# Bridging factor and sparse models

Jianqing Fan  Department of Operations Research and Financial Engineering,
Princeton University<EMAIL_ADDRESS>,  Ricardo Masini  Center for
Statistics and Machine Learning, Princeton University<EMAIL_ADDRESS>and  Marcelo C. Medeiros  Department of Economics, Pontifical Catholic
University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil<EMAIL_ADDRESS>
###### Abstract.

Factor and sparse models are two widely used methods to impose a low-
dimensional structure in high-dimension. They are seemingly mutually
exclusive. We propose a lifting method that combines the merits of these two
models in a supervised learning methodology that allows to efficiently explore
all the information in high-dimensional datasets. The method is based on a
flexible model for high-dimensional panel data, called factor-augmented
regression (FarmPredict) model with both observable or latent common factors,
as well as idiosyncratic components. This model not only includes both
principal component (factor) regression and sparse regression as specific
models but also significantly weakens the cross-sectional dependence and hence
facilitates model selection and interpretability. The methodology consists of
three steps. At each step, the remaining cross-section dependence can be
inferred by a novel test for covariance structure in high-dimensions. We
developed asymptotic theory for the FarmPredict model and demonstrated the
validity of the multiplier bootstrap for testing high-dimensional covariance
structure. This is further extended to testing high-dimensional partial
covariance structures. The theory is supported by an simulation study and
applications to the construction of a partial covariance network of the
financial returns and a prediction exercise for a large panel of macroeconomic
time series from FRED-MD database.

JEL Codes: C22, C23, C32, C33.

Keywords: Principal component analysis, penalized least-squares, high-
dimensional inference, FarmPredict, covariance structure, partial covariance
structure, LASSO.

Acknowledgments: Medeiros gratefully acknowledges the partial financial
support from CNPq and CAPES. We are grateful to Caio Almeida, Gilberto
Boareto, Gustavo Bulhões, Frank Diebold, Bruno Ferman, Marcelo Fernandes,
Claudio Flores, Conrado Garcia, Alexander Giessing, Nathalie Gimenes, Marcelo
J. Moreira, Henrique Pires, Yuri Saporito, Rodrigo Targino for helpful
comments. We also thank seminar participants at Princeton University, the
University of Amsterdam and the Federal University of São Carlos for a number
of valuable comments. Finally, we are deeply grateful to Eduardo F. Mendes and
Michael Wolf for the careful reading of the paper and the many insightful
discussions which led to a much improved version of this manuscript.

##  1\. Introduction

With the emergence of new and large datasets, the correct characterization of
the dependence among variables is of substantial importance. Usually, to
achieve this goal, the literature has followed two seemingly orthogonal tracks
over the last two decades. On the one hand, factor models have become an
essential tool to summarize information in large datasets under the assumption
that the remaining dependence structure is negligible. For instance, panel
factor models are applied now to a wide variety of important applications,
ranging from forecasting (macroeconomic) variables and asset pricing models to
causal inference in applied microeconomics and network analysis. On the other
hand, there have been major advances on parameter estimation in ultra high-
dimensions under the assumption of sparsity or weak-sparsity. That is, a
variable depends only on a (very) small subset of the other variables. For an
overview on these two topics and their exciting developments, see Fan et al.
(2020).

In this paper, we take an alternative route and combine the best of the two
worlds described above in order to better characterize the dependence
structure of high-dimensional data. More specifically, we consider that the
covariance structure of a large set of variables, organized in a panel data
format, is characterized as a combination of a factor structure, where factors
can be either observed, unobserved, or both, and a weakly-sparse idiosyncratic
component. This formulation is general enough in order to accommodate a very
large number of data generating processes of interest in economics, finance,
and related areas. The proposed methodology has two ingredients: a three-step
estimation procedure and a new test for structure in high dimensional
(partial) covariance matrices. The steps of the estimation procedure are as
follows. In the first one, we take the original data and remove the effects of
any observed factors. These factors can be deterministic terms such as
seasonal dummies and/or trends or any other observed covariates. The first
step can be parametric or nonparametric, low or high dimensional. A latent
factor model is then estimated using the residuals from the first stage.
Finally, in a final step we model the dependence among idiosyncratic terms as
a weakly sparse regression estimated by the Least Absolute Shrinkage and
Selection Operator (LASSO). At each step, the null-hypothesis of no remaining
cross-section dependence can be tested by the proposed test for the (partial)
covariance structure in high-dimensions.

Our approach has many downstream econometric applications. It can enhance
high-dimensional prediction, select more interpretable variables, construct
counterfactuals for treatment evaluations, and depict (partial) correlation
networks.

###  1.1. Motivation

Let $\boldsymbol{Y}_{t}:=(Y_{1t},\ldots,Y_{nt})^{\prime}$ be a random vector
generated as
$Y_{it}=\boldsymbol{\lambda}_{i}^{\prime}\boldsymbol{F}_{t}+U_{it}$, for
$i=1,\ldots,n$, $t=1,\ldots,T$, where
$\boldsymbol{\Sigma}:=\boldsymbol{\mathbb{E}}(\boldsymbol{U}_{t}\boldsymbol{U}_{t}^{\prime})$,
with $\boldsymbol{U}_{t}:=(U_{1t},\ldots,U_{nt})^{\prime}$, is not necessarily
diagonal. Fix one component of interest $i\in\\{1,\ldots,n\\}$, which serve as
a response variable. Consider the following prediction models:

$\mathcal{M}_{1}:\,\mathbb{E}(Y_{it}|\boldsymbol{Y}_{-it}),\quad\mathcal{M}_{2}:\,\mathbb{E}(Y_{it}|\boldsymbol{F}_{t}),\quad\textnormal{and}\quad\mathcal{M}_{3}:\,\mathbb{E}(Y_{it}|\boldsymbol{F}_{t},\boldsymbol{U}_{-it}),$
(1.1)

where $\boldsymbol{Y}_{-it}$ and $\boldsymbol{U}_{-it}$ are, respectively,
vectors with the elements of $\boldsymbol{Y}_{t}$ and $\boldsymbol{U}_{t}$
excluding the $i$-th entry. Model $\mathcal{M}_{3}$ is indeed the factor
augmented regression model since it is the same as
$\mathbb{E}(Y_{it}|\boldsymbol{F}_{t},\boldsymbol{Y}_{-it})$.

Suppose that we observe both $\boldsymbol{F}_{t}$ and $\boldsymbol{U}_{-it}$.
Which one of three models above is best in terms of mean square error
($\mathsf{MSE}$) for prediction? Comparison between $\mathcal{M}_{1}$ and
$\mathcal{M}_{2}$ is not clear since it depends, among others, on the
magnitude of $\boldsymbol{\Sigma}$ relative to
$\boldsymbol{\Lambda}^{\prime}\boldsymbol{\Lambda}$, where
$\boldsymbol{\Lambda}:=(\boldsymbol{\lambda}_{1},\ldots,\boldsymbol{\lambda}_{n})^{\prime}$.
However, since the $\sigma$-algebras generated by $\boldsymbol{Y}_{-it}$ and
$\boldsymbol{F}_{t}$ are both included in the $\sigma$-algebra generated by
$(\boldsymbol{F}_{t},\boldsymbol{U}_{-it})$, it is not surprising that
$\mathsf{MSE}(\mathcal{M}_{3})\leq\min[\mathsf{MSE}(\mathcal{M}_{1}),\mathsf{MSE}(\mathcal{M}_{2})]$.
The same will hold true if we replace the models in (1.1) by their best linear
projections, which we denote by $\mathcal{\widetilde{M}}_{j}$ for
$j\in\\{1,2,3\\}$, since the linear space $\widetilde{\mathcal{M}}_{3}$ is the
largest. In fact, since linear span by
$(\boldsymbol{F}_{t},\boldsymbol{U}_{-it})$ is the same as that by
$(\boldsymbol{Y}_{-it},\boldsymbol{F}_{t})$, we expand the space in the
informative factor directions $\boldsymbol{F}_{t}$. In the linear case, we can
explicitly write the “gains” of $\widetilde{\mathcal{M}}_{3}$ when compared to
$\widetilde{\mathcal{M}}_{1}$ and $\widetilde{\mathcal{M}}_{2}$:

$\displaystyle\mathsf{MSE}(\widetilde{\mathcal{M}}_{3})$
$\displaystyle-\mathsf{MSE}(\widetilde{\mathcal{M}}_{1})=-\boldsymbol{\theta}_{i}^{\prime}\boldsymbol{\Sigma}_{-i,-i}\boldsymbol{\theta}_{i}$
$\displaystyle\mathsf{MSE}(\widetilde{\mathcal{M}}_{3})$
$\displaystyle-\mathsf{MSE}(\widetilde{\mathcal{M}}_{2})=-\boldsymbol{\Delta}_{1i}\boldsymbol{\Delta}_{1i}^{\prime}-\boldsymbol{\Delta}_{2i}^{\prime}\boldsymbol{\Sigma}_{-i,-i}\boldsymbol{\Delta}_{2i},$

where $\boldsymbol{\theta}_{i}$ and $\boldsymbol{\beta}_{i}$ are the
coefficients of the projection of $U_{it}$ onto $\boldsymbol{U}_{-it}$ and the
coefficients of the projection of $X_{it}$ onto $\boldsymbol{X}_{-it}$,
respectively; $\boldsymbol{\Sigma}_{-i,-i}$ is $\boldsymbol{\Sigma}$ excluding
the $i$-th row and column;
$\boldsymbol{\Delta}_{1i}:=\boldsymbol{\Lambda}_{i}-\boldsymbol{\beta}_{i}^{\prime}\boldsymbol{\Lambda}_{-i}$
and
$\boldsymbol{\Delta}_{2i}:=\boldsymbol{\beta}_{i}-\boldsymbol{\theta}_{i}$.
From the previous expressions, it becomes evident that both
$\widetilde{\mathcal{M}}_{1}$ and $\widetilde{\mathcal{M}}_{2}$ are
restrictions on $\widetilde{\mathcal{M}}_{3}$. Broadly speaking, whenever one
does not expect to have an _exact_ factor model, there are potential gains of
taking into account the contribution of the idiosyncratic components
$\boldsymbol{U}_{-it}$. Therefore, we use $\widetilde{\mathcal{M}}_{3}$ as the
base model for the estimation methodology described in Section 2.2.

###  1.2. Main Contributions and Comparison with the Literature

The contributions of this paper are multi-fold. First, our methodology bridges
the gap between two apparently competing methods for high-dimensional
modeling; see, for example the discussion in Giannone et al. (2021) and Fan et
al. (2020). This yields a vast number of potential applications and spin-offs.
For instance, in Fan et al. (2020), we apply the methods developed in here to
evaluate the effects of interventions and we contribute to the literature on
synthetic controls and related methods by combining the approaches of Gobillon
and Magnac (2016) and Carvalho et al. (2018). Therefore, in our setup both a
common factor structure and weak sparsity can coexist.111Sparsity and factor
models can also coexist in the framework of sparse principal components; see
Fan et al. (2020).

Second, our results can also serve as a diagnostic and misspecification tool.
For panel data models with interactive fixed effects as in Pesaran (2006), Bai
(2009), Moon and Weidner (2015) and Bai and Liao (2017), our test can be
directly applied to uncover the dependence structure among cross-sectional
units before and after accounting for common factor components. If the factor
structure is informative enough, we expect the idiosyncratic covariance matrix
to be almost sparse. If this is not the case, we may have possibly
underestimated the number of factors. One popular application is in asset
pricing as discussed in Gagliardini et al. (2019) and in the empirical section
of this paper. There are a huge number of proposed factors as described in
Feng et al. (2020), Giglio and Xiu (2020), and Gu et al. (2020). We can apply
our methodology not only to test for omitted factors, but, as well, to
estimate network connections among firms as in Diebold and Yilmaz (2014) and
Brownlees et al. (2020). Finally, as a diagnostic tool, our paper tackle the
same problem as Gagliardini et al. (2019). However, we take an alternative
solution strategy which relies on a much different set of hypothesis; see also
Gagliardini et al. (2020).

Third, the methodology proposed here contributes to the forecasting
literature. For instance, in the second application considered in this paper,
we build forecasting models for a large cross-section of macroeconomic
variables. We call this method the FarmPredict. We show that the combination
of factors and a sparse regression strongly outperforms the traditional
principal component regression as in Stock and Watson (2002a,b). Therefore,
FarmPredict can be an additional contribution to the forecasting and machine
learning toolkit. The method can be easily extended to a multivariate setting
combining factor-augmented vector autoregressions (FAVAR) as in Bernanke et
al. (2005) with sparse vector models as in Kock and Callot (2015) and Masini
et al. (2019). Hence, we contribute to the debate in Giannone et al. (2021).

Fourth, we show consistency of factor estimation based on the residuals of a
first-step regression. Our results hold for both parametric (linear or
nonlinear) and nonparametric first stage. A high-dimensional first stage is
also allowed. Note that, current results in the literature consider that
factors are estimated based on observed data and our derivations favor a much
more flexible and general setup (Bai and Ng, 2002, 2003, 2006). More
specifically, our methodology favors settings where there are both observed
and latent factors, as well as trend-stationary data. In the later, the trend
can be first removed by (nonparametric) first-stage regression. In addition,
whenever the unobserved factors and the observed covariates are correlated,
the method proposed in Pesaran (2006) can be used and all or results follow
directly.

Fifth, we contribute to the LASSO literature. LASSO can not be model selection
consistent for highly correlated variables. Through the decomposition of
covariates into factors and idiosyncratic components, namely the idea of
lifting, we decorrelate the variables and make the model selection condition
much easier to hold; see, for example, (Fan et al., 2020). We show consistency
of the estimates based on residuals of the previous steps. Our results are
derived under restrictions on the population covariance matrix of the data and
not on the estimated one, as it is usual in many papers. See, for example, van
de Geer and Bühlmann (2009). Furthermore, we derive our results under much
mild conditions that the ones considered in (Fan et al., 2020).

Finally, we extend the results in Chernozhukov et al. (2013, 2018) to strong-
mixing data in order to construct hypothesis tests for covariance and partial
covariance structure in high dimensions.222Recently, Giessing and Fan (2020)
also extended the results in Chernozhukov et al. (2013). However, their setup
is very different from ours and the authors only consider the case of
independent and identically distributed data. This step is necessary for
econometrics and financial applications. As side results, in order to develop
the test we first show consistency of kernel-based estimation of a high-
dimensional long-run covariance matrix of dependent process. This is a new
result with important consequences for the theory of high-dimensional
regression with dependent errors. We also establish consistency of a new
estimator of the partial covariance matrix in high-dimensions and strong-
mixing data. Our proposed tests can be used to infer if the (partial)
covariance matrix of a high-dimensional random vector is diagonal or block-
diagonal. More generally, we can test any pre-defined structure. Furthermore,
we show that the test remains valid when we use the residuals from a previous
step estimation to compute the covariance matrix. This result allows us to to
apply the test to the three-stage estimation procedure proposed here. Although
our results are derived under the assumption that the number of factors is
known, simulation results presented in the paper provide evidence that the
test have good finite-sample properties even when the number of factors is
determined by data-driven methods commonly found in the literature. In
addition, due to the factor augmentation, our method is robust to the over-
estimation of the number of factors. Over the past years, a vast number of
papers proposed different methods to test for covariance structure in high
dimensions. See, for example, Ledoit and Wolf (2002), Chen et al. (2010),
Onatski et al. (2013), Cai and Ma (2013), Li and Qin (2014), Zheng et al.
(2019), Cai et al. (2016), Zheng et al. (2019), and Guo and Tang (2020), among
many others.333For a nice recent review, see Cai (2017). To the best of our
knowledge, we complement all the previous papers by simultaneously considering
high-dimensions, strong-mixing data with mild distributional assumptions, and
pre-estimation when constructing tests for both covariance and partial
covariance structure.

Summarizing, our approach provides:

  1. (1)

A systematic way to unify factor and sparse models in order to construct
econometric specifications which use all the available information. These
models can be applied to:

    1. (a)

Forecasting in a high-dimensional setting;

    2. (b)

Construction of counterfactuals to aggregate data;

    3. (c)

Estimation of partial correlation networks;

  2. (2)

An inferential procedure to test for genereal structures in covariance and
partial covariance matrices.

###  1.3. Organization of the Paper

In addition to this Introduction, the paper is organized as follows. We
present the model setup and assumptions in Section 2. The theoretical results
are presented in Section 3 with practical guides given in Section 4. We depict
the results of a simulation experiment in Section 5 and discuss the empirical
application in Section 6. Section 7 concludes. All proofs are deferred to the
Supplementary Material.

