Department of Economics Seminar Series

Zongwu Cai University of Kansas

“A New Semiparametric Quantile Panel Data Model: Theory and Applications”

Friday, September 11, 2015

3:30 p.m. 212 Middlebush

A New Semiparametric Quantile Panel Data Model: Theory and Applications

Zongwu Cai

Department of Economics, University of Kansas, e-mail: caiz@ku.edu WISE, Xiamen University, zongwucai@gmail.com

ABSTRATC: In this paper, we propose a new semiparametric quantile panel data model with correlated random effects in which some of the coefficients are allowed to depend on some smooth economic variables while other coefficients remain constant. A three-stage estimation procedure is proposed to estimate both constant and functional coefficients based on the integrated quasi-likelihood approach and their asymptotic properties are investigated. We show that the estimator of constant coefficients is root-N consistent and the estimator of varying coefficients converges in a nonparametric rate. A Monte Carlo simulation is conducted to examine the finite sample performance of the proposed estimators. Finally, the proposed semiparametric quantile panel data model is applied to estimating the impact of foreign direct investment (FDI) on economic growth using the cross-country data from 1970 to 1999.

This is a join work with Dr. Linna Chen and Dr. Ying Fang.

Keywords: Correlated Random Effect; Foreign Direct Investment; Panel Data; Quantile Regression Model; Quasi-likelihood; Semi-parametric Model; Varying Coefficient Model.

A Quantile Panel Data Model with Partially Varying

Coefficients and Correlated Random Effects∗

Zongwu Caia,b, Linna Chena, and Ying Fangb

aDepartment of Economics, The University of Kansas, Lawrence, KS 66045, USAbWang Yanan Institute for Studies in Economics, MOE Key Laboratory of Econometrics, and

Fujian Key Laboratory of Statistical Science, Xiamen University, Xiamen, Fujian 361005, China

September 1, 2015

Abstract

In this paper, to estimate the impact of FDI on economic growth in host countries,

we propose a new semiparametric quantile panel data model with correlated random

effects, in which some of the coefficients are allowed to depend on some smooth economic

variables while other coefficients remain constant. A three-stage estimation procedure

based on quasi-maximum (local) likelihood estimation (QMLE) is proposed to estimate

both constant and functional coefficients and their asymptotic properties are investigated.

We show that the estimator of constant coefficients is√N consistent and the estimator

of varying coefficients converges in a nonparametric rate. A Monte Carlo simulation is

conducted to examine the finite sample performance of the proposed estimators. Finally,

using the cross-country data from 1970 to 1999, we find a strong empirical evidence of the

existence of the absorptive capacity hypothesis. Moreover, another new finding is that

FDI has a much stronger growth effects for countries with fast economic growth than for

those with slow economic growth.

Keywords: Correlated Random Effect; Foreign Direct Investment; Panel Data; Quantile

Regression Model; Quasi-likelihood; Semiparametric Model; Varying Coefficient Model.

∗Cai’s research is partially supported by the National Natural Science Foundation of China grant #71131008

(Key Project). Fang’s research is partially supported by the National Natural Science Foundation of China

grants #70971113, #71271179 and #71131008 (Key Project).

1

1 Introduction

It is well documented in the growth literature that foreign direct investment (FDI) plays an

important role in the economic growth process in host countries since FDI is often consid-

ered as a vehicle to transfer new ideas, advanced capitals, superior technology and know-how

from developed countries to developing countries and so on. However, the existing empirical

studies provide contradictory results on whether or not FDI promotes an economic devel-

opment in host countries.1 The recent studies in the literature concluded that the mixed

empirical evidences may be due to nonlinearities in FDI effects on the economic growth and

the heterogeneity across countries.

