# Does Principal Component Analysis Preserve the Sparsity in Sparse Weak
Factor Models?

Jie Weia, Yonghui Zhangb
aSchool of Economics, Huazhong University of Science and Technology
bSchool of Economics, Renmin University of China The authors are grateful to
Simon Freyaldenhoven, Cheng Yong Tang, and Yoshimasa Uematsu for sharing their
codes with papers on their websites. They would like to thank Takashi Yamagata
for helpful discussions. An earlier version of this paper was circulated under
the title “Weak Factor Models with Sparse Factor Loadings”. Wei gratefully
acknowledges the financial support from the National Science Foundation of
China (No.71803054), the Program for HUST Academic Frontier Youth Team
(2017QYTD13), and the Fundamental Research Funds for the Central Universities
(HUST: 2022WKYXZX014). Zhang gratefully acknowledges the financial support
from National Natural Science Foundation of China (Projects No.71973141 and
No.71873033). All errors are the authors’ sole responsibilities. Address
correspondence to: Yonghui Zhang, School of Economics, Renmin University of
China. E-mail address<EMAIL_ADDRESS>
###### Abstract

This paper studies the principal component (PC) method-based estimation of
weak factor models with sparse loadings. We uncover an intrinsic near-sparsity
preservation property for the PC estimators of loadings, which comes from the
approximately upper triangular (block) structure of the rotation matrix. It
implies an asymmetric relationship among factors: the rotated loadings for a
stronger factor can be contaminated by those from a weaker one, but the
loadings for a weaker factor is almost free of the impact of those from a
stronger one. More importantly, the finding implies that there is no need to
use complicated penalties to sparsify the loading estimators. Instead, we
adopt a simple screening method to recover the sparsity and construct
estimators for various factor strengths. In addition, for sparse weak factor
models, we provide a singular value thresholding-based approach to determine
the number of factors and establish uniform convergence rates for PC
estimators, which complement Bai and Ng (2023). The accuracy and efficiency of
the proposed estimators are investigated via Monte Carlo simulations. The
application to the FRED-QD dataset reveals the underlying factor strengths and
loading sparsity as well as their dynamic features.

JEL Classification:  C12, C53, C55

Key Words: Factor number, Factor strength, Principal component, Rotation
matrix, Sparse factor loadings, Sparsity preservation, Weak factors

##  1 Introduction

Factor models have been widely used in economics and finance. Until very
recently, a large body of literature on factor models has built on the
assumption that all factors are strong or pervasive in the sense that each
factor can influence almost all cross-sectional units. Under this assumption,
Bai and Ng (2002) and Bai (2003) establish the asymptotics of principal
component (PC) estimators with $N$ units observed with $T$ periods. It is well
known that the space spanned by the factors can be estimated consistently by
the PC method with rate $\min\left(N^{1/2},T^{1/2}\right)$, and both the
estimated factors and factor loadings are consistent up to some rotation
matrices.

However, many empirical findings support the existence of weak factors with
sparse factor loadings. For instance, both Stock and Watson (2002) and
Ludvigson and Ng (2009) find that the extracted PC factors from a large set of
standardized macroeconomic variables can only fit a few variables’ time series
observations well, suggesting a sparsity structure in factor loadings. More
recently, Kristensen (2017), Freyaldenhoven (2022a), and Uematsu and Yamagata
(2023a, b) revisit the same dataset in Stock and Waston (2002) and further
confirm the findings on loadings. In addition, Uematsu and Yamagata (2023a)
find sparse factor loadings in the FRED-MD dataset (McCracken and Ng, 2016)
and in firm security excess returns of the S&P 500 index. Another motivation
for sparse factor loadings comes from the literature on hierarchical or group
factor models, where the local factors have nonzero factor loadings only for
some specific cross-sectional groups. For example, shocks due to oil supply
may only affect industrial production sectors, but not the industry of
service. In empirical asset pricing, it is expected that portfolios formulated
by size or value factor are influenced by such factors, but portfolios
formulated by the momentum factor may not be. For more examples of local/group
factors, see Ando and Bai (2017) and Choi et al. (2021).

Due to the aforementioned reasons, weak factor models have gained immense
popularity and attracted the attention of many researchers. For instance,
Onatski (2012) shows that the PC estimate is inconsistent when the true factor
loadings $\Lambda^{0}$ are weak enough such that
$\Lambda^{0\prime}\Lambda^{0}$ has a positive definite (p.d.) limit as $N$
approaches infinity. More recently, Uematsu and Yamagata (2023a, b) propose a
sparse orthogonal factor regression (SOFAR) estimator that uses $\ell_{1}$
regularization to achieve sparsity, and also provide a new inferential
procedure to determine whether each component of the factor loadings is zero
or not and prove that it controls false discovery rate (FDR) under a pre-
assigned level; Freyaldenhoven (2022a) proposes a new class of estimators to
determine the number of relevant factors in the presence of local factors, and
Guo et al. (2022) investigate information criteria with a new penalty to
determine the number of factors, which can be adaptive to the degree of factor
pervasiveness and favor more parsimonious models; Bai and Ng (2023)
investigate the large sample properties of static PC estimators for
approximate factor models with general weak factor loadings and show that the
PC estimators are still consistent and asymptotically normal; for factor-based
forecasting with large sets of variables, Chao et al. (2022), Chao and Swanson
(2022), and Giglio et al. (2022b) use pre-screening methods to select valid
predictors when factor pervasiveness does not hold.

In this paper, we delve into the study of weak factor models with sparse
factor loadings.111There is another type of weak factor models that involve
nonzero but small factor loadings, i.e., loadings shrinking towards zero as
sample size increases. However, for economic and financial applications,
sparsity-induced weak factor models are more relevant and the estimators with
sparsity tend to have meaningful interpretation. As such, this paper focuses
on weak factor models with sparse loadings. A factor is called “sparsely weak”
when it only affects a small subset of cross-sectional units. This is the same
as the “sparsity-induced” weak factor in Uematsu and Yamagata (2023a, b).
Instead of taking SOFAR of Uematsu and Yamagata (2023a), we follow Bai and Ng
(2023) to consider the PC estimation since it is much simpler and free of
tuning parameters. To recover the sparsity in factor loadings, we adopt a
simple screening on the PC estimator of factor loadings. Our computation
burden is almost negligible compared to the procedures with regularization in
Uematsu and Yamagata (2023a, b).

Under the sparsity assumption on factor loadings, we show that PC estimators
of factors, factor loadings, and the common components are still consistent,
and further establish their asymptotic distributions and uniform convergence
rates. In addition to the usual large sample properties of PC estimators, we
find that the rotation matrix in PC estimation has a special structure that
leads to nearly sparse PC estimates for factor loadings. Based on this
finding, we further sparsify the loading estimator exactly by screening the
estimated PC loadings. Then with the sparsified loading estimator, we can
straightforwardly estimate distinct factor strengths consistently. Finally, we
propose using singular value thresholding (SVT) to determine the number of
factors in sparse weak factor models.

In summary, this paper studies the properties of PC estimates for weak factor
models with sparse loadings and proposes a PC-based screening method to
sparsify the estimator for factor loadings. It has the following additional
contributions to the existing literature on weak factor models. First, under
our sparsity assumption on factor loadings with mixed sparsity degrees, we
establish the large sample properties of PC estimates for factors, factor
loadings, and common components, respectively. In particular, we provide the
uniform convergence rates for PC estimates, which complement Bai and Ng
(2023). Second, we show that the PC estimators for loadings are still sparse,
meaning the sparsity structure in factor loadings is nearly invariant to the
rotation matrix induced by the PC method. To the best of our knowledge, our
paper is the first to reveal this intrinsic sparsity preservation feature of
the PC estimator. This result is also in sharp contrast to the common sense
that the sparsity of factor loadings cannot be reserved due to the matrix
rotation in PC estimation; see Bailey et al. (2021, BKP hereafter) and
Freyaldenhoven (2022b). We show that the invariance of sparsity to rotation
comes from the particular structure of the rotation matrix $Q$, which is an
(approximately) upper triangular matrix. Third, given the invariance of
sparsity of factor loadings to PC rotation, we can easily recover the sparsity
of factor loadings by applying proper screening to the PC estimates of factor
loadings. In some scenarios, it is important and necessary to recover the
sparsity structure of factor loadings when the estimated factor loadings are
used as “generated regressors”. For example, ignoring the sparsity structure
of factor loadings will lead to an inconsistent estimator of risk premia in
the empirical asset pricing model of Giglio and Xiu (2021) and Giglio et al.
(2022a). Fourth, based on the sparse loading estimators, we provide
straightforward estimators for all factor strengths and establish their
consistency. This result improves BKP (2021) where only the strength of the
strongest factor is identified and estimated. Lastly, we make a contribution
to the determination of the number of factors in weak factor models. Research
on selecting the number of factors in weak factor models is still very
limited. We investigate the SVT approach and establish its consistency in
determining the factor number.

It is worth mentioning that our work is also related to the large literature
on sparsity pursuit in factor models. The sparse principal component analysis
(SPCA) pursues a sparse structure in the estimation of factor loadings to
achieve a good interpretation for extracted PC factors. In order to obtain PCs
with sparse factor loadings, different penalties or threshold techniques to
the fitted loadings are usually adopted. For example, Rapach and Zhou (2021)
estimate sparse PCs for 120 monthly macroeconomic variables from the FRED-MD
database by combining the $\ell_{1}$-norm restriction in factor loadings with
the PC objective function; Pelger and Xiong (2022) construct sparse proximate
factors by using the largest, say, $m$ eigenvalues, and show a weaker form of
correlation consistency. They do not focus on whether the factor model being
weak or not but investigate its sparse approximation, while our approach can
be cast as a generalized version of Pelger and Xiong (2022) in terms of
selecting $m$ adaptively to factor strengths. Freyaldenhoven (2022b) explores
possible rotations of estimated loadings to recover sparsity. Based on
consistently estimated factor loadings $\widehat{\Lambda}$, Freyaldenhoven
(2022b) suggests minimizing the $\ell_{0}$-norm of rotated loadings
($\widehat{\Lambda}R)$: $\min_{R}||\widehat{\Lambda}R||_{0}$, subject to the
rotation matrix $R$ being nonsingular, and implements solving the problem with
a more feasible $\ell_{1}$-norm. His basic idea is to search among different
rotations of the estimated loadings to achieve sparsity. If the true factor
loadings are indeed sparse, this approach may identify the sparsity pattern.
For other rotation-based methods of nearly-sparse factor loadings, see Despois
and Doz (2022).

We would also like to compare our paper with the recent one by Bai and Ng
(2023). Both of the two papers show the asymptotic properties of factors,
loadings and common components in weak factor models with possibly
heterogeneous factor strengths.222Bai and Ng (2023) obtain the average errors
in estimating the common components in the squared Frobenius norm only under
homogeneous factor strength, while we generalize the results under
heterogeneous strengths. In fact, to derive these theoretical results we are
inspired by the novel proof in Bai and Ng (2023), e.g., by making use of
several asymptotically equivalent rotation matrices. In spite of the
overlapped targets, there are certain fundamental differences. The setup of
Bai and Ng (2023) is more general allowing that the entries of $\Lambda^{0}$
are non-zero but small, whereas ours only admits sparsity induced weak factor
models. Besides, our requirement of the minimal factor strength for estimation
consistency, while still reasonable, is more stringent than Bai and Ng’s. We
impose the stronger conditions in order to show that the rotation matrix is
upper triangular. The upper triangularity reveals the sparsity preservation
properties by PC estimators, which is our ultimate interest and yet not
pursued by Bai and Ng (2023). Moreover, convergence rates of PC estimators may
also be improved due to the upper triangularity. We also establish uniform
convergence rates of these estimators, propose additional estimators for
factor strengths and number of factors and prove their consistency in our
setup. These questions are not formally addressed in Bai and Ng (2023), and
yet may be of independent interest.

The rest of this paper is organized as follows. We formally introduce weak
factor models with sparse factor loadings and the PC estimation in Section 2\.
In Section 3, we study the asymptotics of PC estimators for factors, factor
loadings and the common components under the weak factor scheme, including
consistency, asymptotic distributions, and uniform convergence rates. Based on
the finding on the rotation matrix in Section 3, we propose a screening method
to sparsify the estimated factor loadings and provide consistent estimators
for factor strengths in Section 4. Section 5 discusses the determination of
the number of factors for weak factor models. The Monte Carlo simulation and
empirical application are reported in Section 6 and 7, respectively. Section 8
concludes. All technical proofs are relegated in the Appendix.

NOTATIONS. For a set $A$, let $|A|$ be its cardinality. Let $a\vee
b=\max\left\\{a,b\right\\}$ and $a\wedge b=\min\left\\{a,b\right\\}$ for two
real numbers $a$ and $b$. For two sequences $\left\\{a_{n}\right\\}$ and
$\left\\{b_{n}\right\\}$, $a_{n}\lesssim b_{n}$ denotes $a_{n}/b_{n}$ is
bounded and $a_{n}\asymp b_{n}$ denotes that both $a_{n}/b_{n}$ and
$b_{n}/a_{n}$ are bounded. For two random sequences $\left\\{a_{n}\right\\}$
and $\left\\{b_{n}\right\\},$ $a_{n}\lesssim_{p}b_{n}$ denotes $a_{n}/b_{n}\
$is stochastically bounded and $a_{n}\asymp_{p}b_{n}$ if
$a_{n}\lesssim_{p}b_{n}$ and $b_{n}\lesssim_{p}a_{n}.$ For a square matrix
$A$, $\mu_{\min}\left(A\right)$ and $\mu_{\max}\left(A\right)$ are the
smallest and largest eigenvalue of $A$, respectively. For a matrix $B,$ let
$\left\|B\right\|=\left[\text{Tr}\left(BB^{\prime}\right)\right]^{1/2}$ be its
Frobenius norm and
$\left\|B\right\|_{sp}=\mu_{\max}^{1/2}\left(B^{\prime}B\right)$ be its
spectral norm. For a $n\times 1$ vector $a$, its $\ell_{0}$-norm is
$\left\|a\right\|_{0}=\sum_{j=1}^{n}\mathbf{1}\left(a_{j}\neq 0\right)$, where
$\mathbf{1}\left(\cdot\right)$ is the usual indicator function. Throughout, we
use the running indices $s$ and $t$ for the $T$ observations, indices $i$, $j$
for the $N$ cross-sectional units, and $k$ and $l$ for the $r$ factors.

##  2 The Weak Factor Models with Sparse Loadings

###  2.1 The model setup

To fix ideas, let us consider a static factor model with the following
representation

$X_{it}=\lambda_{i}^{\prime}F_{t}+e_{it},$ (2.1)

$i=1,\ldots,N$, $t=1,\ldots,T,$ where $X_{it}$ is the observed data for the
$i$th individual at time $t$, $F_{t}$ is a $r\times 1$ vector of latent
factors, $\lambda_{i}$ is the corresponding $r\times 1$ vector of factor
loadings, and $e_{it}$ is the idiosyncratic error with possible weak
dependence across $i$ or/and over $t$. Or we may write the factor model in a
matrix form as

$X=\Lambda F^{\prime}+e,$ (2.2)

where $X$ and $e$ are both $N\times T$ matrices with $X(i,t)=X_{it}$ and
$e(i,t)=e_{it}$, respectively,
$\Lambda=\left(\lambda_{1},\ldots,\lambda_{N}\right)^{\prime}$, and
$F=\left(F_{1},\ldots,F_{T}\right)^{\prime}$. Also for later reference, denote
the $k$th column of $\Lambda$ by
$\Lambda_{\cdot,k}=\left(\Lambda_{1,k},\ldots,\Lambda_{N,k}\right)^{\prime}$
and the $k$th column of $F$ by
$F_{\cdot,k}=\left(F_{1,k},\ldots,F_{T,k}\right)^{\prime}$, $k=1,\ldots,r$.
Throughout the paper, we let $F_{t}^{0}$ be the $r\times 1$ vector of true
factors and $\lambda_{i}^{0}$ be the true loadings, with $F^{0}$ and
$\Lambda^{0}$ being the corresponding matrices.

In a standard factor model, it is typically assumed that all factors are
strong in the sense that the $r\times r$ matrix
$\Lambda^{0\prime}\Lambda^{0}/N$ is a full rank matrix for a sufficiently
large $N$ and it has a positive definite (p.d.) limit. In contrast, we allow
that there are both strong and weak factors. A factor is (sparsely) weak when
it is not persuasive in the sense that the proportion of its non-zero factor
loadings diminishes at a certain rate. The number (in magnitude) of
individuals effectively affected by the $k$th factor $F_{\cdot,k}^{0}$ is of
order $N^{\alpha_{k}},$ $\alpha_{k}\in\left[0,1\right]$, and the strength of
this factor can be represented by the degree parameter $\alpha_{k}$ (BKP,
2021) such that

$\Lambda_{\cdot,k}^{0\prime}\Lambda_{\cdot,k}^{0}\asymp
N^{\alpha_{k}},\;k=1,\ldots,r.$ (2.3)

Obviously, a strong factor has degree $\alpha_{k}=1$. As BKP (2021) point out,
for a factor with an extremely weak signal with $\alpha_{k}\in(0,1/2)$, it is
not identified without priori restrictions and not relevant in most financial
and macroeconomic applications. Moreover, Freyaldenhoven (2022a) discusses the
reasons why only factors affecting proportionally more than $\sqrt{N}$ of
individuals are relevant in arbitrage pricing theory (APT) and aggregate
fluctuations in macroeconomics. Hence we restrict the factor strength so that
$\alpha_{k}\in[\underline{\alpha},1]$ for $k=1,\ldots,r$ with some
$\underline{\alpha}>1/2$.333The factors with the strength parameter bigger
than $1/2$ are named as “semi-strong” factors in BKP (2021). It may be also
worth mentioning that the strength degree $\alpha_{k}$ defined in (2.3) is
$N$-dependent (we suppress $N$ here) and thus may vary with samples.444In this
paper, we aim at estimating $\alpha_{k}$ consistently, but not inferring on
it.

Our setup allows that some or even all of factors in $F^{0}$ are weak in terms
of potential sparsity of $\Lambda^{0}$. In such a model with latent factors, a
standard normalization which one might want to impose is
$T^{-1}F^{0\prime}F^{0}=I_{r}$ as in Uematsu and Yamagata (2023a) and
Freyaldenhoven (2022a). However, this identification restriction is no longer
innocuous as it is not compatible with sparsity of $\Lambda^{0}$ in general.
In other words, a weak factor model (with a particular sparsity structure of
$\Lambda^{0}$) is not invariant to an arbitrary rotation transformation. This
insight is also shared by Bai and Ng (2023, Section 5.2) in terms of an
overidentification restriction on $F^{0}$, implying that the aforementioned
methods would incur a problem when the restriction fails to hold in general,
as verified in our Monte Carlo simulations. Since our proposed methods hinge
on the particular structure of the underlying sparsity, we leave
$T^{-1}F^{0\prime}F^{0}$ unspecified ex ante for coherence (except for its
being p.d.). It is also worth mentioning that the major findings for the weak
factor model in this paper can be readily applied to deal with a rank-
deficient model, as a rank-deficient model is equivalent to a weak factor
model up to a rotation transformation; see Giglio et al. (2022a).

