Low-rank Panel Quantile Regression Estimation and Inference Yiren Wanga Liangjun Suband Yichong Zhanga aSchool of Economics Singapore Management University Singapore

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Low-rank Panel Quantile Regression: Estimation and Inference∗

Yiren Wanga, Liangjun Suband Yichong Zhanga

aSchool of Economics, Singapore Management University, Singapore

bSchool of Economics and Management, Tsinghua University, China

October 21, 2022

Abstract

In this paper, we propose a class of low-rank panel quantile regression models which allow

for unobserved slope heterogeneity over both individuals and time. We estimate the heteroge-

neous intercept and slope matrices via nuclear norm regularization followed by sample splitting,

row- and column-wise quantile regressions and debiasing. We show that the estimators of the

factors and factor loadings associated with the intercept and slope matrices are asymptoti-

cally normally distributed. In addition, we develop two speciﬁcation tests: one for the null

hypothesis that the slope coeﬃcient is a constant over time and/or individuals under the case

that true rank of slope matrix equals one, and the other for the null hypothesis that the slope

coeﬃcient exhibits an additive structure under the case that the true rank of slope matrix

equals two. We illustrate the ﬁnite sample performance of estimation and inference via Monte

Carlo simulations and real datasets.

Key words: Debiasing, heterogeneity, nuclear norm regularization, panel quantile regression,

sample splitting, speciﬁcation test.

JEL Classiﬁcation: C23, C31, C32, C52

∗Su gratefully acknowledges the support from the National Natural Science Foundation of China under Grant

No. 72133002. Zhang acknowledges the ﬁnancial support from a Lee Kong Chian fellowship. Any and all errors are

our own.

arXiv:2210.11062v1 [econ.EM] 20 Oct 2022

1 Introduction

Panel quantile regressions are widely used to estimate the conditional quantiles, which can cap-

ture the heterogeneous eﬀects that may vary across the distribution of the outcomes. Such eﬀects

are usually assumed to be homogeneous across individuals and over time periods. However, in

empirical analyses, it is usually unknown whether the slope coeﬃcients are homogeneous across

individuals and/or time. Mistakenly forcing slopes to be homogeneous across time and individ-

uals may lead to inconsistent estimation and misleading inferences. This prompts two questions

to be answered: how can we estimate the true model at diﬀerent quantiles when we allow for

heterogeneous slopes across individuals and time at the same time? How to conduct speciﬁcation

tests for homogeneous eﬀects over individuals or time and tests for the additive structure of the

slope coeﬃcients?

To answer the ﬁrst question, we propose an estimation procedure for heterogeneous panel

quantile regression models where we allow the ﬁxed eﬀects to be either additive or interactive, and

the slope coeﬃcients to be heterogeneous over both individuals and time. We impose a low-rank

structure for both the intercept and slope coeﬃcient matrices and estimate them via nuclear norm

regularization (NNR) followed by the sample splitting, row- and column-wise quantile regressions

and debiasing steps. The estimation algorithm is inspired by Chernozhukov et al. (2019), where

the main diﬀerence is that we split the full sample into three subsamples rather than two because

we need certain uniform results which require independence of regressors and regressand used in

the debiasing step, and we do not have the closed form for the quantile regression estimates. At

last, we derive the asymptotic distributions for the estimators of the factors and factor loadings

associated with slope coeﬃcient matrices.

To answer the second question, under the case when the rank of slope coeﬃcient matrix equals

one, we conduct sup-type speciﬁcation tests for homogeneous eﬀects over individuals or time

following the lead of Castagnetti et al. (2015) and Lu and Su (2021). We show that our sup-test

statistics follow the Gumbel distribution under the null, and the tests have non-trivial power

against certain classes of local alternatives. Under the case when the rank of slope matrix equals

two, our sup-type test statistic is also shown to follow the Gumbel distribution under the null

that the slope coeﬃcient exhibits an additive structure.

This paper relates to three bunches of literature. First, we contribute to the large literature

on panel quantile regressions (PQRs). Since Koenker (2004) studied the PQRs with individual

ﬁxed eﬀects, there has been an increasing number of papers on PQRs. Galvao and Montes-Rojas

(2010), Kato et al. (2012), Galvao and Wang (2015), Galvao and Kato (2016), Machado and Silva

(2019), and Galvao et al. (2020) study the asymptotics for PQRs with individual ﬁxed eﬀects.

Chen et al. (2021) study quantile factor models and Chen (2022) considers PQRs with interactive

ﬁxed eﬀects (IFEs). We complement the literature by allowing for unobserved heterogeneity in

the slope coeﬃcients of PQRs.