Indeed, it is well recognized by many economists in empirical studies that a standard linear

growth model may be inappropriate for investigating the nonlinear effect of FDI on economic

development. The nonlinearity in FDI effects is mainly due to the so called absorptive capac-

ity in host countries, the fact that host countries need some minimum conditions to absorb

the spillovers from FDI.2 Most existing literature to deal with the nonlinearity issue used

simply some parametric nonlinear models, for example, including an interacted term in the

regression or running a threshold regression. A parametric nonlinear model has the risk of

encountering the model misspecification problem. Misspecified models can lead to biased

estimation and misleading empirical results. Recently, Henderson, Papageorgiou and Parme-

ter (2011) and Kottaridi and Stengos (2010) adopted nonparametric regression techniques

into a growth model. However, due to the curse of dimensionality in a pure nonparametric

estimation, such applications are either restricted by the sample size problem or rely heavily

on the variable selection which is not an easy task.

1For example, Blomstrom and Persson (1983), Blomstrom, Lipsey and Zejan (1992), De Gregorio (1992),

Borensztein, De Gregorio and Lee (1998), De Mello (1999), Ghosh and Wang (2009), Kottaridi and Stengos

(2010) among others found positive effects of FDI on promoting the economic growth in various environments.

On the other hand, many studies including Haddad and Harrison (1993), Aitken and Harrison (1999), Lipsey

(2003), and Carkovic and Levine (2005) failed to find beneficial effects of FDI on the economic growth in

host countries. Grog and Strobl (2001) did a meta analysis of 21 studies using the data from 1974 to 2001

that worked on estimating FDI effects on productivity in host countries, of which 13 studies reported positive

results, 4 studies reported negative effects and the remaining reported inconclusive evidence.2Nunnemkamp (2004) emphasized the importance of the initial condition for host countries to absorb the

positive impacts of FDI, Borensztein, De Gregorio and Lee (1998) found that a threshold stock of human

capital in host countries is necessary for them to absorb beneficial effects of advanced technologies brought

from FDI, and Hermes and Lensink (2003), Alfaroa, Chandab, Kalemli-Ozcan and Sayek (2004) and Durham

(2004) addressed the local financial market conditions of a country’s absorptive capacity.

2

The heterogeneity among countries is another concern in cross-country studies. Grog and

Strobl (2001) found that whether a cross sectional or time series data had been used matters

for estimating the effect of FDI on the economic growth, because both the cross sectional and

time series model cannot control the country-specific heterogeneity. Recent literature focused

on using panel data to estimate growth models, which can control the country-specific un-

observed heterogeneity using individual effects. However, including individual effects which

only allows a location shift for each country, does not have the ability to deal with the hetero-

geneity effect of FDI on the economic growth across the countries. For example, some studies

found that the empirical results changed when the sample included developed countries. The

existing literature to handle this issue is to split sample into groups.3 Generally speaking,

splitting sample can lead to potential theoretical and empirical problems. First, regressing

on the split samples separately may lose other parts of information and degrees of freedom,

which may lead to inefficient estimation. Secondly, the applied researchers often split sample

without following the theoretical guideline of the selection of thresholds.

To deal with the aforementioned two issues (nonlinearities and heterogeneity) in a simul-

taneous fashion, we propose a partially varying-coefficient quantile panel data model with

correlated random effects to estimate the nonlinear effect of FDI on the economic growth

with heterogeneity. Different from the existing literature, we resolve the nonlinearity issue

by employing a partially varying-coefficient model which allows some of coefficients to be

constant but others, reflecting the effects of FDI on the economic growth, to depend on the

country’s initial condition. Compared to a full nonparametric estimation, our model setup

can achieve the dimension reduction and accommodate the well recognized economic theory

such as the absorptive capacity. In addition to using panel data with individual effects which

allows for location shifts for individual countries, we propose a semiparametric conditional

quantile regression model instead of the commonly used conditional mean models. A condi-

tional quantile model can provide more flexible structures than conditional mean models to

characterize heterogeneity among countries. For example, besides including individual effects

allowing country-specific heterogeneity, a conditional quantile model allows different growth

equations for different quantiles. Therefore, we can take the advantage of all sample informa-

3For example, Luiz and De Mello (1999) considered OECD and non-OECD samples and Kottaridi and

Stengos (2010) split the whole sample into high-income and middle-income groups.

