}^{2}\preceq
r_{1}d_{1}^{\alpha_{1,1}}d_{2}^{\alpha_{2,1}}o_{\mathbb{P}}(1)\prec
d_{1}^{\alpha_{1,1}}d_{2}^{\alpha_{2,1}},$
$\displaystyle\textnormal{tr}(\mathbf{S}_{12}^{(1)})+\textnormal{tr}(\mathbf{S}_{21}^{(1)})$
$\displaystyle\leq
2r_{1}\big{\|}\mathbf{D}_{1}^{1/2}\big{\|}\left(\frac{1}{T}\sum_{t=1}^{T}\left\|\mathbf{F}_{t}\right\|\big{\|}\mathbf{D}_{2}^{1/2}\big{\|}\left\|\mathbf{E}_{t}\right\|\right)\leq
r_{1}O_{\mathbb{P}}\left(d_{1}^{\frac{\alpha_{1,1}}{2}}d_{2}^{\frac{\alpha_{2,1}}{2}}\sqrt{d_{2}+\frac{d_{1}}{T}}\right)\prec
d_{1}^{\alpha_{1,1}}d_{2}^{\alpha_{2,1}},$
$\displaystyle\textnormal{tr}(\mathbf{S}_{22}^{(1)})$ $\displaystyle\leq
r_{1}\left\|\frac{1}{T}\sum_{t=1}^{T}\mathbf{E}_{t}\mathbf{E}_{t}^{\mathrm{\scriptscriptstyle
T}}\right\|=r_{1}O_{\mathbb{P}}\left(d_{2}+\frac{d_{1}}{T}\right)\prec
d_{1}^{\alpha_{1,1}}d_{2}^{\alpha_{2,1}},$

where the last two equality follows from Assumption (M5). Therefore, from
(5.33), we have

$\displaystyle\frac{\textnormal{tr}(\widehat{\mathbf{S}}_{1})}{\textnormal{tr}(\mathbf{D}_{1})\textnormal{tr}(\mathbf{D}_{2})}=1+o_{\mathbb{P}}(1),$

which proves the first part of Lemma 1 for $k=1$. We can similarly prove the
result for $k=2$, then we will also have

$\displaystyle\frac{\left[\textnormal{tr}(\widehat{\mathbf{S}}_{1})+\textnormal{tr}(\widehat{\mathbf{S}}_{1})\right]/2}{\textnormal{tr}(\mathbf{D}_{1})\textnormal{tr}(\mathbf{D}_{2})}=1+o_{\mathbb{P}}(1).$

Thus,

$\displaystyle\frac{\widehat{g}_{1}}{g_{1}}=\frac{\sqrt{\frac{\textnormal{tr}(\widehat{\mathbf{S}}_{1})+\textnormal{tr}(\widehat{\mathbf{S}}_{1})}{2}\frac{r_{1}d_{1}}{r_{2}d_{2}}}}{g_{1}}=\frac{\sqrt{g_{1}g_{2}(1+o_{\mathbb{P}}(1))\frac{r_{1}d_{1}}{r_{2}d_{2}}}}{g_{1}}=\frac{\sqrt{g_{1}^{2}(1+o_{\mathbb{P}}(1))}}{g_{1}}=1+o_{\mathbb{P}}(1),$

where the second last step equality follows from the identifiability condition
(3.15) that $\frac{g_{1}}{r_{1}d_{1}}=\frac{g_{2}}{r_{2}d_{2}}$. This
completes the proof of the second part of Lemma 1 for $k=1$. Similar arguments
can be applied to prove the results for $k=2$. Thus, we complete the proof of
Lemma 1. $\square$

Proof of Theorem 3. We prove the results for $k=1$ WLOG, and the results for
$k=2$ will similarly follows. Recall the definition of
$\mathbf{S}_{11,1}^{(1)},\mathbf{S}_{11,2}^{(1)},\mathbf{S}_{11,3}^{(1)},\mathbf{S}_{11,4}^{(1)},\mathbf{S}_{12}^{(1)},\mathbf{S}_{21}^{(1)},\mathbf{S}_{22}^{(1)}$
as from the proof of Lemma 1.