###  1.4. Notation

All random variables (real-valued scalars, vectors and matrices) are defined
in a common probability space $(\Omega,\mathscr{F},\P)$. We denote random
variables by an upper case letter, $X$ for instance, and its realization by a
lower case letter, $X=x$. The expected value operator is with respect to the
$\P$ law such that $\mathbb{E}X:=\int_{\Omega}X(\omega)\mathrm{d}\P(\omega)$.
Matrices and vectors are written in bold letters $\boldsymbol{X}$. Except for
the number of factors, $r$, and number of covariates, $k$, defined below, all
other dimensions are allowed to depend on the sample size ($T$). However, we
omit this dependency throughout the paper to avoid clustering the notation
prematurely.

We use $\|\,\cdot\,\|_{p}$ to denote the $\ell^{p}$ norm for $p\in[1,\infty]$,
such that for a $d-$dimensional (possibly random) vector
$\boldsymbol{X}=(X_{1},\ldots,X_{d})^{\prime}$, we have
$\|\boldsymbol{X}\|_{p}:=(\sum_{i=1}^{d}|X_{i}|^{p})^{1/p}$ for
$p\in[1,\infty)$ and $\|\boldsymbol{X}\|_{\infty}:=\sup_{i\leq d}|X_{i}|$. If
$\boldsymbol{X}$ is a $(m\times n)$ possibly random matrix then
$\|\boldsymbol{X}\|_{p}$ denotes the matrix $\ell^{p}$-induced norm and
$\|\boldsymbol{X}\|_{\max}$ denotes the maximum entry in absolute terms of the
matrix $\boldsymbol{X}$. Note that whenever $\boldsymbol{X}$ is random, then
$\|\boldsymbol{X}\|_{p}$ for $p\in[1,\infty]$ and $\|\boldsymbol{X}\|_{\max}$
are random variables. We also reserve the symbol $\|\,\cdot\,\|$ without
subscript for the Euclidean norm $\|\cdot\|:=\|\cdot\|_{2}$ for both vectors
and matrices.

For any convex function $\psi:\mathbb{R}^{+}\to\mathbb{R}^{+}$ such that
$\psi(0)=0$ and $\psi(x)\to\infty$ as $x\to\infty$ and (real-valued) random
variable $X$, we denote its Orlicz norm by $\|X\|_{\psi}$, which is defined by
$\|X\|_{\psi}:=\inf\left\\{C>0:\mathbb{E}\left[\psi\left(\frac{|X|}{C}\right)\right]\leq
1\right\\}$. Since we are only concerned with polynomial and exponential tails
we restrict ourselves to Orlicz norm induces by the class of function defined
by (3.1). Evidently, as opposed to $\|X\|_{p}$, $\|X\|_{\psi_{p}}$ is always a
non-negative non-random scalar. We do not abide to any convention to apply
Orlicz norm to vector or matrices to avoid confusion.

For any vector $\boldsymbol{X}$, $\mathsf{diag}\,(\boldsymbol{X})$ is a
diagonal matrix whose diagonal is the elements of $\boldsymbol{X}$.
$\mathds{1}(A)$ is an indicator function on the event $A$, i.e,
$\mathds{1}(A)=1$ if $A$ is true and $0$ otherwise. We adopt the Landau
big/small $O,\,o$ notation and the “in probability” $O_{P}$ and $o_{p}$
analogues. We say that $x$ is of the same order of $y$, $x\asymp y$, if both
$x=O(y)$ and $y=O(x)$. We write $X\asymp_{P}Y$ if both $X=O_{P}(Y)$ and
$Y=O_{P}(X)$. Unless stated otherwise, the asymptotics are taken as
$T\rightarrow\infty$, where $T$ is the time dimension, and the $o(1)$ and
$o_{P}(1)$ are with respect to the limit as $T\to\infty$. We denote
convergence in probability and in distribution by
“$\overset{p}{\longrightarrow}$” and “$\Rightarrow$”, respectively.

##  2\. Setup and Method

###  2.1. Data Generating Process

We consider a very general panel data model, which is rich enough in order to
nest several important cases in economics, finance and related areas. We
define the following Data Generating Process (DGP).

######  Assumption 1 (DGP).

The process $\\{Y_{it}:1\leq i\leq n,t\geq 1\\}$ is generated by

$Y_{it}=\boldsymbol{\gamma}_{i}^{\prime}\boldsymbol{X}_{it}+\underbrace{\boldsymbol{\lambda}_{i}^{\prime}\boldsymbol{F}_{t}+U_{it}}_{=:R_{it}}$
(2.1)

where $\boldsymbol{X}_{it}$ is a $k$-dimensional observable (random) vector
which may also include a constant term, $\boldsymbol{F}_{t}$ is a
$r$-dimensional vector of common latent factors, and $U_{it}$ is a zero mean
idiosyncratic shock.444For simplicity, we assume that all the units $i$ have
the same number of covariates ($k$). The framework can certainly accommodate
situations where $k_{i}$ is a function of $i$. The unknown parameters are
$\boldsymbol{\gamma}_{i}\in\mathbb{R}^{k}$, the factor loadings
$\boldsymbol{\lambda}_{i}$, and the covariance matrix of the idiosyncratic
shocks. Finally, we assume that $\boldsymbol{X}_{it}$, $\boldsymbol{F}_{t}$
and $U_{it}$ are mutually uncorrelated.

######  Remark 1.

In Assumption 1 we consider that $k$, the dimension of $\boldsymbol{X}_{it}$
is finite and fixed. Furthermore, the relation between $Y_{it}$ and
$\boldsymbol{X}_{it}$ is linear. This is for the sake of exposition. However,
the theoretical results in this paper are written in terms of the consistency
rate of the first-step estimation. Therefore, the DGP can be made much more
general by just changing the rates.

######  Remark 2.

The assumption that $\boldsymbol{X}_{it}$, $\boldsymbol{F}_{t}$ and $U_{it}$
are mutually uncorrelated can be relaxed. Whenever $\boldsymbol{X}_{it}$ is
correlated with $\boldsymbol{F}_{t}$ and $U_{it}$ are correlated and the
interest lies of the estimation of the parameters $\boldsymbol{\gamma}_{i}$,
$i=1,\ldots,n$, the method proposed by Pesaran (2006) can applied in the
first-stage of the procedure considered in this paper and our theoretical
results will follow. Nevertheless, we provide several examples below where the
assumption that $\boldsymbol{X}_{it}$, $\boldsymbol{F}_{t}$ and $U_{it}$ are
mutually uncorrelated is reasonable.

######  Example 1 (Asset Pricing Models).

Suppose $Y_{it}$ is the return of an asset $i$ at time $t$ and let
$\boldsymbol{X}_{it}:=\boldsymbol{X}_{t}$ be a set of $k$ observable risk
factors, such as the market returns and or Fama-French factors (Fama and
French, 1993, 2015). $\boldsymbol{F}_{t}$ can be a set of additional, non
observable, risk factors. Several asset pricing models, such as the Capital
Asset Pricing Model (CAPM) of the Arbitrage Pricing Theory (APT) model, are
nested into this general framework.

######  Example 2 (Networks).

Model (2.1) also complements the network specifications discussed in Barigozzi
and Hallin (2016,2017b) and Barigozzi and Brownlees (2019). Furthermore, the
test proposed here can be used to detect networks links as in Diebold and
Yilmaz (2014) and Brownlees et al. (2020). For example, $Y_{it}$ can be the
(realized) volatility of financial assets and
$\boldsymbol{X}_{it}:=\boldsymbol{X}_{t}$ can be volatility factors as in
Brito et al. (2018) and Andreou and Ghysels (2021).

######  Example 3 (FAVAR).

In the case where the index $i$ represents a different dependent (endogenous)
variable and $U_{it}$ is a dependent process, model (2.1) turns out to be
equivalent to the Factor Augmented Vector Autoregressive (FAVAR) model of
Bernanke et al. (2005). In this case, $\boldsymbol{X}_{it}$ may also include
lagged dependent variables.

######  Example 4 (Panel Data Models).

Model (2.1) is the panel model with iterative fixed-effects considered in
Gobillon and Magnac (2016), where the authors propose an alternative to the
Synthetic Control method of Abadie and Gardeazabal (2003) and Abadie et al.
(2010) to evaluate the effects of regional policies. Model (2.1) is also in
the heart of the FarmTreat method of Fan et al. (2020).

###  2.2. Three-Stage Method

The method proposed here for estimation, inference and prediction consists of
three stages where at the end of each stage, the covariance structure of the
residuals is tested.

  1. (1)

For each $i\in\\{1,\ldots,n\\}$ run the regression:

$Y_{it}=\boldsymbol{\gamma}_{i}^{\prime}\boldsymbol{X}_{it}+R_{it},\quad\,t\in\\{1,\ldots,T\\},$

and compute
$\widehat{R}_{it}:=Y_{it}-\widehat{\boldsymbol{\gamma}}_{i}^{\prime}\boldsymbol{X}_{it}$.
The first stage may consist of a regression on a constant, a deterministic
time trend and seasonal dummies, for instance, or, as in Example 1, a
regression on observed factors. After removing the contribution from the
observables, we can use the test for the null hypothesis of no remaining
(partial) covariance structure to check if the (partial) covariance of
$R_{it}$ is dense or sparse. If it is dense we move to Step 2. Otherwise, we
jump directly to Step 3. This first parametric, low dimensional step can be
replaced by a nonlinear/nonparametric regression or by a high-dimensional
model, when, for example, the number of observed factors is large. As pointed
out in Remark 2, the Pesaran’s (2006) estimator can be also used whenever
correlation between $\boldsymbol{X}_{it}$ and $R_{it}$ is allowed. This will
be discussed more in the subsequent sections.

  2. (2)

Write $\boldsymbol{R}_{t}:=(R_{1t},\ldots,R_{nt})^{\prime}$ and
$\boldsymbol{R}_{t}=\boldsymbol{\Lambda}\boldsymbol{F}_{t}+\boldsymbol{U}_{t}$.
The second step consists of estimating $\boldsymbol{\Lambda}$ and
$\boldsymbol{F}_{t}$ for $t=1,\ldots,T$ using $\widehat{\boldsymbol{R}}_{t}$
through principal component analysis (PCA) and compute

$\widehat{\boldsymbol{U}}_{t}=\widehat{\boldsymbol{R}}_{t}-\widehat{\boldsymbol{\Lambda}}\widehat{\boldsymbol{F}}_{t}.$

After estimating the factors and loadings, we apply our testing procedure to
check for remaining covariance structure in $\boldsymbol{U}_{t}$. The second-
step estimation can be carried out also by dynamic factor models as in
Barigozzi and Hallin (2016,2017,2020) or Barigozzi et al. (2020). In Section 4
we discuss the determination of the number of factors.

  3. (3)

Now, define
$\widehat{\boldsymbol{U}}_{-it}:=(\widehat{U}_{1t},\ldots,\widehat{U}_{i-1,t},\widehat{U}_{i+1,t}\dots\widehat{U}_{nt})^{\prime}$.
The third step consists of a sparse regression to estimate the following model
for each $i\in\\{1,\ldots,n\\}$:

$\widehat{U}_{it}=\boldsymbol{\theta}_{i}^{\prime}\widehat{\boldsymbol{U}}_{-it}+V_{it},\qquad
t\in\\{1,\ldots,T\\}.$

  4. (4)

The regression in Step 3 provides useful augmentation for predicting the
response variable $Y_{it}$ (if $Y_{it}$ is a variable to be predicted; see
(2.3)), which reduces the error further from $U_{it}$ to $V_{it}$. The
sparsity of estimated $\boldsymbol{\theta}_{i}$ is useful for constructing
partial correlation network or graphica model.

At the end of Steps 2 and 3, we can conduct the relevant inference on the
structures of the covariance or partial covariance matrices. We can also
provide updated prediction future outcomes. We detail those in the next
subsection. Also note that the nonzero estimates of $\boldsymbol{\theta}_{i}$
shed light on the links among idiosyncratic components.555The three-stage
procedure described here could be replaced by a single-step joint estimation.
However, not only the computational burden will be much higher, but also the
technical challenges will be greater. I believe that the simplicity of the
method is more a blessing than a curse.

###  2.3. Estimators and Inference Procedure

In a pure prediction exercise one is usually interested in the linear
projection of $Y_{it}$ onto
$(\boldsymbol{X}_{it}^{\prime},\boldsymbol{F}_{t}^{\prime},\boldsymbol{U}_{-it}^{\prime})^{\prime}$,
which results in the factor-augmented regression model (FARM)

${Y}_{it}={\boldsymbol{\gamma}_{i}}^{\prime}\boldsymbol{X}_{it}+{\boldsymbol{\lambda}_{i}}^{\prime}{\boldsymbol{F}}_{t}+{\boldsymbol{\theta}_{i}}^{\prime}{\boldsymbol{U}}_{-it}+\varepsilon_{it},\qquad
t\in\\{1,\ldots,T\\},$ (2.2)

for each given $i$, and can be predicted by

$\widehat{Y}_{it}:=\widehat{\boldsymbol{\gamma}_{i}}^{\prime}\boldsymbol{X}_{it}+\widehat{\boldsymbol{\lambda}_{i}}^{\prime}\widehat{\boldsymbol{F}}_{t}+\widehat{\boldsymbol{\theta}_{i}}^{\prime}\widehat{\boldsymbol{U}}_{-it};\qquad
i\in\\{1,\ldots,n\\}.$ (2.3)

This will be called FarmPredict. Note that model (2.2) is equivalent to using
the predictors $X_{it},{\boldsymbol{Y}}_{-it}$ and ${\boldsymbol{F}}_{t}$,
which augment predictors $X_{it},{\boldsymbol{Y}}_{-it}$ by using the common
factors ${\boldsymbol{F}}_{t}$. The form in (2.2) mitigates the collinearity
issues in high dimensions.

Model (2.2) also bridges factor regression ($\boldsymbol{\theta}_{i}=0$) on
one end and (sparse) regression on the other end with
$\boldsymbol{\lambda}_{i}=\boldsymbol{\Lambda}_{-i}^{\prime}\boldsymbol{\theta}_{i}$,
where $\boldsymbol{\Lambda}_{-i}$ is the loading matrix without the $i^{th}$
row. In the latter case, model (2.2) becomes a (sparse) regression model:

${Y}_{it}={\boldsymbol{\gamma}_{i}}^{\prime}\boldsymbol{X}_{it}+{\boldsymbol{\theta}_{i}}^{\prime}{\boldsymbol{R}}_{-it}+\varepsilon_{it},\qquad
t\in\\{1,\ldots,T\\}.$ (2.4)

In this case, FARM specification as in (2.2) decorrelates the variables
$\boldsymbol{R}_{-i}$ in (2.4). It makes the model selection consistency much
easier to satisfy and forms the basis of FarmSelect in (Fan et al., 2020). Our
contribution in this specific task is to allow heteroscadestic adjustments,
resulting in the estimated data $\boldsymbol{R}_{t}$. In general, for FARM
(2.2) with sparsity, FarmPredict chooses additional idiosyncratic components
to enhance the prediction of the factor regression.

In other cases, the structure of the idiosyncratic components
$\boldsymbol{U}=(U_{1},\ldots,U_{n})^{\prime}$ is the objective of interest.
An estimator for
$\boldsymbol{\Sigma}=\mathbb{E}(\boldsymbol{U}_{t}\boldsymbol{U}_{t}^{\prime})$
could be simply given by

$\widehat{\boldsymbol{\Sigma}}:=\frac{1}{T}\sum_{t=1}^{T}\widehat{\boldsymbol{U}}_{t}\widehat{\boldsymbol{U}}_{t}^{\prime}.$
(2.5)

In order to proper understand the (linear) relation between a pair
$(U_{it},U_{jt})$ of $\boldsymbol{U}_{t}$, a simple covariance estimate
sometimes is not enough. In applications, it is often desirable to have a
direct measure of how $U_{it}$ and $U_{jt}$ are connected. By direct
connection, we meant the relation between those units removing the
contribution of other variables of $\boldsymbol{U}_{t}$. For this purpose, we
use the partial covariance between $U_{it}$ and $U_{jt}$, defined for any pair
$i,j\in\\{1,\ldots,n\\}$ as $\pi_{ij}:=\mathbb{E}(V_{ijt}V_{jit})$, where
$V_{ijt}:=U_{it}-\mathsf{Proj}(U_{it}|\boldsymbol{U}_{-ij,t})$ and
$\mathsf{Proj}(U_{it}|\boldsymbol{U}_{-ij,t})$ denotes the linear projection
of $U_{it}$ onto the space spanned by all the units except $i$ and $j$, which
we denote by $\boldsymbol{U}_{-ij,t}$. We suggest to estimate the partial
covariance matrix $\boldsymbol{\Pi}:=(\pi_{ij})$ by

$\widehat{\boldsymbol{\Pi}}:=(\widehat{\pi}_{ij})\quad\textnormal{and}\quad\widehat{\pi}_{ij}:=\frac{1}{T}\sum_{t=1}^{T}\widehat{V}_{ij,t}\widehat{V}_{ji,t},$
(2.6)

where $\widehat{V}_{ijt}$ is the residual of the LASSO regression of
$\widehat{U}_{it}$ onto $\widehat{\boldsymbol{U}}_{-ij,t}$ for
$i,j\in\\{1,\ldots,n\\}$.