3

tion to identify the effect of FDI on the economic growth without splitting sample according

to development stages. Moreover, estimating all quantiles can provide a whole picture of the

conditional distribution and avoid the possibly misleading conditional mean results. In other

words, using the quantile approach can characterize the different roles of FDI in the economic

growth for different types of countries.

The application of conditional quantile model to analyze economic and financial data has

a long history that can be traced to the seminal paper by Koenker and Bassett (1978, 1982);

see the book by Koenker (2005) for more details. Recently, many studies have focused on

nonparametric or semiparametric quantile regression models for either independently iden-

tically distributed (iid) data or time series data. For example, Chaudhuri (1991) studied

nonparametric quantile estimation and derived its local Bahadur representation, He, Ng and

Portnoy (1998), He and Ng (1999), and He and Portnoy (2000) considered nonparametric

estimation using splines, De Gooijer and Zerom (2003), Yu and Lu (2004), and Horowitz and

Lee (2005) focused on additive quantile models, and Honda (2004) and Cai and Xu (2008)

studied varying-coefficient quantile models for time series data. In particular, semiparametric

quantile models have attracted increasing research interests during the recent years due to

their flexibility. For example, He and Liang (2000) investigated the quantile regression of

a partially linear errors-in-variable model, Lee (2003) discussed the efficient estimation of a

partially linear quantile regression, and Cai and Xiao (2012) proposed a partially varying-

coefficient dynamic quantile regression model, among others.

Due to the fact that the approach of taking a difference, which is commonly used in con-

ditional mean panel data (linear) models to eliminate individual effects, is invalid in quantile

regression settings, even for linear quantile regression model, the literature on quantile panel

data models is relatively small. To the best of our knowledge, the paper by Koenker (2004)

is the first paper to consider a linear quantile panel data model with fixed effects, where the

fixed effects are assumed to have pure location shift effects on the conditional quantiles of

the dependent variable but the effects of regressors are allowed to be dependent on quantiles.

Koenker (2004) proposed two methods to estimate such a panel data model with fixed effects.

The first method is to solve a piecewise linear quantile loss function by using interior point

methods and the second one is the penalized quantile regression method, in which the quan-

tile loss function is minimized by adding L1 penalty on fixed effects. Recently, in a penalized

4

quantile panel data regression model as in Koenker (2004), Lamarche (2010) discussed how

to select the tuning parameter, which can control the degree of shrinkage for fixed effects,

whereas Galvao (2011) extended the quantile regression to a dynamic panel data model with

fixed effects by employing the lagged dependent variables as instrumental variables and by ex-

tending Koenker’s first method to Chernozhukov and Hansen (2006)’s quantile instrumental

variable framework. Finally, Canay (2011) proposed a simple two-stage method to estimate

a quantile panel data model with fixed effects. However, the consistency of the estimator in

Canay (2011) relies on the assumption of T going to infinity and the existence of an initial√NT -consistent estimator in the conditional mean model.

An alternative way to deal with individual effects in a panel data model is to treat them

as correlated random effects initiated by Chamberlain (1982, 1984) for the mean regression

model. Under the framework of Chamberlain (1982, 1984), to estimate the effect of birth

inputs on birth weight, Abrevaya and Dahl (2008) employed a linear quantile panel data

model with correlated random effects which are viewed as a linear projection onto some

covariates plus an error term. The identification of the effects of covariates only requires two-

period information. Furthermore, Gamper-Rabindram, Khan and Timmins (2008) estimated

the impact of piped water provision on infant mortality by adopting a linear quantile panel

data model with random effects where the random effects were allowed to be correlated with

covariates nonparametrically. The model can be estimated through a two-step procedure, in

which some conditional quantiles were estimated nonparametrically in the first step and in the

second step, the coefficients are estimated by regressing the differenced estimated quantiles

on the differenced covariates.