Second, our paper also pertains to slope heterogeneity in panel data models. Latent group

structures across individuals and structural changes over time are two common types of slope

heterogeneity that have received vast attention in the literature. To recover the unobserved group

structures, various methods have been proposed. For example, Lin and Ng (2012), Bonhomme

and Manresa (2015) and Ando and Bai (2016) use the K-means algorithm; Su et al. (2016)

propose the C-lasso algorithm which is further studied and extended by Su and Ju (2018), Su

et al. (2019) and Wang et al. (2019); Wang et al. (2018) propose an clustering algorithm in

regression via data-driven segmentation called CARDS; Wang and Su (2021) propose a sequential

binary segmentation algorithm to identify the latent group structures in nonlinear panels. Recent

literature on the estimation with structural changes in panel data models includes, but is not

limited to, Chen (2015), Cheng et al. (2016), Ma and Su (2018), Baltagi et al. (2021). In addition,

Galvao et al. (2018) and Zhang et al. (2019) consider individual heterogeneity in PQRs while they

assume homogeneity across time. To allow for both latent groups and structural breaks, Okui and

Wang (2021) study a linear panel data model with individual ﬁxed eﬀects where each latent group

has common breaks and the breaking points can be diﬀerent across diﬀerent groups, and they

propose a grouped adaptive group fused lasso (GAGFL) approach to estimate slope coeﬃcients.

Lumsdaine et al. (2021) consider a linear panel data model with a grouped pattern of heterogeneity

where the latent group membership structure and/or the values of slope coeﬃcients can change

at a breaking point, and they propose a K-means-type estimation algorithm and establish the

asymptotic properties of the resulting estimators. Compared with the models studied above,

our model combines both individual and time heterogeneity and only requires certain low-rank

structure in the slope coeﬃcient matrix. So the unobserved heterogeneity takes a more ﬂexible

form in our model than those in the literature such as Okui and Wang (2021) and Lumsdaine

et al. (2021).

Last, our paper also connects with the burgeoning literature on nuclear norm regularization.

Such a method has been widely adopted to study panel and network models. See, Alidaee et al.

(2020), Athey et al. (2021), Bai and Ng (2019), Belloni et al. (2022), Chen et al. (2020), Cher-

nozhukov et al. (2019), Feng (2019), Hong et al. (2022), Miao et al. (2022), among others. In

the least squares panel framework, Moon and Weidner (2018) consider a homogeneous panel with

IFEs by using NNR-based estimator as an initial estimator to construct iterative estimators that

are asymptotically equivalent to the least squares estimators; Chernozhukov et al. (2019) study

a heterogenous panel where both the intercept and slope coeﬃcient matrices exhibit a low-rank

structure and establish the asymptotic distribution theory based on NNR. In the presence of

endogeneity, Hong et al. (2022) proposes a proﬁle GMM method to estimate panel data models

with IFEs. In the panel quantile regression setting, Feng (2019) develops error bounds for the

low-rank estimates in terms of Frobenius norms under independence assumption; Belloni et al.

(2022) relaxes the independence assumption to the β-mixing condition along the time dimension.

Our paper extends Chernozhukov et al. (2019) from the least squares framework to the PQR

framework, derives the asymptotic distribution theory and develops various speciﬁcation tests

under some strong mixing conditions along the time dimension that is weaker than the β-mixing

condition. We also rely on the sequential symmetrization technique developed by Rakhlin et al.

(2015) to obtain the convergence rates of the nuclear norm regularized estimators.

The rest of the paper is organized as follows. We ﬁrst introduce the low-rank structure PQR

model and the estimation algorithm in Section 2. We study the asymptotic properties of our

estimators in Section 3. In Section 4, we propose two speciﬁcation tests: one for the no-factor

structure and one for the additive structure, and study the asymptotic properties of the test

statistics. In Section 5, we show the ﬁnite sample performance of our method via Monte Carlo

simulations. In Section 6, we apply our method to two datasets: one is to study how Tobin’s q

and cash ﬂows aﬀect corporate investment and whether ﬁrm’s external investment to its internal

ﬁnancing exhibits heterogeneity structure, and the other is to study the relationship between

economics growth, foreign direct investment and unemployment. Section 7 concludes. All proofs

are related to the online supplement.

Notation. k·k1,k·kop,k·k∞,k·kmax k·k2,k·kF,k·k∗denote the matrix norm induced by 1-norms,

the matrix norm induced by 2-norms, the matrix norm induced by ∞-norms, the maximum norm,

the Euclidean norm, the Frobenius norm and the nuclear norm. is the element-wise product.

b·c and d·e denote the ﬂoor and ceiling functions, respectively. a∨band a∧breturn the max and

the min of aand b, respectively. The symbol .means “the left is bounded by a positive constant

times the right”. Let A={Ait}i∈[n],t∈[T]be a matrix with its (i, t)-th entry denoted as Ait, where

[n] to denote the set {1,··· , n}for any positive integer n. Let {Aj}p

j=0 denote the collection of

matrices Ajfor all j∈ {0,··· , p}. When Ais symmetric, λmax(A) and λmin(A) denote its largest

and smallest eigenvalues, respectively. The operators and p

→denote convergence in distribution

and in probability, respectively. Besides, we use w.p.a.1 and a.s. to abbreviate “with probability

approaching 1” and “almost surely”, respectively.