We also would like to conduct formal test on the population structure of
$\boldsymbol{U}_{t}$. Specifically, we propose a test for the following null
hypothesis:

$\H{_}\mathcal{D}^{\Sigma}:\boldsymbol{\Sigma}_{\mathcal{D}}=\boldsymbol{\Sigma}_{\mathcal{D}}^{0},\qquad\mathcal{D}\subseteq\\{1,\ldots,n\\}\times\\{1,\ldots,n\\},$
(2.7)

for a given subset $\mathcal{D}$, where $\boldsymbol{\Sigma}_{\mathcal{D}}$
denotes the elements of $\boldsymbol{\Sigma}$ indexed by $\mathcal{D}$ and we
allow $d:=|\mathcal{D}|$ to diverge as $n,T\to\infty$. For example, to test if
$\boldsymbol{\Sigma}$ is diagonal, $\mathcal{D}$ consists of all off diagonal
elements and $\boldsymbol{\Sigma}_{\mathcal{D}}^{0}=\boldsymbol{0}$. To test
if $\boldsymbol{\Sigma}$ is block diagonal, $\mathcal{D}$ can be taken to the
corresponding off-diagonal blocks. Similarly, for testing the structure on the
partial covariance matrix

$\H{_}{\mathcal{D}}^{\Pi}:\boldsymbol{\Pi}_{\mathcal{D}}=\boldsymbol{\Pi}^{0}_{\mathcal{D}},\qquad\mathcal{D}\subseteq\\{1,\ldots,n\\}\times\\{1,\ldots,n\\}.$
(2.8)

The null hypotheses (2.7) and (2.8) nest several cases of interest. The most
common would be to test for a diagonal or a block diagonal structure in
$\boldsymbol{\Sigma}$ and/or $\boldsymbol{\Pi}$. But it also accommodates
other structures.666With minor changes, the proposed test can also be used to
test the null
$\boldsymbol{M}\mathsf{vec}\,(\boldsymbol{\Sigma})=\boldsymbol{m}$ for some
$(d\times n^{2})$ matrix $\boldsymbol{M}$ and $d$-dimensional vector
$\boldsymbol{m}$ where $d:=d_{T}$ is also a function of $T$. The task of
estimating $\boldsymbol{\Sigma}$ is well documented in literature even in
high-dimensions; see, for example, Ledoit and Wolf (2004,2012,2017,2020), Fan
et al. (2008), Lam and Fan (2009), or Fan et al. (2013).777See Ledoit and Wolf
(2021a) for a recent survey or the book by Fan et al. (2020).

The challenges for testing (2.7) and (2.8) can be summarized as follows:

  1. (1)

As we allow for both $n$ and $d$ to diverge to infinite as $T$ grows,
sometimes at a faster rate, we have a high-dimensional test where some sort of
Gaussian approximation result for dependent data must be deployed as we also
allow the number covariances to be tested ($d$) to diverge. In this case, a
high-dimensional long-run covariance matrix must be estimated if one expects
to get (asymptotic) correct test size.

  2. (2)

We do not observe $\\{\boldsymbol{U}_{t}\\}$ or $\\{V_{ij,t}\\}$. Instead, we
have an estimate of both from a postulated model on observable random
variables. Therefore, the estimation error must be taken into account to claim
some sort of asymptotic properties of the test. In fact, it is not uncommon to
obtain estimates of both $\\{\boldsymbol{U}_{t}\\}$ and $\\{V_{ij,t}\\}$ from
a multi-stage estimation procedure as we illustrate later.

We propose to test (2.7) using the statistic

$S_{\mathcal{D}}^{\Sigma}:=\|\sqrt{T}(\widehat{\boldsymbol{\Sigma}}_{\mathcal{D}}-\boldsymbol{\Sigma}_{\mathcal{D}}^{0})\|_{\max}.$
(2.9)

The quantiles of $S_{\mathcal{D}}^{\Sigma}$ are approximated by a Gaussian
bootstrap. To describe the procedure, let $\boldsymbol{\Upsilon}_{\Sigma}$
denote the $(d\times d)$ covariance matrix for the vectorized submatrix
$(\widetilde{\sigma}_{ij})_{(i,j)\in\mathcal{D}}$, where
$\widetilde{\sigma}_{ij}:=\frac{1}{T}\sum_{t=1}^{T}U_{i,t}U_{j,t}$. Since the
process $\\{\boldsymbol{U}_{t}\\}$ might present some form of temporal
dependence (refer to Assumption 3(c)) we estimate
$\boldsymbol{\Upsilon}_{\Sigma}$ using a Newey-West-type estimator. For a
given $\mathcal{K}\in\K$, where $\K$ is a class of kernel functions described
below in (3.11) and bandwidth $h>0$ , $\boldsymbol{\Upsilon}_{\Sigma}$ is
estimated by

$\widehat{\boldsymbol{\Upsilon}}_{\Sigma}:=\sum_{|\ell|<T}\mathcal{K}(\ell/h)\widehat{\boldsymbol{M}}_{\Sigma,\ell}\qquad\textnormal{and}\qquad\widehat{\boldsymbol{M}}_{\Sigma,\ell}:=\frac{1}{T}\sum_{t=\ell+1}^{T}\widehat{\boldsymbol{D}}_{\Sigma,t}\widehat{\boldsymbol{D}}_{\Sigma,t-\ell}^{\prime},$
(2.10)

where $\widehat{\boldsymbol{D}}_{\Sigma,t}$ is a $d$-dimensional vector with
entries given by $\widehat{U}_{it}\widehat{U}_{jt}-\widehat{\sigma}_{ij}$ for
$(i,j)\in\mathcal{D}$, where $\widehat{\sigma}_{ij}$ is the $(i,j)$ element of
$\widehat{\boldsymbol{\Sigma}}$ defined in (2.5). Finally, let
$c^{*}_{\Sigma}(\tau)$ be the $\tau$-quantile of the Gaussian bootstrap
$S^{*}_{\mathcal{D}}:=\|\boldsymbol{Z}^{*}_{\Sigma}\|_{\infty}$ and
$\boldsymbol{Z}^{*}_{\Sigma}|\boldsymbol{X},\boldsymbol{Y}\sim\mathcal{N}(\boldsymbol{0},\widehat{\boldsymbol{\Upsilon}}_{\Sigma})$.

Theorem 4 demonstrates the validity of Gaussian bootstrap procedure described
above, i.e., it states conditions under which the $\tau$-quantile of the test
statistic (2.9) can be approximated by $c^{*}_{\Sigma}(\tau)$ in the
appropriate sense.

Similarly, the test statistic for (2.8) is given by

$S_{\mathcal{D}}^{\Pi}:=\|\sqrt{T}(\widehat{\boldsymbol{\Pi}}_{\mathcal{D}}-\boldsymbol{\Pi}_{\mathcal{D}}^{0})\|_{\max}.$
(2.11)

Let $\boldsymbol{\Upsilon}_{\Pi}$ denote the $(d\times d)$ covariance matrix
of $(\widetilde{\pi}_{ij})_{(i,j)\in\mathcal{D}}$ where
$\widetilde{\pi}_{ij}:=\frac{1}{T}\sum_{t=1}^{T}V_{ij,t}V_{ji,t}$.
$\boldsymbol{\Upsilon}_{\Pi}$ is estimated by

$\widehat{\boldsymbol{\Upsilon}}_{\Pi}:=\sum_{|\ell|<T}\mathcal{K}(\ell/h)\widehat{\boldsymbol{M}}_{\Pi,\ell};\qquad\widehat{\boldsymbol{M}}_{\Pi,\ell}:=\frac{1}{T}\sum_{t=\ell+1}^{T}\widehat{\boldsymbol{D}}_{\Pi,t}\widehat{\boldsymbol{D}}_{\Pi,t-\ell}^{\prime},$
(2.12)

where $\widehat{\boldsymbol{D}}_{\Pi,t}$ is a $d$-dimensional vector with
entries given by $\widehat{V}_{ij,t}\widehat{V}_{ji,t}-\widehat{\pi}_{ij}$ for
$(i,j)\in\mathcal{D}$. Also, let $c^{*}_{\Pi}(\tau)$ be the $\tau$-quantile of
the Gaussian bootstrap
$S^{*}_{\mathcal{D}}:=\|\boldsymbol{Z}^{*}_{\Pi}\|_{\infty}$ and
$\boldsymbol{Z}^{*}_{V}|\boldsymbol{X},\boldsymbol{Y}\sim\mathcal{N}(\boldsymbol{0},\widehat{\boldsymbol{\Upsilon}}_{V})$.

Theorem 5 demonstrate the validity of Gaussian bootstrap, i.e., it states
conditions under which the $\tau$-quantile of the test statistic (2.11) can be
approximated by $c^{*}_{\Pi}(\tau)$ in the appropriate sense.

##  3\. Theoretical Results

In this section we collect all the theoretical guarantees for the estimation
of the model (2.1) by using the proposed three-stage method described above.
Specifically, Section 3.1 deals with estimation and Section 3.2 with inference
on the (partial) covariance structure of $\boldsymbol{\Pi}$.

To present the next results it is convenient to use a more compact notation.
For each $i=1,\ldots,n$, we define the $T$-dimensional vectors
$\boldsymbol{Y}_{i}:=(Y_{i1},\ldots,Y_{iT})^{\prime}$ and
$\boldsymbol{U}_{i}:=(U_{i1},\ldots,U_{iT})^{\prime}$. We also define the
$(T\times k)$ matrix of covariates
$\boldsymbol{X}_{i}:=(\boldsymbol{X}_{i1},\ldots,\boldsymbol{X}_{iT})^{\prime}$
for each $i=1,\ldots,n$ and the $(T\times r)$ matrix of factors
$\boldsymbol{F}:=(\boldsymbol{F}_{1},\dots,\boldsymbol{F}_{T})^{\prime}$ such
that (2.1) can be represented as

$\begin{split}\boldsymbol{Y}_{i}&=\boldsymbol{X}_{i}\boldsymbol{\gamma}_{i}+\boldsymbol{F}\boldsymbol{\lambda}_{i}+\boldsymbol{U}_{i},\qquad
i=1,2,\ldots,n,\\\
&=\boldsymbol{X}_{i}\boldsymbol{\gamma}_{i}+\boldsymbol{R}_{i},\end{split}$
(3.1)

for each cross-sectional unit $i$, where
$\boldsymbol{R}_{i}:=\boldsymbol{F}\boldsymbol{\lambda}_{i}+\boldsymbol{U}_{i}$.

When no confusion is likely to arise, we also define for each $t=1,\ldots,T$,
the $n$-dimensional vectors
$\boldsymbol{Y}_{t}:=(Y_{1t},\ldots,Y_{nt})^{\prime}$ and
$\boldsymbol{U}_{t}:=(U_{1t},\ldots,U_{nt})^{\prime}$; and the
$nk$-dimensional vector
$\boldsymbol{X}_{t}:=(\boldsymbol{X}_{1t}^{\prime},\ldots,\boldsymbol{X}_{nt}^{\prime})^{\prime}$.
Also, set the $(n\times nk)$ block diagonal matrix $\boldsymbol{\Gamma}$ whose
block diagonal is given by
$(\boldsymbol{\gamma}_{1}^{\prime},\dots,\boldsymbol{\gamma}_{n}^{\prime})$
and the $(n\times r)$ loading matrix
$\boldsymbol{\Lambda}:=(\boldsymbol{\lambda}_{1},\dots,\boldsymbol{\lambda}_{n})^{\prime}$.
Then, (2.1) can also be represented as panel time series

$\begin{split}\boldsymbol{Y}_{t}&=\boldsymbol{\Gamma}\boldsymbol{X}_{t}+\boldsymbol{\Lambda}\boldsymbol{F}_{t}+\boldsymbol{U}_{t},\qquad
t=1,2,\ldots,T\\\
&=\boldsymbol{\Gamma}\boldsymbol{X}_{t}+\boldsymbol{R}_{t},\end{split}$ (3.2)

where
$\boldsymbol{R}_{t}:=\boldsymbol{\Lambda}\boldsymbol{F}_{t}+\boldsymbol{U}_{t}$.

###  3.1. Estimation

We consider the following set of assumptions

######  Assumption 2 (Factor Model).

Assume:

  1. (a)

$\mathbb{E}(\boldsymbol{F}_{t})=\boldsymbol{0}$,
$\mathbb{E}\left(\boldsymbol{F}_{t}\boldsymbol{F}_{t}^{\prime}\right)=\boldsymbol{I}_{r}$
and $\boldsymbol{\Lambda}^{\prime}\boldsymbol{\Lambda}$ is a diagonal matrix;

  2. (b)

All eigenvalues of $\boldsymbol{\Lambda}^{\prime}\boldsymbol{\Lambda}/n$ are
bounded away from zero and infinity as $n\to\infty$;

  3. (c)

$\|\boldsymbol{\Sigma}-\boldsymbol{\Lambda}\boldsymbol{\Lambda}^{\prime}\|=O(1)$;
and

  4. (d)

$\|\boldsymbol{\Lambda}\|_{\max}\leq C$.

######  Remark 3.

Assumption 2 is standard in the literature. Note also that the assumption
$\mathbb{E}(\boldsymbol{F}_{t})=\boldsymbol{0}$ is not restrictive as our
approach considers a first-step estimation which may include a constant in the
set of regressors. It is also needed for identifiability.

In order to present the results in a unified manner for both light and heavy
tail distributions, we state the next assumption in terms the Orlicz norm of
the random variables. Specifically, since we are only concerned with
polynomial and exponential tails we define the following subset of unbounded,
convex, real-valued functions that vanish at the origin:

$\displaystyle\Psi$
$\displaystyle:=\\{\psi_{p}:\mathbb{R}^{+}\to\mathbb{R}^{+}:\psi_{p}(x)=x^{p},p\geq
6$ $\displaystyle\qquad\qquad\text{or }\psi_{p}(x)=x\mathds{1}[0\leq
x^{p}<(1-p)/p]+[\exp(x^{p})-1]\mathds{1}[x^{p}\geq(1-p)/p],p>0\\}.$ (3.3)

Also, for each $\psi_{p}\in\Psi$, we define $\psi_{p+}(x):=x^{p+\epsilon}$ for
some $\epsilon>0$ if $\psi_{p}(x)=x^{p}$; otherwise (for the exponential case)
$\psi_{p+}:=\psi_{p}$.

######  Assumption 3 (Moments and Dependency).

There exists a constant $C<\infty$ and function $\psi_{p}\in\Psi$ defined in
(3.1) such that, for all $i=1,\ldots,n$; $\ell=1,\dots,k$; $s,t=1,\ldots,T$;
and $j=1,\dots,r$:

  1. (a)

$\|X_{it\ell}\|_{\psi_{p+}}\leq C$, $\|U_{it}\|_{\psi_{p+}}\leq C$,
$\|F_{j,t}\|_{\psi_{p+}}\leq C$;

  2. (b)

$\|\|(\boldsymbol{X}_{i}^{\prime}\boldsymbol{X}_{i}/T)^{-1}\|_{\max}\|_{\psi_{p+}}\leq
C$;

  3. (c)

The process
$\\{(\boldsymbol{X}_{S,t}^{\prime},\boldsymbol{F}_{t}^{\prime},\boldsymbol{U}_{t}^{\prime})^{\prime},t\in\Z\\}$
is weakly stationary with strong mixing coefficient $\alpha$ satisfying
$\alpha(m)\leq\exp(-2cm)$ for some $c>0$ and for all $m\in\Z$, where
$\boldsymbol{X}_{S,t}$ denotes the vector $\boldsymbol{X}_{t}$ after excluding
all deterministic (non-random) components.

  4. (d)

$\|n^{-1/2}\left[\boldsymbol{U}_{s}^{\prime}\boldsymbol{U}_{t}-\mathbb{E}(\boldsymbol{U}_{s}^{\prime}\boldsymbol{U}_{t})\right]\|_{\psi_{p+}}\leq
C$;

  5. (e)

$\|n^{-1/2}\sum_{i=1}^{n}\lambda_{j,i}U_{it}\|_{\psi_{p+}}\leq C$; and

  6. (f)

$\log n=o\left(\frac{T^{p/4}}{[\log T]^{2}}\right)$.