###  2.2 PC estimation

To begin with, we assume that the true number of factors $r$ is known, and
leave the determination of $r$ in Section 5. The estimation of factors and
their loadings is via the method of PC in minimizing

$\underset{\Lambda,F}{\min}\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}\left(X_{it}-\lambda_{i}^{\prime}F_{t}\right)^{2},$

subject to the usual identification restriction that $F^{\prime}F/T=I_{r}$ and
$\Lambda^{\prime}\Lambda$ being diagonal.555Since it is known that the PC
estimator $\widetilde{F}$ is identified up to a full rank rotation, the
identification restriction $F^{\prime}F/T=I_{r}$ is only employed to pin down
$\widetilde{F}$ for the purpose of estimation, and should not be understood as
a restriction imposed on the underlying true factors $F^{0}$ in the weak
factor model. The estimated factors, denoted by $\widetilde{F}$, is $\sqrt{T}$
times the eigenvectors corresponding to the $r$ largest eigenvalues of the
$T\times T$ matrix $\frac{X^{\prime}X}{NT}$ in decreasing order. Then
$\widetilde{\Lambda}=X\widetilde{F}/T$, and
$\widetilde{e}=X-\widetilde{\Lambda}\widetilde{F}^{\prime}$. Also, let
$\widetilde{V}=\operatorname{diag}\left(\widetilde{V}_{1},\ldots,\widetilde{V}_{r}\right)$
be the $r\times r$ diagonal matrix consisting of the $r$ largest eigenvalues
of $\frac{X^{\prime}X}{NT}$ in decreasing order. We also define the common
component estimator $\widetilde{C}=\widetilde{\Lambda}\widetilde{F}^{\prime}$
as the estimator for $C^{0}=\Lambda^{0}F^{0\prime}.$

Without loss of generality, we assume that the degrees of factor strength are
arranged such that
$1\geq\alpha_{1}\geq\ldots\geq\alpha_{r}\geq\underline{\alpha}$. Given the
possibility that two factors may have the same degree of strength, e.g., there
may be two strong factors or two weak ones with an equal strength, we can
distribute $\alpha_{j}$’s into $G$ groups so that

$\displaystyle\alpha^{[1]}$
$\displaystyle=\left\\{\alpha_{1},\ldots,\alpha_{m_{1}}:\alpha_{1}=\cdots=\alpha_{m_{1}}\right\\},$
$\displaystyle\text{ }\vdots$ $\displaystyle\alpha^{[G]}$
$\displaystyle=\left\\{\alpha_{m_{G-1}+1},\ldots,\alpha_{r}:\alpha_{m_{G-1}+1}=\cdots=\alpha_{r}\right\\},$

with $G\leq r$, and $G=r$ when every factor’s strength is unique. For ease of
notation, let us also define the cardinality of $\alpha^{[g]}$ by
$r_{g}\equiv\lvert\alpha^{[g]}\rvert,$ for $g=1,\ldots,G.$ Clearly,
$r=\sum_{g=1}^{G}r_{g}$. In addition, let $\alpha_{[j]}$ be the same strength
shared in group $g$, i.e., $\alpha_{[g]}=\alpha_{m_{g}}$ for $g=1,\ldots,G.$
It is obvious that $\alpha_{1}=\alpha_{[1]}.$

##  3 Large Sample Properties for PC Estimators

###  3.1 Main assumptions and a key result on the eigenvalue matrix

We first define a scale matrix $A$ as follows,

$A=\operatorname{diag}\left(N^{\alpha_{1}},\ldots,N^{\alpha_{r}}\right),$

which will be frequently used in this paper. To study the large sample
properties of PC estimators, we make the following assumptions on the weak
factor models.

Assumption 1. $E\|F_{t}^{0}\|^{4}<\infty$ for $t=1,...,T$ and
$T^{-1}F^{0\prime}F^{0}\rightarrow_{p}\Sigma_{F}$ as $T\rightarrow\infty$ for
some p.d. matrix $\Sigma_{F}$.

Assumption 2. For the factor loadings,

(i)
$0<\underline{\lambda}\leq\left\|\lambda_{i}^{0}\right\|\leq\overline{\lambda}<\infty$
for all $i$’s and some constants $\underline{\lambda}$ and
$\overline{\lambda}$;

(ii) For the $k$th factor, the number of non-zero factor loading is
$n_{k}\equiv\left\|\Lambda_{\cdot,k}^{0}\right\|_{0}\asymp N^{\alpha_{k}}$ for
$\alpha_{k}\in[\underline{\alpha},1]$ with $\underline{\alpha}>1/2,$ for
$k=1,\ldots,r;$

(iii)
$A^{-1/2}\Lambda^{0\prime}\Lambda^{0}A^{-1/2}\rightarrow\Sigma_{\Lambda}^{\ast}$
as $N\rightarrow\infty$ for some p.d. matrix $\Sigma_{\Lambda}^{\ast}$.

Assumption 3. There exists a positive constant $M\leq\infty$, such that for
all $N$ and $T$,

(i) $E(e_{it})=0$ and $E(e_{it}^{2})<M$ for all $i$ and $t;$

(ii) $E(e_{s}^{\prime}e_{t}/N)=\gamma_{N}(s,t)$,
$\left|\gamma_{N}(s,s)\right|\leq M$ for all $s$, and
$T^{-1}\sum_{s=1}^{T}\sum_{t=1}^{T}\left|\gamma_{N}(s,t)\right|\leq M$;

(iii) $E(e_{it}e_{jt})=\tau_{ij,t}$ with
$\left|\tau_{ij,t}\right|\leq\tau_{ij}$ for all $t$ with some $\tau_{ij}>0$.
In addition, for $\forall\mathcal{S}\subset\\{1,\ldots,N\\}$,
$\sum_{j=1}^{N}\tau_{ij}\mathbf{1}\left(j\in\mathcal{S}\right)\leq M$ and
$\left|\mathcal{S}\right|^{-1/2}T^{-1/2}\sum_{j=1}^{N}\sum_{t=1}^{T}\left(e_{it}e_{jt}-\tau_{ij,t}\right)\mathbf{1}\left(j\in\mathcal{S}\right)=O_{p}\left(1\right);$

(iv) For every $(t,s)$,
$E\left|N^{-1/2}\sum_{i=1}^{N}\left[e_{is}e_{it}-E(e_{is}e_{it})\right]\right|^{4}\leq
M$;

(v)
$\left(n_{k}T\right)^{-1/2}\sum_{i=1}^{N}\sum_{t=1}^{T}F_{t}^{0}\lambda_{ik}^{0}e_{it}=O_{p}(1)$
for $k=1,\ldots,r$;

(vi) Define $\gamma_{N,F}(s,t)=N^{-1}E(F_{t}^{0}e_{t}^{\prime}e_{s})$ and
$\varpi(s)=\left(NT\right)^{-1/2}\sum_{i=1}^{N}\sum_{t=1}^{T}\left[F_{t}^{0}e_{it}e_{is}-E(F_{t}^{0}e_{it}e_{is})\right]$.
$\max_{s}\sum_{t=1}^{T}\left\|\gamma_{N,F}(s,t)\right\|^{2}\leq M$ and
$\max_{s}E\left\|\varpi(s)\right\|^{2}\leq M$;

(vii)
$E\left(N^{-1}\sum_{i=1}^{N}\left\|T^{-1/2}\sum_{t=1}^{T}F_{t}^{0}e_{it}\right\|^{2}\right)\leq
M$;

(viii) The eigenvalues of $\Sigma_{\Lambda}^{\ast}\Sigma_{F}$ are distinct.

Assumption 4.
$\left\|e\right\|_{sp}^{2}=O_{p}\left(\max\left\\{N,T\right\\}\right).$

Assumptions 1-2 are standard in the literature on panel factor models.
Assumption 1 imposes a moment condition on the true factors and requires the
existence of a p.d. probability limit of $T^{-1}F^{0\prime}F^{0}$. Assumption
2 imposes a boundedness condition on the norm of factor loading vectors, and
specifies the sparse structure in factor loadings. The deterministic factor
loadings can be relaxed to be stochastic with some moment conditions.
Assumption 3 provides some conditions on factors, factor loadings, and errors.
3(i)-(iii) impose moment conditions on errors and allow for weak cross-
sectional/serial dependence as Bai (2003). Note that Assumption 3(iii) is
weaker than Bai (2003) and thus generalizes the counterpart under a strong
factor model. Assumption 3(vi) is used by Su and Wang (2017), and it is not
redundant since we do not assume $F_{t}^{0}$ and $e_{s}$ are independent.
Assumption 4 is also adopted by Bai and Ng (2002, 2023) and Moon and Weidner
(2017). It surely holds for independently identically distributed (iid) data
with uniformly bounded $4$th moments, and may also hold for weakly dependent
data across $i$ and $t.$

We first present one of the key interesting results for weak factor models via
PC: the eigenvalue matrix $\widetilde{V}$ preserves the magnitude of factor
strength, as stated in Proposition 3.1 below.

###### Proposition 3.1

Under Assumptions 1-4, we have $\widetilde{V}_{k}\asymp_{p}N^{\alpha_{k}-1}$
for $k=1,\ldots,r.$

Remark 1. Unlike in a strong factor model, the diagonal elements of matrix
$\widetilde{V}$ vanish at various rates determined by their corresponding
factor strengths. It raises more challenges to our asymptotic theory later
involving $\widetilde{V}^{-1}$ frequently, whereas in a strong factor model
all these terms are $O_{p}(1)$.

###  3.2 Consistency and asymptotic distributions

In this subsection, we establish the consistency and asymptotic distributions
for PC estimators. To reveal the asymptotic properties of our estimators, it
is necessary to introduce several rotation matrices as below. We have
suppressed the dependence on sample sizes for these matrices to ease burden of
notations. Define

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{\Lambda^{0\prime}\Lambda^{0}}{N}\frac{F^{0\prime}\widetilde{F}}{T}\widetilde{V}^{-1},\text{
}H_{1}=\Lambda^{0\prime}\Lambda^{0}(\widetilde{\Lambda}^{\prime}\Lambda^{0})^{-1},\text{
}H_{2}=\left(F^{0\prime}F^{0}\right)^{-1}F^{0\prime}\widetilde{F},\text{ }$
$\displaystyle H_{3}$ $\displaystyle=$ $\displaystyle Q^{-1}\text{ with\
}Q=\frac{\widetilde{F}^{\prime}F^{0}}{T},\text{ and
}H_{4}=\Lambda^{0\prime}\widetilde{\Lambda}(\widetilde{\Lambda}^{\prime}\widetilde{\Lambda})^{-1}.$

We will show the asymptotic equivalence of the matrices $H$ and $H_{j}$ for
$j=1,2,3,4,$ which also generalizes Lemma 3 of Bai and Ng (2023). These
rotation matrices and the equivalence results will play an indispensable role
in establishing the convergence rates and the asymptotic distributions of our
PC estimators. Before doing so, we would like to first introduce an
interesting finding related to the matrix $Q$ defined above. One more
additional condition is needed.

Assumption 5. $N^{1-\alpha_{r}}/T^{1/2}\rightarrow 0$.

It is worth mentioning that Bai and Ng (2023) impose a weaker condition
$N^{1-\alpha_{r}}/T$ $\rightarrow 0$ for consistency. They do not impose
$N^{1-\alpha_{r}}/T^{1/2}\rightarrow 0$ until in proving distributional
theory. However, we impose Assumption 5 at an early stage in this paper to
obtain our main result stated in Proposition 3.2, which is fundamental in
sparsity recovery results in Section 4. Besides, a sharper result may be
obtained for consistency rate, e.g., Proposition 3.4, by making use of
Proposition 3.2 valid under Assumption 5. Also note that under $N/T\rightarrow
c\in\left(0,\infty\right)$ as assumed by Freyaldenhoven (2022a), Assumption 5
is trivial.

###### Proposition 3.2

Under Assumptions 1-5, the $r\times r$ matrix
$Q=T^{-1}\widetilde{F}^{\prime}F^{0}$ is a full rank matrix in probability and

$Q_{lk}\equiv Q\left(l,k\right)\asymp_{p}N^{\alpha_{l}-\alpha_{k}}\text{ for
}1\leq k\leq l\leq r.$

Remark 2. Proposition 3.2 implies a very important property of $Q$ serving as
the rotation matrix: $Q$ is (block) upper triangular asymptotically. Such a
property is very helpful in working with the sparsity-induced weak factor
models.

Next, we state the convergence rate of $\widetilde{F}$ in the following
proposition.

###### Proposition 3.3

Under Assumptions 1-5,

$\frac{1}{T}\left\|\widetilde{F}-F^{0}H\right\|^{2}=O_{p}\left(N^{\alpha_{1}-2\alpha_{r}}+N^{2\left(1-\alpha_{r}\right)}T^{-2}\right)=o_{p}\left(1\right).$

Remark 3. The associated convergence rate for Proposition 3.3 from Bai and Ng
(2023, Proposition 6.i) is
$O_{p}\left(N^{-\alpha_{r}}+N^{2\left(1-\alpha_{r}\right)}T^{-2}\right)=o_{p}\left(1\right)$,
which is a bit better than ours. The reason is that they use a slightly
different rotation matrix from ours $H$, as indicated by the comment below
their Proposition 6. The reason behind our choice of the rotation matrix $H$
is that it is more closely related to $Q$ playing a major role in our later
analysis.

The convergence rate for factor loading estimate is provided in the next
proposition.

###### Proposition 3.4

Under Assumptions 1-5,

$\frac{1}{N}\left\|\widetilde{\Lambda}-\Lambda^{0}Q^{\prime}\right\|^{2}=O_{p}\left(N^{-\alpha_{r}-1}+T^{-1}\right)=o_{p}\left(1\right).$

Remark 4. Note that our rate in Proposition 3.4 is better than what is stated
in Bai and Ng (2023, Proposition 6.ii). The reason is that we utilize the
upper triangularity of rotation $Q$ in our proof, which helps sharpen the
bound for estimation errors.

Before proceeding, we present here the equivalence results of $H$ and $H_{j}$
for $j=1,2,3,4.$ Define
$\gamma_{NT}\equiv\max\left(N^{\frac{\alpha_{1}}{2}-\alpha_{r}}T^{-1/2},N^{1-\alpha_{r}}T^{-1},N^{-\alpha_{r}}\right).$

###### Lemma 3.5

Under Assumptions 1-5,

$H_{j}=H+O_{p}\left(\gamma_{NT}\right),\text{ for }j=1,2,3,4.$

Recall that $H_{3}=Q^{-1},$ and Lemma 3.5 thus implies that
$Q^{-1}=H+O_{p}\left(\gamma_{NT}\right).$ This equivalence immediately implies
the consistent estimation of the common component $\widetilde{C}.$ Lemma 3.5
generalizes Lemma 1 of Bai and Ng (2019) for strong factor models, and Lemma 3
of Bai and Ng (2023) for weak factor models with a single strength.

###### Proposition 3.6

Under Assumptions 1-5,

$\frac{1}{NT}\left\|\widetilde{C}-C^{0}\right\|^{2}=O_{p}\left(N^{2\left(\alpha_{1}-\alpha_{r}\right)-1}+T^{-1}\right)=o_{p}\left(1\right).$

Remark 5.  It is noted in Proposition 3.6 that the convergence rate of
$\widetilde{C}$ in general depends on weak factor strengths, or precisely, the
discrepancy of strengths between the strongest and weakest factors. It is only
when all factor strengths are the same that the convergence rate is maximal
and coincides with that under strong factor models. Also note that this result
agrees with Bai and Ng (2023, Proposition 3) in considering homogeneous factor
strength.

Now we turn to the asymptotic distributions of PC estimators. We impose the
following assumption for establishing the central limit theorem (CLT), which
is adaptive to weak factor models.

Assumption 6. The following hold for each $i$ and $t$ as
$N,T\rightarrow\infty:$

(i)
$A^{-1/2}\Lambda^{0\prime}e_{t}\overset{d}{\rightarrow}N\left(0,\Gamma_{t}\right),$
where
$\Gamma_{t}=\lim_{N\rightarrow\infty}A^{-1/2}\sum_{i=1}^{N}\sum_{j=1}^{N}\lambda_{i}^{0}\lambda_{j}^{0\prime}E\left(e_{it}e_{jt}\right)A^{-1/2}.$

(ii)
$T^{-1/2}\sum_{t=1}^{T}F_{t}^{0}e_{it}\overset{d}{\rightarrow}N\left(0,\Phi_{i}\right),$where
$\Phi_{i}=\lim_{T\rightarrow\infty}T^{-1}\sum_{s=1}^{T}\sum_{t=1}^{T}E\left(F_{t}^{0}F_{s}^{0\prime}e_{it}e_{is}\right).$

One more regularity condition is imposed on sample size ($N,T$) and the
weakest factor strength $\alpha_{r}$ to guarantee the distributional theory,
which is also used by Bai and Ng (2023, Assumption C’(iv)).

Assumption 7.  $N^{\frac{3}{2}-\alpha_{r}}T^{-1}\rightarrow 0$.

###### Theorem 3.7

Under Assumptions 1-7,

$A^{1/2}\left(\widetilde{F}_{t}-H_{4}F_{t}^{0}\right)\overset{d}{\rightarrow}N\left(0,\Psi^{0}\Gamma_{t}\Psi^{0\prime}\right),$

where
$\Psi^{0}=\mathop{\mathrm{p}lim}_{N,T\rightarrow\infty}N^{-1}A^{1/2}\widetilde{V}^{-1}QA^{1/2}.$

Remark 6. (i) As shown in the proof of Theorem 3.7, the matrix$\ \Psi^{0}$ is
(block) diagonal. In particular, if no two factors have the same strength,
i.e., $\alpha_{k}\neq\alpha_{l}$ for $k\neq l,$ then $\Psi^{0}$ is an exactly
diagonal matrix so that $\Psi^{0\prime}=\Psi^{0}.$ (ii) Theorem 3.7 also
reveals that the $k$th factor $\widetilde{F}_{tk}$ is asymptotically normally
distributed with convergence rate of $N^{\alpha_{k}/2}.$

Assumption 8. $\sqrt{T}N^{-\alpha_{r}}\rightarrow 0$.

###### Theorem 3.8

Under Assumptions 1-8,

$\sqrt{T}\left(\widetilde{\lambda}_{i}-Q\lambda_{i}^{0}\right)\overset{d}{\rightarrow}N\left(0,Q^{\prime-1}\Phi_{i}Q^{-1}\right).$

Remark 7. Theorem 3.8 reveals that the asymptotic distribution of
$\widetilde{\lambda}_{i}$ is the same under both strong and weak factors and
invariant to factor strengths.

With Theorems 3.7 and 3.8, we come to the limiting distribution of
$\widetilde{C}_{it}.$ To this end, recall $r_{G}$ from Section 2.2, and define
the $r\times r$ matrix
$S^{\dagger}=\operatorname{diag}(\underbrace{0,\ldots,0}_{r-r_{G}},\underbrace{1\ldots,1}_{r_{G}})$.

###### Theorem 3.9

Under Assumptions 1-8,

$\frac{\widetilde{C}_{it}-C_{it}^{0}}{\sqrt{N^{-\alpha_{r}}V_{it}+T^{-1}U_{it}}}\overset{d}{\rightarrow}N\left(0,1\right),$

where
$V_{it}=\lambda_{i}^{0\prime}S^{\dagger}\Sigma_{\Lambda}^{\ast-1}\Gamma_{t}\Sigma_{\Lambda}^{\ast-1}S^{\dagger}\lambda_{i}^{0}$
and $U_{it}=F_{t}^{0\prime}\Sigma_{F}^{-1}\Phi_{i}\Sigma_{F}^{-1}F_{t}^{0}.$

Remark 8. Theorem 3.9 thus implies that
$\widetilde{C}_{it}-C_{it}^{0}=O_{p}\left(\max\left\\{N^{-\alpha_{r}/2},T^{-1/2}\right\\}\right).$
As for the asymptotic covariance matrix for $\widetilde{F}_{t}$
($\widetilde{\lambda}_{i}),$ Bai and Ng (2023) have proposed consistent
estimators assuming cross-section (serial) independence for
$\left\\{e_{it}\right\\}.$ Under weakly serial dependence, Bai (2003) proposes
a consistent Newey-West HAC estimator for the asymptotic covariance of
$\widetilde{\lambda}_{i}.$ For estimating the asymptotic covariance of
$\widetilde{F}_{t}$ under weakly cross-section (CS) independence, Bai and Ng
(2006) propose a consistent CS-HAC estimator under covariance stationarity
with $E\left(e_{it}e_{jt}\right)=\sigma_{ij}$ for all $t$’s. One could follow
the aforementioned approaches to formulate consistent estimators of the
asymptotic covariance matrices for $\widetilde{F}_{t}$ and
$\widetilde{\lambda}_{i},$ which would lead to a consistent variance estimator
for $\widetilde{C}_{it}$. For hypothesis testing, with $\widetilde{F}_{t}$ for
instance, there is no need to know factor strengths, as the feasible estimator
for the variance of $\widetilde{F}_{t}$ automatically accommodates factor
strengths, and is thus adequate for such a purpose.666See the discussion under
Proposition 4 of Bai and Ng (2023) for the case with a homogeneous factor
strength, and one can easily show that the argument also works under
heterogeneous factor strength.

###  3.3 Uniform convergence rates

In this section, we will establish uniform convergence rate results for
$\widetilde{F}_{t}$, $\widetilde{\lambda}_{i},$ and $\widetilde{C}_{it}$ over
$i$ or (and) $t.$ These results can be exploited in recovering model sparsity
in Section 4, in the potential application of factor-augmented forecast
regression, and are perhaps also of independent interest.