2 Model and Estimation

In this section, we introduce the PQR model and estimation algorithm.

2.1 Model

Consider the PQR model

QτYit{Xj,it}j∈[p],t∈[T],Θ0

j,it (τ)j∈[p]∪{0},t∈[T]= Θ0

0,it(τ) +

j=1

Xj,itΘ0

j,it(τ),(2.1)

where i∈[N], t ∈[T], τ ∈(0,1) is the quantile index, Yit is the dependent variable, Xj,it is the

j-th regressor for individual iat time t,{Θ0

j,it}j∈[p]is the corresponding slope coeﬃcient, Θ0

0,it

is the intercept, and QτYit{Xj,it}j∈[p],t∈[T],nΘ0

j,it (τ)oj∈[p]∪{0},t∈[T]denotes the conditional

τ-quantile of Yit given the regressors {Xj,it}j∈[p],t∈[T]and the parameters nΘ0

j,it (τ)oj∈[p]∪0,t∈[T].1

Alternatively, we can rewrite the above model as

Y= Θ0

0(τ) +

j=1

XjΘ0

j(τ) + (τ) and

Qτit(τ){Xj,it}j∈[p],t∈[T],Θ0

j,it (τ)j∈[p]∪{0},t∈[T]= 0,(2.2)

where (τ) is the idiosyncratic error matrix with the (i, t)-th entry being it(τ). Similarly, Xj,

Θj(τ), and Yare matrices with the (i, t)-th entry being Xj,it, Θj,it (τ), and Yit, respectively. In

this model, we assume p, the number of regressors, is ﬁxed and both Nand Tpass to inﬁnity. In

Assumption 1below, we characterize the dependence of the data, under which (2.1) holds.

In the paper, we focus on the panel quantile regression for a ﬁxed τand thus suppress the

dependence of Θ0

j(τ) and (τ) on τfor notation simplicity. In addition, we impose low-rank

structures for the intercept and slope matrices, i.e., rank(Θ0

j) = Kjfor some positive constant Kj

and for each j∈ {0,··· , p}. By the singular value decomposition (SVD), we have

Θ0

j=√NT U0

jΣ0

jV00

j=U0

jV00

j∀j= 0,··· , p,

where U0

j∈RN×Kj,V0

j∈RT×Kj, Σ0

j= diag(σ1,j,··· , σKj,j), U0

j=√NU0

jΣ0

jwith each row being

u00

i,j, and V0

j=√TV0

jwith each row being v00

t,j.

The low-rank structure assumption includes several popular cases. For the intercept term,

one commonly assumes that Θ0

0,it to take the forms α0

i, µ0

t,or α0

i+µ0

tin classical PQRs. Then

the matrix Θ0

0has rank 1, 1, and 2, respectively. It is also possible to assume Θ0

0,it to take an

interactive form, say, Θ0

0,it =λ00

0,if0

0,t,where both λ0

0,i and f0

0,t are K0-vectors. For the slope matrix

Θ0

j,j∈[p],the early PQR models frequently assume that Θ0

j,it is a constant across (i, t) to yield

a homogenous PQR model. Obviously, such a model is very restrictive by assuming homogenous

slope coeﬃcients. It is possible to allow the slope coeﬃcients to change over either i, or t, or both.

See the following examples for diﬀerent low-rank structures.

Example 1. When Θ0

j,it = Θ0

j,i ∀t∈[T],or Θ0

j,it = Θ0

j,t ∀i∈[N],or Θ0

j,it = Θ0

j∀(i,t)∈[N]×[T],

and this holds for all j∈[p],we have the PQR models with only individual heterogeneity, with

only time heterogeneity, and with homogeneity, respectively. We observe that Kj= 1 for these

three cases.

1We will assume that both the intercept term Θ0

0,it and the slope coeﬃcients {Θ0

j,it}j∈[p]have low-rank struc-

tures, and follow the convention in the panel data literature by treating the factors to be random. Therefore,

{Θ0

j,it}j∈[p]∪{0}are random as well.

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Low-rankPanelQuantileRegression:EstimationandInference*YirenWanga,LiangjunSubandYichongZhangaaSchoolofEconomics,SingaporeManagementUniversity,SingaporebSchoolofEconomicsandManagement,TsinghuaUniversity,ChinaOctober21,2022AbstractInthispaper,weproposeaclassoflow-rankpanelquantileregressionmodelswhich...

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Low-rank Panel Quantile Regression Estimation and Inference Yiren Wanga Liangjun Suband Yichong Zhanga aSchool of Economics Singapore Management University Singapore

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