A few words about Assumption 3 are in order. Assumptions (3.a) and (3.c) allow
us to apply a Marcinkiewicz-Zygmund type inequality for partial sums to deal
with the polinomial tails (Rio (1994) and Doukhan and Louhichi (1999)) and a
Bernstein inequality (Merlevède et al. (2009) \- Theorem 2) to control
exponential tails. Moreover, Assumption (3.c) excludes the deterministic
component of $\boldsymbol{X}_{t}$ to accommodate possibly non-random non-
stationary (but uniformly bounded by (a)) covariates. Assumption (3.d) is only
used to prove results for the first-stage estimation in case it is performed
by ordinary least-squares (Theorem 1). Assumption (3.d) controls for the level
of cross-sectional dependence among the units. As we allow the number of units
to diverge with $T$, some sort of control on this quantity is necessary which
is not implied by $(3.c)$. Assumption (3.e) has a similar role to (3.d) but in
terms of linear combinations of the idiosyncratic components. Assumption (3.e)
only bounds the growth rate of the number of units $n$ to be sub-exponential
with respect to $T$. As a matter of fact, this assumption is only binding in
the exponential tail case, otherwise the rate conditions imposed in the
theorems below imply (3.e).

For each $i=1,\ldots,n$, let
$\boldsymbol{R}_{i}:=\boldsymbol{F}\boldsymbol{\lambda}_{i}+\boldsymbol{U}_{i}$
denote the unobservable error term in (3.1),
$\widehat{\boldsymbol{\gamma}}_{i}$ the least-squares estimator of
$\boldsymbol{\gamma}_{i}$ and
$\widehat{\boldsymbol{R}}_{i}:=\boldsymbol{Y}_{t}-\boldsymbol{X}_{t}\widehat{\boldsymbol{\gamma}}_{i}$
the vector of residuals. Also set
$\widehat{\boldsymbol{R}}:=(\widehat{\boldsymbol{R}}_{1},\dots,\widehat{\boldsymbol{R}}_{n})^{\prime}$
and $\boldsymbol{R}:=(\boldsymbol{R}_{1},\ldots,\boldsymbol{R}_{n})^{\prime}$.
We must control for the least-squares estimation error in the first step of
the proposed methodology. The next result gives a bound for the maximum entry
of the $(n\times T)$ matrix $\widehat{\boldsymbol{R}}-\boldsymbol{R}$ when the
first-stage is conducted by OLS in a linear setup. Note that in this case we
assume that $\boldsymbol{X}_{it}$, $\boldsymbol{F}_{t}$ and $U_{it}$ are
mutually uncorrelated.

######  Theorem 1.

Under Assumption 3(a)-(d)

$\max_{i,t}\|\widehat{R}_{it}-R_{it}\|_{\psi_{p/4}}\leq\frac{C_{k,\psi}}{\sqrt{T}}\quad\textnormal{and}\quad\|\widehat{\boldsymbol{R}}-\boldsymbol{R}\|_{\max}=O_{P}\left[\frac{\psi^{-1}_{p/4}(nT)}{\sqrt{T}}\right],$

where the $C_{k,\psi}$ is a constant only depending on $k$ and $\psi_{p}$.

######  Remark 4.

In case the first step of the method involves more complicated estimation,
such as Pesaran’s (2006) method, instrumental variables, or LASSO, for
example, we write
$\|\widehat{\boldsymbol{R}}-\boldsymbol{R}\|_{\max}=O_{P}(\omega)$, where
$\omega:=\omega_{n,T}$ is a non-negative sequence. This will be used in the
next theorems.

Define the $(n\times T)$ matrices
$\boldsymbol{Y}:=(\boldsymbol{Y}_{1},\ldots,\boldsymbol{Y}_{T})$ and
$\boldsymbol{U}:=(\boldsymbol{U}_{1},\ldots,\boldsymbol{U}_{T})$; and the
$(nk\times T)$ matrix
$\boldsymbol{X}:=(\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{T})$. We can write
(2.1) in the matrix form as

$\boldsymbol{Y}=\boldsymbol{\Gamma}\boldsymbol{X}+\boldsymbol{\Lambda}\boldsymbol{F}^{\prime}+\boldsymbol{U}.$
(3.4)

Notice that
$\widehat{\boldsymbol{R}}=\boldsymbol{\Lambda}\boldsymbol{F}^{\prime}+\widetilde{\boldsymbol{U}}$
where
$\widetilde{\boldsymbol{U}}:=\boldsymbol{U}+\widehat{\boldsymbol{R}}-\boldsymbol{R}$
and $(\boldsymbol{\Lambda},\boldsymbol{F})$ can be estimated by Principal
Component Analysis (PCA), which minimizes

$q(\boldsymbol{\Lambda},\boldsymbol{F}):=\|\widehat{\boldsymbol{R}}-\boldsymbol{\Lambda}\boldsymbol{F}^{\prime}\|_{F}^{2},$
(3.5)

with respect to $\boldsymbol{\Lambda}$ and $\boldsymbol{F}$, subject to the
normalization $\boldsymbol{F}^{\prime}\boldsymbol{F}/T=\boldsymbol{I}_{r}$.
The solution $\widehat{\boldsymbol{F}}$ is the matrix whose columns are
$\sqrt{T}$ times $r$ eigenvectors of the top $r$ eigenvalues of
$\widehat{\boldsymbol{R}}^{\prime}\widehat{\boldsymbol{R}}$ and
$\widehat{\boldsymbol{\Lambda}}=\widehat{\boldsymbol{R}}\widehat{\boldsymbol{F}}/T$.

Since we do not directly observe $\boldsymbol{U}$, in the third step of our
estimation procedure we use
$\widehat{\boldsymbol{U}}:=\widehat{\boldsymbol{R}}-\widehat{\boldsymbol{\Lambda}}\widehat{\boldsymbol{F}}^{\prime}$
instead. Therefore, we must control of the estimation error in the factor
model given by $(n\times T)$ matrix $\widehat{\boldsymbol{U}}-\boldsymbol{U}$
which is the main purpose of Theorem 2 below. Also, it is well know fact that
the loading matrix $\boldsymbol{\Lambda}$ and the factors $\boldsymbol{F}$ are
not separably identified since
$\boldsymbol{\Lambda}\boldsymbol{F}_{t}=\boldsymbol{\Lambda}\boldsymbol{H}^{\prime}\boldsymbol{H}\boldsymbol{F}_{t}$
for any matrix $\boldsymbol{H}$ such that
$\boldsymbol{H}^{\prime}\boldsymbol{H}=\boldsymbol{I}_{r}$. If we let
$\boldsymbol{H}:=T^{-1}\boldsymbol{V}^{-1}\widehat{\boldsymbol{F}}^{\prime}\boldsymbol{F}\boldsymbol{\Lambda}^{\prime}\boldsymbol{\Lambda}$,
where $\boldsymbol{V}$ is the $(r\times r)$ diagonal matrix containing the $r$
largest eigenvalues of
$\widehat{\boldsymbol{R}}^{\prime}\widehat{\boldsymbol{R}}/T$ in decreasing
order, we have that $\boldsymbol{H}\boldsymbol{F}_{t}$ is identified as
$\boldsymbol{\Lambda}\boldsymbol{F}_{t}$ is identified.

The result below first appeared in Bai (2003) for the case of fixed $(n,T)$,
and was further extended to hold uniformly in $(i\leq n,t\leq T)$ by Fan et
al. (2013). Fan et al. (2020) makes the conditions modular. However, both
consider the case when the factor model is estimated using the true data as
opposed to an “estimated” one as in our case. Therefore, the next result is a
generalization that takes into account that pre-estimation error term and
quantifies how the error impact on the precision of factor analysis.

######  Theorem 2.

Let $\omega:=\omega_{n,T}$ be a non-negative sequence such that
$\|\widehat{\boldsymbol{R}}-\boldsymbol{R}\|_{\max}=O_{P}(\omega)$. Then,
under Assumptions 1 -3 and
$\psi^{-1}_{p}(n^{2})/\sqrt{T}+\psi^{-1}_{p}(nT)\omega=O(1)$, we have that

  1. (a)

$\max_{t\leq
T}\|\widehat{\boldsymbol{F}}_{t}-\boldsymbol{H}\boldsymbol{F}_{t}\|_{2}=O_{P}\left[\frac{1}{\sqrt{T}}+\frac{\psi^{-1}_{p}(T)}{\sqrt{n}}+\omega\psi^{-1}_{p/2}(nT)\right],$

  2. (b)

$\max_{i\leq
n}\|\widehat{\boldsymbol{\lambda}}_{i}-\boldsymbol{H}\boldsymbol{\lambda}_{i}\|_{2}=O_{P}\left[\frac{\psi^{-1}_{p/2}(n)}{\sqrt{T}}+\frac{1}{\sqrt{n}}+\omega\right],$

  3. (c)

$\|\widehat{\boldsymbol{U}}-\boldsymbol{U}\|_{\max}=O_{P}\left[\frac{\psi_{p}^{-1}(n)\psi^{-1}_{p}(T)}{\sqrt{T}}+\frac{\psi^{-1}_{p}(T)}{\sqrt{n}}+\omega\psi_{p/2}^{-1}(nT)\right].$

By setting $\omega=0$, i.e., no estimation error in the first step, we recover
Theorem 4 and Corollary 1 in Fan et al. (2013) under sub-Gaussian assumption.
Note that in order to have the error
$\|\widehat{\boldsymbol{U}}-\boldsymbol{U}\|_{\max}$ vanishing in probability
we must have the pre-estimation error
$\|\widehat{\boldsymbol{R}}-\boldsymbol{R}\|_{\max}$ of order (in probability)
smaller than $1/\psi_{p/2}^{-1}(nT)$.

We have decided not to replace $\omega$ in Theorem 2 with the rate obtained in
Theorem 1 as the latter only applies to the least square estimator. In some
applications, however, the first step of the procedure could be done using a
different type of estimator. For instance a penalized adaptive Huber
regression (Fan et al., 2017) if the number of features $k$ is comparable or
even larger than $T$ and the tail of the distribution is heavy. By stating the
Theorem 2 in terms of a generic rate, it is easier to account for the effect
of a different estimator. By combining Theorems 1 and 2 we have the following
corollary

######  Corollary 1.

Under the same assumptions of Theorems 1 and 2, for the OLS used in the first-
stage to obtain $\widehat{\boldsymbol{R}}$, we have

$\|\widehat{\boldsymbol{U}}-\boldsymbol{U}\|_{\max}=O_{P}\left[\frac{\psi^{-1}_{p/6}(nT)}{\sqrt{T}}+\frac{\psi^{-1}_{p}(T)}{\sqrt{n}}\right].$

In particular for the sub-Gaussian case ($\psi(x)=\exp(x^{2})-1$) we have

$\|\widehat{\boldsymbol{U}}-\boldsymbol{U}\|_{\max}=O_{P}\left[\frac{[\log(nT)]^{3}}{\sqrt{T}}+\sqrt{\frac{\log
T}{n}}\right],$

and for polynomial tails ($\psi(x)=x^{p}$)

$\|\widehat{\boldsymbol{U}}-\boldsymbol{U}\|_{\max}=O_{P}\left[\frac{n^{6/p}}{T^{1/2-6/p}}+\frac{T^{1/p}}{\sqrt{n}}\right].$

For notational convenience, for each $i\in\\{1,\ldots,n\\}$, consider the
split $\boldsymbol{U}^{\prime}=(\boldsymbol{U}_{i},\boldsymbol{U}_{-i})$ where
$\boldsymbol{U}_{i}$ is a $T$-dimensional vector and $\boldsymbol{U}_{-i}$ a
$T\times(n-1)$-dimensional matrix. Analogously, we split
$\widehat{\boldsymbol{U}}^{\prime}=(\widehat{\boldsymbol{U}}_{i},\widehat{\boldsymbol{U}}_{-i})$.
Then for a the penalized parameter $\xi\geq 0$, the LASSO objective function
can be written for each $i\in\\{1,\ldots,n\\}$

$L(\boldsymbol{\theta})+\text{Penalty}(\boldsymbol{\theta}):=\frac{1}{T}\|\widehat{\boldsymbol{U}}_{i}-\widehat{\boldsymbol{U}}_{-i}\boldsymbol{\theta}\|^{2}+\xi\|\boldsymbol{\theta}\|_{1}.$
(3.6)

To ensure a consistent estimation of $\boldsymbol{\theta}$, a sort of
restricted strong convexity of the objective function is required when $n>T$.
This in turns is ensured, in the case of a quadratic loss, by bounding the
minimum eigenvalue on
$\widehat{\boldsymbol{U}}_{-i}^{\prime}\widehat{\boldsymbol{U}}_{-i}/T$ away
from zero restrict to a cone (refer to Negahban et al. (2012) or Fan et al.
(2020) for a thorough discussion). Here, we adopt the compatibility constant
defined in van de Geer and Bühlmann (2009). For an index
$\mathcal{S}\subseteq\\{1,\ldots,n\\}$ and any $n$-dimensional vector
$\boldsymbol{v}$, let $\boldsymbol{v}_{\mathcal{S}}$ be the vector containing
only the elements of the vector $\boldsymbol{v}$ indexed by $\mathcal{S}$.
Thus, $\\#\boldsymbol{v}_{\mathcal{S}}=\\#\mathcal{S}$ and
$\mathcal{S}^{c}:=\mathcal{S}\setminus\\{1,\ldots,n\\}$ is the complement of
$\mathcal{S}$.

######  Definition 1.

For an $n\times n$ matrix $\boldsymbol{M}$, a set
$\mathcal{S}\subseteq\\{1,\ldots,n\\}$ and a scalar $\zeta\geq 0$, the
compatibility constant is given by

$\kappa(\boldsymbol{M},\mathcal{S},\zeta):=\inf\left\\{\frac{\|\boldsymbol{x}\|_{\boldsymbol{M}}\sqrt{|\mathcal{S}|}}{\|\boldsymbol{x}_{\mathcal{S}}\|_{1}}:\boldsymbol{x}\in\mathbb{R}^{n}:\|\boldsymbol{x}_{\mathcal{S}^{c}}\|_{1}\leq\xi\|\boldsymbol{x}_{\mathcal{S}}\|_{1}\right\\},$
(3.7)

where
$\|\boldsymbol{x}\|_{\boldsymbol{M}}=\boldsymbol{x}^{\prime}{\boldsymbol{M}}\boldsymbol{x}$.
Moreover, we say that $(\boldsymbol{M},\mathcal{S},\zeta)$ satisfies the
compatibility condition if $\kappa(\boldsymbol{M},\mathcal{S},\zeta)>0$.

Notice that the square of the compatibility constant is close related to the
minimum of the $\ell_{1}$-norm of the eigenvalues of $\boldsymbol{\Sigma}$
restricted to a cone in $\mathbb{R}^{n}$. Let t
$\mathcal{S}_{0,i}:=\\{j:\theta_{i,j}\neq 0\\}$ and
$\kappa_{i}:=\kappa\left[\mathbb{E}(\boldsymbol{U}_{-i}^{\prime}\boldsymbol{U}_{-i})/T),\mathcal{S}_{0,i},3\right]$.
Set the maximum non-sparsity level and minimum compatibility constant

$s_{0}:=\max_{i\leq
n}|\mathcal{S}_{0,i}|\quad\mbox{and}\quad\kappa_{0}:=\min_{i\leq
n}\kappa_{i}.$ (3.8)

The next result shows the $\ell_{1}$ and $\ell_{2}$ rate results for the Lasso
estimator (3.6) based on the “estimated data” and quantifies how the
estimation errors impact on the choice of regularization parameter and rates
of convergence.

######  Theorem 3.

Let $\eta:=\eta_{n,T}$ be a non-negative sequence such that
$\|\widehat{\boldsymbol{U}}-\boldsymbol{U}\|_{\max}=O_{P}(\eta)$. Assume that
Assumption 3 holds and

$s_{0}=O\left[\kappa_{0}\left(\eta\left[\psi^{-1}_{p}(nT)+\eta\right]+\frac{\psi^{-1}_{p/2}(n^{2})}{\sqrt{T}}\right)^{-1}\right].$
(3.9)

For every $\epsilon>0$ there is a constant $0<C_{\epsilon}<\infty$ such that
if the penalty parameter is set to

$\xi=C_{\epsilon}\left[\frac{\psi^{-1}_{p/2}(n)}{\sqrt{T}}+\eta\psi^{-1}_{p}(T)\right],$
(3.10)

then for any minimizer $\widehat{\boldsymbol{\theta}_{i}}$ of (3.6), with
probability at least $1-\epsilon$:

$T^{-1}(\widehat{\boldsymbol{\theta}}_{i}-\boldsymbol{\theta}_{i})^{\prime}\boldsymbol{U}_{-i}^{\prime}\boldsymbol{U}_{-i}(\widehat{\boldsymbol{\theta}}_{i}-\boldsymbol{\theta}_{i})+\xi\|\widehat{\boldsymbol{\theta}}_{i}-\boldsymbol{\theta}_{i}\|_{1}\leq
8\frac{\xi^{2}s_{0}}{\kappa_{0}}.\quad i\in\\{1,\ldots,n\\}.$

######  Remark 5.

Notice that we apply the compatibility condition on the non-random covariance
matrix $\mathbb{E}(\boldsymbol{U}_{-i}^{\prime}\boldsymbol{U}_{-i})/T$ instead
of the estimated random covariance matrix
$\widehat{\boldsymbol{U}}_{-i}^{\prime}\widehat{\boldsymbol{U}}_{-i}/T$ or the
“unobservable” random matrix
$\boldsymbol{U}_{-i}^{\prime}\boldsymbol{U}_{-i}/T$. Careful review of the
proofs reveals that the same is true for the gradient of the objective
function that defines our parameter via a first order condition.