Given that dependence is allowed across both $i$ and $t$, we first define a
strong mixing condition, generalized over $i$ and $t,$ similar to Ma et al.
(2021). Suppose that there is some labeling of the cross-sectional units
$i_{l_{1}},\ldots,i_{l_{N}},$ whose generic index we denote by $i^{\ast},$
such that the CS dependence decays with distance
$\left|i^{\ast}-j^{\ast}\right|.$ Then we will define a mixing rate applied
for the random field $\left\\{G_{i^{\ast}t}:1\leq i^{\ast}\leq N,1\leq t\leq
T\right\\},$ where
$G_{i^{\ast}t}\equiv\left(F_{t}^{0\prime},e_{i^{\ast}t}\right)^{\prime}.$ For
$S_{1},S_{2}\subset\left[1,\ldots,N\right]\times\left[1,\ldots,T\right],$ let

$\alpha\left(S_{1},S_{2}\right)\equiv\sup\left\\{\left|P\left(A\right)P\left(B\right)-P\left(A\cap
B\right)\right|:A\in\sigma\left(G_{i^{\ast}t},\left(i^{\ast},t\right)\in
S_{1}\right),B\in\sigma\left(G_{i^{\ast}t},\left(i^{\ast},t\right)\in
S_{2}\right)\right\\},$

where $\sigma\left(\cdot\right)$ denotes a sigma-field. Then the
$\alpha$-mixing coefficient of $\left\\{G_{i^{\ast}t}\right\\}$ is defined as

$\alpha\left(k\right)\equiv\sup\left\\{\alpha\left(S_{1},S_{2}\right):d\left(S_{1},S_{2}\right)\geq
k\right\\},$

where
$d\left(S_{1},S_{2}\right)\equiv\min\left\\{\sqrt{\left(t-s\right)^{2}+\left(i^{\ast}-j^{\ast}\right)^{2}}:\left(i^{\ast},t\right)\in
S_{1},\left(j^{\ast},s\right)\in S_{2}\right\\}.$

The definition of $\alpha\left(k\right)$ generalizes the usual one in the time
series context. In particular, when $\alpha\left(k\right)$ is applied to to a
(single or vector of) time series, it coincides with the usual one defined by,
e.g., Fan et al. (2011). (ii) For the purpose of estimation, we do not need to
know the true labeling $\left\\{i^{\ast}\right\\}.$ Ma et al. (2021) show that
their inference is valid as long as the number of mis-assigned indices is
$o\left(N^{1/2}\right).$ In conducting inference, our approach is effective
with the true labeling $\left\\{i^{\ast}\right\\}$ completely unknown, and
thus further relaxes the assumption by Ma et al. (2021).

We now specify the additional assumptions for the uniform results.

Assumption 9. (i) $\left\\{e_{t}\right\\}_{t\geq 1}$ and
$\left\\{F_{t}^{0}\right\\}_{t\geq 1}$ are both stationary and ergodic;

(ii) There exists $s_{2}>0$ and $K>0$ such that $\forall t\in\mathbb{Z}^{+},$
$\alpha\left(t\right)\leq\exp\left(-Kt^{s_{2}}\right);$

(iii) There exists $s_{1}>0,$ $b_{1}>0,$ $s_{3}>0,$ $b_{2}>0,$ satisfying
$3s_{1}^{-1}+s_{2}^{-1}>1$ and $3s_{3}^{-1}+s_{2}^{-1}>1$, such that $\forall
w>0,$
$P\left(\left|e_{it}\right|>w\right)\leq\exp\left(-\left(w/b_{1}\right)^{s_{1}}\right)$
and
$P\left(\left|F_{tk}^{0}\right|>w\right)\leq\exp\left(-\left(w/b_{2}\right)^{s_{3}}\right)$
for $k=1,...,r;$

(iv) $\left(\ln N\right)^{2/\gamma_{2}-1}=o\left(T\right),$ where
$\gamma_{2}=1.5s_{1}^{-1}+1.5s_{3}^{-1}+s_{2}^{-1}.$

From the previous section, we see that the PC estimators $\widetilde{F}_{t}$
and $\widetilde{\lambda}_{i}$ are both consistent up to a certain rotation
matrix. So to better state the uniform convergence result, we define the
rotated factor and factor loading by $\lambda_{i}^{\ast}\equiv
Q\lambda_{i}^{0}$ and $F_{t}^{\ast}=H_{4}^{\prime}F_{t}^{0},$ respectively.

###### Theorem 3.10

Under Assumptions 1-9,

(i)
$\sup_{i}\left|\widetilde{\lambda}_{ik}-\lambda_{ik}^{\ast}\right|=O_{p}\left(\sqrt{(\ln
N)/T}\right)$ for $k=1,...,r;$

(ii)
$\sup_{t}\left|\widetilde{F}_{tk}-F_{tk}^{\ast}\right|=O_{p}\left(N^{-\alpha_{k}/2}\sqrt{\ln
T}\right)$ for $k=1,...,r;$

(iii) $\sup_{i,t}\left|C_{it}-C_{it}^{0}\right|=O_{p}\left(\left(\ln
T\right)^{1/s_{3}}\sqrt{(\ln N)/T}+N^{-\alpha_{r}/2}\sqrt{\ln T}\right).$

Remark 9. In Theorem 3.10, result (i) implies that the estimation errors for
factor loadings are uniformly dominated by $\sqrt{(\ln N)/T}$. This result is
very useful to explore loading sparsity and factor strength in the next
section. For each factor, result (ii) provides the uniform convergence rate
which depends on the factor strength. Result (iii) establishes the uniform
convergence rate for the common component estimators and the rate is
determined by the smallest factor strength $\alpha_{r}$ and a parameter
$s_{3}$ which controls the probability tail bound of factors.

##  4 Revelation of the Sparsity in Loadings with PC Estimators

It is well known that latent factor models are subject to a rotational
indeterminacy. This identification issue is unwanted to reveal the loading
sparsity structure. Specifically, our results in Section 2 indicates that
loadings are identified only up to the rotation $Q$, and it is believed that a
rotation in general plays a deterrent role in revealing sparsity. As a result,
interpretation of estimated factors becomes intimidating, as it is usually
drawn in a manner by relating to observables which an individual factor loads
on primarily. For example, Ludvigson and Ng (2009) write,

“Moreover, we caution that any labeling of the factors is imperfect, because
each is influenced to some degree by all the variables in our large dataset
and the orthogonalization means that no one of them will correspond exactly to
a precise economic concept like output or unemployment, which are naturally
correlated.”

However, we show that even in the presence of the rotation $Q$ brought in by
PC, the sparsity in factor loadings can still be preserved. This justifies the
ad hoc manner of interpretation of PC factors with only a small set of
variables or individuals in many empirical applications; see Pelger and Xiong
(2022).

###  4.1 Preservation of sparsity degree by the PC rotation

The success of sparsity preservation by PC largely hinges on the property of
the particular rotation matrix $Q$. Proposition 3.2 implies that $Q$ is a non-
strictly (block) upper triangular matrix with full rank in probability.777A
(lower, upper) triangular matrix is strictly (lower, upper) triangular if its
diagonal elements are zero, see Abadir and Magnus (2005, page 17). In
particular,

$Q_{lk}\begin{cases}\asymp_{p}N^{\alpha_{l}-\alpha_{k}},\;\text{for }1\leq
k<l\leq r\\\ \asymp_{p}1,\;\text{for }1\leq k=l\leq r\\\ =O_{p}(1),\;\text{for
}1\leq l<k\leq r\end{cases}$

The particular formation of matrix $Q$ gives rise to sparsity recovery as $N$
grows. To illustrate, consider a simple two-factor model with factor strengths
$\alpha_{1}=0.9$ and $\alpha_{2}=0.7$. Now

$Q=\left(\begin{array}[]{cc}Q_{11}&Q_{12}\\\
Q_{21}&Q_{22}\end{array}\right)\asymp_{p}\left(\begin{array}[]{cc}1&O(1)\\\
N^{-0.2}&1\end{array}\right).$ (4.1)

According to Proposition 3.4, the PC estimated factor loading matrix converges
to a rotation of the true loading matrix, i.e.,
$\Lambda^{\ast}=\Lambda^{0}Q^{\prime}$, which can be written as

$\displaystyle\Lambda^{\ast}$ $\displaystyle=$
$\displaystyle\left(\Lambda_{\cdot 1}^{\ast},\Lambda_{\cdot 2}^{\ast}\right)$
$\displaystyle=$ $\displaystyle\left(\Lambda_{\cdot 1}^{0},\Lambda_{\cdot
2}^{0}\right)\left(\begin{array}[]{cc}Q_{11}&Q_{21}\\\
Q_{12}&Q_{22}\end{array}\right)$ $\displaystyle=$
$\displaystyle\left(Q_{11}\Lambda_{\cdot 1}^{0}+Q_{12}\Lambda_{\cdot
2}^{0},Q_{21}\Lambda_{\cdot 1}^{0}+Q_{22}\Lambda_{\cdot 2}^{0}\right).$

Given the approximately upper triangular structure of $Q$ in (4.1), we have

$\displaystyle\left\|\Lambda_{\cdot 1}^{\ast}\right\|_{0}$ $\displaystyle=$
$\displaystyle\left\|Q_{11}\Lambda_{\cdot 1}^{0}+Q_{12}\Lambda_{\cdot
2}^{0}\right\|_{0}\asymp_{p}N^{0.9}+N^{0.7}\asymp_{p}N^{0.9}.$ (4.3)

So the rotated loadings for factor 1 preserve its sparsity degree. Given that
$\Lambda_{\cdot 2}^{\ast}=Q_{21}\Lambda_{\cdot 1}^{0}+Q_{22}\Lambda_{\cdot
2}^{0}$, the sparsity degree of factor 2 is contaminated by
$Q_{21}\Lambda_{\cdot 1}^{0}$. Nevertheless, the contamination is non-
essential due to diminishing $Q_{21}$. So, for $\kappa=N^{-\varrho}$ with
$0<\varrho<0.2$, if we adopt a slightly adaptive measure for sparsity degree
for a generic $n\times 1$ vector $a$ as
$\left\|a\right\|_{\kappa_{N},0}=\sum_{j=1}^{n}\mathbf{1}\left(|a_{j}|\geq\kappa_{N}\right)$,
then it follows that

$\displaystyle\left\|\Lambda_{\cdot 2}^{\ast}\right\|_{\kappa_{N},0}$
$\displaystyle=$ $\displaystyle\left\|Q_{21}\Lambda_{\cdot
1}^{0}+Q_{22}\Lambda_{\cdot 2}^{0}\right\|_{\kappa_{N},0}\asymp_{p}N^{0.7}.$
(4.4)

The statement in (4.3) will also certainly hold if the $\ell_{0}$-norm is
replaced with the norm $\left\|\cdot\right\|_{\kappa_{N},0}$. This result
suggests that the PC rotation will preserve sparsity degree, up to negligible
terms, so that the factor strengths remain unchanged.

To have a better understanding of how the PC rotation acts upon the sparse
loading $\Lambda^{0}$, we continue to illustrate with the two-factor model as
above via a simulation. The data generating process is set the same as in
Section 6 where $(N,T)=(200,200)$. Define the non-null set with regard to
$\lambda_{ik}^{0}$’s, for $k=1,\ldots,r$, as

$\mathcal{L}_{k}^{0}=\\{i:\lambda_{ik}^{0}\neq 0,i=1,\ldots,N\\}.$

Figure 1: Illustration of sparsity degree preservation by PC rotation

Part (a) of Figure 1 shows underlying $\Lambda^{0}$ in which
$\mathcal{L}_{1}^{0}=\\{i:i=1,\ldots,117\\}$, and
$\mathcal{L}_{2}^{0}=\\{i:i=97,\ldots,136\\}$. The rotation matrix is
calculated as $Q=\left(\begin{array}[]{rr}-1.07&1.46\\\
-0.19&-0.67\end{array}\right).$ Panel (b) illustrates a common concern that
$\Lambda^{0}$ rotated by an arbitrary nonsingular matrix, say $Q^{(b)}$, would
become less sparse. To make an example, we randomly generate each element of
$Q^{(b)}$ as $\mathcal{N}(0,1)$ so that
$Q^{(b)}=\left(\begin{array}[]{rr}-1.44&-0.98\\\
-1.16&-0.75\end{array}\right)$. It is obvious in part (b) that
$\mathcal{L}_{2}^{0}$ is “inflated” considerably (by the shaded area) such
that the two factors turn out to have the same strength. Part (c) displays the
rotation effect by a stylized exactly upper triangular matrix $Q^{(c)}$.
$Q^{(c)}$ here is set the same as $Q$ except $Q_{21}^{(c)}=0$, i.e.,
$Q^{(c)}=\left(\begin{array}[]{cc}-1.07&1.46\\\ 0&-0.67\end{array}\right)$.
Clearly, the upper triangular $Q^{(c)}$ resolves inflation to
$\mathcal{L}_{2}^{0}$. On the other hand, we see in part (c) that
$\mathcal{L}_{1}^{0}$ is somewhat inflated due to the loading on
$\\{i:i=118,\ldots,136\\}$ (by the shaded area) from factor 2. Nevertheless,
the distortion is much less severe and negligible relative to factor 1’s
pervasiveness, since factor 2 is relatively weak. Finally, part (d) displays
$\Lambda^{\ast}$ rotated by the real $Q$ in the simulation. The absolute value
of $Q_{21}$ is indeed very small as expected in a finite sample
($Q_{21}\asymp_{p}200^{-0.2}$), though it is not exactly 0. By virtue of this
feature, the inflated part of $\Lambda_{\cdot,2}^{\ast}$ (by the shaded area)
is uniformly small and vanishing at a certain rate, which will be screened off
together with estimation errors to reveal sparsity. We formalize the idea in
the next subsection.

###  4.2 Sparsity recovery with PC estimators

Our previous analysis in subsection 4.1 provides a positive identification
result for $\Lambda^{\ast}$ rotated by $Q$. In practice, $\Lambda^{\ast}$ is
not observable, and the available is the PC estimator $\widetilde{\Lambda}$.
In this subsection, we will present how the recovery of sparsity, not just its
degree, is achieved by working with $\widetilde{\Lambda}$. Undoubtedly we need
to take into account of the estimation error consisting of
$\widetilde{\lambda}_{i}-\lambda_{i}^{\ast}=T^{-1}H^{\prime}F^{0\prime}e_{i}+T^{-1}(\widetilde{F}-F^{0}H)^{\prime}e_{i}$.
Theorem 3.10 (i) lends us a hand stating that the estimation errors are also
uniformly vanishing, and thus justifies the sparsity recovery via screening
the PC estimator $\widetilde{\lambda}_{i}$.

We shall show that the set $\mathcal{L}_{k}^{0}=\\{i:\lambda_{ik}^{0}\neq
0\\}$ characterizing sparsity for factor $k$ can be recovered well
approximately with regulated PC estimators. When the absolute value of
$\widetilde{\lambda}_{ik}$ is small, we can set the factor loading as 0. So we
choose the non-zero factor loadings with a threshold value
$c_{\lambda,NT}$:888With the limiting distribution for PC loadings estimators
in Theorem 3.8, we can also follow Uematsu and Yamagata (2023b) to select the
nonzero factor loadings based on multiple testing approach. We leave this as
future research.

$\widehat{\lambda}_{ik}=\widetilde{\lambda}_{ik}\mathbf{1}\left\\{\left|\widetilde{\lambda}_{ik}\right|>c_{\lambda,NT}\right\\}\
\text{where }c_{\lambda,NT}=\frac{1}{\sqrt{\ln\left(NT\right)}}.\text{ }$

Similarly define the estimated non-zero set for the $k$th factor loading by
$\widehat{\mathcal{L}}_{k}=\\{i:\widehat{\lambda}_{ik}\neq 0\\}.$ To evaluate
the sparsity recovery accuracy of $\widehat{\mathcal{L}}_{k}$, we denote the
symmetric difference between the true non-zero set $\mathcal{L}_{k}^{0}$ and
its estimator $\widehat{\mathcal{L}}_{k}$ by

$\mathcal{L}_{k}^{0}\triangle\widehat{\mathcal{L}}_{k}=\left(\mathcal{L}_{k}^{0}\backslash\widehat{\mathcal{L}}_{k}\right)\cup\left(\widehat{\mathcal{L}}_{k}\backslash\mathcal{L}_{k}^{0}\right).$

We present the main results on sparsity recovery of factor loadings in the
following Proposition.

###### Proposition 4.1

Under Assumptions 1-9, for $k=1,\ldots,r$, we have:

(i) if $\alpha_{k}$ is unique in
$\\{\alpha_{1},\alpha_{2},\ldots,\alpha_{r}\\}$, then

$\frac{\lvert\mathcal{L}_{k}^{0}\triangle\widehat{\mathcal{L}}_{k}\rvert}{N^{\alpha_{k}}}=o_{p}(1);$

(ii) if $\alpha_{k}$ is not unique in
$\\{\alpha_{1},\alpha_{2},\ldots,\alpha_{r}\\}$, then

$\frac{\lvert\mathcal{L}_{k}^{0}\triangle\widehat{\mathcal{L}}_{k}\rvert}{N^{\alpha_{k}}}=O_{p}(1).$

Remark 10.  The approximation of sparsity for the $k$th factor is
characterized by bounding the symmetric difference between
$\mathcal{L}_{k}^{0}$ and $\widehat{\mathcal{L}}_{k}$
($\mathcal{L}_{k}^{0}\triangle\widehat{\mathcal{L}}_{k}$) well relative to its
factor strength. In particular, the $k$th factor’s sparsity, measured in
$\mathcal{L}_{k}^{0}$, is recovered consistently relatively to its
pervasiveness when $\alpha_{k}$ is unique; when $\alpha_{k}$ is not unique,
the difference $\mathcal{L}_{k}^{0}\triangle\widehat{\mathcal{L}}_{k}$ is
still well bounded relatively—a condition implying consistency of the factor
strength estimator in Section 4.3.

Remark 11. (i) In the sparsity recovery of factor loadings, $c_{\lambda,NT}$
plays the key role of screening off noises due to rotation and estimation
errors that are negligible in size. We show that the noises are
$O_{p}\left(N^{-\zeta}+\left[(\ln N)/T\right]^{1/2}\right)$ uniformly, where
$\zeta=\min_{1\leq k\leq G-1}\left(\alpha_{[k]}-\alpha_{[k+1]}\right),$ i.e.,
the minimum discrepancy between distinct factor strengths. So we set
$c_{\lambda,NT}=\left[\ln\left(NT\right)\right]^{-1/2}$ to dominate the noise.
(ii) Alternatively, We can consider an alternative threshold value
$\widetilde{c}_{\lambda,NT}=c\left[\ln\left(NT\right)\right]^{-1/2}$ with a
tuning parameter $c$. The optimal value of $c$ can be determined by some
adaptive methods such as cross validation. Given that each (time) series will
be demeaned and standardized to have unit variance, as recommended by Bai and
Ng (2002), we focus on the simple threshold $c_{\lambda,NT}$. In both
simulations and the empirical application, our tuning parameter-free threshold
value $c_{\lambda,NT}$ works reasonably well. (iii) The choice of a threshold
value simply by the relevant dominant rate is also adopted in Fan et al.
(2015) to construct a screening statistic.

Figure 2 shows the intuition behind Proposition 4.1 using a three-factor
model.999For illustration, parts (a)-(d) in Figure 2 are based on stylized
manifestation but not simulation. By the path from part (a) to (b) and then to
(c), we see that the distortion to model sparsity by rotation and estimation
errors is either (i) dominated in absolute values relative to our screening
threshold $c_{\lambda,NT}$, or (ii) negligible in size relative to factor
pervasiveness. Consequently, sparsity is well approximated by regulation in
part (d).

Figure 2: Illustration of the loading sparsity revelation

Note that Uematsu and Yamagata’s (2023a) SOFAR estimator recovers sparsity
alternatively by $\ell_{1}$ regularization. Interestingly, if we contrast the
estimate of factor loadings by PC (their Figures 7 and 2) with that by SOFAR
(their Figures 9 and 11) in their empirical results, we immediately realize
that PC and SOFAR estimates are almost identical, except for many small noises
introduced by PC. This observation suggests the validity of combining PC
estimate with proper screening to recover sparsity, which in fact underlines
the method used in our paper.

###  4.3 Estimating factor strengths

BKP (2021) consider the pervasiveness of the unobserved macroeconomic shocks
by using the estimated factor strength. They can only identify and estimate
the strength of the strongest factor, and the results are thus limited. We can
consistently estimate factor strengths for all weak factors with various
strengths, which helps understand pervasiveness of factors completely. We hope
that our proposed estimator of factor strength is of independent interest.