The motivation of this study is to examine the role of FDI in the economic growth process

based on the cross-country data from 1970 to 1999 by using the proposed partially varying-

coefficient quantile regression model for panel data with correlated random effects. Indeed,

this model includes the models in Lee (2003), Cai and Xu (2008), and Cai and Xiao (2012) as

special cases. In contrast to Koenker (2004), Galvao (2011), and Canay (2011) by requiring

that both N and T go to infinity in their asymptotics, our model requires only N going

to infinity with T possibly fixed. Actually, T ≥ 2 is required, and indeed, it is an impor-

tant assumption for identification; see Abrevaya and Dahl (2008) and Assumption A7 later.

Also, different from Abrevaya and Dahl (2008) and Gamper-Rabindram, Khan and Timmins

5

(2008), we use a partially varying-coefficient structure in the conditional quantile model to

provide more flexibility in model specification than a linear model. Based on this empirical

study, there some novel findings. For the countries with fast economic growth, there is a

strong evidence that the initial condition of GDP benefits to the effect of FDI on economic

growth when Ui is bigger than 8.2 for the host country. However, on the other hand, for

the countries with slow economic growth, the initial condition of GDP benefits to the effect

of FDI on economic growth when Ui is smaller than 8.2 for the host country. The detailed

analysis of this empirical example is reported in Section 4.

The rest of the paper is organized as follows. In Section 2, we introduce a partially

varying-coefficient quantile panel data model with correlated random effects and propose a

three-stage estimation procedure. Furthermore, the asymptotic properties of our estimators

are established. In Section 3, a simulation study is conducted to examine the finite sample

performance. Section 4 is devoted to reporting the empirical results of the cross-country

panel data growth model. Section 5 concludes.

2 Econometric Modeling

2.1 Model Setup

2.1.1 The Model

In this paper, we consider the following partially varying-coefficient panel data quantile model

with correlated random effects, in which there are both constant coefficients and varying

coefficients. Let Yit, a scalar dependent variable, be the observation on ith individual at time

t for 1 ≤ i ≤ N and 1 ≤ t ≤ T . The conditional quantile model is given by

Qτ (Yit | Uit,Xit, αi) = X ′it,1γτ +X ′it,2βτ (Uit) + αi, (1)

where Qτ (Yit | Uit,Xit, αi) is the τth quantile of Yit given Uit, Xit, and αi. Here, Xit,1 and

Xit,2 are regressors with K1×1 and K2×1 dimensions, respectively, Xit = (X ′it,1,X′it,2)

′, γτ

denotes a K1×1 vector of constant coefficients, βτ (Uit) denotes a K2×1 vector of functional

coefficients, Uit is an observable scalar smoothing variable4 , and αi is an individual effect. In

the quantile panel data literature, Koenker (2004) treated αi as a fixed effect and proposed

4For simplicity, we only consider the univariate case for the smoothing variable. The estimation procedure

and asymptotic results still hold for the multivariate case with much complicated notation.

6

a penalized quantile regression method which requires both N and T go to infinity. In this

paper, following Abrevaya and Dahl (2008) and Gamper-Rabindran, Khan and Timmins

(2008), we view the individual effect as a correlated random effect which is allowed to be

correlated with covariates Xi = (X ′i1, · · · ,X ′iT )′ and UitTt=1; that is,

αi = α(Xi, Ui1, · · · , UiT ) + vi, (2)

where α(·) is an unknown function of Xi and UitTt=1, and vi is a random error.

A fully nonparametric model of α(·) may lead to the problem of the curse of dimensionality

and become infeasible in practice. Compared with a linear projection in Chamberlain (1982)

and Abrevaya and Dahl (2008), an additive model with functional coefficients can accommo-

date more flexibility. Thus, we approximate the unknown function α(Xi, Ui1, · · · , UiT ) by a

functional-coefficient model5 such that

α(Xi, Ui1, · · · , UiT ) =T∑t=1

X ′itδt(Uit), (3)

where δ(Uit) is a K × 1 vector of unknown functional coefficients.