Once again, we purposely avoided to replace $\eta$ in Theorem 3 with the rate
of Corollary 1 to make it readily applicable to the case when a different type
of factor models was used or, as a matter of fact, any other pre-estimation
procedure. By plugging the rate of Corollary 1 into $\eta$ we have the next
corollary

######  Corollary 2.

If $\eta$ defined in Theorem 3 is taken to be rate given by Corollary 1 and
the compatibility condition holds, i.e.: $\kappa_{0}\geq C>0$ then under the
conditions of the Theorem 3:

$\max_{i\leq
n}\|\widehat{\boldsymbol{\theta}}_{i}-\boldsymbol{\theta}_{i}\|_{1}=O_{P}\left[\left(\frac{\psi^{-1}_{p}(T)\psi^{-1}_{p/6}(nT)}{\sqrt{T}}+\frac{\psi^{-1}_{p/2}(T)}{\sqrt{n}}\right)s_{0}\right].$

###  3.2. Inference on Covariance and Precision Matrices

We now obtain the null distributions of our test statistics for the structures
of the covariance and the partial covariance. Recall the setup and notation of
section 2.3. In particular, we consider the kernel $\mathcal{K}(\cdot)$
appearing in the covariance estimator defined by (2.10) belongs to the class
defined in Andrews (1991) which we reproduce below for convenience

$\K:=\\{f:\mathbb{R}\to[-1,1]:f(0)=1,f(x)=f(-x),\forall x\in\mathbb{R},\int
f^{2}(x)dx<\infty,f\text{ is continuous}\\}.$ (3.11)

This includes most of the well-known kernel used in the literature. To avoid
confusion, it is worth to point out that our tuning parameter $h$, also called
bandwidth parameter by Andrews (1991), is supposed to diverge, as opposed to
the bandwidth in the density kernel estimation setup, which is expected to
shrink to zero.

The next result shows how accurately the elements of the covariance matrix are
estimated and validates the bootstrap method.

######  Theorem 4.

Let $\eta:=\eta_{n,T}$ and $\nu:=\nu_{n,T}$ be non-negative sequence such that
$\|\widehat{\boldsymbol{U}}-\boldsymbol{U}\|_{\max}=O_{P}(\eta)$ and
$\max_{i,t}\|\widehat{R}_{it}-R_{it}\|_{\psi_{p}}=O(\nu)$ and
$\mathcal{K}(\cdot)\in\K$. Under Assumptions 1–3, if further

  1. (a)

$\\{\boldsymbol{U}_{t}\\}$ is fourth-order stationary process

  2. (b)

$\|\mathsf{diag}(\boldsymbol{\Upsilon}_{\Sigma})\|_{\infty}\geq\underline{c}$
for some $\underline{c}>0$

  3. (c)

As $h,n,T\to\infty$:

    1. (c.1)

$\frac{(\log n)^{7/6}\psi_{p/2}^{-1}(n)}{T^{1/6}}+\frac{\sqrt{\log T\log
n}\psi_{p/2}^{-1}(n)\psi^{-1}_{p/2}(T^{1/4})}{T^{1/4}}=o(1)$

    2. (c.2)

$(\log
n)^{3}h\left[\eta(\psi^{-1}_{p}(nT))^{3}+\psi_{p/4}^{-1}(n^{4})/\sqrt{T}\right]=o(1)$

    3. (c.3)

$(\log
n)^{3}\left(\sqrt{T}\eta^{2}+\frac{r_{1}}{\sqrt{T}}+\frac{r_{2}}{\sqrt{n}}+r_{3}\nu\right)=o(1)$,

where the rates $r_{1},r_{2},r_{3}$ are defined in Lemma 10 and $h>0$ is the
bandwidth parameter of the covariance estimator defined in (2.10); then

$\|\widehat{\boldsymbol{\Upsilon}}_{\Sigma}-\boldsymbol{\Upsilon}_{\Sigma}\|_{\max}=O_{P}\left[h\left(\eta\left[\psi^{-1}_{p}(nT)\right]^{3}+\psi_{p/4}^{-1}(n^{4})/\sqrt{T}\right)\right]=o(1),$

and
$\sup_{\mathcal{D}}\sup_{\tau\in(0,1)}|\P\left[S_{\mathcal{D}}^{\Sigma}\leq
c^{*}_{\Sigma}(\tau)\right]-\tau|=o(1)$, where the first supremum is over all
null hypotheses of the form (2.7) indexed by
$\mathcal{D}\in\\{1:n\\}\times\\{1:n\\}$.

######  Remark 6.

The rate assumptions (c.1)–(c.3) in Theorem 4 seem over complicated. However,
they are a direct consequence of having the first and second step estimation
error rates, $\nu$ and $\eta$ respectively, explicitly appearing in the final
rate and the general tail condition through the $\psi_{p}(\cdot)$ function. It
allows the practitioner to directly adjust the final rate should (s)he prefer
to employ different intermediate estimators. For instance, a LASSO estimator
in the first step in case the number of covariates $k$ is large enough or
estimate the factor model by PCA variants.

######  Remark 7.

Careful review of Theorem 4’s proof reveals that (c.1) traces back to the
Gaussian Approximation of the (unobservable) process
$\left\\{\frac{1}{\sqrt{T}}\sum_{t=1}^{T}\boldsymbol{U}_{t}\boldsymbol{U}_{t}^{\prime}-\mathbb{E}\boldsymbol{U}_{t}\boldsymbol{U}_{t}^{\prime}\right\\}_{T\geq
1}$; whereas (c.3) controls for the difference between
$\boldsymbol{U}_{t}-\widehat{\boldsymbol{U}}_{t}$ and, therefore, takes into
account the estimation error of the first and second steps. Note the presence
of $\nu$ and $\eta$ in (c.3) which are absent in (c.1). Finally, (c.2) make
sure that the bootstrap constructed in terms of the estimated covariance
matrix is close to the bootstrap based in the true covariance. Note the
presence of the bandwidth parameter $h$ in (c.2).

If we were to specialized Theorem 4 to the sub-Gaussian case and incorporate
the rates obtain in Theorem 1 and Corollary 1 we have the following cleaner
Corollary.

######  Corollary 3.

Consider the sub-Gaussian where $\psi_{2}(x)=\exp(x^{2})$. Suppose that the
Assumptions 1-3 and conditions $(a)$ and $(b)$ of Theorem 4 hold. If the rates
$\nu$ and $\eta$ are set to be rates given by Theorem 1 and Corollary 1,
respectively, then the conclusion of Theorem 4 holds provided that as
$h,n,T\to\infty$:

  1. (a)

$\log n=o(T^{1/18})$

  2. (b)

$h\left[\frac{(\log n)^{15/2}}{\sqrt{T}}+\frac{(\log
n)^{5}}{\sqrt{n}}\right]=o(1)$

  3. (c)

$\frac{(\log n)^{3}(\log T)\sqrt{T}}{n}=o(1)$.

######  Remark 8.

In order to establish the rate of convergence in the last result of Theorem 4
we need an upper bound on the tails of the pre-estimation error namely
$\|\widehat{\boldsymbol{Z}}-\boldsymbol{Z}\|_{max}$. In fact, we need to
control the tails of the factor model estimation to establish uniform bounds
on $\|\widehat{U}_{it}-U_{it}\|_{\psi}$, which translate into obtain bounds on
$\max_{jt}\|\widehat{F}_{jt}-F_{jt}\|_{\psi}$ and
$\max_{ji}\|\widehat{\lambda}_{ji}-\lambda_{ji}\|_{\psi}$.

The next theorem shows how well the elements of the partial autocovariance
matrix is estimated and gives the conditions under which the bootstrap test
works. Note that in calculation the partial covariance (2.6), we used the
residual $\widehat{V}_{ijt}$ of the LASSO regression of $\widehat{U}_{it}$
onto $\widehat{\boldsymbol{U}}_{-ij,t}$ for $i,j\in\\{1,\ldots,n\\}$. This
adds extra technical challenges to the proof. Let
$\widetilde{\mathcal{S}}_{0,ij}$ denote the active set of the projection of
$U_{it}$ onto $\boldsymbol{U}_{-ij,t}$ for $i,j\in\\{1,\ldots,n\\}$ and
$\widetilde{\kappa}_{ij}:=\kappa\left[\mathbb{E}(\boldsymbol{U}_{-ij}^{\prime}\boldsymbol{U}_{-ij})/T),\widetilde{\mathcal{S}}_{0,ij},3\right]$.
Set the maximum non-sparsity level and minimum compatibility constant

$\widetilde{s}_{0}:=\max_{1\leq i\neq j\leq
n}|\widetilde{\mathcal{S}}_{0,ij}|\quad\mbox{and}\quad\widetilde{\kappa}_{0}:=\min_{1\leq
i\neq j\leq n}\widetilde{\kappa}_{ij}.$ (3.12)

We assume the same sparsity structure as in Theorem 3.

######  Theorem 5.

Let $\eta:=\eta_{n,T}$ and $\nu:=\nu_{n,T}$ be non-negative sequence such that
$\|\widehat{\boldsymbol{U}}-\boldsymbol{U}\|_{\max}=O_{P}(\eta)$ and
$\max_{i,t}\|\widehat{R}_{it}-R_{it}\|_{\psi_{p}}=O(\nu)$, $\mathcal{K}\in\K$
where $\mathcal{K}$ appears in (2.12) and $\K$ defined by (3.11), the LASSO
regularization parameter $\widetilde{\xi}$ set as in (3.10), and the sparsity
level obeying (3.9) with $s_{0}$ and $\kappa_{0}$ replaced by
$\widetilde{s}_{0}$ and $\widetilde{\kappa}_{0}$, respectively. Under
Assumptions $2$-$4$, if further

  1. (a)

$\\{\boldsymbol{U}_{t}\\}$ is fourth-order stationary process

  2. (b)

$\|\mathsf{diag}(\boldsymbol{\Upsilon}_{\Pi})\|_{\infty}\geq\underline{c}$ for
some $\underline{c}>0$

  3. (c)

As $n,T\to\infty$:

    1. (c.1)

$\frac{(\log n)^{7/6}\psi_{p/2}^{-1}(n)}{T^{1/6}}+\frac{\sqrt{\log T\log
n}\psi_{p/2}^{-1}(n)\psi^{-1}_{p/2}(T^{1/4})}{T^{1/4}}=o(1)$

    2. (c.2)

$(\log
n)^{3}h\left(\widetilde{s}_{0}\left[\eta+\xi\psi^{-1}_{p}(n)\right]\left[\widetilde{s}_{0}\psi^{-1}_{p}(nT)\right]^{3}+\widetilde{s}_{0}\frac{\psi_{p/4}^{-1}(n^{4})}{\sqrt{T}}\right)]=o(1)$

    3. (c.3)

$(\log
n)^{3}\left(\widetilde{s}_{0}^{2}\left\\{\frac{r_{1}}{\sqrt{T}}+\frac{r_{2}}{\sqrt{n}}+r_{3}\nu+\widetilde{\xi}\psi^{-1}_{p}(n)+\sqrt{T}\left[\eta+\xi\psi^{-1}_{p}(n)\right]^{2}\right\\}\right)=o(1)$,

where the rates $r_{1},r_{2},r_{3}$ are defined in Lemma 10 and $h>0$ is the
bandwidth parameter of the covariance estimator defined in (2.12); then

$\|\widehat{\boldsymbol{\Upsilon}}_{\Pi}-\boldsymbol{\Upsilon}_{\Pi}\|_{\max}=O_{P}\left(h\left\\{\widetilde{s}_{0}[\eta+\widetilde{\xi}\psi^{-1}_{p}(n)][\widetilde{s}_{0}\psi^{-1}_{p}(nT)]^{3}+\widetilde{s}_{0}\frac{\psi_{p/4}^{-1}(n^{4})}{\sqrt{T}}\right\\}\right)=o(1)$

and $\sup_{\mathcal{D}}\sup_{\tau\in(0,1)}|\P(S_{\mathcal{D}}^{\Pi}\leq
c^{*}_{\Pi}(\tau))-\tau|=o(1)\quad\text{under $\H{_}0^{\Pi}$}$, where the
first supremum is over all null hypotheses of the form (2.8) indexed by
$\mathcal{D}\in\\{1\times n\\}\times\\{1\times n\\}$.

Similar comments as in Remarks 6-8 and Corollary 3 apply to Theorem 5 as well.

######  Remark 9.

As opposed to the case of testing covariance, when testing partial covariance
in high-dimensional setup, the sparse structure plays a role in terms of
$\widetilde{s}_{0}$ appearing in the rates (c.2) and (c.3). Therefore, these
assumptions restricts the cases when the proposed partial covariance test has
the correct asymptotic size. For instance, in the case of a complete dense
partial covariance structure, i.e, all the regressors are active in all LASSO
regressions we are likely to have $\widetilde{s}_{0}$ of order of $n$ and,
therefore, $(c.2)$ and $(c.3)$ are not expected to hold.

##  4\. Guide to Practice

The methodology in this paper involves three steps. The first step consists of
identifying known covariates that we may want to control for. It may involve
the removal of deterministic trends and seasonal effects, for instance. This
can be done either by parametric or nonparametric regressions. It is important
to notice, however, that the convergence rates of the estimations in the
subsequent steps will be influenced by the convergence rate of the estimation
in the first part of the procedure.

After the data is filtered in the first step, one can test for remaining
covariance structure. For instance, if the covariance matrix of the filtered
data is (almost) diagonal, there is no need to estimate a latent factor
structure and the practitioner may jump directly to the third step of the
method.

On the other hand, if the covariance of the first-step filtered data is dense,
a latent factor model should be considered and the number of factors must be
determined. To determine the number of factors we consider either the
eigenvalue ratio test of Horenstein (2013) or the information criteria put
forward in Bai and Ng (2002). The factors can be estimated by the usual
methods.

The last step involves a sparse regression in order to estimate any remaining
links between idiosyncratic components. Before running the last step, we may
test for a diagonal covariance matrix of the idiosyncratic terms. If the null
is not rejected, there is no need for additional estimation. In case of
rejection, we can proceed with a LASSO regression. We recommend that the
penalty term is selected by Bayesian Information Criterion (BIC) as advocated
by Medeiros and Mendes (2016).

Finally, concerning the estimation of the long-run matrices, the usual methods
discussed in the literature can be used here to select the kernel and the
bandwidth. We use the simple Bartlett kernel with bandwidth given as $\lfloor
T/3\rfloor$.

##  5\. Simulation

In this section we report simulation results divided into two parts. In the
first one, we evaluate the finite-sample properties of the test for remaining
covariance structure. In the second part, we highlight the informational gains
when considering both the common factors and the idiosyncratic component. We
simulate 1,000 replications of the following model for various combinations of
sample size ($T$) and number of variables ($n$):

$\displaystyle\boldsymbol{Y}_{it}$
$\displaystyle=\boldsymbol{\Lambda}_{i}^{\prime}\boldsymbol{F}_{t}+W_{it},$
(5.1) $\displaystyle\boldsymbol{F}_{t}$
$\displaystyle=0.8\boldsymbol{F}_{t-1}+\boldsymbol{E}_{t},$ (5.2)
$\displaystyle W_{it}$ $\displaystyle=\phi W_{it-1}+U_{it},$ (5.3)
$\displaystyle U_{it}$
$\displaystyle=I(i=1)(\theta_{12}U_{2t}+\theta_{13}U_{3t}+\theta_{14}U_{4t}+\theta_{15}U_{5t})+V_{it}$
(5.4)

where $I(\cdot)$ is the indicator function, $\\{V_{it}\\}$ is a sequence of
independent Gaussian random variables with zero mean and variance equal to
0.25, and $\\{\boldsymbol{E}_{t}\\}$ is a sequence of $r$-dimensional
independent random vectors normally distributed with zero mean and identity
covariance. Furthermore, $\\{V_{it}\\}$ and $\\{\boldsymbol{E}_{t}\\}$ are
mutually independent for all time periods, factors and variables. For each
Monte Carlo replication, the vector of loadings is sampled from a Gaussian
distribution with mean -6 and standard deviation 0.2 for $i=1$ and mean 2 and
unit variance for $i=2,\ldots,n$. The value of $\phi$ is either 0 or 0.5. The
coefficients $\theta_{12}$, $\theta_{13}$, $\theta_{14}$, and $\theta_{15}$
are equal to zero or 0.8, 0.9, -0.7, and 0.5, respectively. We set $r=3$.