We can understand how influential each factor is by studying to its strength.
Recall that $\alpha_{k}$ is such that
$\sum_{i=1}^{N}\lambda_{ik}^{0\prime}\lambda_{ik}^{0}\asymp N^{\alpha_{k}}.$
Now for $k=1,...,r,$ define

$\widehat{D}_{k}=\sum_{i=1}^{N}\mathbf{1}\left\\{\left|\widetilde{\lambda}_{ik}\right|>c_{\lambda,NT}\right\\},$

where $c_{\lambda,NT}=\left[\ln\left(NT\right)\right]^{-1/2}$ as before. Our
estimator for $\alpha_{k}$ is simply

$\widehat{\alpha}_{k}=\frac{\ln\widehat{D}_{k}}{\ln N}.$

To consistently estimate all factor strengths, one additional restriction is
needed. Define

$\Theta(\Lambda^{0})=\left\\{\Lambda:\Lambda=\Lambda^{0}R,\;\text{for a fixed
nonsingular $r\times r$ matrix $R$}\right\\}.$

Assumption 10.  $\forall\breve{\Lambda}\in\Theta(\Lambda^{0})$,
$\|\breve{\Lambda}_{\cdot,k}\|_{0}\leq\|\Lambda_{\cdot,k}\|_{0}+M$, for some
constant $M<\infty$.

The above assumption implies that $\Lambda^{0}$ is the sparsest
representation, up to a constant, among all observationally equivalent loading
matrices. $\Lambda^{0}$ is of most interest as it provides the most concise
interpretation of latent factors. Assumption 10 excludes, for instance, two
factors with highly correlated loadings.

###### Theorem 4.2

Under Assumptions 1-10, for $k=1,...,r,$
$\widehat{\alpha}_{k}\overset{p}{\rightarrow}\alpha_{k}.$

Remark 12. Theorem 4.2 shows that our proposed estimator can consistently
estimate strengths for all factors, not just for the strongest one as in BKP
(2021). The intuition behind this consistency result is similar to that behind
Proposition 4.1. Again, the key is to realize that the rotated loading matrix
$\Lambda_{\cdot,k}^{0\ast}$ is able to preserve strength as $\alpha_{k}$ up to
the given threshold $c_{\lambda,NT}$, and also that the errors induced by
estimation are also uniformly dominated by $c_{\lambda,NT}$.

##  5 The Determination of the Number of Factors

The determination of the number of factors has been of long-standing interest
in the literature of factor models. Various selection criteria have been
proposed, e.g., Bai and Ng (2002), Onatski (2010), Ahn and Horenstein (2013),
and Wei and Chen (2020). As for consistent selection in weak factor models,
Uematsu and Yamagata (2023a) show that the edge distribution (ED) estimator by
Onatski (2010) is consistent, and Onatski (2015) proposes selecting the number
of factors based on the approximations to the squared error of the least
squares estimator of the common component under both strong and weak factors
asymptotics. Freyaldenhoven (2022a) devises a statistic in combining both
eigenvalues and eigenvectors of the covariance matrix, to enhance its
discriminatory power in distinguishing factors stronger than a certain
threshold, assuming that $N$ and $T$ grows proportionally. Guo et al. (2022)
exploit a data-driven adaptive penalty of factor strength for information
criteria, to select weak factors and meanwhile avoid overfitting.

In this paper, we propose determining the number of factors based on SVT, as
discussed by Bai and Ng (2019) and Freyaldenhoven (2022a). The procedure is
very simple and works in the same spirit as Bai and Ng (2002). To determine
the number of factors in $F^{0}$, we use the following estimator of $r$,

$\widehat{r}=\max\left\\{k:\widetilde{V}_{k}^{r_{\max}}\geq
C_{r}\widehat{\sigma}^{2}N^{-1/2}\left(\ln\ln N\right)^{1/2}\right\\},$

where $\widehat{\sigma}^{2}$ is a consistent estimator of
$(NT)^{-1}\sum_{i=1}^{N}\sum_{t=1}^{T}E\left(e_{it}^{2}\right)$, $C_{r}$ is a
constant independent of $N$ and $T$, $r_{\max}$ is a large bounded positive
integer such that $r\leq r_{\max}$, and $\widetilde{V}_{k}^{r_{\max}}$ is the
$k$th diagonal element of
$\widetilde{V}^{r_{\max}}\equiv\operatorname{diag}\left(\widetilde{V}_{1},\ldots,\widetilde{V}_{r_{\max}}\right)$
being an $r\times r$ diagonal matrix consisting of the $r_{\max}$ largest
eigenvalues of $X^{\prime}X/(NT)$ in decreasing order. As for
$\widehat{\sigma}^{2}$ in practice, we compute it similar to Bai and Ng (2002)
as
$\widehat{\sigma}^{2}=\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}\left(X_{it}-\widetilde{\lambda}_{i}^{r_{\max}\prime}\widetilde{F}_{t}^{r_{\max}}\right)^{2}$,
where the superscript $r_{\max}$ signifies the allowance of $r_{\max}$ factors
in the estimation.

Remark 13. To implement, we choose the value of $C_{r}$ via twice K-fold cross
validation as proposed by Wei and Chen (2020). Specifically, for a given
$C_{r}$ and the consequent estimator of $r$, in step one we run a “leave one
of $K$ folds out” (across columns of $X$) estimation by PC to obtain estimated
loadings for all $N$ units, which are then sparsified by $c_{\lambda,NT}$. In
step two we further run a “leave one of $J$ folds out” (across rows of $X$)
estimation based on the fold left over from step one, to obtain estimated
factors. Lastly, we evaluate the performance of estimation by out-of-sample
mean squared errors averaged over all possible folds, and choose the $C_{r}$
to minimize the average. One can refer to Wei and Chen (2020) for
implementation details. In practice, we choose $J=K=5$ and find the results
are very robust against various values of $J$ and $K$.

###### Theorem 5.1

Under Assumptions 1-7, $\widehat{r}\overset{p}{\rightarrow}r.$

Remark 14. Proving Theorem 5.1, detailed in the Appendix, is to check two
conditions to hold in probability: (i) $\widehat{r}\geq r$ and (ii)
$\widehat{r}\leq r.$ Condition (i) is actually implied by Proposition 3.1. We
prove condition (ii) by contradiction, and this relies on a sharper bound
obtained under
$\left\|e\right\|_{sp}^{2}=O_{p}\left(\max\left\\{N,T\right\\}\right)$ imposed
by Assumption 4. This assumption is also needed for Bai and Ng (2002) in
determining the number of factors via information criteria even in strong
factor models.

Remark 15. The ED estimator proposed by Onatski (2010) is based on the fact
that all the “systematic” eigenvalues diverge to infinity, whereas any finite
number of the largest “idiosyncratic” eigenvalues cluster around a single
point. Onatski (2010) determines the number of factors by separating the
diverging eigenvalues with a wedge $\delta$, and proves the consistency of the
ED estimator only requiring that $\min_{1\leq k\leq r}\alpha_{k}>0$. However,
the ED estimator hinges on the idiosyncratic terms being guassian, or
independent cross-sectionally or over time in case of nonguassian, and is thus
restrictive in applications with macroeconomics and finance. As for choosing
$\delta$, Onatski (2010) approximates the upper bound of eigenvalue
differences by an OLS estimate which is then doubled to formulate $\delta$.
His final estimator of $r$ is obtained via iterations given $\delta$. In
contrast, our estimator of $r$ is more straightforward to use.

##  6 Monte Carlo Simulations

In this section, we study the finite sample performance of our proposed
estimators for sparsity-induced weak factor models. It includes comparison of
simple PC estimators with Lasso type estimators, estimation of number of
factors, as well as factor strength.

###  6.1 The data generating process

We consider the following data generating process (DGP):

$\displaystyle X_{it}$ $\displaystyle=$
$\displaystyle\lambda_{i}^{0\prime}F_{t}^{0}+e_{it},\text{ }i=1,...,N,\text{
}t=1,...,T$ $\displaystyle F_{1t}^{0}$ $\displaystyle=$ $\displaystyle
0.5F_{1,t-1}^{0}+u_{1t},\text{ and
}F_{kt}^{0}=\left(-0.8\right)^{k}F_{1t}^{0}+u_{kt},\text{ }k=2,...,r$

The simulated factors are correlated with each other with various degrees. We
let $u_{kt}$ be mutually uncorrelated $\mathcal{N}\left(0,1\right)$ for
$k=1,...,r$. To specify the factor loadings $\lambda_{ik}^{0},$ for each
$k=1,...,r,$ we first randomly select $\lfloor N^{\alpha_{k}}\rfloor$ of
$\left\\{\lambda_{ik}^{0}\right\\}_{i=1}^{N}$ and specify them as iid
$\mathcal{N}\left(0,1\right)$, and then set the rest of
$\left\\{\lambda_{ik}^{0}\right\\}_{i=1}^{N}$ as zero.

For the $N\times 1$ vector $e_{t},$ we specify the (marginal) distribution of
$e_{it}$ as the student-$t\left(5\right)$ to allow for heavy tail. The cross
sectional dependence across $e_{it}$ is admitted through the $N\times N$
covariance matrix $\Sigma_{e}$ as follows.
$\Sigma_{e}=\operatorname{diag}\left\\{\Sigma_{1},...,\Sigma_{N/4}\right\\}$
as a block-diagonal matrix with $4\times 4$ blocks located along the main
diagonal. Each $\Sigma_{i}$ is assumed to be $I_{4}$ initially. We then
randomly choose $\lfloor N^{0.3}\rfloor$ blocks among them and make them non-
diagonal by setting $\Sigma_{i}\left(m,n\right)=0.5^{\left|m-n\right|}$. The
design of cross sectional dependence follows Fan et al. (2015) except that the
dependence is stronger here.

We have tried simulations with the number of factors $r=3$ and $5.$ For $r=3,$
we set $\alpha=\left(0.9,0.75,0.6\right);$ while for $r=5,$ we set
$\alpha=\left(1,0.9,0.8,0.7,0.6\right)$. The number of simulations is 2000.

###  6.2 Simulation results

We first compare different methods to determine the number of factors with
ours (WZ). The alternative selection rules range from the $IC_{p1}$ by Bai and
Ng (2002, BN), Guo et al. (2022, GCT), Freyaldenhoven (2022b, FR), ED by
Onatski (2010), and Ahn and Horenstein (2013, AH). $r_{\max}$ is set to be 8
if needed. The root mean square error (RMSE) and bias of the estimated number
of factors by each method are reported in Tables 1 and 2. It is obvious that
GCT, FR and AH are all subject to overestimation of $r$ in the presence of
weak factors. ED and BN are not very bad but they are outperformed by our
proposed method, since they tend to over- and under-estimate $r$,
respectively. The ED estimator is not as effective as found previously in weak
factor models, e.g., by Uematsu and Yamagata (2023a), implying that the ED
performance may be sensitive to the choice of the wedge parameter $\delta$, as
remarked in Section 5. Our proposed estimator of factor numbers is outstanding
against all alternatives at all sample sizes.

Table 1: Estimating the number of factors when $r=3$

|  | RMSE |  | Bias
---|---|---|---|---
$N$ | $T$ | WZ | BN | GCT | FR | ED | AH |  | WZ | BN | GCT | FR | ED | AH
100 | 100 | 0.273 | 0.673 | 0.939 | 1.962 | 0.400 | 1.998 |  | -0.059 | -0.443 | -0.882 | -1.943 | 0.030 | -1.998
| 200 | 0.132 | 0.446 | 0.866 | 1.964 | 0.425 | 1.999 |  | -0.011 | -0.190 | -0.748 | -1.945 | 0.090 | -1.999
| 400 | 0.084 | 0.314 | 0.783 | 1.978 | 0.964 | 1.998 |  | -0.006 | -0.095 | -0.611 | -1.968 | 0.330 | -1.997
200 | 100 | 0.102 | 0.600 | 0.916 | 1.973 | 0.284 | 2.000 |  | -0.002 | -0.355 | -0.839 | -1.955 | 0.059 | -2.000
| 200 | 0.055 | 0.241 | 0.739 | 1.952 | 0.354 | 2.000 |  | -0.001 | -0.056 | -0.543 | -1.909 | 0.075 | -2.000
| 400 | 0.000 | 0.120 | 0.445 | 1.952 | 0.448 | 1.999 |  | 0.000 | -0.010 | -0.180 | -1.909 | 0.091 | -1.998
400 | 100 | 0.081 | 0.687 | 0.932 | 1.946 | 0.244 | 2.000 |  | 0.004 | -0.468 | -0.870 | -1.895 | 0.058 | -2.000
| 200 | 0.000 | 0.180 | 0.610 | 1.767 | 0.248 | 2.000 |  | 0.000 | -0.032 | -0.370 | -1.562 | 0.057 | -2.000
| 400 | 0.000 | 0.039 | 0.209 | 1.584 | 0.297 | 1.997 |  | 0.000 | 0.001 | -0.011 | -1.255 | 0.061 | -1.995

Table 2: Estimating the number of factors when $r=5$

|  | RMSE |  | Bias
---|---|---|---|---
$N$ | $T$ | WZ | BN | GCT | FR | ED | AH |  | WZ | BN | GCT | FR | ED | AH
100 | 100 | 0.273 | 0.742 | 2.708 | 3.975 | 0.455 | 4.000 |  | -0.060 | -0.499 | -2.654 | -3.962 | -0.032 | -4.000
| 200 | 0.122 | 0.443 | 2.408 | 3.970 | 0.362 | 4.000 |  | -0.014 | -0.190 | -2.321 | -3.950 | 0.059 | -4.000
| 400 | 0.081 | 0.305 | 2.114 | 3.981 | 0.803 | 4.000 |  | -0.006 | -0.089 | -2.011 | -3.970 | 0.230 | -4.000
200 | 100 | 0.120 | 0.590 | 2.465 | 3.960 | 0.278 | 4.000 |  | -0.001 | -0.333 | -2.387 | -3.925 | 0.052 | -4.000
| 200 | 0.059 | 0.228 | 1.843 | 3.909 | 0.254 | 4.000 |  | -0.001 | -0.046 | -1.749 | -3.822 | 0.049 | -4.000
| 400 | 0.032 | 0.095 | 1.371 | 3.911 | 0.257 | 3.998 |  | -0.001 | -0.007 | -1.289 | -3.826 | 0.041 | -3.996
400 | 100 | 0.087 | 0.640 | 2.299 | 3.800 | 0.239 | 4.000 |  | 0.006 | -0.404 | -2.220 | -3.610 | 0.052 | -4.000
| 200 | 0.039 | 0.166 | 1.548 | 3.256 | 0.235 | 4.000 |  | 0.002 | -0.025 | -1.462 | -2.650 | 0.051 | -4.000
| 400 | 0.022 | 0.045 | 1.051 | 2.889 | 0.249 | 3.996 |  | 0.001 | 0.001 | -1.033 | -2.085 | 0.054 | -3.992

To study the finite sample performance of PC estimators under weak factor
models, we compare PC regression with the sparse orthogonal factor regression
(SOFAR) proposed by Uematsu and Yamagata (2023a, b), assuming the true number
of factors to be known. Based on Lasso penalization on factor loadings for
sparsity, Uematsu and Yamagata develop the Adaptive (Ada) SOFAR estimator.
They later improve the SOFAR estimator by introducing the Debiased (Deb) SOFAR
estimator to recover its asymptotic normality. Furthermore, they address the
multiple testing problem of loading sparsity and construct the Resparsifed
(Res) SOFAR estimator to fulfill FDR control. Note that the various SOFAR
estimators proposed by Uematsu and Yamagata result from penalized regression
which targets sparsity explicitly. Thus it seems natural to suggest that one
should be better off using their approaches to deal with weak factor models
than using PC. However, as we have mentioned before, the simple PC estimators
enjoy a nice property of automatic sparsity recognition, so that PC could even
outperform the complicated SOFAR.

To measure the performance of factor and loading estimators, we follow Doz et
al. (2012) to use the trace statistics:101010Uematsu and Yamagata (2023a)
evaluate performance of estimators by the $\ell_{2}-$norm losses:
$\left\|\sum_{k=1}^{r}N_{k}^{-1/2}\left[\mathop{\mathrm{a}bs}\left(\widehat{\Lambda}_{\cdot,k}\right)-\mathop{\mathrm{a}bs}\left(\Lambda_{\cdot,k}^{0}\right)\right]\right\|_{2}$
and
$\left\|\sum_{k=1}^{r}T^{-1/2}\left[\mathop{\mathrm{a}bs}\left(\widehat{F}_{\cdot,k}\right)-\mathop{\mathrm{a}bs}\left(F_{\cdot,k}^{0}\right)\right]\right\|_{2}$.
Such norm losses are more relevant when the factors and loadings are
identified up to column-wise sign indeterminacy, rather than just rotation
indeterminacy, and additional restrictions are required, as we explain right
below. We instead employ the trace statistics whose validity does not rely on
such restrictions, and they demonstrate how effectively estimators of factors
(loadings) span the same space as latent factors (loadings).

$TR^{F}=\frac{\mathop{\mathrm{T}r}\left(F^{0\prime}\widetilde{F}\left(\widetilde{F}^{\prime}\widetilde{F}\right)^{-1}\widetilde{F}^{\prime}F^{0}\right)}{\mathop{\mathrm{T}r}\left(F^{0\prime}F^{0}\right)},\quad
TR^{\Lambda}=\frac{\mathop{\mathrm{T}r}\left(\Lambda^{0\prime}\widetilde{\Lambda}\left(\widetilde{\Lambda}^{\prime}\widetilde{\Lambda}\right)^{-1}\widetilde{\Lambda}^{\prime}\Lambda^{0}\right)}{\mathop{\mathrm{T}r}\left(\Lambda^{0\prime}\Lambda^{0}\right)}.$

We also report the root mean squared errors in estimating the common component
$C_{it}$ (RMSEC). All results are included in Tables 3 and 4. The Res
estimator is under the FDR rate $q=0.1$.111111We have calculated the Res
estimator when $q=0.05$, and the results are very close to reported here. It
is interesting to see that the PC estimators of factors and common components
are almost always better than any of SOFAR type estimators. Although the
Resparsifed SOFAR estimator for $\Lambda$ outperforms PC under bigger sample
sizes, the margin is small. Recall that the SOFAR estimator are constructed
under the restriction on the true DGP that $\Lambda^{0^{\prime}}\Lambda^{0}$
being diagonal and $F^{0^{\prime}}F^{0}/T=I_{r}$ (or
$E(F^{0^{\prime}}F^{0}/T)=I_{r}$), which implies that the rotation matrix $H$
will be asymptotically $I_{r}$. However, the restriction is violated under our
DGP with correlated underlying factors, which might account for the worse
performance of SOFAR. In an additional experiment whose results are not
reported here, we modify our DGP with factors being indeed independent, and
find that the SOFAR estimators are much closer to or even outperforms the PC
estimator in estimating factors evaluated by $TR^{F}$, though their $RMSE^{C}$
are still bigger in general. Hence regression by PC seems a better choice
given its robustness and easy implementation.121212The estimators obtained via
$\ell_{1}$ rotation suggested by Freyaldenhoven (2022b) render exactly the
same results as PC in Tables 3 and 4, as the trace statistics and estimators
for common components are rotation-invariant.