Finally, in the case of estimating FDI effect on the economic growth in our empirical studies,

the smoothing variable varies only across different individual units but keeps constant over

time periods.6 Therefore, in this paper, we focus on the simple case where Uit = Ui for all

1 ≤ t ≤ T . Model (1) can be rewritten as

Qτ (Yit | Ui,Xi, vi) = X ′it,1γτ +X ′it,2βτ (Ui) +

T∑t=1

X ′itδt(Ui) + vi. (4)

2.1.2 Model identification

The identification issue of panel data quantile model with fixed effects for large N and

short T has received increasing attentions in the recent literature. Canay (2011) discussed

the identification condition for T ≥ 2. The identification of the parameters of interest is

obtained from the conditional distribution of Yit using deconvolution. The key condition of

5As elaborated by Cai, Das, Xiong and Wu (2006) and Cai (2010), a functional-coefficient model can be a

good approximation to a general fully nonparametric model, g(X,Z) =d∑j=0

gj(Z)Xj = X ′g(Z).

6When Uit1 6= Uit2 6= Ui for any t1 6= t2, two estimation approaches can be employed. We can apply the

series estimation or adopt a single index method Ui = ω1Ui1 + · · · + ωTUiT using the iterative backfiiting

method proposed by Fan, Yao and Cai (2003).

7

the identification up to location is a conditional independence restriction which implies that

αi does not change across quantiles. Rosen (2012) considered a different set of identification

conditions for the same model. In addition to the usual conditional quantile restriction, Rosen

(2012) showed further that a weak conditional independence helps to identify the coefficients

of interest.

We consider a semiparametric quantile model with correlated random effects, which is a

generalization of Abrevaya and Dahl (2008). The identification of the model can be regarded

as a special case of Canay (2011). Two additional assumptions are required to identify the

model up to a location:

I1. vi is independently and identically distributed (iid) and independent of Ui,Xi.

I2. Qτ (Yit|Ui,Xi, vi) = Qτ (Yit|Ui,Xit, vi).

Remark 1. The independence restriction in Assumption I1 is a common assumption used

in the literature of correlated random effects; see Abrevaya and Dahl (2008). It restricts that

the quantiles of vi do not depend upon Xi. Assumption I2 allows for arbitrary forms of

heteroscedasticity only through Xit. Note that unfortunately, Assumption I2 rules out the

case of dynamic panel data models.

It is easy to see by Assumptions I1 and I2 that

Qτ (Yit|Ui,Xi, vi) = Qτ (Yit|Ui,Xit, vi) ≡ πτ (Ui,Xit, vi)

where πτ (Ui,Xit, vi) is a nonlinear unknown form. We can approximate πτ (Ui,Xit, vi) by a

semiparametric form as in (4). As argued in Cai (2010), such a semiparametric specification

of πτ (·) is flexible enough for many real applications.

2.2 Estimation Procedures

2.2.1 Pooling Regression Strategy

From model (4), one can observe that the conditional quantile effects ofXit on Yit are through

two channels: a direct effect γτ for constant coefficients or βτ (Ui) for varying coefficients,

and an indirect effect δt(Ui) working through the correlated random effects. Assuming that

T ≥ 2, to identify the direct effects γτ or βτ (Ui), one has to estimate at least two conditional

8

quantile models Qτ (Yit | Ui,Xi, vi) and Qτ (Yis | Ui,Xi, vi) given by

Qτ (Yit|Ui,Xi, vi) = X ′it,1[γτ + δ1t(Ui)] +X ′it,2[βτ (Ui) + δ2t(Ui)] +∑l 6=tX ′ilδl(Ui) + vi

and

Qτ (Yis|Ui,Xi, vi) = X ′it,1δ1t(Ui) +X ′it,2δ2t(Ui) +X ′is,1γτ +X ′is,2βτ (Ui) +∑l 6=tX ′ilδl(Ui) + vi,

respectively, where t 6= s, δ1t(Ui) is the vector which contains the first K1 components of