###  5.1. Test for Remaining Covariance Structure

We start by reporting results for the test of no remaining structure on the
covariance matrix of $\boldsymbol{U}_{t}=(U_{1t},\ldots,U_{nt})^{\prime}$. The
null hypothesis considered is that all the covariances between the first
variable ($i=1$) and the remaining ones are all zero. For size simulations we
set $\theta_{12}=\theta_{13}=\theta_{14}=\theta_{15}=0$ in the DGP. In order
to evaluate the effects of factor estimation as well as the methods in
selecting the number of factors, we consider the following scenarios: (1)
factors are known and there is no estimation involved; (2) factors are
estimated by principal components but the number of factors is known; (3) the
number of factors is determined by the eigenvalue ratio procedure of
Horenstein (2013); (4)-(7) the number of factors is determined by one of the
four information criteria proposed by Bai and Ng (2002) as defined by

$\begin{matrix}[l]\textnormal{IC}_{1}=\log[S(r)]+r\frac{n+T}{nT}\log\left(\frac{nT}{n+T}\right)&\textnormal{IC}_{2}=\log[S(r)]+r\frac{n+T}{nT}\log
C_{nT}^{2}\\\ \textnormal{IC}_{3}=\log[S(r)]+r\frac{\log
C_{nT}^{2}}{C_{nT}^{2}}&\textnormal{IC}_{4}=\log[S(r)]+r\frac{(n+T-k)\log(nT)}{nT}.\end{matrix}$

where
$S(r)=\frac{1}{nT}\|\boldsymbol{R}-\widehat{\boldsymbol{\Lambda}}_{r}\widehat{\boldsymbol{F}}_{r}\|_{2}^{2}$
and $C_{nT}:=\sqrt{\min(n,T)}$.

Tables 1 and 5 report the results of the empirical size of test for different
significance levels. We consider the case of $\phi=0$ in Table 1 and
$\phi=0.5$ in Table 5 in the Supplementary Material. The tables present the
results when the factors are known in panel (a), the factors are unknown but
the number of factors is known in panel (b), or the number of factors are
estimated either by the information criterion $\textnormal{IC}_{1}$ in panel
(c) or the eigenvalue ratio procedure in panel (d). Table 7 in the
Supplementary Material shows the results when the number of factors are
determined by $\textnormal{IC}_{2}-\textnormal{IC}_{4}$.

A number of facts emerge from the inspection of the results in the Table 1.
First, size distortions are small when the factors are known. In this case,
the test is undersized when the pair $(n,T)$ is small. When the factor are not
known but the true number of factors is available, the size distortions are
high only when $T=100$ and $n=50$ due to inaccurate estimation of factors.
However, the distortions disappear when the pair $(T,n)$ grows. In this case,
the empirical size is similar to the situation reported in Panel (a). The
finite performance of test in the case where the number of factors is selected
by information criterion $\textnormal{IC}_{1}$ is almost indistinguishable to
the case reported in Panel (b). However, the results with the eigenvalue ratio
procedure are much worse when $T=100$ and $n=50$. In this case, the procedure
selects less factors than true number $r=3$. For instance, the procedure
selects 2 or less factors in 36% of the replications. Just as comparison, for
$T=100$ and $n=50$, $\textnormal{IC}_{1}$ underdetermines the number of
factors only in 3.10% of the cases. The latter also confirms that over
estimation of the number of factors will not have big adversorial effect. For
all the other combinations of $T$ and $n$ all the data-driven methods selects
the correct number of factors in almost all replications.

When the idiosyncratic components are autocorrelated the size distortions are
higher, as reported in Table 5. This is mainly caused by the well-known
difficulties in the estimation of the long-run covariance matrix.

Tables 2 and 6 report the results of the empirical power with
$\beta_{12}=0.8$, $\beta_{13}=0.9$, $\beta_{14}=-0.7$, and $\beta_{15}=-0.5$
in the DGP. When the factors are known, the test always rejects the null and
the empirical power is one for any significance level. On the other hand, when
factors must be estimated but the number of factors are known, the power
decreases as depicted in panel (b) in the tables. Nevertheless for $T=500,700$
the power is reasonably high, specially when test is conducted at a $10\%$
significance level. For $T=100$, the performance deteriorates as $n$ grows.
The results are similar when data-driven procedures are used to determine the
number of factors and the conclusions are mostly the same if $\phi=0$ or
$\phi=0.5$.

The main message of the simulation exercise is that the finite-sample
performance of the proposed tests depend on the correct selection of factors.
Nevertheless, for the DGP considered here, the usual data-driven methods
available in the literature to determine the true number of factors seem to
work reasonably well.

###  5.2. Informational Gains

The goal of this simulation is to compare, in a prediction environment, the
three-stage method developed in the paper by evaluating the information gains
in predicting $Y_{1t}$ by three different methods. First, the predictions are
computed from a LASSO regression of $Y_{1t}$ on all the other $n-1$ variables.
This is the Sparse Regression (SR) approach. Second, we consider a principal
component regression (PCR), i.e., an ordinary least squares (OLS) regression
of the variable of interest on factors computed from the pool of other
variables. Finally, we consider predictions constructed from the method
proposed here, the FarmPredict methodology. Table 3 presents the results. The
table presents the average mean squared error (MSE) over 5-fold cross-
validation (CV) subsamples. As in the size and power simulations, we consider
different combinations of $T$ and $n$. We report results for the case where
$\theta_{12}=0.8$, $\theta_{13}=0.9$, $\theta_{14}=-0.7$, and
$\theta_{15}=-0.5$ in the DGP.

According to the DGP, the theoretical MSE is 0.25 when all the information is
used. When just a factor is used, the MSE is 2.21. From the table is clear
that there are significant informational gains when we consider both factors
and the cross-dependence between idiosyncratic components. Several conclusions
emerge from the table. First, it is clear that when the sample size increases
the MSE reduces. This is expected. Second, the PCR’s MSE and FarmPredict’s MSE
are close to their theoretical values of 2.21 and 0.25 when the sample
increases. The performance of the FarmPredict is quite remarkable when $T=500$
or $T=700$ and is always superior to Sparse Regression and PCR.

##  6\. Applications

In this section we consider two applications with real data to illustrate the
benefits of the methodology developed in the paper.

###  6.1. Factor Models and Network Structure in Asset Returns

####  6.1.1. The Dataset

We illustrate the methodology developed in this paper by studying the factor
structure of asset returns. We consider monthly close-to-close excess returns
from a cross-section of 9,456 firms traded in the New York Stock Exchange. The
data starts on November 1991 and runs until December 2018. There are 326
monthly observations in total. In addition to the returns we also consider 16
monthly factors: Market ($\mathsf{MKT}$), Small-minus-Big ($\mathsf{SMB}$),
High-minus-Low ($\mathsf{HML}$), Conservative-minus-Aggressive
($\mathsf{CMA}$), Robust-minus-Weak ($\mathsf{RMW}$), earning/price ratio,
cash-flow/price ratio, dividend/price ratio, accruals, market beta, net share
issues, daily variance, daily idiosyncratic variance, 1-month momentum, and
36-month momentum. The firms are grouped according to 20 industry sectors as
in Moskowitz and Grinblatt (1999). The following sectors are considered:888The
number between parenthesis indicate the number of firms in our sample that
belong to each sector. Mining (602), Food (208), Apparel (161), Paper (81),
Chemical (513), Petroleum (48), Construction (68), Primary Metals (133),
Fabricated Metals (186), Machinery (710), Electrical Equipment (782),
Transportation Equipment (166), Manufacturing (690), Railroads (25), Other
transportation (157), Utilities (411), Department Stores (67), Retail (1018),
Financial (3419), and Other (11).

####  6.1.2. Results

We start the analysis by looking at the correlation matrix of a sample of nine
different sectors, namely: Mining, Food, Petroleum, Construction,
Manufacturing, Utilities, Department Stores, Retail, and Financial. Figure 1
plots the correlations that are larger than 0.15 in absolute value. We also
test for the null of diagonal covariance matrix. The null hypothesis is
strongly rejected with $p$-value much lower than 1%. To conduct the test of
the covariance matrix we use the simple sample estimator as described in the
paper. However, the correlations plotted in Figure 1 and in the subsequent
ones are based on the nonlinear shrinkage estimator proposed by Ledoit and
Wolf (2020).

We proceed by regressing the daily returns on the observed 16 factors. These
three factors explain most of the variation of the returns. Figure 2 shows the
empirical distribution of the OLS estimates of factor loadings over the 9,456
regressions. Figure 3 presents the estimated correlations for the first-stage
residuals. We focus on the nine sectors as before. The first-stage regression
as efficient in removing the correlation within specific sectors in some
cases. The most notable ones are Financial and Retail, followed by
Construction, Petroleum, and Manufacturing. Nevertheless, the tests for
diagonal covariance matrix reject the null even in these specific cases.

The second step is to conduct a principal component analysis on the residuals
of the first-stage. The eigenvalue ratio procedure selects two factors, while
all four information criteria points to a single factor. We proceed with two
factors. Note that, by construction, the principal component factors are
orthogonal to all the 16 risk factors considered in the first stage. Figure 4
shows the estimated correlations for the residuals of the second-stage. The
latent factor are not able to reduce the correlations within each sector.
However, when we consider the partial correlations the conclusions are much
different. As can be seen from Figure 5 that the partial correlation matrices
are (almost) diagonal. In addition, we are not able to reject the null of a
diagonal covariance matrix at a 5% significance level.

To shed some light on the links among different sectors, we report how often
variables from sector $i$ are selected in the third-stage LASSO regression for
firms in sector $j$. The numbers are normalized by the total number of firms
in each sector and are presented in Figure 6. The most interesting fact is
that covariates from the financial sector are the ones most frequently
selected for all the other sectors. This may indicate that there is a
“financial factor” that was unmodeled in the first two stages.

The results presented here can be useful in applications where forecasting
future returns is the goal, for instance. The results indicate that the
inclusion of the returns of firms belonging to the financial sector may
improve the performance of forecasting models. For example, if we run a
regression of the residuals of the first-stage regression of firms that do not
belong to the financial sector on the first principal component computed with
the first-stage residuals only from the financial sector, we find a
statistically significant coefficient in 28% of the cases.

###  6.2. Macroeconomic Forecasting

The second application consists of forecasting of a large set of monthly
macroeconomic variables. We compare four different models: (1) Autoregressive
model; (2) Sparse LASSO Regression (SR); (3) Principal Component Regression
(PCR); and (4) FarmPredict.

####  6.2.1. The Dataset

Our data consist of variables from the FRED-MD database, which is a large
monthly macroeconomic dataset designed for empirical analysis in data-rich
macroeconomic environments. The dataset is updated in real time through the
FRED database and is available from Michael McCraken’s
webpage.999https://research.stlouisfed.org/econ/mccracken/fred-databases/. For
further details, we refer to McCracken and Ng (2016).

We use the vintage as of October 2020. Our sample extends from January 1960 to
December 2019 (719 monthly observations), and only variables with all
observations in the sample period are used (119 variables). The dataset is
divided into eight groups: (i) output and income; (ii) labor market; (iii)
housing; (iv) consumption, orders and inventories; (v) money and credit; (vi)
interest and exchange rates; (vii) prices; and (viii) stock market. Finally,
all series are transformed in order to become approximately stationary as in
McCracken and Ng (2016).

####  6.2.2. Setup and Methodology

In order to highlight the gains of exploring all relevant information in the
the dataset, we construct one-step forecasts for each one of the 119 variables
in the dataset according to the following models:

  1. (1)

Autoregressive model (AR):

$\widehat{Y}_{i,t+1|t}^{(\texttt{AR})}=\widehat{\phi}_{i0}+\widehat{\phi}_{i1}\widehat{Y}_{i,t}+\ldots+\widehat{\phi}_{ip}\widehat{Y}_{i,t-p+1},\quad
i=1,\ldots,n,$

where $\widehat{\phi}_{i0},\widehat{\phi}_{i1},\ldots,\widehat{\phi}_{ip}$,
$i=1,\ldots,n$, are OLS estimates. This model will be also the first-stage
model in our methodology.

  2. (2)

AR + Sparse regression (SR):

$\widehat{Y}_{i,t+1|t}^{(\texttt{SR})}=\widehat{Y}_{i,t+1|t}^{(\texttt{AR})}+\widehat{R}_{i,t+1|t},$

where $\widehat{R}_{i,t}=Y_{i,t}-\widehat{Y}_{i,t|t-1}^{(\texttt{AR})}$,
$i=1,\ldots,n$,
$\widehat{\boldsymbol{R}}_{t}=\left(\widehat{R}_{1,t},\ldots,\widehat{R}_{n,t}\right)$,
and

$\widehat{R}_{i,t+1|t}=\widehat{\beta}_{0i}+\widehat{\boldsymbol{\beta}}_{1i}^{\prime}\widehat{\boldsymbol{R}}_{t}+\ldots+\widehat{\boldsymbol{\beta}}_{pi}^{\prime}\widehat{\boldsymbol{R}}_{t-p+1},\quad
i=1,\ldots,n,$

$\widehat{\beta}_{0i},\widehat{\boldsymbol{\beta}}_{1i}\ldots,\widehat{\boldsymbol{\beta}}_{pi}$,
$i=1,\ldots,n$, are LASSO estimates. The parameters are estimated equation-
wise for each one of the 119 variables in the dataset. The penalty parameter
is selected by BIC as discussed in Section 4.

  3. (3)

AR + Principal Component Regression (PCR):

$\widehat{Y}_{i,t+1|t}^{(\texttt{PCR})}=\widehat{Y}_{i,t+1|t}^{(\texttt{AR})}+\widehat{\boldsymbol{\lambda}}_{i}^{\prime}\widehat{\boldsymbol{F}}_{t},$

where $\widehat{\boldsymbol{F}}_{t}$ is the estimate of the $(k\times 1)$
vector of factors $\boldsymbol{F}_{t}$ given by principal component analysis
of $\boldsymbol{\widehat{R}}_{t}$, the residuals of the first-stage
regression. The parameter $\boldsymbol{\lambda}_{i}$ is computed by OLS
regression of $\widehat{R}_{i,t}$ on $\widehat{\boldsymbol{F}}_{t}$ in-sample.

  4. (4)

AR + Full Information (FarmPredict):

$\widehat{Y}_{i,t+1|t}^{(\texttt{FarmPredict})}=\widehat{Y}_{i,t+1|t}^{(\texttt{PCR})}+\widehat{U}_{i,t+1|t}$

where

$\widehat{U}_{i,t+1|t}=\widehat{\theta}_{0i}+\widehat{\boldsymbol{\theta}}_{1i}^{\prime}\widehat{\boldsymbol{U}}_{t}+\ldots+\widehat{\boldsymbol{\theta}}_{pi}^{\prime}\widehat{\boldsymbol{U}}_{t-p+1},$

$\widehat{\boldsymbol{U}}_{t}=\left(\widehat{U}_{1,t},\ldots,\widehat{U}_{n,t}\right)^{\prime}$
and $\widehat{U}_{i,t}=Y_{i,t}-\widehat{Y}_{i,t|t-1}^{(\texttt{PCR})}$,
$i=1,\ldots,n$. The estimates
$\widehat{\theta}_{0i},\widehat{\boldsymbol{\theta}}_{1i}\ldots,\widehat{\boldsymbol{\theta}}_{pi}$,
$i=1,\ldots,n$, are given by LASSO.

The forecasts are based on a rolling-window framework of fixed length of 480
observations, starting in January 1960. Therefore, the forecasts start on
January 1990. The last forecasts are for December 2019. Note that the AR model
only considers information concerning the own past of the variable of
interest. SR and PCR expand the information by two opposing routes. While SR
uses a sparse combination of the set of variables, PCR considers only a factor
structure (dense model). FarmPredict combines these two approaches and uses
the full information available.

####  6.2.3. Brief In-Sample Analysis

We start by looking at the full sample in order to analyze the structure of
dependence among the variables. We first estimate an autoregressive model of
order 4, AR($p$), for each transformed series. Panel (a) in Figure 7 reports
the empirical distribution of the OLS estimators of the AR coefficients and
Panel (b) in the figure shows the distribution of the absolute value of the
sum of the estimates. This gives an idea of the persistence of each series.
Only one series has estimated persistence above one. This is the case for
_NONBORRES: Reserves of Depositary Institutions_ , which belongs to group (v):
Money and Credit. The reason for such high persistence if due to a major
structural break present in the second half of the series. However, 82.35% of
the series have estimated persistence below 0.9.101010Conventional unit-root
tests also reject the null of unit-root for all but one of the series.