Table 3: Estimation of factor models when $r=3$

|  | $TR^{F}$ |  | $TR^{\Lambda}$ |  | $RMSE^{C}$
---|---|---|---|---|---|---
$N$ | $T$ | PC | Ada | Deb | Res |  | PC | Ada | Deb | Res |  | PC | Ada | Deb | Res
100 | 100 | 0.924 | 0.898 | 0.909 | 0.909 |  | 0.718 | 0.622 | 0.709 | 0.713 |  | 0.973 | 1.020 | 1.005 | 1.005
| 200 | 0.936 | 0.918 | 0.924 | 0.924 |  | 0.786 | 0.720 | 0.783 | 0.788 |  | 0.953 | 0.988 | 0.967 | 0.967
| 400 | 0.943 | 0.935 | 0.938 | 0.938 |  | 0.830 | 0.806 | 0.833 | 0.837 |  | 0.948 | 0.970 | 0.958 | 0.957
200 | 100 | 0.955 | 0.934 | 0.942 | 0.942 |  | 0.745 | 0.647 | 0.765 | 0.786 |  | 0.886 | 0.941 | 0.894 | 0.893
| 200 | 0.964 | 0.953 | 0.957 | 0.957 |  | 0.811 | 0.752 | 0.826 | 0.841 |  | 0.881 | 0.910 | 0.879 | 0.879
| 400 | 0.969 | 0.964 | 0.966 | 0.966 |  | 0.852 | 0.838 | 0.861 | 0.871 |  | 0.872 | 0.892 | 0.877 | 0.877
400 | 100 | 0.969 | 0.951 | 0.956 | 0.956 |  | 0.750 | 0.660 | 0.767 | 0.803 |  | 0.813 | 0.866 | 0.820 | 0.818
| 200 | 0.976 | 0.966 | 0.970 | 0.970 |  | 0.816 | 0.764 | 0.828 | 0.853 |  | 0.806 | 0.844 | 0.812 | 0.812
| 400 | 0.980 | 0.975 | 0.977 | 0.977 |  | 0.858 | 0.846 | 0.864 | 0.879 |  | 0.802 | 0.822 | 0.805 | 0.804

Table 4: Estimation of factor models when $r=5$

|  | $TR^{F}$ |  | $TR^{\Lambda}$ |  | $RMSE^{C}$
---|---|---|---|---|---|---
$N$ | $T$ | PC | Ada | Deb | Res |  | PC | Ada | Deb | Res |  | PC | Ada | Deb | Res
100 | 100 | 0.946 | 0.919 | 0.922 | 0.922 |  | 0.774 | 0.685 | 0.769 | 0.766 |  | 1.612 | 1.653 | 1.620 | 1.622
| 200 | 0.953 | 0.938 | 0.941 | 0.941 |  | 0.820 | 0.761 | 0.818 | 0.817 |  | 1.599 | 1.633 | 1.610 | 1.611
| 400 | 0.956 | 0.948 | 0.950 | 0.950 |  | 0.844 | 0.814 | 0.845 | 0.845 |  | 1.594 | 1.615 | 1.603 | 1.603
200 | 100 | 0.969 | 0.947 | 0.949 | 0.949 |  | 0.794 | 0.699 | 0.796 | 0.800 |  | 1.500 | 1.574 | 1.527 | 1.529
| 200 | 0.973 | 0.960 | 0.963 | 0.963 |  | 0.837 | 0.777 | 0.839 | 0.843 |  | 1.496 | 1.541 | 1.513 | 1.514
| 400 | 0.976 | 0.969 | 0.971 | 0.971 |  | 0.860 | 0.829 | 0.862 | 0.864 |  | 1.497 | 1.516 | 1.500 | 1.501
400 | 100 | 0.980 | 0.961 | 0.963 | 0.963 |  | 0.798 | 0.704 | 0.804 | 0.816 |  | 1.438 | 1.497 | 1.444 | 1.446
| 200 | 0.983 | 0.972 | 0.974 | 0.974 |  | 0.841 | 0.780 | 0.844 | 0.852 |  | 1.430 | 1.467 | 1.434 | 1.435
| 400 | 0.985 | 0.979 | 0.980 | 0.980 |  | 0.866 | 0.833 | 0.867 | 0.872 |  | 1.425 | 1.447 | 1.428 | 1.429

We further investigate the estimated factor strength by various methods in
Tables 5 and 6. All estimates become less accurate as the true strength degree
decreases, as expected. It is admitted that our proposed strength estimate
(PC+Screening) suffers more of overestimation with fairly weak factors, e.g.,
those with $\alpha_{k}=0.6$. Otherwise, our estimate is comparable to, and
sometimes even better than, those from SOFAR which is designed deliberately
for sparsity recovery, especially for strength $\alpha_{k}\in[0.7,0.9]$. The
message here again delivers the usefulness of our factor strength estimator,
which works reasonably well while avoids involving heavy computation as in
SOFAR. Moreover, we find that SOFAR tends to underestimate the number of
factors in our simulation, and hence the relative advantage of our estimate is
conservative as we assume $r$ is known here.

Table 5: Estimation of factor strength when $r=3$ with $\alpha=\left(0.9,0.75,0.6\right)$ |  | RMSE |  | Bias
---|---|---|---|---
$N$ | $T$ | $\widehat{\alpha}_{1}$ | $\widehat{\alpha}_{2}$ | $\widehat{\alpha}_{3}$ |  | $\widehat{\alpha}_{1}$ | $\widehat{\alpha}_{2}$ | $\widehat{\alpha}_{3}$
|  | Panel A: PC+Screening
100 | 100 | 0.014 | 0.047 | 0.138 |  | 0.000 | 0.006 | 0.100
| 200 | 0.014 | 0.048 | 0.169 |  | 0.002 | 0.003 | 0.073
| 400 | 0.014 | 0.049 | 0.208 |  | 0.003 | 0.007 | 0.048
200 | 100 | 0.010 | 0.048 | 0.126 |  | 0.002 | 0.028 | 0.111
| 200 | 0.009 | 0.045 | 0.138 |  | 0.002 | 0.023 | 0.091
| 400 | 0.009 | 0.045 | 0.166 |  | 0.004 | 0.026 | 0.075
400 | 100 | 0.007 | 0.053 | 0.114 |  | 0.002 | 0.044 | 0.101
| 200 | 0.006 | 0.048 | 0.103 |  | 0.002 | 0.040 | 0.079
| 400 | 0.006 | 0.052 | 0.115 |  | 0.003 | 0.042 | 0.065
|  | Panel B: SOFAR_Debiased
100 | 100 | 0.017 | 0.094 | 0.085 |  | 0.004 | -0.060 | -0.019
| 200 | 0.023 | 0.065 | 0.106 |  | 0.019 | 0.028 | 0.067
| 400 | 0.030 | 0.103 | 0.162 |  | 0.028 | 0.096 | 0.146
200 | 100 | 0.012 | 0.095 | 0.059 |  | 0.000 | -0.071 | -0.021
| 200 | 0.017 | 0.052 | 0.084 |  | 0.014 | 0.025 | 0.062
| 400 | 0.025 | 0.095 | 0.148 |  | 0.024 | 0.091 | 0.138
400 | 100 | 0.010 | 0.102 | 0.050 |  | -0.004 | -0.085 | -0.033
| 200 | 0.013 | 0.043 | 0.054 |  | 0.011 | 0.023 | 0.036
| 400 | 0.020 | 0.088 | 0.114 |  | 0.019 | 0.086 | 0.105
|  | Panel C: SOFAR_Resparsified
100 | 100 | 0.024 | 0.067 | 0.095 |  | 0.020 | 0.044 | 0.064
| 200 | 0.030 | 0.103 | 0.130 |  | 0.028 | 0.095 | 0.106
| 400 | 0.036 | 0.137 | 0.179 |  | 0.035 | 0.135 | 0.164
200 | 100 | 0.017 | 0.062 | 0.096 |  | 0.015 | 0.057 | 0.085
| 200 | 0.025 | 0.099 | 0.129 |  | 0.024 | 0.098 | 0.121
| 400 | 0.031 | 0.130 | 0.162 |  | 0.031 | 0.129 | 0.157
400 | 100 | 0.013 | 0.057 | 0.090 |  | 0.012 | 0.054 | 0.081
| 200 | 0.021 | 0.093 | 0.104 |  | 0.020 | 0.092 | 0.098
| 400 | 0.027 | 0.123 | 0.140 |  | 0.026 | 0.123 | 0.136
Table 6: Estimation of factor strength when $r=5$ with
$\alpha=\left(1,0.9,0.8,0.7,0.6\right)$

|  | RMSE |  | Bias
---|---|---|---|---
$N$ | $T$ | $\widehat{\alpha}_{1}$ | $\widehat{\alpha}_{2}$ | $\widehat{\alpha}_{3}$ | $\widehat{\alpha}_{4}$ | $\widehat{\alpha}_{5}$ |  | $\widehat{\alpha}_{1}$ | $\widehat{\alpha}_{2}$ | $\widehat{\alpha}_{3}$ | $\widehat{\alpha}_{4}$ | $\widehat{\alpha}_{5}$
|  | Panel A: PC+Screening
100 | 100 | 0.036 | 0.045 | 0.041 | 0.076 | 0.156 |  | -0.034 | -0.035 | 0.020 | 0.050 | 0.111
| 200 | 0.034 | 0.043 | 0.043 | 0.076 | 0.187 |  | -0.033 | -0.034 | 0.021 | 0.047 | 0.081
| 400 | 0.033 | 0.040 | 0.043 | 0.078 | 0.223 |  | -0.032 | -0.030 | 0.023 | 0.045 | 0.058
200 | 100 | 0.031 | 0.032 | 0.047 | 0.083 | 0.139 |  | -0.030 | -0.023 | 0.035 | 0.065 | 0.115
| 200 | 0.030 | 0.030 | 0.047 | 0.076 | 0.159 |  | -0.029 | -0.021 | 0.035 | 0.052 | 0.095
| 400 | 0.029 | 0.027 | 0.049 | 0.076 | 0.172 |  | -0.028 | -0.018 | 0.037 | 0.053 | 0.079
400 | 100 | 0.027 | 0.020 | 0.053 | 0.089 | 0.117 |  | -0.027 | -0.009 | 0.045 | 0.077 | 0.101
| 200 | 0.026 | 0.018 | 0.052 | 0.084 | 0.107 |  | -0.026 | -0.008 | 0.042 | 0.070 | 0.065
| 400 | 0.025 | 0.015 | 0.053 | 0.082 | 0.124 |  | -0.025 | -0.006 | 0.044 | 0.067 | 0.051
|  | Panel B: SOFAR_Debiased
100 | 100 | 0.029 | 0.056 | 0.045 | 0.055 | 0.062 |  | -0.027 | -0.045 | -0.009 | 0.003 | 0.021
| 200 | 0.020 | 0.024 | 0.061 | 0.089 | 0.106 |  | -0.019 | 0.000 | 0.050 | 0.077 | 0.088
| 400 | 0.015 | 0.037 | 0.097 | 0.142 | 0.157 |  | -0.013 | 0.033 | 0.093 | 0.136 | 0.144
200 | 100 | 0.027 | 0.055 | 0.041 | 0.046 | 0.046 |  | -0.026 | -0.047 | -0.013 | -0.016 | 0.018
| 200 | 0.019 | 0.020 | 0.059 | 0.070 | 0.089 |  | -0.018 | 0.002 | 0.048 | 0.058 | 0.076
| 400 | 0.013 | 0.037 | 0.097 | 0.127 | 0.136 |  | -0.013 | 0.035 | 0.094 | 0.123 | 0.127
400 | 100 | 0.027 | 0.053 | 0.039 | 0.045 | 0.031 |  | -0.026 | -0.048 | -0.023 | -0.031 | -0.002
| 200 | 0.018 | 0.017 | 0.051 | 0.051 | 0.061 |  | -0.017 | 0.002 | 0.041 | 0.040 | 0.046
| 400 | 0.012 | 0.039 | 0.095 | 0.107 | 0.103 |  | -0.012 | 0.037 | 0.092 | 0.103 | 0.095
|  | Panel C: SOFAR_Resparsified
100 | 100 | 0.024 | 0.021 | 0.053 | 0.092 | 0.100 |  | -0.023 | 0.003 | 0.043 | 0.080 | 0.084
| 200 | 0.017 | 0.036 | 0.087 | 0.136 | 0.135 |  | -0.016 | 0.033 | 0.082 | 0.130 | 0.122
| 400 | 0.012 | 0.056 | 0.121 | 0.181 | 0.179 |  | -0.011 | 0.055 | 0.119 | 0.178 | 0.168
200 | 100 | 0.022 | 0.018 | 0.048 | 0.081 | 0.098 |  | -0.022 | 0.010 | 0.040 | 0.074 | 0.092
| 200 | 0.016 | 0.039 | 0.076 | 0.116 | 0.115 |  | -0.015 | 0.038 | 0.071 | 0.111 | 0.110
| 400 | 0.011 | 0.058 | 0.111 | 0.163 | 0.153 |  | -0.011 | 0.058 | 0.108 | 0.160 | 0.148
400 | 100 | 0.021 | 0.019 | 0.041 | 0.068 | 0.087 |  | -0.021 | 0.016 | 0.034 | 0.063 | 0.085
| 200 | 0.015 | 0.042 | 0.066 | 0.095 | 0.098 |  | -0.014 | 0.041 | 0.062 | 0.092 | 0.095
| 400 | 0.010 | 0.060 | 0.097 | 0.139 | 0.123 |  | -0.010 | 0.059 | 0.095 | 0.137 | 0.121

##  7 Empirical Application

We apply our approach to explore potential weak factors on macroeconomic
indicators. We use data from the FRED-QD as a quarterly database for
macroeconomic research (McCracken and Ng, 2021). The original data set
consists of 248 quarterly frequency series dating back to 1959:Q2. By
disregarding series with missing observations, we end up with $N=181$ series.
The series are classified into 13 groups: NIPA; Industrial Production;
Employment and Unemployment; Housing; Inventories, Orders, and Sales; Prices;
Earnings and Productivity; Interest Rates; Money and Credit; Household Balance
Sheets; Exchange Rates; Stock Markets; and Non-Household Balance
Sheets.131313The original FRED-QD data have one additional group named
“Other”, containing only two series. However, both series are dropped due to
missing observations, leaving us 13 groups in use.

For each of the series used in our sample, after having made the decision that
the series should be managed in levels or log-levels, the transformation codes
are first and second differences based on whether the series is $I(0)$,
$I(1$), or $I(2)$ tested and suggested by McCracken and Ng (2021). Due to the
transformation, two initial observations are dropped so each series starts
from 1959:$Q4$ actually. We also normalize each series to have zero mean and
unit variance.

We depict the numbers of weak factors estimated by WZ, BN ($IC_{p1}$) and ED
in Figure 3. The estimation is implemented under a rolling window scheme with
length fixed at 120. The time on the x-axis denotes the right end point of the
window interval. By WZ, the large panel is with five factors in most of the
time considered here. There are a few windows taking four factors, in which BN
also reports the same number. On the other hand, BN results in only three
factors for the first half of rolling windows. For the second half, BN finds
five factors in only a few window intervals. Onatski (2010) suggests that the
ED method is expected to detect present weak factors. However, the result does
not seem to agree with this: ED outputs three factors most of time, and the
number may drop to two or even one at times. There are only three times when
ED reports estimated factor numbers bigger than three.

Figure 3: Estimated factor numbers by rolling windows

To have an idea of estimated factor strength and its dynamic, for each of our
identified weak factors we further draw its associated factor strength over
rolling windows in Figure 4 in a fashion similar to that in Figure 3. For most
of window intervals ending from $1989$:$Q4$ to $2021$:$Q4$, we have five weak
factors. Figure 4 demonstrates a clear sparsity structure of latent factors.
The first two strongest factors seem to have very close strength around 0.8
most of the time, although $\widehat{\alpha}_{1}$ may spike to being close to
0.9 a few times. $\widehat{\alpha}_{3}$ ranges from 0.7 to 0.8.
$\widehat{\alpha}_{4}$ fluctuates between 0.6 to 0.7, and interestingly it
seems to spike simultaneously with $\widehat{\alpha}_{1}$, while reach bottom
simultaneously with $\widehat{\alpha}_{3}$. $\widehat{\alpha}_{5}$ is around
0.6 and moves close to $\widehat{\alpha}_{4}$ up to 2014. However,
$\widehat{\alpha}_{5}$ drops even to 0 (a reduced factor) during the last 6
years.

Figure 4: Estimated factor strength by rolling windows

We are also interested in factor pervasiveness in the cross sectional
dimension, that is, what are those specific series exposed to each weak factor
and the relative influence each factor exerts over space. We can infer the
degree of influence roughly by looking at the sparsity of
$\widehat{\lambda}_{ik}$ over $i$, given the implication by Proposition 4.1.
We represent distribution of $\lvert\widehat{\lambda}_{ik}\rvert$ via a heat
map of Figure 5. We consider two subperiods of roughly the same length:
1959:$Q3$-1989:$Q2$ and 1989:$Q3$-2021:$Q4$. For each subperiod, we detect 5
weak factors. Each row ($i$) represents a series and we add its associated
group number ($\\#1-\\#13$) in front of its name. Each column ($k$) represents
a principle component extracted as a latent factor, and we also signify its
rank by factor strength. In particular, the estimated factor strengths are
0.831, 0.815, 0.756, 0.648, and 0.566 for 1959:$Q3$-1989:$Q2$, and 0.888,
0.784, 0.771, 0.654, 0.576 for 1989:$Q3$-2021:$Q4$, as put in the parenthesis.
The darkness of each cell indicates the absolute value of
$\widehat{\lambda}_{ik}$. Given that $\lvert\widehat{\lambda}_{ik}\rvert\leq
3$ mostly across $i$ and $k$, we right censor $\widehat{\lambda}_{ik}$ at 3 to
obtain a sharper visualization of the heat map.

Figure 5: Heat map for recovered factor loading sparsity

Some interesting results are the following. While the top two factor strengths
are close during the first subperiod, they move farther away from each other
during the second subperiod. The rest of three factor strengths are fairly
stable over the two subperiods. The 1st PC extracted as a factor is not
necessarily the strongest one: it ranks the 3rd by the number of influenced
series on the second subperiod. The 2nd PC ranks 2nd in strength in the first
subperiod, yet rises to the 1st in strength in the second subperiod, as it is
loaded by many new series most of which belong to group 6 (Prices). Meanwhile,
the 3rd PC also gains additional influence, mostly from groups 1, 2 and 3
(NIPA; Industrial Production; Employment and Unemployment), giving rise to its
incremental strength and rank enhancement, although it loses influence on a
few series belonging to groups 8 (Interest Rates) and 10 (Household Balance
Sheets). It may be also worth noticing that while some series are not
influenced by any weak factors in subperiod 1, e.g., many of those from group
6, they start to load on some factors in subperiod 2.

##  8 Conclusion

This paper proposes a novel approach to deal with weak latent factor models
with sparse factor loadings. It reveals an interesting and fundamental fact
that the PC estimators can preserve the sparsity in estimated factor loadings
for sparsity-induced weak factor models. This fact facilitates the derivation
of asymptotic properties of PC estimators, enables us to recover the sparsity
of loadings, and estimate the strengths of each factor. In addition, the
determination of the number of factors in weak factor models is also
investigated. The numerical studies confirm that our proposed approach works
reasonably well in finite sample, and an empirical application to FRED-QD data
set shows that our method is useful to detect factor strengths, loading
sparsity and their dynamics.

Our estimators of weak factor models belong to unsupervised PCA. Conceptually,
one could think of applying supervised PCA (e.g., Huang et al., 2022) to weak
factor models and obtain more efficient results for estimation and inference.
We agree that supervision would further improve the performance of our
proposed estimators by better exploiting information available, and yet also
raise up additional complexity. So we leave it to future research.

References

Abadir, K. and Magnus, J. (2005) Matrix Algebra. Cambridge University Press.

Ahn, S.C. and Horenstein, A.R. (2013) Eigenvalue ratio test for the number of
factors. Econometrica, 81(3), 1203–1227.

Ando, T. and Bai, J. (2017) Clustering Huge Number of Financial Time Series: A
Panel Data Approach With High-Dimensional Predictors and Factor Structures.
Journal of the American Statistical Association, 112(519), 1182–1198.

Bai, J. (2003) Inferential theory for factor models of large dimensions.
Econometrica, 71(1), 135–171.

Bai, J. and Ng, S. (2002) Determining the number of factors in approximate
factor models. Econometrica, 70(1), 191–221.

Bai, J., and Ng, S. (2006). Confidence intervals for diffusion index forecasts
and inference for factor-augmented regressions. Econometrica, 74(4),
1133-1150.

Bai, J. and Ng, S. (2019) Rank regularized estimation of approximate factor
models. Journal of Econometrics, 212(1), 78–96.

Bai, J. and Ng, S. (2023) Approximate factor models with weaker loadings.
Forthcoming at Journal of Econometrics.

Bailey, N., Kapetanios, G., and Pesaran, M.H. (2021) Measurement of factor
strength: Theory and practice. Journal of Applied Econometrics, 36(5),
587–613.

Chao J.C., Liu, Y., and Swanson, N.R. (2022) Consistent factor estimation and
forecasting in factor-augmented VAR models. Working Paper, Maryland
University.

Chao J.C. and Swanson, N.R. (2022) Selecting the relevant variables for factor
estimation in FAVAR models. Working Paper, Maryland University.

Choi, I., Lin, R., and Shin, Y. (2023) Canonical correlation-based model
selection for the multilevel factors. Journal of Econometrics, 233(1), 22-44.