For each $j=1,\cdots,r_{1}$, we have $(\mathbf{S}_{11,1}^{(1)})_{jj}$ contains
the true signal part, and from (5.35), we have

$\displaystyle(\mathbf{S}_{11,1}^{(1)})_{jj}=d_{1}^{\alpha_{1,j}}\left(\textnormal{tr}(\mathbf{D}_{2})C+[\textnormal{tr}(\mathbf{D}_{2})]^{1/2}O_{\mathbb{P}}(T^{-1/2})\right),$

and we can further define

$\displaystyle(\widetilde{\mathbf{S}}_{11,1}^{(1)})_{jj}:=(\mathbf{S}_{11,1}^{(1)})_{jj}-\textnormal{tr}(\mathbf{D}_{2})d_{1}^{\alpha_{1,j}}=(C-1)\textnormal{tr}(\mathbf{D}_{2})d_{1}^{\alpha_{1,j}}+O_{\mathbb{P}}\left(d_{1}^{\alpha_{1,j}}T^{-\frac{1}{2}}[\textnormal{tr}(\mathbf{D}_{2})]^{1/2}\right).$

Next, we bound the terms
$\mathbf{S}_{11,2}^{(1)},\mathbf{S}_{11,3}^{(1)},\mathbf{S}_{11,4}^{(1)},\mathbf{S}_{12}^{(1)},\mathbf{S}_{21}^{(1)},\mathbf{S}_{22}^{(1)}$
accordingly. Similar to the proof of Theorem 1, we have

$\displaystyle\big{\|}\mathbf{S}_{11,2}^{(1)}+\mathbf{S}_{11,3}^{(1)}\big{\|}$
$\displaystyle\leq
2\big{\|}\mathbf{M}_{1}\big{\|}\big{\|}\mathbf{D}_{1}\big{\|}\left\|\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right\|\big{\|}\mathbf{M}_{1}-\mathbf{I}_{r_{1}}\big{\|}$
$\displaystyle\preceq
d_{1}^{\alpha_{1,1}}d_{2}^{\alpha_{2,1}}\big{\|}\mathbf{M}_{1}-\mathbf{I}_{r_{1}}\big{\|}$
$\displaystyle\preceq
o_{\mathbb{P}}(d_{1}^{\alpha_{1,r_{1}}}d_{2}^{\alpha_{2,1}}),$

where the last step follows as
$\big{\|}\mathbf{M}_{1}-\mathbf{I}_{r_{1}}\big{\|}=\big{\|}\widehat{\mathbf{Q}}_{1}^{\mathrm{\scriptscriptstyle
T}}(\mathbf{Q}_{1}-\widehat{\mathbf{Q}}_{1})\big{\|}\leq\big{\|}\widehat{\mathbf{Q}}_{1}-\mathbf{Q}_{1}\big{\|}=o_{\mathbb{P}}(d_{1}^{\alpha_{1,r_{1}}-\alpha_{1,1}})$.
Thus,
$\max_{j}|(\mathbf{S}_{11,2}^{(1)}+\mathbf{S}_{11,3}^{(1)})_{jj}|=o_{\mathbb{P}}(d_{1}^{\alpha_{1,r_{1}}}d_{2}^{\alpha_{2,1}})$.
Next,

$\displaystyle\big{\|}\mathbf{S}_{11,4}^{(1)}\big{\|}$
$\displaystyle\leq\big{\|}\mathbf{D}_{1}\big{\|}\left\|\frac{1}{T}\sum_{t=1}^{T}\mathbf{F}_{t}\mathbf{D}_{2}\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\right\|\big{\|}\mathbf{M}_{1}-\mathbf{I}_{r_{1}}\big{\|}^{2}$
$\displaystyle\preceq
d_{1}^{\alpha_{1,1}}d_{2}^{\alpha_{2,1}}\big{\|}\mathbf{M}_{1}-\mathbf{I}_{r_{1}}\big{\|}^{2}$
$\displaystyle\preceq
o_{\mathbb{P}}(d_{1}^{\alpha_{1,r_{1}}}d_{2}^{\alpha_{2,1}}),$

and thus
$\max_{j}|(\mathbf{S}_{11,4}^{(1)})_{jj}|=o_{\mathbb{P}}(d_{1}^{\alpha_{1,r_{1}}}d_{2}^{\alpha_{2,1}})$.
Similarly,