δt(Ui), and δ2t(Ui) is the vector which contains the last K2 components of δt(Ui). Hence, the

estimates of γτ and β(Ui) are respectively given by

γτ =∂Qτ (Yit | Ui,Xi, vi)

∂Xit,1− ∂Qτ (Yis | Ui,Xi, vi)

∂Xit,1,

and

βτ (Ui) =∂Qτ (Yit | Ui,Xi, vi)

∂Xit,2− ∂Qτ (Yis | Ui,Xi, vi)

∂Xit,2.

However, in order to avoid running two separating conditional quantile models, we adopt

Abrevaya and Dahl (2008)’s pooling regression strategy by stacking covariates. From model

(4), Qτ (Yit | Ui,Xi, vi) and Qτ (Yis | Ui,Xi, vi) can be expressed as

Qτ (Yit|Ui,Xi, vi) = X ′it,1γτ +X ′it,2βτ (Ui) +X ′i1δ1(Ui) + · · ·+X ′iTδT (Ui) + vi,

and

Qτ (Yis|Ui,Xi, vi) = X ′is,1γτ +X ′is,2βτ (Ui) +X ′i1δ1(Ui) + · · ·+X ′iTδT (Ui) + vi.

Hence, we treat

Y11...

Y1T...

Yi1...

YiT...

YN1

...

YNT

and

X ′11,1 X ′11,2 X ′11 · · · X ′1T...

...... · · ·

...

X ′1T,1 X ′1T,2 X ′11 · · · X ′1T...

...... · · ·

...

X ′i1,1 X ′i1,2 X ′i1 · · · X ′iT...

...... · · ·

...

X ′iT,1 X ′iT,2 X ′i1 · · · X ′iT...

...... · · ·

...

X ′N1,1 X ′N1,2 X ′N1 · · · X ′NT...

...... · · ·

...

X ′NT,1 X ′NT,2 X ′N1 · · · X ′NT

9

as the dependent variable and the right-side explanatory variables, respectively. This pooled

regression directly estimates (γ ′τ ,β′τ (Ui), δ

′1(Ui), · · · , δ′T (Ui))

′. We now consider the following

transformed model from (4),

Qτ (Ui,Zit, vi) = Z ′it,1γτ +Z ′it,2θτ (Ui) + vi, (5)

where θτ (Ui) = (β′τ (Ui), δ′1(Ui), · · · , δ′T (Ui))

′, Zit,1 denotes the corresponding variables in

the first column in the above design matrix, Zit,2 represents those entries in the remaining

columns, and Zit = (Z ′it,1,Z′it,2)

′.

2.2.2 Quasi-Likelihood Function

For a conditional quantile regression model, according to Koenker and Bassett (1978), the

estimation of parameters can be obtained by minimizing the following objective (loss) function

minθLKB(θ) =

T∑t=1

ρτ (yt − qτ (wt, θ)),

where qτ (wt, θ) is the regression function with unknown parameters θ, ρτ (x) = x(τ − Ix<0)

is called the check function, and IA is the indicator function of any set A. Komunjer (2005)

generalized the estimation of Koenker and Bassett (1978) by proposing a class of quasi-

maximum likelihood estimations (QMLEs), θ, obtained by solving

maxθLQ(θ) =

T∑t=1

ln lt(yt, qτ (wt, θ)),

where lt(·) is a period-t conditional quasi-likelihood. As pointed out by Komunjer (2005),

if lt(yt, qτ (wt, θ)) is taken to be C(yt, wt) exp(−ρτ (yt − qτ (wt, θ))), the QMLEs become the

estimators of Koenker and Bassett (1978).