Doz, C., Giannone, D., and Reichlin, L. (2012) A quasi-maximum likelihood
approach for large, approximate dynamic factor models. The Review of Economics
and Statistics, 94(4), 1014–1024.

Despois, T. and Doz, C. (2022). Identifying and interpreting the factors in
factor models via sparsity: Different approaches. Forthcoming at Journal of
Applied Econometrics.

Fan, J., Liao, Y., and Mincheva, M. (2011) High-dimensional covariance matrix
estimation in approximate factor models. Annals of Statistics, 39(6),
3320–3356.

Fan, J., Liao, Y., and Yao, J. (2015) Power enhancement in high-dimensional
cross sectional tests. Econometrica, 83(4), 1497–1541.

Freyaldenhoven, S. (2022a) Factor models with local factors determining the
number of relevant factors. Journal of Econometrics, 229(1), 80–102.

Freyaldenhoven, S. (2022b) Identification through sparsity in factor models:
The $\ell_{1}$-rotation criterion. Working Paper, Federal Reserve Bank of
Philadelphia.

Giglio, S. and Xiu, D. (2021) Asset pricing with omitted factors. Journal of
Political Economy, 129(7), 1947–1990.

Giglio, S., Xiu, D., and Zhang, D. (2022a) Test assets and weak factors.
Working Paper, NBER.

Giglio, S., Xiu, D., and Zhang, D. (2022b) Prediction when factors are weak.
Working Paper, NBER.

Guo, X., Chen, Y., and Tang, C. Y. (2023) Information criteria for latent
factor models: A study on factor pervasiveness and adaptivity. Journal of
Econometrics, 233(1), 237-250.

Huang, D., Jiang, F., Li, K., Tong, G., and Zhou, G. (2022) Scaled PCA: A new
approach to dimension reduction. Management Science, 68(3), 1678–1695.

Kristensen, J.T. (2017) Diffusion indexes with sparse loadings. Journal of
Business & Economic Statistics 35(3), 434–451

Ludvigson, S.C. and Ng, S. (2009) Macro factors in bond risk premia. The
Review of Financial Studies, 22(12), 5027-5067.

Ma, S., Linton, O., and Gao, J. (2021) Estimation and inference in
semiparametric quantile factor models. Journal of Econometrics, 222(1),
295–323.

McCracken, M., and Ng, S. (2016). FRED-MD: A Monthly database for
macroeconomic research. Journal of Business and Economic Statistics, 34(4),
574–589.

McCracken, M., and Ng, S. (2021). FRED-QD: A quarterly database for
macroeconomic research. Federal Reserve Bank of St. Louis Review, 103(1),
1–44.

Merlevede, F., Peligrad, M., and Rio, E., (2011) A Bernstein type inequality
and moderate deviations for weakly dependent sequences. Probability Theory and
Related Fields, 151, 435–474

Moon, H.R. and Weidner, M. (2017) Dynamic linear panel regression models with
interactive fixed effects. Econometric Theory, 33(1), 158–195.

Onatski, A. (2010) Determining the number of factors from empirical
distribution of eigenvalues. The Review of Economics and Statistics, 92(4),
1004–1016.

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, A. (2015) Asymptotic analysis of the squared estimation error in
misspecified factor models. Journal of Econometrics, 186(2), 388–406.

Pelger, M. and Xiong, R. (2022) Interpretable Sparse Proximate Factors for
Large Dimensions. Journal of Business and Economic Statistic, 40(4),
1642–1664.

Rapach, D. and Zhou, G. (2021) Sparse macro factors. Working paper, Washington
University in St. Louis.

Stock, J.H. and Watson, M.W. (2002) Macroeconomic forecasting using diffusion
indexes. Journal of Business & Economic Statistics, 20(2), 147–162.

Su, L. and Wang, X. (2017) On time-varying factor models: Estimation and
testing. Journal of Econometrics, 198(1), 84–101.

Uematsu, Y. and Yamagata, T. (2023a) Estimation of sparsity-induced weak
factor models. Journal of Business & Economic Statistics, 41(1), 126–139.

Uematsu, Y. and Yamagata, T. (2023b) Inference in sparsity-induced weak factor
models. Journal of Business & Economic Statistics, 41(1), 213–227.

Wei, J., and Chen, H. (2020) Determining the number of factors in approximate
factor models by twice K-fold cross validation. Economics Letters, 191,
109149.

APPENDIX

(Online)

The appendix provides formal proofs of main results in Sections 3-5 as well as
auxiliary lemmas. We will use the fact that
$\|B\|_{sp}\leq\|B\|_{F}\leq\|B\|_{sp}\mathop{\mathrm{r}ank}(B)^{1/2}$ over
several places in our proofs.

##  Appendix A Proofs of the main results in Section 3

To start with, we present several useful lemmas which will be used frequently
in the proofs of main results. Their proofs can be found in Section D.

###### Lemma A.1

Under Assumptions 1-5, $\Xi=\widetilde{V}^{-1}QAN^{-1}$ is a full rank (block)
lower triangular matrix, and $\Xi=O_{p}(1).$

###### Lemma A.2

Under Assumptions 1-5,
$\frac{1}{NT}\left\|e^{\prime}\Lambda^{0}F^{0\prime}\widetilde{F}\widetilde{V}^{-1}\right\|=O_{p}\left(N^{-\frac{\alpha_{r}}{2}}T^{\frac{1}{2}}\right).$

###### Lemma A.3

Under Assumptions 1-5,
$\frac{1}{NT}ee^{\prime}\Lambda^{0}F^{0\prime}\widetilde{F}\widetilde{V}^{-1}=A^{-\frac{1}{2}}O_{p}(N+T).$

###### Lemma A.4

Under Assumptions 1-5, the $r\times r$ matrix $G\equiv
F^{0\prime}\widetilde{F}/T$ is of full rank $r$ in probability.

Next we provide the proofs for the main results on PC estimator.

Proof of Proposition 3.1. Given $X=\Lambda^{0}F^{0\prime}+e,$ we get

$\frac{1}{NT}X^{\prime}X=\frac{1}{NT}F^{0}\Lambda^{0\prime}\Lambda^{0}F^{0\prime}+\frac{1}{NT}F^{0}\Lambda^{0\prime}e+\frac{1}{NT}e^{\prime}\Lambda^{0}F^{0\prime}+\frac{1}{NT}e^{\prime}e.$

Multiplying both sides by $N/N^{\alpha_{r}}$,

$\frac{1}{N^{\alpha_{r}}T}X^{\prime}X=\frac{1}{N^{\alpha_{r}}T}F^{0\prime}\Lambda^{0\prime}\Lambda^{0}F^{0\prime}+\frac{1}{N^{\alpha_{r}}T}F^{0}\Lambda^{0\prime}e+\frac{1}{N^{\alpha_{r}}T}e^{\prime}\Lambda^{0}F^{0\prime}+\frac{1}{N^{\alpha_{r}}T}e^{\prime}e.$

Note that the Frobenius norm of the second and third terms on the RHS is
$o_{p}\left(1\right)$ as

$\displaystyle\frac{1}{N^{\alpha_{r}}T}\left\|F^{0}\Lambda^{0\prime}e\right\|$
$\displaystyle\leqslant$
$\displaystyle\frac{1}{N^{\alpha_{r}}T}\left\|F^{0}\right\|\left\|\Lambda^{0\prime}e\right\|=\frac{1}{N^{\alpha_{r}}T}O_{p}\left(T^{\frac{1}{2}}\right)O_{p}\left(N^{\frac{\alpha_{1}}{2}}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(N^{\frac{\alpha_{1}}{2}-\alpha_{r}}T^{-\frac{1}{2}}\right)=o_{p}(1).$

This implies that

$\frac{1}{N^{\alpha_{r}}T}\left\|F^{0}\Lambda^{0\prime}e\right\|_{sp}=\frac{1}{N^{\alpha_{r}}T}\left\|e^{\prime}\Lambda^{0}F^{0\prime}\right\|_{sp}=o_{p}(1).$

For the fourth term,

$\frac{1}{N^{\alpha_{r}}T}\left\|e^{\prime}e\right\|_{sp}=\frac{1}{N^{\alpha_{r}}T}O_{p}(N+T)=O_{p}\left(N^{1-\alpha_{r}}T^{-1}+N^{-\alpha_{r}}\right)=o_{p}(1).$

Hence, the $r$ eigenvalues of $\frac{1}{N^{\alpha_{r}}T}X^{\prime}X$ are equal
to the $r$ eigenvalues of
$\frac{1}{N^{\alpha_{r}}T}F^{0}\Lambda^{0\prime}\Lambda^{0}F^{0\prime}$ in
probability. Given that the eigenvalues of matrix $AB$ and those of $BA$ are
identical, the $r$ eigenvalue of $\frac{1}{N^{\alpha_{r}}T}X^{\prime}X$ are
determined in probability by

$\left(\frac{1}{N^{\alpha_{r}}}\Lambda^{0\prime}\Lambda^{0}\right)\left(\frac{1}{T}F^{0\prime}F^{0}\right)=\frac{1}{N^{\alpha_{r}}}A^{1/2}\left(A^{-1/2}\Lambda^{0}\Lambda^{0}A^{-1/2}\right)A^{1/2}\left(\frac{1}{T}F^{0\prime}F^{0}\right).$

By Assumptions 1-2, $A^{-1/2}\Lambda^{0\prime}\Lambda^{0}A^{-1/2}$ and
$T^{-1}F^{0\prime}F^{0}$ are both converging to p.d. matrices. It follows that
the $k$th eigenvalue of $\frac{1}{N^{\alpha_{r}}T}X^{\prime}X$

$\mu_{k}\left(\frac{1}{N^{\alpha_{r}}T}X^{\prime}X\right)\asymp_{p}N^{-\alpha_{r}}A_{k}=N^{-\alpha_{r}}N^{\alpha_{k}},\text{
for }k=1,\cdots,r.$

Thus,

$\widetilde{V}_{k}=\mu_{k}\left(\frac{1}{NT}X^{\prime}X\right)\asymp_{p}N^{\alpha_{r}-1}N^{-\alpha_{r}}A_{k}=N^{\alpha_{k}-1},\text{
for }k=1,\cdots,r.\hskip 5.0pt\hbox{\hskip 5.0pt\vrule
width=4.0pt,height=6.0pt,depth=1.5pt\hskip 1.0pt}$

Proof of Proposition 3.2. For the so defined $G=F^{0\prime}\widetilde{F}/T,$
recall that we have already shown in proving Lemma A.4 that (a) $H$ is of full
rank $r$ in probability, and (b) $H=O_{p}(1)$. Also recall that
$\Sigma_{N,\Lambda}(k,k)\asymp N^{\alpha_{k}-1}$,
$\Sigma_{N,\Lambda}(l,k)=O\left(N^{\alpha_{l\vee k}-1}\right)$,141414Recall
that $\alpha_{l}\leq\alpha_{k}$ iff $l\geq k$. and
$\widetilde{V}_{k}\asymp_{p}N^{\alpha_{k}-1}$. Hence for
$H=\Sigma_{N,\Lambda}G\widetilde{V}^{-1}$ to hold with the asymptotic
properties (a) and (b) just mentioned above, the $r\times r$ matrix $G$ must
be such that

$G(k,l)=O_{p}\left(N^{\alpha_{l}-\alpha_{k}}\right),\text{ for }1\leq k<l\leq
r.$

This clearly demonstrates that matrix $G$ is a (block) lower triangular matrix
in probability. Specifically, for $1\leq k<l\leq r$ such that
$\alpha_{l}<\alpha_{k},$ it must be that
$G(k,l)=O_{p}\left(N^{\alpha_{l}-\alpha_{k}}\right)=o_{p}(1).$ Moreover, since
$G$ is also of full rank of $r$ asymptotically as proved in Lemma A.4, we also
have that $G(k,k)\asymp_{p}1$. Therefore, $G$ is a non-strictly (block) lower
triangular matrix. Noticing that $Q=G^{\prime},$ we are done.

Proof of Proposition 3.3. Note that

$\widetilde{F}-F^{0}H=\left(\frac{F^{0}\Lambda^{0\prime}e\widetilde{F}}{NT}+\frac{e^{\prime}\Lambda^{0}F^{0\prime}\widetilde{F}}{NT}+\frac{e^{\prime}e\widetilde{F}}{NT}\right)\widetilde{V}^{-1}.$

This implies

$\displaystyle\frac{1}{\sqrt{T}}\left\|\widetilde{F}-F^{0}H\right\|$
$\displaystyle\leq\left\\{\left(\frac{\left\|F^{0}\right\|\|\widetilde{F}\|}{T}\right)\left(\frac{\left\|\Lambda^{0\prime}e\right\|}{\sqrt{T}N}\right)\left\|\widetilde{V}^{-1}\right\|+\frac{\left\|e^{\prime}\Lambda^{0}F^{0\prime}\widetilde{F}\widetilde{V}^{-1}\right\|}{NT\sqrt{T}}+\frac{\left\|e^{\prime}e\widetilde{F}\right\|}{NT^{3/2}}\left\|\widetilde{V}^{-1}\right\|\right\\}$
$\displaystyle=O_{p}\left(\frac{\left\|\Lambda^{0\prime}e\right\|}{\sqrt{T}N}\left\|\widetilde{V}^{-1}\right\|\right)+O_{p}\left(\frac{\left\|e^{\prime}\Lambda^{0}F^{0\prime}\widetilde{F}\widetilde{V}^{-1}\right\|}{NT\sqrt{T}}\right)+O_{p}\left(\frac{\left\|e^{\prime}e\widetilde{F}\right\|}{NT^{3/2}}\left\|\widetilde{V}^{-1}\right\|\right).$

Note that
$\frac{1}{\sqrt{T}N}\left\|\Lambda^{0\prime}e\right\|=O_{p}\left(N^{\alpha_{1}/2-1}\right).$
Also,
$\frac{\left\|ee^{\prime}\widetilde{F}\right\|}{NT^{3/2}}\leq\frac{\left\|ee^{\prime}\right\|_{sp}\|\widetilde{F}\|}{NT^{3/2}}=O_{p}\left(\frac{1}{N}\right)+O_{p}(\frac{1}{T}).$
Together with Lemma A.2 and Proposition 3.1, it leads to that

$\frac{1}{T}\left\|\widetilde{F}-F^{0}H\right\|^{2}=O_{p}\left(N^{2\left(1-\alpha_{r}\right)}\right)O_{p}\left(N^{\alpha_{1}-2}+T^{-2}\right)=O_{p}\left(N^{\alpha_{1}-2\alpha_{r}}+N^{2\left(1-\alpha_{r}\right)}T^{-2}\right).\hskip
5.0pt\hbox{\hskip 5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip
1.0pt}$

Proof of Proposition 3.4. First, note that

$\widetilde{\Lambda}-\Lambda^{0}Q^{\prime}=\frac{1}{T}e\left(\widetilde{F}-F^{0}H\right)+\frac{1}{T}eF^{0}H.$

Then let

$\displaystyle L_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{T}e\left(\widetilde{F}-F^{0}H\right)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{NT}ee^{\prime}\Lambda^{0}\right)\left(\frac{1}{T}F^{0^{\prime}}\widetilde{F}\right)\widetilde{V}^{-1}+\frac{1}{NT^{2}}eF^{0}\Lambda^{0\prime}e\widetilde{F}\widetilde{V}^{-1}+\frac{1}{NT^{2}}ee^{\prime}e\widetilde{F}\widetilde{V}^{-1}$
$\displaystyle\equiv$ $\displaystyle L_{11}+L_{12}+L_{13}.$

Note

$L_{11}=\frac{1}{NT}\left(ee^{\prime}\right)\Lambda^{0}Q^{\prime}\widetilde{V}^{-1}=\frac{1}{NT}O_{p}(N+T)\left(\Lambda^{0}A^{-\frac{1}{2}}\right)\left(A^{-\frac{1}{2}}N\right)\left(AN^{-1}Q^{\prime}\widetilde{V}^{-1}\right).$

By Assumption 2, $\left\|\Lambda^{0}A^{-\frac{1}{2}}\right\|=O\left(1\right).$
By Lemma A.1, $AN^{-1}Q^{\prime}\widetilde{V}^{-1}=O_{p}\left(1\right).$ So,

$\left\|L_{11}\right\|\leq\frac{1}{NT}O_{p}(N+T)O_{p}\left(A^{-\frac{1}{2}}N\right)=O_{p}\left(N^{-\alpha_{r}}+N^{1-\frac{\alpha_{r}}{2}}T^{-1}\right).$

Next for $L_{12},$
$L_{12}=\frac{1}{NT^{2}}\left(eF^{0}\right)\left(\Lambda^{0^{\prime}}eF^{0}H\right)\widetilde{V}^{-1}+\frac{1}{NT^{2}}eF^{0}\Lambda^{0^{\prime}}e\left(\widetilde{F}-F^{0}H\right)\widetilde{V}^{-1},$
so

$\left\|L_{12}\right\|=\frac{1}{NT^{2}}O_{p}(\sqrt{NT})O_{p}\left(\sqrt{N^{\alpha_{1}}T}\right)O_{p}\left(N^{1-\alpha_{r}}\right)\left[1+O_{p}\|\widetilde{F}-F^{0}H\|\right]$

Given that
$\left\|\widetilde{F}-F^{0}H\right\|=O_{p}\left(\sqrt{TN^{\alpha_{1}-2\alpha_{r}}}+\sqrt{N^{2(1-2\alpha_{r})}T^{-1}}\right)$
by Proposition 3.3, we have

$\left\|L_{12}\right\|=O_{p}\left(N^{\frac{\alpha_{1}}{2}-\alpha_{r}+\frac{1}{2}}T^{-1}+N^{\alpha_{1}-2\alpha_{r}+\frac{1}{2}}T^{-\frac{1}{2}}\right).$

Lastly for $L_{13},$

$\left\|L_{13}\right\|=\frac{1}{NT^{2}}\left\|e\right\|_{sp}^{3}O_{p}\left(T^{\frac{1}{2}}\right)O_{p}\left(N^{1-\alpha_{r}}\right)=O_{p}\left(T^{-\frac{3}{2}}N^{\frac{3}{2}-\alpha_{r}}+N^{-\alpha_{r}}\right).$

Given Assumption 5$,\quad N^{\frac{3}{2}-\alpha_{r}}T^{-\frac{3}{2}}\ll
N^{1-\frac{\alpha_{r}}{2}}T^{-1}.$ Also, it’s easy to see
$N^{1-\frac{\alpha_{r}}{2}}\gg N^{\alpha_{1}-2\alpha_{r}+\frac{1}{2}}.$ So it
follows that

$L_{1}=O_{p}\left(N^{-\frac{\alpha_{r}}{2}}+N^{1-\frac{\alpha_{r}}{2}}T^{-1}+N^{\alpha_{1}-2\alpha_{r}+\frac{1}{2}}T^{-\frac{1}{2}}\right).$
(A.1)

Meanwhile, as
$T^{-1}eF^{0}H=O_{P}\left(N^{\frac{1}{2}}T^{-\frac{1}{2}}\right)$ implied by
Assumption 3 (vii), and $H=O_{p}\left(1\right)$ as proved in Lemma A.4,

$\left\|\widetilde{\Lambda}-\Lambda^{0}Q^{\prime}\right\|=O_{p}\left(N^{-\frac{\alpha_{r}}{2}}+N^{1-\frac{\alpha_{r}}{2}}T^{-1}+N^{\frac{1}{2}}T^{-\frac{1}{2}}\right)=O_{p}\left(N^{-\frac{\alpha_{r}}{2}}+N^{\frac{1}{2}}T^{-\frac{1}{2}}\right).$

So we have
$N^{-1}\left\|\widetilde{\Lambda}-\Lambda^{0}Q^{\prime}\right\|^{2}=O_{p}\left(N^{-\alpha_{r}-1}+T^{-1}\right).$

Proof of Lemma 3.5. First, note that from
$\frac{1}{NT}X^{\prime}X\widetilde{F}=\widetilde{F}\widetilde{V},$ we have
$\frac{1}{NT}\left(\Lambda^{0}F^{0\prime}+e\right)^{\prime}(\Lambda^{0}F^{0\prime}+e)\widetilde{F}=\widetilde{F}\widetilde{V}.$
That is,

$\widetilde{F}\widetilde{V}=\frac{1}{NT}F^{0}\Lambda^{0\prime}\Lambda^{0}\widetilde{F}+\frac{1}{NT}F^{0}\Lambda^{0\prime}e\widetilde{F}+\frac{1}{NT}e^{\prime}\Lambda^{0}F^{0\prime}\widetilde{F}+\frac{1}{NT}e^{\prime}e\widetilde{F}.$