We continue by estimating the number of factors when the full sample is used.
We consider two different situations. In the first, we do not include any lag
in the basket of variables used to compute the factors. In the second
approach, we include four lags of each variable as well. The eigenvalue ratio
procedure selects either two (no lags) or a single factor (with lags). The
four information criteria of Bai and Ng (2002) estimate for the case with no
lags (with lags) the following number of factors: six (one), five (one), nine
(one), and one (one). Note that the factors are estimated for the residuals of
the first-step AR filter. If we remove the _NONBORRES_ variable from the
sample the results to not change for the eigenvalue ratio procedure. On the
other hand, the new numbers of factors selected by the information criteria
are as follows: seven (one), six (one), eleven (one), and one (one).

Finally, we apply the testing approach developed in this paper to check for
remaining (partial) covariance structure in the data. The tests strongly
reject the null of a diagonal matrix when applied to the residuals either of
the first or the second stages of the methodology. This serves as evidence
that FarmPredict may be a useful modeling approach for this macroeconomic
dataset.

####  6.2.4. Forecasting Results

For each of the models above, we report a number of performance metrics in
Table 4. The table presents the frequency each model has the best performance
among the four alternatives. Numbers between parentheses indicates the
frequency each model is the second, third, and fourth best. We report the
results for each one of the eight sectors as well as for the set of all 119
variables. We show the results for two methods to determine the number of
factors. Panel (a) reports the results for the eigenvalue ratio method while
Panel (d) presents the results for the information criterion
$\textnormal{IC}_{4}$. Criteria $\textnormal{IC}_{1}$, $\textnormal{IC}_{2}$,
and $\textnormal{IC}_{3}$ select a very large number of factors. Panels (c)
and (d) in the table show the results for the cases where the number of
factors are kept fixed ($r=1$ or $r=2$).

FarmPredict is the model which is ranked first more frequently when all the
series are considered. It is also the best model for the following groups:
output and income, labor market, housing, and consumption, orders and
inventories. The AR model is best for the following groups: money and credit
and stock market. The sparse regression is superior also for two groups:
interest and exchange rates and prices.

##  7\. Conclusions

In this paper we propose a new methodology which bridges the gap between
sparse regressions and factor models and evaluate the gains of increasing the
information set via factor augmentation. Our proposal consists in several
steps. In the first one, we filter the data for known factors (trends,
seasonal adjustments, covariates). In the second step, we estimate a latent
factor structure. Finally, in the last part of the procedure we estimate a
sparse regression for the idiosyncratic components. We also propose a new test
for remaining structure in both high-dimensional covariance and partial
covariance matrices. Our test can be used to evaluate the benefits of adding
more structure in the model. Our paper has also a number of important side
results. First, we proved consistency of kernel estimation of long-run
covariance matrices in high-dimensions where both the number of observations
and variables grows. Second, we derive the theoretical properties of factor
estimation on the residuals of a first step process. Third, the proposed test
can be used as a diagnostic tool for factor models.

We evaluate our methodology with simulations and real data. The simulations
show the test has good size and power properties even when the true number of
factors is unknown and must be determined from the data. If the number of
factors is underestimated, we observe size distortions. This is specially the
case when the eigenvalue ratio test is used to determine the number of latent
factors. The simulations also show that there are major informational gains
when combining factor models and sparse regressions in a forecasting exercise.
Two applications are considered.

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Table 1. Simulation Results: Size with $\phi=0$.

The table reports the empirical size of the test of remaining covariance
structure. Panel (a) reports the case where the factors are known, whereas
Panel (b) considers that the factors are unknown but the number of factors is
known. Panels (c) and (d) present the results when the number of factors are
determined, respectively, by the eigenvalue ratio test and the information
criterion $IC_{1}$. Factors are estimated by the usual principal component
algorithm. Three nominal significance levels are considered: 0.01, 0.05, and
0.10. The table reports the results for the case where $\phi=0$ in (5.3).

|  | Panel(a): Known factors
---|---|---
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 0.08 | 0.03 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.09 | 0.04 | 0.01
$n=1\times T$ |  | 0.06 | 0.02 | 0.00 |  | 0.07 | 0.03 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=2\times T$ |  | 0.07 | 0.02 | 0.00 |  | 0.07 | 0.02 | 0.00 |  | 0.08 | 0.04 | 0.00
$n=3\times T$ |  | 0.05 | 0.01 | 0.00 |  | 0.08 | 0.04 | 0.01 |  | 0.07 | 0.04 | 0.01
|  | Panel(b): Known number of factors
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 0.23 | 0.13 | 0.02 |  | 0.14 | 0.06 | 0.02 |  | 0.11 | 0.05 | 0.01
$n=1\times T$ |  | 0.13 | 0.06 | 0.01 |  | 0.09 | 0.04 | 0.01 |  | 0.12 | 0.05 | 0.01
$n=2\times T$ |  | 0.09 | 0.04 | 0.01 |  | 0.08 | 0.04 | 0.01 |  | 0.09 | 0.04 | 0.00
$n=3\times T$ |  | 0.06 | 0.02 | 0.00 |  | 0.08 | 0.04 | 0.01 |  | 0.07 | 0.03 | 0.01
|  | Panel(c): Information criterion ($\textnormal{IC}_{1}$)
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 0.24 | 0.14 | 0.03 |  | 0.14 | 0.06 | 0.02 |  | 0.11 | 0.05 | 0.01
$n=1\times T$ |  | 0.14 | 0.07 | 0.02 |  | 0.09 | 0.04 | 0.01 |  | 0.12 | 0.05 | 0.01
$n=2\times T$ |  | 0.10 | 0.05 | 0.01 |  | 0.08 | 0.04 | 0.01 |  | 0.09 | 0.04 | 0.00
$n=3\times T$ |  | 0.07 | 0.03 | 0.01 |  | 0.08 | 0.04 | 0.01 |  | 0.07 | 0.03 | 0.01
|  | Panel(d): Eigenvalue ratio
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 0.47 | 0.38 | 0.25 |  | 0.14 | 0.06 | 0.02 |  | 0.11 | 0.05 | 0.01
$n=1\times T$ |  | 0.14 | 0.07 | 0.02 |  | 0.09 | 0.04 | 0.01 |  | 0.12 | 0.05 | 0.01
$n=2\times T$ |  | 0.09 | 0.04 | 0.01 |  | 0.08 | 0.04 | 0.01 |  | 0.09 | 0.04 | 0.00
$n=3\times T$ |  | 0.06 | 0.02 | 0.00 |  | 0.08 | 0.04 | 0.01 |  | 0.07 | 0.03 | 0.01

Table 2. Simulation Results: Power ($\phi=0$).

The table reports the empirical power of the test of remaining covariance
structure. Panel (a) reports the case where the factors are known, whereas
Panel (b) considers that the factors are unknown but the number of factors is
known. Factors are estimated by the usual principal component algorithm. Three
nominal significance levels are considered: 0.01, 0.05, and 0.10.

|  | Panel(a): Known factors
---|---|---
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 1 | 1 | 1 |  | 1 | 1 | 1 |  | 1 | 1 | 1
$n=1\times T$ |  | 1 | 1 | 1 |  | 1 | 1 | 1 |  | 1 | 1 | 1
$n=2\times T$ |  | 1 | 1 | 1 |  | 1 | 1 | 1 |  | 1 | 1 | 1
$n=3\times T$ |  | 1 | 1 | 1 |  | 1 | 1 | 1 |  | 1 | 1 | 1
|  | Panel(b): Known number of factors
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 0.33 | 0.18 | 0.03 |  | 0.99 | 0.97 | 0.83 | 0.99 | 0.99 | 0.94 |
$n=1\times T$ |  | 0.20 | 0.08 | 0.01 |  | 0.84 | 0.60 | 0.13 | 0.95 | 0.81 | 0.33 |
$n=2\times T$ |  | 0.16 | 0.07 | 0.01 |  | 0.83 | 0.55 | 0.11 | 0.94 | 0.82 | 0.34 |
$n=3\times T$ |  | 0.08 | 0.03 | 0.00 |  | 0.82 | 0.56 | 0.10 | 0.95 | 0.81 | 0.34 |
|  | Panel(c): Eigenvalue ratio
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 0.15 | 0.09 | 0.02 |  | 0.99 | 0.97 | 0.83 |  | 0.99 | 0.99 | 0.94
$n=1\times T$ |  | 0.19 | 0.07 | 0.01 |  | 0.84 | 0.60 | 0.13 |  | 0.95 | 0.81 | 0.33
$n=2\times T$ |  | 0.16 | 0.07 | 0.01 |  | 0.83 | 0.55 | 0.11 |  | 0.94 | 0.82 | 0.34
$n=3\times T$ |  | 0.08 | 0.03 | 0.00 |  | 0.82 | 0.56 | 0.10 |  | 0.95 | 0.81 | 0.34
|  | Panel(d): Information criterion ($\textnormal{IC}_{1}$)
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 0.15 | 0.09 | 0.02 |  | 0.99 | 0.97 | 0.83 |  | 0.99 | 0.99 | 0.94
$n=1\times T$ |  | 0.19 | 0.07 | 0.01 |  | 0.84 | 0.60 | 0.13 |  | 0.95 | 0.81 | 0.33
$n=2\times T$ |  | 0.16 | 0.07 | 0.01 |  | 0.83 | 0.55 | 0.11 |  | 0.94 | 0.82 | 0.34
$n=3\times T$ |  | 0.08 | 0.03 | 0.00 |  | 0.82 | 0.56 | 0.10 |  | 0.95 | 0.81 | 0.34

Table 3. Simulation Results: Informational Gains

The table reports the average mean squared error (MSE) of three different
prediction models over 5-fold cross-validation subsamples. The goal is to
predict the first variable using information from the remaining $n-1$. Panel
(a) considers the case of Sparse Regression (SR) where $Y_{1t}$ is LASSO-
regressed on all the other variables. Panel (b) shows the results of Principal
Component Regression (PCR). Finally, Panel (c) presents the results of
FarmPredict. “N/A” means “not available”. Note that there is no factor
selection for Sparse Regression. “Known Number” means that the number of
factors is known.

|  | Panel(a): Sparse Regression (SR)
---|---|---
|  | Known Number |  | Eigenvalue Ratio |  | Information Criterion ($\textnormal{IC}_{1}$)
|  | $T=100$ | $T=500$ | $T=700$ |  | $T=100$ | $T=500$ | $T=700$ |  | $T=100$ | $T=500$ | $T=700$
$n=0.5\times T$ |  | 0.57 | 0.35 | 0.34 |  | N/A | N/A | N/A |  | N/A | N/A | N/A
$n=1\times T$ |  | 0.40 | 0.36 | 0.32 |  | N/A | N/A | N/A |  | N/A | N/A | N/A
$n=2\times T$ |  | 0.39 | 0.33 | 0.31 |  | N/A | N/A | N/A |  | N/A | N/A | N/A
$n=3\times T$ |  | 0.35 | 0.32 | 0.30 |  | N/A | N/A | N/A |  | N/A | N/A | N/A
|  | Panel(b): Principal Component Regression (PCR)
|  | Known Number |  | Eigenvalue Ratio |  | Information Criterion ($\textnormal{IC}_{1}$)
|  | $T=100$ | $T=500$ | $T=700$ |  | $T=100$ | $T=500$ | $T=700$ |  | $T=100$ | $T=500$ | $T=700$
$n=0.5\times T$ |  | 3.82 | 3.12 | 3.01 |  | 4.69 | 3.12 | 3.01 |  | 3.26 | 3.04 | 2.34
$n=1\times T$ |  | 3.09 | 2.35 | 2.34 |  | 4.05 | 3.35 | 3.34 |  | 3.22 | 3.02 | 2.32
$n=2\times T$ |  | 3.14 | 2.97 | 2.21 |  | 4.13 | 3.97 | 2.21 |  | 3.29 | 3.21 | 2.27
$n=3\times T$ |  | 3.83 | 3.00 | 2.33 |  | 3.83 | 3.00 | 2.33 |  | 3.12 | 3.00 | 2.28
|  | Panel(c): FarmPredict
|  | Known Number |  | Eigenvalue Ratio |  | Information Criterion ($\textnormal{IC}_{1}$)
|  | $T=100$ | $T=500$ | $T=700$ |  | $T=100$ | $T=500$ | $T=700$ |  | $T=100$ | $T=500$ | $T=700$
$n=0.5\times T$ |  | 0.50 | 0.33 | 0.31 |  | 0.52 | 0.33 | 0.31 |  | 0.50 | 0.34 | 0.30
$n=1\times T$ |  | 0.32 | 0.29 | 0.28 |  | 0.37 | 0.29 | 0.28 |  | 0.53 | 0.28 | 0.27
$n=2\times T$ |  | 0.27 | 0.27 | 0.26 |  | 0.28 | 0.27 | 0.26 |  | 0.32 | 0.28 | 0.28
$n=3\times T$ |  | 0.22 | 0.21 | 0.21 |  | 0.22 | 0.21 | 0.21 |  | 0.34 | 0.27 | 0.27

Table 4. Forecasting Results.

The table reports the frequency each model is ranked the first, second, third
and fourth best model among the four alternatives. Panel (a) considers the
case when the factors are selected by the eigenvalue ratio procedure. Panel
(b) presents the results when factors are selected by the information
criterion $\textnormal{IC}_{1}$. Panels (c) and (d) consider the cases when
the number of factors are pre-specified as either one or two. We present the
results for each individual group of variables as well as for the full set of
macroeconomic variables.