$\displaystyle\big{\|}\mathbf{S}_{22}^{(1)}\big{\|}\leq\left\|\frac{1}{T}\sum_{t=1}^{T}\mathbf{E}_{t}\mathbf{E}_{t}^{\mathrm{\scriptscriptstyle
T}}\right\|=O_{\mathbb{P}}\left(d_{2}+\frac{d_{1}}{T}\right).$

and

$\displaystyle\big{\|}\mathbf{S}_{12}^{(1)}+\mathbf{S}_{21}^{(1)}\big{\|}$
$\displaystyle\leq
2\big{\|}\mathbf{D}_{1}^{1/2}\big{\|}\left(\frac{1}{T}\sum_{t=1}^{T}\left\|\mathbf{F}_{t}\right\|\big{\|}\mathbf{D}_{2}^{1/2}\big{\|}\left\|\mathbf{E}_{t}\right\|\right)$
$\displaystyle\leq
2\big{\|}\mathbf{D}_{1}^{1/2}\big{\|}\big{\|}\mathbf{D}_{2}^{1/2}\big{\|}\sqrt{\left\|\frac{1}{T}\sum_{t=1}^{T}\frac{\mathbf{F}_{t}^{\mathrm{\scriptscriptstyle
T}}\mathbf{F}_{t}}{T}\right\|}\sqrt{\left\|\frac{1}{T}\sum_{t=1}^{T}\frac{\mathbf{E}_{t}\mathbf{E}_{t}^{\mathrm{\scriptscriptstyle
T}}}{T}\right\|}$ $\displaystyle\preceq
O_{\mathbb{P}}\left(d_{1}^{\frac{\alpha_{1,1}}{2}}d_{2}^{\frac{\alpha_{2,1}}{2}}\sqrt{d_{2}+\frac{d_{1}}{T}}\right),$

Thus,
$\max_{j}|(\mathbf{S}_{12}^{(1)}+\mathbf{S}_{21}^{(1)})_{jj}|=O_{\mathbb{P}}\left(d_{1}^{\frac{\alpha_{1,1}}{2}}d_{2}^{\frac{\alpha_{2,1}}{2}}\left[d_{2}+\frac{d_{1}}{T}\right]^{1/2}\right)$
and
$\max_{j}|(\mathbf{S}_{22}^{(1)})_{jj}|=O_{\mathbb{P}}\left(d_{2}+\frac{d_{1}}{T}\right)$.

Now, we have
$\alpha_{1,j}=\textnormal{log}((\widehat{\mathbf{S}}_{1})_{jj}/\widehat{g}_{2})/\textnormal{log}(d_{1})$.
Thus,

$\displaystyle|\widehat{\alpha}_{1,j}-\alpha_{1,j}|$
$\displaystyle=\left|\frac{\textnormal{log}((\widehat{\mathbf{S}}_{1})_{jj}/\widehat{g}_{2})-\textnormal{log}(d^{\alpha_{1,j}})}{\textnormal{log}(d_{1})}\right|$
$\displaystyle=\left|\textnormal{log}\left(\frac{(\mathbf{S}_{11,1}^{(1)})_{jj}+(\mathbf{S}_{11,2}^{(1)}+\mathbf{S}_{11,3}^{(1)})_{jj}+(\mathbf{S}_{11,4}^{(1)})_{jj}+(\mathbf{S}_{12}^{(1)}+\mathbf{S}_{21}^{(1)})_{jj}+(\mathbf{S}_{22})^{(1)}_{jj}}{\widehat{g}_{2}d_{1}^{\alpha_{1,j}}}\right)\right|\frac{1}{\textnormal{log}(d_{1})}$
$\displaystyle=\left|\textnormal{log}\left(\frac{g_{2}d_{1}^{\alpha_{1,j}}\left[1+r_{s}^{(1)}\right]}{\widehat{g}_{2}d_{1}^{\alpha_{1,j}}}\right)\right|\frac{1}{\textnormal{log}(d_{1})}$

where

$\displaystyle
r_{s}^{(1)}:=\frac{(\widetilde{\mathbf{S}}_{11,1}^{(1)})_{jj}}{g_{2}d_{1}^{\alpha_{1,j}}}+\frac{(\mathbf{S}_{11,2}^{(1)}+\mathbf{S}_{11,3}^{(1)})_{jj}}{g_{2}d_{1}^{\alpha_{1,j}}}+\frac{(\mathbf{S}_{11,4}^{(1)})_{jj}}{g_{2}d_{1}^{\alpha_{1,j}}}+\frac{(\mathbf{S}_{12}^{(1)}+\mathbf{S}_{21}^{(1)})_{jj}}{g_{2}d_{1}^{\alpha_{1,j}}}+\frac{(\mathbf{S}_{22}^{(1)})_{jj}}{g_{2}d_{1}^{\alpha_{1,j}}}.$
(5.36)