Thus, multiplying $\widetilde{F}^{\prime}/T$ on both sides of the above
equation leads to

$\widetilde{V}=\frac{1}{T}\widetilde{F}^{\prime}F^{0}\frac{1}{N}\Lambda^{0\prime}\Lambda^{0}\frac{1}{T}F^{0\prime}\widetilde{F}+\frac{1}{NT^{2}}\widetilde{F}^{{}^{\prime}}F^{0\prime}\Lambda^{0\prime}e\widetilde{F}+\frac{1}{NT^{2}}\widetilde{F}^{\prime}e^{\prime}\Lambda^{0}F^{0\prime}\widetilde{F}+\frac{1}{NT^{2}}\widetilde{F}^{\prime}e^{\prime}e\widetilde{F},$

which gives rise to

$\displaystyle I_{r}$
$\displaystyle=QH+\frac{1}{NT^{2}}\widetilde{F}^{\prime}F^{0\prime}\Lambda^{0\prime}e\widetilde{F}\widetilde{V}^{-1}+\frac{1}{NT^{2}}\widetilde{F}^{\prime}e^{\prime}\Lambda^{0}F^{0\prime}\widetilde{F}\widetilde{V}^{-1}+\frac{1}{NT^{2}}\widetilde{F}^{\prime}e^{\prime}e\widetilde{F}\widetilde{V}^{-1}$
$\displaystyle\equiv QH+b_{1}+b_{2}+b_{3},\text{ say.}$

It is easy to show that
$b_{1}=O_{p}\left(N^{\frac{\alpha_{1}}{2}-1}\right)O_{p}\left(N^{1-a_{r}}\right)O_{p}\left(T^{-1/2}\right)=O_{p}\left(N^{\frac{\alpha_{1}}{2}-a_{r}}T^{-1/2}\right).$
The same result also holds for $b_{2}.$ Meanwhile,

$b_{3}=O_{p}\left(1\right)\frac{1}{NT}\left\|e^{\prime}e\right\|_{sp}\widetilde{V}^{-1}=O_{p}\left(N^{-1}+T^{-1}\right)O_{p}\left(N^{1-a_{r}}\right)=O_{p}\left(N^{-a_{r}}+N^{1-a_{r}}T^{-1}\right).$

So it follows that

$I_{r}=QH+O_{p}\left(N^{\frac{\alpha_{1}}{2}-a_{r}}T^{-1/2}+N^{-a_{r}}+N^{1-a_{r}}T^{-1}\right)=QH+O_{p}\left(\gamma_{NT}\right),$
(A.2)

implying that $Q=H^{-1}+O_{p}\left(\gamma_{NT}\right).$

Second, notice that (A.2) also implies that
$H=Q^{-1}+O_{p}\left(\gamma_{NT}\right)=H_{3}+O_{p}\left(\gamma_{NT}\right).$

Third, recall by definition,
$\widetilde{\Lambda}=\frac{1}{T}\Lambda^{0}F^{0^{\prime}}\widetilde{F}+\frac{1}{T}e\widetilde{F}.$
Therefore, we have

$\frac{\Lambda^{0\prime}\widetilde{\Lambda}}{N}=\frac{\Lambda^{0\prime}\Lambda^{0}}{N}\frac{F^{0\prime}\widetilde{F}}{T}+\frac{\Lambda^{0\prime}e\widetilde{F}}{NT}.$

So it is followed by

$\left(\Lambda^{0\prime}\Lambda^{0}\right)^{-1}\Lambda^{0\prime}\widetilde{\Lambda}=F^{0\prime}\widetilde{F}/T+\left(\Lambda^{0\prime}\Lambda^{0}\right)^{-1}\Lambda^{0\prime}e\widetilde{F}/T=F^{0\prime}\widetilde{F}/T+\mathcal{R}_{1},$

where
$\mathcal{R}_{1}\equiv\left(\Lambda^{0^{\prime}}\Lambda^{0}\right)^{-1}\Lambda^{0\prime}e\widetilde{F}/T=\left(\Lambda^{0\prime}\Lambda^{0}A^{-1/2}\right)^{-1}A^{-1/2}\Lambda^{0\prime}e\widetilde{F}/T.$
We further note that
$A^{-1/2}\Lambda^{0\prime}e\widetilde{F}/T=A^{-\frac{1}{2}}\Lambda^{0\prime}e(\widetilde{F}-F^{0}H)/T+A^{-\frac{1}{2}}\Lambda^{0\prime}eF^{0}H/T.$
It is followed by

$\displaystyle\left\|A^{-1/2}\Lambda^{0\prime}e\widetilde{F}/T\right\|$
$\displaystyle\leq$
$\displaystyle\left\|A^{-\frac{1}{2}}\Lambda^{0\prime}e\right\|\left\|(\widetilde{F}-F^{0}H)/T\right\|+\left\|A^{-\frac{1}{2}}\Lambda^{0\prime}eF^{0}/T\right\|\left\|H\right\|$
$\displaystyle=$ $\displaystyle
O_{p}\left(N^{\frac{\alpha_{1}}{2}-\alpha_{r}}T^{-1/2}+N^{1-\alpha_{r}}T^{-\frac{3}{2}}\right)+O_{p}\left(T^{-1/2}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(N^{\frac{\alpha_{1}}{2}-\alpha_{r}}T^{-1/2}+N^{1-\alpha_{r}}T^{-\frac{3}{2}}\right)=O_{p}\left(\gamma_{NT}\right).$

The results above imply that
$T^{-1}\widetilde{F}^{\prime}F^{0}=\widetilde{\Lambda}^{\prime}\Lambda^{0}\left(\Lambda^{0\prime}\Lambda^{0}\right)^{-1}+O_{p}\left(\gamma_{NT}\right).$
Again, recall the definitions that $Q=T^{-1}\widetilde{F}^{\prime}F^{0},$ and
$H_{1}=\left(\Lambda^{0\prime}\Lambda^{0}\right)^{-1}\Lambda^{0\prime}\widetilde{\Lambda}$,
together with $Q=H^{-1}+O_{p}\left(\gamma_{NT}\right)$ proved previously, we
come to that $H_{1}=H+O_{p}\left(\gamma_{NT}\right).$

Fourth, notice immediately that
$\Lambda^{0\prime}\widetilde{\Lambda}/N=\left(\Lambda^{0\prime}\Lambda/N\right)\left(F^{0\prime}\widetilde{F}/T\right)+\frac{1}{NT}\Lambda^{0\prime}e\widetilde{F}$,
and we post-multiply
$(\widetilde{\Lambda}^{\prime}\widetilde{\Lambda}/N)^{-1}$ on both sides of
the equation, making use of
$\widetilde{\Lambda}^{\prime}\widetilde{\Lambda}/N=\widetilde{V}$, to get that

$\displaystyle H_{4}$ $\displaystyle\equiv$
$\displaystyle\left(\Lambda^{0\prime}\widetilde{\Lambda}\right)\left(\widetilde{\Lambda}^{\prime}\widetilde{\Lambda}\right)^{-1}$
$\displaystyle=$ $\displaystyle
H+\frac{1}{NT}\Lambda^{0\prime}e\widetilde{F}\left(\frac{1}{N}\widetilde{\Lambda}^{\prime}\widetilde{\Lambda}\right)^{-1}$
$\displaystyle=$ $\displaystyle
H+\left[\frac{1}{NT}\Lambda^{0\prime}e\left(\widetilde{F}-F^{0}H\right)+\frac{1}{NT}\Lambda^{0^{\prime}}eF^{0}H\right]O_{p}\left(N^{1-\alpha_{r}}\right)$
$\displaystyle=$ $\displaystyle
H+O_{p}\left(N^{\frac{\alpha_{1}}{2}-1}N^{\frac{\alpha_{1}}{2}-a_{r}}T^{-1/2}+N^{\frac{\alpha_{1}}{2}-1}N^{1-a_{r}}T^{-3/2}+N^{\frac{\alpha_{1}}{2}-1}T^{-1/2}\right)O_{p}\left(N^{1-\alpha_{r}}\right)$
$\displaystyle=$ $\displaystyle
H+O_{p}\left(N^{\frac{\alpha_{1}}{2}-a_{r}}T^{-1/2}\right)=H+O_{p}\left(\gamma_{NT}\right).$

Lastly, to deal with $H_{2}$, we again start by definition

$\widetilde{F}=\frac{1}{N}X^{\prime}\widetilde{\Lambda}\widetilde{V}^{-1}=\frac{1}{N}F^{0}\Lambda^{0\prime}\widetilde{\Lambda}\widetilde{V}^{-1}+\frac{1}{N}e^{\prime}\widetilde{\Lambda}\widetilde{V}^{-1}.$

Now, if we pre-multiply $\frac{1}{T}F^{0\prime}$ on both sides, we get

$\displaystyle\frac{1}{T}F^{0\prime}\widetilde{F}$ $\displaystyle=$
$\displaystyle\frac{1}{T}F^{0\prime}F^{0}\left(\frac{1}{N}\Lambda^{0\prime}\widetilde{\Lambda}\right)\widetilde{V}^{-1}+\frac{1}{NT}F^{0\prime}e^{\prime}\widetilde{\Lambda}\widetilde{V}^{-1}$
$\displaystyle=$
$\displaystyle\left(\frac{1}{T}F^{0\prime}F^{0}\right)\left(\frac{1}{N}\Lambda^{0\prime}\widetilde{\Lambda}\right)\widetilde{V}^{-1}+\mathcal{R}_{2},$

where

$\displaystyle\mathcal{R}_{2}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{NT}F^{0\prime}e^{\prime}\left(\widetilde{\Lambda}-\Lambda^{0}Q^{\prime}\right)\widetilde{V}^{-1}+\frac{1}{NT}F^{0\prime}e^{\prime}\Lambda^{0}Q^{\prime}\widetilde{V}^{-1}$
$\displaystyle=$
$\displaystyle\frac{1}{NT}F^{0\prime}e^{\prime}\left(\widetilde{\Lambda}-\Lambda^{0}Q^{\prime}\right)\widetilde{V}^{-1}+\frac{1}{T}F^{0\prime}e^{\prime}\Lambda^{0}A^{-1}\left(AN^{-1}Q^{\prime}\widetilde{V}^{-1}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(N^{-\frac{1}{2}}T^{-\frac{1}{2}}\right)O_{p}\left(N^{-\frac{\alpha_{r}}{2}}+N^{\frac{1}{2}}T^{-\frac{1}{2}}\right)O_{p}\left(N^{1-\alpha_{r}}\right)+O_{p}\left(N^{-\frac{\alpha_{r}}{2}}T^{-\frac{1}{2}}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(N^{\frac{1}{2}-\frac{3}{2}\alpha_{r}}T^{-\frac{1}{2}}+N^{1-\alpha_{r}}T^{-1}\right)+O_{p}\left(N^{-\frac{\alpha_{r}}{2}}T^{-\frac{1}{2}}\right)=O_{p}\left(\gamma_{NT}\right),$

and the third equality is by Theorem 3.4, Lemma A.1 and Assumption 3 (v). It
follows that

$\underbrace{\left(F^{0\prime}F^{0}\right)^{-1}F^{0\prime}\widetilde{F}}_{:=H_{2}}=\underbrace{\left(\Lambda^{0\prime}\widetilde{\Lambda}\right)\left(\widetilde{\Lambda}^{\prime}\widetilde{\Lambda}\right)^{-1}}_{:=H_{4}}+O_{p}\left(\gamma_{NT}\right).$

Putting things together, we have shown (i)
$H=H_{3}+O_{P}\left(\gamma_{NT}\right),$ (ii)
$H_{1}=H+O_{p}\left(\gamma_{NT}\right),$ (iii)
$H_{4}=H+O_{p}\left(\gamma_{NT}\right),$ (iv)
$H_{2}=H_{4}+O_{p}\left(\gamma_{NT}\right).$ In all, we conclude that
$H_{l}=H+O_{p}\left(\gamma_{NT}\right),$ $l=1,\cdots,4.$

Proof of Proposition 3.6. Now we have

$\displaystyle\frac{1}{NT}\left\|\widetilde{C}-C^{0}\right\|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{NT}\left\|\widetilde{\Lambda}\widetilde{F}^{\prime}-\Lambda^{0}H^{\prime-1}H^{\prime}F^{0\prime}\right\|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{NT}\left\|\left(\widetilde{\Lambda}-\Lambda^{0}H^{\prime-1}+\Lambda^{0}H^{\prime-1}\right)\widetilde{F}^{\prime}-\Lambda^{0}H^{\prime-1}H^{\prime}F^{0\prime}\right\|^{2}$
$\displaystyle\leq$
$\displaystyle\frac{2}{NT}\left\|\Lambda^{0}H^{\prime-1}\left(\widetilde{F}-F^{0}H\right)^{\prime}\right\|^{2}+\frac{2}{NT}\left\|\widetilde{\Lambda}-\Lambda^{0}H^{\prime-1}\right\|^{2}\|\widetilde{F}\|^{2}$
$\displaystyle=$ $\displaystyle
O_{p}\left(\frac{1}{N}\left\|\Lambda^{0}\right\|^{2}\right)O_{p}\left(\frac{1}{T}\left\|\widetilde{F}-F^{0}H\right\|^{2}\right)$
$\displaystyle+O_{p}\left(\frac{1}{N}\left\|\widetilde{\Lambda}-\Lambda^{0}Q^{{}^{\prime}}\right\|^{2}\right)+O_{p}\left(\left\|Q-H^{-1}\right\|^{2}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(N^{\alpha_{1}-1}\right)O_{p}\left(N^{\alpha_{1}-2\alpha_{r}}+N^{2\left(1-\alpha_{r}\right)}T^{-2}\right)$
$\displaystyle+O_{p}\left(N^{-\alpha_{r}-1}+T^{-1}\right)+O_{p}\left(N^{\alpha_{1}-2\alpha_{r}}T^{-1}+N^{2\left(1-\alpha_{r}\right)}T^{-2}+N^{-2\alpha_{r}}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(N^{2\left(\alpha_{1}-\alpha_{r}\right)-1}+T^{-1}\right).$

The second last equality is due to Propositions 3.3, 3.4, and Lemma 3.5. The
last equality is due to Assumption 5.

Proof of Theorem 3.7. Let us start from the definition,
$\widetilde{F}=\frac{1}{N}X^{\prime}\widetilde{\Lambda}\widetilde{V}^{-1}.$
Then by plugging $X=\Lambda^{0}F^{0\prime}+e,$ we get
$\widetilde{F}=\frac{1}{N}F^{0}\Lambda^{0\prime}\widetilde{\Lambda}\widetilde{V}^{-1}+\frac{1}{N}e^{\prime}\widetilde{\Lambda}\widetilde{V}^{-1}.$
That is,
$\widetilde{F}_{t}=H_{4}^{\prime}F_{t}^{0}+\widetilde{V}^{-1}\frac{1}{N}\widetilde{\Lambda}^{\prime}e_{t}$.
So

$\widetilde{F}_{t}-H_{4}^{\prime}F_{t}^{0}=\widetilde{V}^{-1}\frac{1}{N}\left(\widetilde{\Lambda}-\Lambda^{0}Q\right)^{\prime}e_{t}+\widetilde{V}^{-1}\frac{1}{N}Q\Lambda^{0\prime}e_{t}.$

We first consider the first term on the RHS of the above equation.

$\widetilde{V}^{-1}\frac{1}{N}\left(\widetilde{\Lambda}-\Lambda^{0}Q\right)^{\prime}e_{t}=\widetilde{V}^{-1}\frac{1}{N}\left[\frac{1}{T}e\left(\widetilde{F}-F^{0}H\right)\right]^{\prime}e_{t}+\widetilde{V}^{-1}\frac{1}{N}\left(\frac{1}{T}eF^{0}H\right)^{\prime}e_{t}\equiv
W_{t}+Z_{t},\text{ say.}$

For $W_{t},$ we have a further decomposition as

$\displaystyle W_{t}$ $\displaystyle=$
$\displaystyle\widetilde{V}^{-1}\frac{1}{N}\left(\frac{eF^{0}\Lambda^{0\prime}e\widetilde{F}\widetilde{V}^{-1}}{NT^{2}}\right)^{\prime}e_{t}+\widetilde{V}^{-1}\frac{1}{N}\left(\frac{ee^{\prime}\Lambda^{0}F^{0\prime}\widetilde{F}\widetilde{V}^{-1}}{NT^{2}}\right)^{\prime}e_{t}+\widetilde{V}^{-1}\frac{1}{N}\left(\frac{ee^{\prime}e\widetilde{F}\widetilde{V}^{-1}}{NT^{2}}\right)^{\prime}e_{t}$
$\displaystyle\equiv$ $\displaystyle
W_{t}^{(a)}+W_{t}^{(b)}+W_{t}^{(c)},\text{ say.}$

For
$W_{t}^{(a)}=\frac{1}{N^{2}T^{2}}\widetilde{V}^{-2}\widetilde{F}^{\prime}e^{\prime}\Lambda^{0}F^{0\prime}e^{\prime}e_{t},$
we first have
$\frac{1}{NT}F^{0\prime}e^{\prime}e_{t}=O_{p}\left(\left(NT\right)^{-1/2}+T^{-1}\right)$
by Assumption 3 (vi). Also recall that
$\left\|e^{\prime}\Lambda^{0}\right\|=O_{p}\left(N^{\alpha_{1}/2}T^{1/2}\right).$
Hence,

$\displaystyle W_{t}^{(a)}$ $\displaystyle=$
$\displaystyle\frac{1}{NT}\widetilde{V}^{-2}O_{p}\left(T^{\frac{1}{2}}\right)O_{p}\left(N^{\frac{\alpha_{1}}{2}}T^{\frac{1}{2}}\right)O_{p}\left(\frac{1}{\sqrt{NT}}+\frac{1}{T}\right)$
$\displaystyle=$
$\displaystyle\widetilde{V}^{-2}O_{p}\left(N^{\frac{\alpha_{1}-3}{2}}T^{-\frac{1}{2}}+N^{\frac{\alpha_{1}}{2}-1}T^{-1}\right).$

For $W_{t}^{(b)}$, by Lemma A.3,

$W_{t}^{(b)}=\widetilde{V}^{-1}\frac{1}{NT}A^{-\frac{1}{2}}O_{p}(N+T)O_{p}(\sqrt{N})=A^{-\frac{1}{2}}\widetilde{V}^{-1}O_{p}\left(N^{-\frac{1}{2}}+N^{\frac{1}{2}}T^{-1}\right).$

For $W_{t}^{(c)}$,

$\displaystyle W_{t}^{(c)}$ $\displaystyle=$
$\displaystyle\widetilde{V}^{-2}\frac{1}{N^{2}T^{2}}\left\|\widetilde{F}\right\|\left\|e\right\|_{sp}^{3}\left\|e_{t}\right\|=\widetilde{V}^{-2}\frac{1}{N^{2}T^{2}}O_{p}\left(T^{\frac{1}{2}}\right)O_{p}\left(N^{\frac{3}{2}}+T^{\frac{3}{2}}\right)O_{p}\left(N^{\frac{1}{2}}\right)$
$\displaystyle=$
$\displaystyle\widetilde{V}^{-2}O_{p}\left(N^{-\frac{3}{2}}+T^{-\frac{3}{2}}\right).$

Turning to $Z_{t},$ given that
$\left(NT\right)^{-1}F^{0\prime}e^{\prime}e_{t}=O_{p}\left(\left(NT\right)^{-1/2}+T^{-1}\right)$
and $H=O_{p}\left(1\right),$ it follows that
$Z_{t}=\widetilde{V}^{-1}O_{p}\left(\left(NT\right)^{-1/2}+T^{-1}\right).$
Therefore,

$\displaystyle A^{\frac{1}{2}}Z_{t}$ $\displaystyle=$ $\displaystyle
O_{p}\left(N^{1-\frac{\alpha_{r}}{2}}\right)O_{p}\left(\frac{1}{\sqrt{NT}}+\frac{1}{T}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(N^{\frac{1-\alpha_{r}}{2}}T^{-\frac{1}{2}}\right)+O_{p}\left(N^{1-\frac{\alpha_{r}}{2}}T^{-1}\right)=o_{p}\left(1\right),$

as $N^{1-\frac{\alpha_{r}}{2}}T^{-1}\rightarrow 0$ by the assumption of
$N^{1-\alpha_{r}}T^{-1}\rightarrow 0$ and
$N^{\frac{3}{2}-\alpha_{r}}T^{-1}\rightarrow 0.$