We consider a class of QMLEs for conditional quantile like the model defined in (5), ob-

tained by solving the maximization of a quasi-likelihood function for the τth conditional

quantile

maxϑ

QLτ (ϑ) = maxϑ

N∑i=1

T∑t=1

ln lτ (Yit, qτ (Ui,Zit,ϑ)), (6)

where lτ (Yit, qτ (Ui,Zit,ϑ)) is a quasi-likelihood function for the τth conditional quantile on

individual i at time t and ϑ′ = (γ ′τ ,θ′τ (U1), · · · ,θ′τ (UN )). For simplicity, by assuming that

10

vi is iid as normal7 with mean zero and variance σ2, lτ (y, q) is generated from the integrated

quasi-likelihood function for the τth conditional quantile,

lτ (y, a1) =1√2πσ

∞∫−∞

qlτ (y, qτ (a1 + v))e−v2

2σ2 dv,

where qlτ (y, q) is the quasi-likelihood function for the τth conditional quantile. It is empha-

sized by Komunjer (2005) that different choices of qlτ (·, ·) affect the asymptotic theory of

the QMLE for quantile, similar to the case that different choices of likelihood function would

affect the asymptotic theory of QMLE for mean model when the object of interest is the

conditional mean. In this paper, for simplicity, we define qlτ (y, q) as

qlτ (y, q) = e−ρτ (y−q), (7)

This definition makes qlτ (·, ·) belongs to the so-called tick-exponential family defined by

Komunjer (2005). Substituting (7) into the integrated quasi-likelihood function for the τth

conditional quantile, we have

lτ (y, a) =1√2πσ

∞∫−∞

exp[−ρτ (a− v)− v2

2σ2]dv,

where a = y − a1. By a simple calculation,

lτ (a, σ) = e−ρτ (a)λτ (a, σ)(Ia≥0 + e−aIa<0). (8)

Thus,

ln lτ (a, σ) = −ρτ (a) + ln(λτ (a, σ)) + ln(Ia≥0 + e−aIa<0), (9)

where λτ (a, σ) = eτ2σ2

2 Φ( aσ − τσ) + e(τ−1)2σ2

2 Φ(− aσ + (τ − 1)σ)ea and Φ(·) is the standard

normal distribution function. Clearly, the last two terms in (9) can be regarded as a penalty

due to the randomness of vi. Also, it is easily to show that the penalty goes to zero when

σ goes to zero. When σ = 0, (9) is exactly same as the case without vi. Therefore, the

quasi-likelihood function QLτ (ϑ, σ) is given by

QLτ (ϑ, σ) =N∑i=1

T∑t=1

[−ρτ (ait) + ln(λτ (ait, σ)) + ln(Iait≥0 + e−aitIait<0)], (10)

7The normality assumption on vi here is just for simplicity and of course, it can be relaxed but the quasi-

likelihood function would be very complex. It would be very interesting to explore this issue in the future

research.

11

where ϑ is the vector of all the parameters to be estimated. Hence, for a given σ satisfying

the following equation

∂QLτ (ϑ, σ)

∂σ|σ=σ

= NTτ2σ +N∑i=1

T∑t=1

−φ(aitσ − τ σ

)+ (1− 2τ)σeait+

(1−2τ)σ2

2 Φ(− ait

σ + (τ − 1)σ)

Φ(aitσ − τ σ

)+ eait+

(1−2τ)σ2

2 Φ(− ait

σ + (τ − 1)σ) = 0, (11)

the QMLE of ϑ, ϑ, is obtained by

ϑ = (γ ′τ , θ′τ (U1), · · · , θ′τ (UN ))′ = arg max

ϑQLτ (ϑ, σ). (12)

Remark 2. We can simply estimate ϑ and σ by iterating (11) and (12) until convergence.

Given an initial value of σ, we can estimate ϑ by maximizing QLτ (ϑ, σ). Alternatively, σ

can be obtained by the following

σ2 =1

TVar

(Yit −Z ′it,1γ0.5 −Z ′it,2θ0.5(Ui)

)− 1

12NT 2

N∑i=1

T∑t=1

[Z ′it,1ˆ∂γτ∂τ|τ=0.5+Z

′it,2

ˆ∂θτ (Ui)

∂τ|τ=0.5]

2.