For each term in (5.36), since $g_{2}\asymp d_{2}^{\alpha_{2,1}}$, we can
obtain

$\displaystyle\frac{(\widetilde{\mathbf{S}}_{11,1}^{(1)})_{jj}}{g_{2}d_{1}^{\alpha_{1,j}}}$
$\displaystyle=O_{\mathbb{P}}(d_{2}^{-\frac{\alpha_{2,1}}{2}}T^{-1/2})+C-1\preceq
o_{\mathbb{P}}(1)+C-1,$
$\displaystyle\frac{(\mathbf{S}_{11,2}^{(1)}+\mathbf{S}_{11,3}^{(1)})_{jj}}{g_{2}d_{1}^{\alpha_{1,j}}}$
$\displaystyle=o_{\mathbb{P}}(d_{1}^{\alpha_{1,r_{1}}-\alpha_{1,j}})\preceq
o_{\mathbb{P}}(1)$
$\displaystyle\frac{(\mathbf{S}_{11,4}^{(1)})_{jj}}{g_{2}d_{1}^{\alpha_{1,j}}}$
$\displaystyle=o_{\mathbb{P}}(d_{1}^{\alpha_{1,r_{1}}-\alpha_{1,j}})\preceq
o_{\mathbb{P}}(1),$
$\displaystyle\frac{(\mathbf{S}_{12}^{(1)}+\mathbf{S}_{21}^{(1)})_{jj}}{g_{2}d_{1}^{\alpha_{1,j}}}$
$\displaystyle=O_{\mathbb{P}}\left(d_{1}^{\frac{\alpha_{1,1}}{2}-\alpha_{1,j}}d_{2}^{-\frac{\alpha_{2,1}}{2}}\left[d_{2}+d_{1}/T\right]^{1/2}\right)\preceq
o_{\mathbb{P}}(1),$
$\displaystyle\frac{(\mathbf{S}_{22}^{(1)})_{jj}}{g_{2}d_{1}^{\alpha_{1,j}}}$
$\displaystyle=O_{\mathbb{P}}\left(d_{1}^{-\alpha_{1,j}}d_{2}^{-\alpha_{2,1}}(d_{2}+\frac{d_{1}}{T})\right)\preceq
o_{\mathbb{P}}(1),$

where the last two formulas follow from Assumption (M5). Finally, if $C\neq
1$, then the constant $C-1$ dominates $r_{s}^{(1)}$, and thus

$\displaystyle|\widehat{\alpha}_{1,j}-\alpha_{1,j}|=\left|\textnormal{log}\left(\frac{g_{2}\left[C+o_{\mathbb{P}}(1)\right]}{\widehat{g}_{2}}\right)\right|\textnormal{log}(d_{1})^{-1}=O_{\mathbb{P}}\left(\frac{\textnormal{log}(C)}{\textnormal{log}(d_{1})}\right)=O_{\mathbb{P}}\left(\frac{1}{\textnormal{log}(d_{1})}\right).$

If $C=1$, then we have $r_{s}^{(1)}=o_{\mathbb{P}}(1)$, thus
$\textnormal{log}(1+r_{s})=r_{s}+O(r_{s}^{2})$, and we have a faster
convergence rate of

$\displaystyle|\widehat{\alpha}_{1,j}-\alpha_{1,j}|$
$\displaystyle=O_{\mathbb{P}}\left(\frac{r_{s}^{(1)}}{\textnormal{log}(d_{1})}\right)$
$\displaystyle=o_{\mathbb{P}}\left(\frac{d_{1}^{\alpha_{1,r_{1}}-\alpha_{1,j}}}{\textnormal{log}(d_{1})}\right)+O_{\mathbb{P}}\left(\frac{d_{2}^{-\frac{\alpha_{2,1}}{2}}T^{-1/2}+d_{1}^{\frac{\alpha_{1,1}}{2}-\alpha_{1,j}}d_{2}^{-\frac{\alpha_{2,1}}{2}}\left[d_{2}+d_{1}/T\right]^{1/2}+d_{1}^{-\alpha_{1,j}}d_{2}^{-\alpha_{2,1}}(d_{2}+\frac{d_{1}}{T})}{\textnormal{log}(d_{1})}\right).$

This completes the proof of Theorem 3. $\square$