We next show that $A^{\frac{1}{2}}W_{t}=o_{p}\left(1\right)$ by showing
$A^{\frac{1}{2}}W_{t}^{(k)}=o_{p}\left(1\right),$ for $k=a,b,c.$ First,

$\displaystyle A^{\frac{1}{2}}W_{t}^{(a)}$
$\displaystyle=O_{P}\left(A\widetilde{V}^{-2}\right)O_{P}\left(N^{\frac{\alpha_{1}-3}{2}}T^{-\frac{1}{2}}+N^{\frac{\alpha_{1}}{2}-1}T^{-1}\right)$
$\displaystyle=O_{p}\left(N^{2-\frac{3}{2}\alpha_{r}}\right)O_{p}\left(N^{\frac{\alpha_{1}-3}{2}}T^{-\frac{1}{2}}+N^{\frac{\alpha_{1}}{2}-1}T^{-1}\right)$
$\displaystyle=O_{p}\left(N^{\frac{1}{2}+\frac{1}{2}\alpha_{1}-\frac{3}{2}\alpha_{r}}T^{-\frac{1}{2}}\right)+O_{p}\left(N^{1+\frac{1}{2}\alpha_{1}-\frac{3}{2}\alpha_{r}}T^{-1}\right)$
$\displaystyle=O_{p}\left(N^{\frac{1-\alpha_{r}}{2}}T^{-\frac{1}{2}}N^{\frac{1}{2}\alpha_{1}-\alpha_{r}}\right)+O_{p}\left(N^{\frac{1}{2}\alpha_{1}-\alpha_{r}}N^{1-\frac{1}{2}\alpha_{r}}T^{-1}\right)$
$\displaystyle=o_{p}(1),$

where we have used $N^{1-\alpha_{r}}T^{-1}\rightarrow 0$ and
$N^{1-\frac{\alpha_{r}}{2}}T^{-1}\rightarrow 0$ again. Second,

$A^{\frac{1}{2}}W_{t}^{(b)}=O_{p}\left(\widetilde{V}^{-1}\right)O_{p}\left(N^{-\frac{1}{2}}+N^{\frac{1}{2}}T^{-1}\right)=O_{p}\left(N^{\frac{1}{2}-\alpha_{r}}\right)+O_{p}\left(N^{\frac{3}{2}-\alpha_{r}}T^{-1}\right)=o_{p}\left(1\right)\text{,}$

by Assumption 7. Third,

$\displaystyle A^{\frac{1}{2}}W_{t}^{(c)}$
$\displaystyle=O_{p}\left(A^{\frac{1}{2}}\widetilde{V}^{-2}\right)O_{p}\left(T^{-\frac{3}{2}}+N^{-\frac{3}{2}}\right)$
$\displaystyle=O_{p}\left(N^{2-\frac{3}{2}\alpha_{r}}\right)O_{p}\left(T^{-\frac{3}{2}}+N^{-\frac{3}{2}}\right)$
$\displaystyle=O_{p}\left(N^{2-\frac{3}{2}\alpha_{r}}T^{-\frac{3}{2}}+N^{\frac{1}{2}-\frac{3}{2}\alpha_{r}}\right)$
$\displaystyle=O_{p}\left[\left(N^{\frac{3}{2}-\alpha_{r}}T^{-1}\right)\left(N^{\frac{1}{2}-\frac{1}{2}\alpha_{r}}T^{-\frac{1}{2}}\right)\right]+O_{p}\left(N^{\frac{1}{2}(1-3\alpha_{r})}\right)=o_{p}(1).$

So we have come to that

$\displaystyle
A^{\frac{1}{2}}\left(\widetilde{F}_{t}-H_{4}^{\prime}F_{t}^{0}\right)$
$\displaystyle=A^{\frac{1}{2}}\left(W_{t}+Z_{t}\right)+A^{\frac{1}{2}}\widetilde{V}^{-1}\frac{1}{N}Q\Lambda^{0\prime}e_{t}$
$\displaystyle=A^{\frac{1}{2}}\widetilde{V}^{-1}\frac{1}{N}Q\Lambda^{0\prime}e_{t}+o_{p}(1).$
(A.3)

As for the term $\psi_{t}\equiv
A^{\frac{1}{2}}\widetilde{V}^{-1}Q\frac{1}{N}\Lambda^{0\prime}e_{t}=\underbrace{A^{\frac{1}{2}}\widetilde{V}^{-1}Q\frac{1}{N}A^{\frac{1}{2}}}_{:=\Psi}\left(A^{-\frac{1}{2}}\Lambda^{0\prime}e_{t}\right),$
given Assumption 8, we come to that
$\psi_{t}\overset{d}{\rightarrow}\Psi\times N\left(0,\Gamma_{t}\right)$.

Here it may be interesting to investigate the $r\times r$ matrix $\Psi$. Note
that
$A^{\frac{1}{2}}\widetilde{V}^{-1}/N\asymp_{p}\mathop{\mathrm{d}iag}\left(N^{-\frac{\alpha_{1}}{2}},\cdots,N^{-\frac{\alpha_{r}}{2}}\right).$
Also recall that $Q$ is a (block) upper triangular matrix (in probability)
such that

$Q(l,k)\asymp_{p}N^{\alpha_{l}-\alpha_{k}}\text{ for }1\leq k\leq l\leq r,$

and $Q(l,k)=O_{p}(1),$ for $r\geq k>l\geq 1.$ It then follows that
$\Psi\left(l,k\right)\asymp_{p}N^{-\frac{\alpha_{l}}{2}}Q(l,k)N^{\frac{\alpha_{k}}{2}}=N^{\frac{\alpha_{k}-\alpha_{l}}{2}}Q(l,k).$
Hence,

$\Psi\left(l,k\right)\begin{cases}\asymp_{p}N^{\frac{\alpha_{k}-\alpha_{l}}{2}}N^{\alpha_{l}-\alpha_{k}}=N^{\frac{\alpha_{l}-\alpha_{k}}{2}},\text{
for }1\leq k\leq l\leq r,\\\ \asymp_{p}1,\text{ for }1\leq k=l\leq r,\\\
=O_{p}\left(N^{\frac{\alpha_{k}-\alpha_{l}}{2}}\right),\text{ for}\ r\geq
k>l\geq 1.\end{cases}\ $

So obviously, $\Psi$ is an asymptotically (block) diagonal matrix with full
rank.

Finally, we conclude that
$A^{\frac{1}{2}}\left(\widetilde{F}_{t}-H_{4}^{\prime}F_{t}^{0}\right)\overset{d}{\rightarrow}N\left(0,\Psi^{0}\Gamma_{t}\Psi^{0\prime}\right).$

Proof of Theorem 3.8. Recall that
$\widetilde{\lambda}_{i}-Q\lambda_{i}^{0}=T^{-1}H^{\prime}F^{0\prime}e_{i}+T^{-1}\left(\widetilde{F}-F^{0}H\right)^{\prime}e_{i}.$
As for the second term,

$\frac{1}{T}\left(\widetilde{F}-F^{0}H\right)^{\prime}e_{i}=\widetilde{V}^{-1}\left(\frac{1}{T}\widetilde{F}^{\prime}F^{0}\right)\frac{\Lambda^{0\prime}ee_{i}}{NT}+\widetilde{V}^{-1}\widetilde{F}^{\prime}e^{\prime}\frac{\Lambda^{0}F^{0\prime}e_{i}}{NT^{2}}+\widetilde{V}^{-1}\widetilde{F}^{\prime}e^{\prime}\frac{ee_{i}}{NT^{2}}.$

The first term on the RHS is

$\left(\widetilde{V}^{-1}QAN^{-1}\right)\left(A^{-1}\frac{\Lambda^{0\prime}ee_{i}}{T}\right)=O_{p}\left(1\right)\left(A^{-1}\frac{\Lambda^{0\prime}ee_{i}}{T}\right)=O_{p}\left(N^{-\alpha_{r}}+N^{-\alpha_{r}/2}T^{-1/2}\right),$

where is first equality is by Lemma A.1 and the second equality is by
Assumption 2 and 3(iii). So we have

$\displaystyle\left\|\frac{1}{T}\left(\widetilde{F}-F^{0}H\right)^{\prime}e_{i}\right\|$
$\displaystyle=$ $\displaystyle
O_{p}\left(N^{-\alpha_{r}}+N^{-\alpha_{r}/2}T^{-1/2}\right)+O_{p}\left(N^{{1-\alpha_{r}}}\right)O_{p}\left(\sqrt{T}\right)O_{p}\left(N^{\frac{{}^{a_{1}}}{2}-1}\right)O_{p}\left(\frac{1}{T^{\frac{3}{2}}}\right)$
$\displaystyle+O_{p}\left(N^{1-\alpha_{r}}\right)O_{p}\left(\sqrt{T}\right)O_{p}\left(\frac{N+T}{NT}\right)\frac{1}{T}O_{p}\left(\sqrt{T}\right)$
$\displaystyle=$ $\displaystyle
O_{p}\left(N^{-\alpha_{r}}+N^{-\alpha_{r}/2}T^{-1/2}\right)+O_{p}\left(N^{\frac{\alpha_{1}}{2}-\alpha_{r}}T^{-1}\right)+O_{p}\left(N^{-\alpha_{r}}+N^{1-\alpha_{r}}T^{-1}\right).$

Together with Assumption 8, we come to that
$\sqrt{T}\left(\widetilde{\lambda}_{i}-Q\lambda_{i}^{0}\right)=H^{\prime}T^{-1/2}\sum_{t=1}^{T}F_{t}^{0}e_{it}+o_{p}\left(1\right).$
Given Assumption 6(ii) and the proved result that
$H^{\prime}=\left(Q^{\prime}\right)^{-1}+o_{p}\left(1\right),$ we conclude
that
$\sqrt{T}\left(\widetilde{\lambda}_{i}-Q\lambda_{i}^{0}\right)\overset{d}{\rightarrow}N\left(0,Q^{\prime-1}\Phi_{i}Q^{-1}\right).$

Proof of Theorem 3.9. By definition of $C_{it}$, we have the following
decomposition:

$\displaystyle\widetilde{C}_{it}-C_{it}^{0}$ $\displaystyle=$
$\displaystyle\left(\widetilde{F}_{t}-H^{\prime}F_{t}^{0}\right)^{\prime}H^{-1}\lambda_{i}^{0}+F_{t}^{0\prime}H\left(\widetilde{\lambda}_{i}-H^{-1}\lambda_{i}^{0}\right)+\left(\widetilde{F}_{t}-H^{\prime}F_{t}^{0}\right)^{\prime}\left(\widetilde{\lambda}_{i}-H^{-1}\lambda_{i}^{0}\right)$
$\displaystyle\equiv$ $\displaystyle a_{it}+b_{it}+\varphi_{it},\text{ say.}$

For $a_{it},$ define $d_{t}\equiv
N^{-1}\left(\widetilde{\Lambda}-\Lambda^{0}Q^{\prime}\right)^{\prime}e_{t},$
and then

$a_{it}=\left(\widetilde{V}^{-1}Q\frac{\Lambda^{0\prime}e_{t}}{N}\right)^{\prime}H^{-1}\lambda_{i}^{0}+\left(\widetilde{V}^{-1}d_{t}\right)^{\prime}H^{-1}\lambda_{i}^{0}=\frac{e_{t}^{\prime}\Lambda^{0}}{N}Q^{\prime}\widetilde{V}^{-1}H^{-1}\lambda_{i}^{0}+\left(\widetilde{V}^{-1}d_{t}\right)^{\prime}H^{-1}\lambda_{i}^{0}.$

Notice that
$Q^{\prime}\widetilde{V}^{-1}H^{-1}=Q^{\prime}\widetilde{V}^{-1}\widetilde{V}\left(Q^{\prime}\right)^{-1}\left(\Lambda^{0\prime}\Lambda^{0}/N\right)^{-1}=\left(\frac{\Lambda^{0^{\prime}}\Lambda^{0}}{N}\right)^{-1}.$
So,

$\displaystyle a_{it}$ $\displaystyle=$
$\displaystyle\frac{e_{t}^{\prime}\Lambda^{0}}{N}\left(\Lambda^{0\prime}\Lambda^{0}/N\right)^{-1}\lambda_{i}^{0}+\left(\widetilde{V}^{-1}d_{t}\right)^{\prime}H^{-1}\lambda_{i}^{0}$
$\displaystyle=$
$\displaystyle\lambda_{i}^{0\prime}A^{-1/2}\left(A^{-1/2}\Lambda^{0\prime}\Lambda^{0}A^{-1/2}\right)^{-1}\left(A^{-\frac{1}{2}}\Lambda^{0\prime}e_{t}\right)+\lambda_{i}^{0\prime}H^{\prime-1}\widetilde{V}^{-1}d_{t}.$

Consider

$N^{\frac{\alpha_{r}}{2}}a_{it}=\lambda_{i}^{0\prime}\left(N^{\frac{\alpha_{r}}{2}}A^{-1/2}\right)\left(A^{-1/2}\Lambda^{0\prime}\Lambda^{0}A^{-1/2}\right)^{-1}\left(A^{-\frac{1}{2}}\Lambda^{0\prime}e_{t}\right)+\lambda_{i}^{0\prime}H^{\prime-1}N^{\frac{\alpha_{r}}{2}}\widetilde{V}^{-1}d_{t.}$

Note that $N^{\frac{\alpha_{r}}{2}}A^{-1/2}\rightarrow
S^{\dagger}\equiv\operatorname{diag}(\underbrace{0,\ldots,0}_{r-r_{G}},\underbrace{1\ldots,1}_{r_{G}}).$
Also note that

$\lambda_{i}^{0\prime}H^{\prime-1}N^{\frac{\alpha_{r}}{2}}\widetilde{V}^{-1}d_{t}\leq\left\|\lambda_{i}^{0\prime}H^{\prime-1}\right\|N^{1-\frac{\alpha_{r}}{2}}\left\|d_{t}\right\|=o_{p}\left(1\right),$

where the last equality is due to $\widetilde{V}^{-1}d_{t}=W_{t}+Z_{t},$ and
$A^{1/2}\left(W_{t}+Z_{t}\right)=o_{p}\left(1\right)\ $which has been proved
as in (A.3) when proving 3.7. So it follows that
$N^{\frac{\alpha_{r}}{2}}a_{it}\overset{d}{\rightarrow}N\left(0,\lambda_{i}^{0\prime}S^{\dagger}\Sigma_{\Lambda}^{\ast-1}\Gamma_{t}\Sigma_{\Lambda}^{\ast-1}S^{\dagger}\lambda_{i}^{0}\right).$

For $b_{it},$

$\displaystyle b_{it}$ $\displaystyle=$ $\displaystyle
F_{t}^{0\prime}H\left(\widetilde{\lambda}_{i}-H^{-1}\lambda_{i}^{0}\right)$
$\displaystyle=$ $\displaystyle
F_{t}^{0\prime}H\left(\widetilde{\lambda}_{i}-Q\lambda_{i}^{0}\right)+F_{t}^{0\prime}H\left(Q-H^{-1}\right)\lambda_{i}^{0}$
$\displaystyle=$ $\displaystyle
F_{t}^{0\prime}H\frac{1}{T}H^{\prime}F^{0\prime}e_{i}+F_{t}^{0\prime}H\frac{1}{T}\left(\widetilde{F}-F^{0}H\right)^{\prime}e_{i}+F_{t}^{0\prime}H\left(Q-H^{-1}\right)\lambda_{i}^{0}$
$\displaystyle\equiv$ $\displaystyle
F_{t}^{0\prime}HH^{\prime}\frac{1}{T}F^{0\prime}e_{i}+\nu_{it}+\zeta_{it},\text{
say.}$

Given that we have shown (a)
$\sqrt{T}\left\|\frac{1}{T}\left(\widetilde{F}-F^{0}H\right)^{\prime}e_{i}\right\|=o_{p}\left(1\right),$
(b)
$\left\|Q-H^{-1}\right\|=O_{p}\left(\gamma_{NT}\right)=o_{p}\left(T^{-\frac{1}{2}}\right),$
and (c)

$\displaystyle HH^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\left(H-H_{3}+H_{3}\right)\left(H-H_{2}+H_{2}\right)^{\prime}$
$\displaystyle=$ $\displaystyle
H_{3}H_{2}^{\prime}+H_{3}\left(H-H_{2}\right)^{\prime}+\left(H-H_{3}\right)H_{2}^{\prime}+\left(H-H_{3}\right)\left(H-H_{2}\right)^{\prime}$
$\displaystyle=$
$\displaystyle\left(\frac{\widetilde{F}^{\prime}F^{0}}{T}\right)^{-1}\frac{\widetilde{F}^{\prime}F^{0}}{T}\left(\frac{F^{0\prime}F^{0}}{T}\right)^{-1}+O_{p}\left(\gamma_{NT}\right)$
$\displaystyle=$
$\displaystyle\Sigma_{F}^{-1}+O_{p}\left(\gamma_{NT}\right)=\Sigma_{F}^{-1}+o_{p}\left(T^{-\frac{1}{2}}\right),$

it then follows that
$\sqrt{T}b_{it}=F_{t}^{0\prime}\Sigma_{F}^{-1}\left(T^{-1/2}F^{0\prime}e_{i}\right)+o_{p}\left(1\right)\overset{d}{\rightarrow}N\left(0,F_{t}^{0\prime}\Sigma_{F}^{-1}\Phi_{i}\Sigma_{F}^{-1}F_{t}^{0}\right).$

Lastly, under the weak dependence assumption over both $i$ and $t,$
$N^{\frac{\alpha_{r}}{2}}a_{it}$ and $\sqrt{T}b_{it}$ are asymptotically
independent. Therefore, by a similar argument in proving Theorem 3 of Bai
(2003), we come to that

$\frac{\widetilde{C}_{it}-C_{it}^{0}}{\sqrt{N^{-\alpha_{r}}V_{it}+T^{-1}U_{it}}}\overset{d}{\rightarrow}N\left(0,1\right),$

where
$V_{it}=\lambda_{i}^{0\prime}S^{\dagger}\Sigma_{\Lambda}^{\ast-1}\Gamma_{t}\Sigma_{\Lambda}^{\ast-1}S^{\dagger}\lambda_{i}^{0},$
and $U_{it}=F_{t}^{0\prime}\Sigma_{F}^{-1}\Phi_{i}\Sigma_{F}^{-1}F_{t}^{0}.$

Proof of Theorem 3.10. (1) We first prove the uniform convergence rate of
$\widetilde{\lambda}_{i}$. Recall

$\widetilde{\lambda}_{i}-Q\lambda_{i}^{0}=H^{\prime}\frac{1}{T}F^{0\prime}e_{i}+\frac{1}{T}\left(\widetilde{F}-F^{0}H\right)^{\prime}e_{i}$

As for the second term,

$\displaystyle\frac{1}{T}\left(\widetilde{F}-F^{0}H\right)^{\prime}e_{i}$
$\displaystyle=$
$\displaystyle\widetilde{V}^{-1}\left(\frac{1}{T}\widetilde{F}^{\prime}F^{0}\right)\frac{\Lambda^{0\prime}ee_{i}}{NT}+\widetilde{V}^{-1}\widetilde{F}^{\prime}\frac{e^{\prime}\Lambda^{0}F^{0\prime}e_{i}}{NT^{2}}+\widetilde{V}^{-1}\frac{\widetilde{F}^{\prime}e^{\prime}ee_{i}}{NT^{2}}$
$\displaystyle\equiv$
$\displaystyle\varkappa_{1i}+\varkappa_{2i}+\varkappa_{3i}.$

For $\varkappa_{1i},$

$\displaystyle\varkappa_{1i}$ $\displaystyle=$
$\displaystyle\widetilde{V}^{-1}\left(\frac{1}{T}\widetilde{F}^{\prime}F^{0}\right)\frac{1}{NT}\sum_{j=1}^{N}\sum_{t=1}^{T}\lambda_{j}^{0}e_{jt}e_{it}$
$\displaystyle=$
$\displaystyle\widetilde{V}^{-1}\frac{\widetilde{F}^{\prime}F^{0}}{T}\frac{1}{NT}\sum_{j=1}^{N}\sum_{t=1}^{T}\lambda_{j}^{0}\left[e_{jt}e_{it}-E\left(e_{jt}e_{it}\right)\right]+\widetilde{V}^{-1}\frac{\widetilde{F}^{\prime}F^{0}}{T}\frac{1}{NT}\sum_{j=1}^{N}\sum_{t=1}^{T}\lambda_{j}^{0}E\left(e_{jt}e_{it}\right)$