Panel (a): Optimal Factor Selection (eigenvalue ratio) |  |
---|---|---
|  | AR |  | SR |  | PCR |  | FarmPredict
Group |  | 1st | 2nd | 3rd | 4th |  | 1st | 2nd | 3rd | 4th |  | 1st | 2nd | 3rd | 4th |  | 1st | 2nd | 3rd | 4th
(i) output and income |  | 0.125 | 0.000 | 0.250 | 0.625 |  | 0.000 | 0.125 | 0.625 | 0.250 |  | 0.375 | 0.500 | 0.125 | 0.000 |  | 0.500 | 0.375 | 0.000 | 0.125
(ii) labor market |  | 0.032 | 0.097 | 0.290 | 0.581 |  | 0.226 | 0.065 | 0.516 | 0.194 |  | 0.194 | 0.516 | 0.097 | 0.194 |  | 0.548 | 0.323 | 0.097 | 0.032
(iii) housing |  | 0.100 | 0.100 | 0.300 | 0.500 |  | 0.400 | 0.400 | 0.100 | 0.100 |  | 0.000 | 0.200 | 0.400 | 0.400 |  | 0.500 | 0.300 | 0.200 | 0.000
(iv) consumption, orders and inventories |  | 0.000 | 0.000 | 0.333 | 0.667 |  | 0.000 | 0.000 | 0.667 | 0.333 |  | 0.333 | 0.667 | 0.000 | 0.000 |  | 0.667 | 0.333 | 0.000 | 0.000
(v) money and credit |  | 0.429 | 0.357 | 0.143 | 0.071 |  | 0.214 | 0.214 | 0.357 | 0.214 |  | 0.214 | 0.286 | 0.357 | 0.143 |  | 0.143 | 0.143 | 0.143 | 0.571
(vi) interest and exchange rates |  | 0.368 | 0.211 | 0.263 | 0.158 |  | 0.526 | 0.316 | 0.158 | 0.000 |  | 0.053 | 0.263 | 0.211 | 0.474 |  | 0.053 | 0.211 | 0.368 | 0.368
(vii) prices |  | 0.150 | 0.150 | 0.600 | 0.100 |  | 0.650 | 0.100 | 0.200 | 0.050 |  | 0.050 | 0.200 | 0.100 | 0.650 |  | 0.150 | 0.550 | 0.100 | 0.200
(viii) stock market |  | 0.667 | 0.000 | 0.000 | 0.333 |  | 0.000 | 0.000 | 0.667 | 0.333 |  | 0.000 | 1.000 | 0.000 | 0.000 |  | 0.333 | 0.000 | 0.333 | 0.333
(ix) all |  | 0.185 | 0.134 | 0.311 | 0.370 |  | 0.311 | 0.160 | 0.378 | 0.151 |  | 0.160 | 0.387 | 0.168 | 0.286 |  | 0.345 | 0.319 | 0.143 | 0.193
Panel (b): Optimal Factor Selection ($\textnormal{IC}_{4}$) |  |
|  | AR |  | SR |  | PCR |  | FarmPredict
Group |  | 1st | 2nd | 3rd | 4th |  | 1st | 2nd | 3rd | 4th |  | 1st | 2nd | 3rd | 4th |  | 1st | 2nd | 3rd | 4th
(i) output and income |  | 0.125 | 0.125 | 0.188 | 0.563 |  | 0.063 | 0.250 | 0.500 | 0.188 |  | 0.250 | 0.375 | 0.313 | 0.063 |  | 0.563 | 0.250 | 0.000 | 0.188
(ii) labor market |  | 0.032 | 0.097 | 0.258 | 0.613 |  | 0.226 | 0.032 | 0.548 | 0.194 |  | 0.226 | 0.581 | 0.065 | 0.129 |  | 0.516 | 0.290 | 0.129 | 0.065
(iii) housing |  | 0.000 | 0.000 | 0.400 | 0.600 |  | 0.200 | 0.500 | 0.100 | 0.200 |  | 0.200 | 0.100 | 0.500 | 0.200 |  | 0.600 | 0.400 | 0.000 | 0.000
(iv) consumption, orders and inventories |  | 0.000 | 0.000 | 0.333 | 0.667 |  | 0.167 | 0.000 | 0.500 | 0.333 |  | 0.167 | 0.667 | 0.167 | 0.000 |  | 0.667 | 0.333 | 0.000 | 0.000
(v) money and credit |  | 0.571 | 0.286 | 0.071 | 0.071 |  | 0.143 | 0.429 | 0.357 | 0.071 |  | 0.143 | 0.286 | 0.429 | 0.143 |  | 0.143 | 0.000 | 0.143 | 0.714
(vi) interest and exchange rates |  | 0.316 | 0.105 | 0.105 | 0.474 |  | 0.368 | 0.158 | 0.368 | 0.105 |  | 0.158 | 0.263 | 0.474 | 0.105 |  | 0.158 | 0.474 | 0.053 | 0.316
(vii) prices |  | 0.100 | 0.150 | 0.650 | 0.100 |  | 0.500 | 0.300 | 0.150 | 0.050 |  | 0.100 | 0.150 | 0.100 | 0.650 |  | 0.300 | 0.400 | 0.100 | 0.200
(viii) stock market |  | 0.667 | 0.000 | 0.000 | 0.333 |  | 0.000 | 0.667 | 0.333 | 0.000 |  | 0.000 | 0.333 | 0.667 | 0.000 |  | 0.333 | 0.000 | 0.000 | 0.667
(ix) all |  | 0.176 | 0.118 | 0.277 | 0.429 |  | 0.252 | 0.227 | 0.378 | 0.143 |  | 0.176 | 0.353 | 0.269 | 0.202 |  | 0.395 | 0.303 | 0.076 | 0.227
Panel (c): Fixed Number of Factors ($r=1$) |  |
|  | AR |  | SR |  | PCR |  | FarmPredict
Group |  | 1st | 2nd | 3rd | 4th |  | 1st | 2nd | 3rd | 4th |  | 1st | 2nd | 3rd | 4th |  | 1st | 2nd | 3rd | 4th
(i) output and income |  | 0.125 | 0.125 | 0.188 | 0.563 |  | 0.063 | 0.250 | 0.500 | 0.188 |  | 0.250 | 0.375 | 0.313 | 0.063 |  | 0.563 | 0.250 | 0.000 | 0.188
(ii) labor market |  | 0.032 | 0.097 | 0.258 | 0.613 |  | 0.226 | 0.032 | 0.548 | 0.194 |  | 0.226 | 0.581 | 0.065 | 0.129 |  | 0.516 | 0.290 | 0.129 | 0.065
(iii) housing |  | 0.000 | 0.000 | 0.400 | 0.600 |  | 0.200 | 0.500 | 0.100 | 0.200 |  | 0.200 | 0.100 | 0.500 | 0.200 |  | 0.600 | 0.400 | 0.000 | 0.000
(iv) consumption, orders and inventories |  | 0.000 | 0.000 | 0.333 | 0.667 |  | 0.167 | 0.000 | 0.500 | 0.333 |  | 0.167 | 0.667 | 0.167 | 0.000 |  | 0.667 | 0.333 | 0.000 | 0.000
(v) money and credit |  | 0.571 | 0.286 | 0.071 | 0.071 |  | 0.143 | 0.429 | 0.357 | 0.071 |  | 0.143 | 0.286 | 0.429 | 0.143 |  | 0.143 | 0.000 | 0.143 | 0.714
(vi) interest and exchange rates |  | 0.316 | 0.105 | 0.105 | 0.474 |  | 0.368 | 0.158 | 0.368 | 0.105 |  | 0.158 | 0.263 | 0.474 | 0.105 |  | 0.158 | 0.474 | 0.053 | 0.316
(vii) prices |  | 0.100 | 0.150 | 0.650 | 0.100 |  | 0.500 | 0.300 | 0.150 | 0.050 |  | 0.100 | 0.150 | 0.100 | 0.650 |  | 0.300 | 0.400 | 0.100 | 0.200
(viii) stock market |  | 0.667 | 0.000 | 0.000 | 0.333 |  | 0.000 | 0.667 | 0.333 | 0.000 |  | 0.000 | 0.333 | 0.667 | 0.000 |  | 0.333 | 0.000 | 0.000 | 0.667
(ix) all |  | 0.176 | 0.118 | 0.277 | 0.429 |  | 0.252 | 0.227 | 0.378 | 0.143 |  | 0.176 | 0.353 | 0.269 | 0.202 |  | 0.395 | 0.303 | 0.076 | 0.227
Panel (d): Fixed Number of Factors ($r=2$) |  |
|  | AR |  | SR |  | PCR |  | FarmPredict
Group |  | 1st | 2nd | 3rd | 4th |  | 1st | 2nd | 3rd | 4th |  | 1st | 2nd | 3rd | 4th |  | 1st | 2nd | 3rd | 4th
(i) output and income |  | 0.063 | 0.125 | 0.250 | 0.563 |  | 0.063 | 0.063 | 0.625 | 0.250 |  | 0.250 | 0.625 | 0.063 | 0.063 |  | 0.625 | 0.188 | 0.063 | 0.125
(ii) labor market |  | 0.065 | 0.129 | 0.226 | 0.581 |  | 0.226 | 0.097 | 0.516 | 0.161 |  | 0.226 | 0.452 | 0.097 | 0.226 |  | 0.484 | 0.323 | 0.161 | 0.032
(iii) housing |  | 0.200 | 0.200 | 0.000 | 0.600 |  | 0.500 | 0.400 | 0.100 | 0.000 |  | 0.000 | 0.100 | 0.700 | 0.200 |  | 0.300 | 0.300 | 0.200 | 0.200
(iv) consumption, orders and inventories |  | 0.167 | 0.167 | 0.167 | 0.500 |  | 0.167 | 0.167 | 0.500 | 0.167 |  | 0.333 | 0.333 | 0.333 | 0.000 |  | 0.333 | 0.333 | 0.000 | 0.333
(v) money and credit |  | 0.500 | 0.357 | 0.071 | 0.071 |  | 0.214 | 0.357 | 0.143 | 0.286 |  | 0.143 | 0.286 | 0.429 | 0.143 |  | 0.143 | 0.000 | 0.357 | 0.500
(vi) interest and exchange rates |  | 0.316 | 0.368 | 0.000 | 0.316 |  | 0.368 | 0.263 | 0.263 | 0.105 |  | 0.105 | 0.316 | 0.263 | 0.316 |  | 0.211 | 0.053 | 0.474 | 0.263
(vii) prices |  | 0.100 | 0.100 | 0.100 | 0.700 |  | 0.500 | 0.150 | 0.250 | 0.100 |  | 0.200 | 0.150 | 0.550 | 0.100 |  | 0.200 | 0.600 | 0.100 | 0.100
(viii) stock market |  | 0.667 | 0.000 | 0.000 | 0.333 |  | 0.000 | 0.667 | 0.333 | 0.000 |  | 0.000 | 0.333 | 0.667 | 0.000 |  | 0.333 | 0.000 | 0.000 | 0.667
(ix) all |  | 0.193 | 0.193 | 0.126 | 0.487 |  | 0.286 | 0.202 | 0.361 | 0.151 |  | 0.176 | 0.345 | 0.311 | 0.168 |  | 0.345 | 0.261 | 0.202 | 0.193

Figure 1. Correlations of returns.

We estimate the correlations between all pairs of returns from specific
sectors. The correlations that are higher than 0.15 in absolute value are
shown as black dots. We consider the sectors: mining, food, petroleum,
construction, manufacturing, utilities, department stores, retail, and
financial.

Figure 2. First-stage coefficient estimates.

The figure shows the empirical distribution of the first-stage regression
where each excess returns are linearly regressed on 16 risk factors.

Figure 3. Correlations of first-stage residuals.

We estimate the correlations between all pairs of residuals from the first-
stage OLS regression on 16 observed risk factors from specific sectors. The
correlations that are higher than 0.15 in absolute value are shown as black
dots. We consider the sectors: mining, food, petroleum, construction,
manufacturing, utilities, department stores, retail, and financial.

Figure 4. Correlations of second-stage residuals.

We estimate the correlations between all pairs of residuals from the second-
stage principal component analysis from specific sectors. The correlations
that are higher than 0.15 in absolute value are shown as black dots. We
consider the sectors: mining, food, petroleum, construction, manufacturing,
utilities, department stores, retail, and financial.

Figure 5. Partial correlations of second-stage residuals.

We estimate the partial correlations between all pairs of residuals from the
second-stage LASSO regression from specific sectors. The correlations that are
higher than 0.15 in absolute value are shown as black dots. We consider the
sectors: mining, food, petroleum, construction, manufacturing, utilities,
department stores, retail, and financial.

Figure 6. Variable Selection Frequency.

We report how often that variables from column sectors are selected in the
third-stage LASSO regression for firms on row sectors . The numbers are
normalized by the total number of firms in each sector.

(a) AR coefficients. (b) Absolute sum of AR coefficients.

Figure 7. Distribution of AR estimates

Panel (a) illustrates the empirical distribution of the ordinary least squares
(OLS) estimation of the coefficients of an fourth-order autoregressive, AR(4),
model across the 119 macroeconomic time series. Panel (b) illustrates the
empirical distribution of the absolute sum of the AR(4) coefficients across
the 119 series.

Figure 8. Estimated number of factors.

The figure illustrates the number of selected factors over the estimation
windows. The figure reports the results for the eigenvalue ratio procedure and
the four information criteria discussed in the paper.

##  Appendix A Additional Simulation Results

Tables 5 and 6 show, respectively, size and power simulations with $\phi=0.5$.
Table 7 presents additional size results. The table reports the empirical size
of the test of remaining covariance structure. Panels (a)–(c) consider that
the number of factors are determined by information criteria $IC_{2}$,
$IC_{3}$ and $IC_{4}$. Factors are estimated by the usual principal component
algorithm. Three nominal significance levels are considered: 0.01, 0.05, and
0.10.

##  Appendix B Proof of the Theorems

Throughout the proofs we use the equivalence
$\|X\|_{\psi_{p}}<\infty\iff\P(|X|>x)=O[\psi^{-1}_{p}(x)]$, as $x\to\infty$,
for any random variable $X$ and $\psi_{p}\in\Psi$ (refer to Lemma 17 below)
combined with Lemma 6 in Carvalho et al. (2018) and Lemma 1 in Masini and
Medeiros (2021). The key ingredients of the lemmas are a Marcinkiewicz-Zygmund
type inequality for strong mixing sequences to deal with the polynomial tails
(Rio, 1994; Doukhan and Louhichi, 1999) and a Bernstein inequality under
strong mixing conditions to control exponential tails (Merlevède et al. (2009)
\- Theorem 2).

Table 5. Simulation Results: Size with $\phi=0.5$.

The table reports the empirical size of the test of remaining covariance
structure. Panel (a) reports the case where the factors are known, whereas
Panel (b) considers that the factors are unknown but the number of factors is
known. Panels (c) and (d) present the results when the number of factors are
determined, respectively, by the eigenvalue ratio test and the information
criterion $IC_{1}$. Factors are estimated by the usual principal component
algorithm. Three nominal significance levels are considered: 0.01, 0.05, and
0.10. The table reports the results for the case where $\phi=0.5$ in (5.3).

|  | Panel(a): Known factors
---|---|---
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 0.09 | 0.03 | 0.01 |  | 0.11 | 0.06 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=1\times T$ |  | 0.07 | 0.03 | 0.00 |  | 0.07 | 0.03 | 0.01 |  | 0.11 | 0.06 | 0.01
$n=2\times T$ |  | 0.08 | 0.02 | 0.00 |  | 0.08 | 0.03 | 0.00 |  | 0.09 | 0.05 | 0.00
$n=3\times T$ |  | 0.05 | 0.02 | 0.00 |  | 0.09 | 0.04 | 0.01 |  | 0.07 | 0.04 | 0.01
|  | Panel(b): Known number of factors
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 0.24 | 0.14 | 0.02 |  | 0.15 | 0.07 | 0.02 |  | 0.12 | 0.06 | 0.01
$n=1\times T$ |  | 0.13 | 0.07 | 0.01 |  | 0.09 | 0.04 | 0.01 |  | 0.14 | 0.06 | 0.02
$n=2\times T$ |  | 0.09 | 0.04 | 0.01 |  | 0.08 | 0.04 | 0.01 |  | 0.09 | 0.05 | 0.00
$n=3\times T$ |  | 0.07 | 0.02 | 0.00 |  | 0.08 | 0.04 | 0.01 |  | 0.08 | 0.03 | 0.01
|  | Panel(c): Information criterion ($\textnormal{IC}_{1}$)
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 0.49 | 0.42 | 0.29 |  | 0.15 | 0.07 | 0.02 |  | 0.11 | 0.06 | 0.01
$n=1\times T$ |  | 0.15 | 0.09 | 0.02 |  | 0.10 | 0.04 | 0.01 |  | 0.14 | 0.06 | 0.01
$n=2\times T$ |  | 0.09 | 0.04 | 0.01 |  | 0.09 | 0.04 | 0.01 |  | 0.09 | 0.05 | 0.00
$n=3\times T$ |  | 0.07 | 0.03 | 0.00 |  | 0.10 | 0.04 | 0.01 |  | 0.08 | 0.03 | 0.01
|  | Panel(d): Eigenvalue ratio
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 0.25 | 0.14 | 0.04 |  | 0.14 | 0.07 | 0.02 |  | 0.13 | 0.06 | 0.01
$n=1\times T$ |  | 0.15 | 0.07 | 0.02 |  | 0.10 | 0.04 | 0.01 |  | 0.13 | 0.06 | 0.02
$n=2\times T$ |  | 0.11 | 0.05 | 0.01 |  | 0.08 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.00
$n=3\times T$ |  | 0.08 | 0.03 | 0.01 |  | 0.09 | 0.04 | 0.01 |  | 0.08 | 0.03 | 0.01

Table 6. Simulation Results: Power ($\phi=0.5$).

The table reports the empirical power of the test of remaining covariance
structure. Panel (a) reports the case where the factors are known, whereas
Panel (b) considers that the factors are unknown but the number of factors is
known. Factors are estimated by the usual principal component algorithm. Three
nominal significance levels are considered: 0.01, 0.05, and 0.10.

|  | Panel(a): Known factors
---|---|---
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 1 | 1 | 1 |  | 1 | 1 | 1 |  | 1 | 1 | 1
$n=1\times T$ |  | 1 | 1 | 1 |  | 1 | 1 | 1 |  | 1 | 1 | 1
$n=2\times T$ |  | 1 | 1 | 1 |  | 1 | 1 | 1 |  | 1 | 1 | 1
$n=3\times T$ |  | 1 | 1 | 1 |  | 1 | 1 | 1 |  | 1 | 1 | 1
|  | Panel(b): Known number of factors
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 0.36 | 0.18 | 0.03 |  | 1.00 | 1.00 | 0.91 |  | 1.00 | 1.00 | 1.00
$n=1\times T$ |  | 0.20 | 0.09 | 0.02 |  | 0.89 | 0.69 | 0.13 |  | 1.00 | 0.92 | 0.39
$n=2\times T$ |  | 0.18 | 0.07 | 0.01 |  | 0.98 | 0.59 | 0.13 |  | 1.00 | 0.96 | 0.36
$n=3\times T$ |  | 0.10 | 0.03 | 0.00 |  | 0.91 | 0.66 | 0.11 |  | 1.00 | 0.92 | 0.39
|  | Panel(c): Eigenvalue ratio
|  | $T=100$ |  | $T=500$ |  | $T=700$
|  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01 |  | 0.10 | 0.05 | 0.01
$n=0.5\times T$ |  | 0.15 | 0.11 | 0.03 |  | 1.00 | 1.00 | 0.96 |  | 1.00 | 1.00 | 1.00
$n=1\times T$ |  | 0.22 | 0.08 | 0.01 |  | 0.99 | 0.70 | 0.16 |  | 1.00 | 0.95 | 0.38
$n=2\times T$ |  | 0.17 | 0.07 | 0.01 |  | 0.94 | 0.62 | 0.12 |  | 0.98 | 0.88 | 0.37
$n=3\times T$ |  | 0.09 | 0.03 | 0.00 |  | 0.89 | 0.59 | 0.11 |  | 1.00 | 0.87 | 0.40
|  | Panel(d): Information criterion ($\textnormal{IC}_{1}$)
|  | $T=100$ |  | $T=500$ |  | $T=700$