(13)

Then, we let σ = σ and iterate (12) and (13) until convergence.

2.2.3 Three-stage Estimation Procedure

To estimate the semiparametric model (5), similar to Cai and Xiao (2012), we propose a

three-stage estimation procedure to the panel data model.

At the first stage, we treat all coefficients as functional coefficients depending on Ui, such as

γτ = γτ (Ui). It is assumed throughout that γτ (·) and θτ (·) are both twice continuously dif-

ferentiable, and then we apply the local constant approximations to γτ (·) and the local linear

approximations to θτ (·) respectively. Hence, model (5) is estimated as a fully functional-

coefficient model and following Cai and Xu (2008), the localized quasi-likelihood function is

given by

maxγ0,θ0,θ1

N∑i=1

T∑t=1

[−ρτ (ait,1) + ln(λτ (ait,1, σ)) + ln(Iait,1≥0 + e−ait,1Iait,1<0)]Kh(Ui − u0), (14)

where ait,1 = Yit−Z ′it,1γ0−Z ′it,2θ0−Z ′it,2θ1(Ui−u0), γ0 = γτ (u0), θ0 = θτ (u0), θ1 = θτ (u0),

Kh(u) = K(u/h)/h, and K(·) is the kernel function. Note that A and A denote the first

order and second order partial derivatives of A throughout the paper.

12

Since γτ is a global parameter, in order to utilize all sample information to estimate γτ ,

at the second stage, we employ the average method to achieve the√N consistent estimator

of γτ , which is given by

γτ =1

N

N∑i=1

γτ (Ui). (15)

Theorem 1 (see later) shows that indeed, γτ is a√N consistent estimator.

Remark 3. First, it is worth pointing out that the well known profile least squares type of

estimation approach (Robinson (1988) and Speckman (1988)) for classical semiparametric

regression models may not be suitable to quantile setting due to lack of explicit normal equa-

tions. Secondly, the estimator γτ given in (15) has the advantage that it is easy to construct

and also achieves the√N -rate of convergence (see Theorem 1 later). In addition to this

simple estimator, other√N consistent estimators of γτ can be constructed. For example, to

estimate the parameter γτ without being overly influenced by the tail behavior of the distri-

bution of Ui, one might use a trimming function wi = IUi∈D with a compact subset D of R;

see Cai and Masry (2000) for details. Then, (15) becomes the weighted average estimator as

γw,τ =

[1

N

N∑i=1

wi

]−1 [1

N

N∑i=1

wiγτ (Ui)

].

Indeed, this type of estimator was considered by Lee (2003) for a partially linear quantile

regression model. To estimate γτ more efficiently, a general weighted average approach can

be constructed as follows

γw,τ =

[1

N

N∑i=1

W (Ui)

]−1 [1

N

N∑i=1

W (Ui)γτ (Ui)

],

where W (·) is a weighting function (a symmetric matrix) which can be chosen optimally by

minimizing the asymptotic variance; see Cai and Xiao (2012) for details. For simplicity, our

focus is on γτ given in (15).

At the last step, to estimate the varying coefficients, for a given√N -consistent estimator

γτ of γτ which may be obtained from (15), we plug γτ into model (5) and obtain the partial

residual denoted by Y ∗it = Yit − Z ′it,1γτ . Hence, the functional coefficients can be estimated

by using the local linear quantile estimation which is given by

maxθ0,θ1

N∑i=1

T∑t=1

[−ρτ (ait,2) + ln(λτ (ait,2, σ)) + ln(Iait,2≥0 + e−ait,2Iait,2<0)]Kh(Ui − u0), (16)

13

where ait,2 = Y ∗it − Z ′it,2θ0 − Z ′it,2θ1(Ui − u0). By moving u0 along the domain of Ui, the

entire estimated curve of the functional coefficient is obtained.

2.3 Asymptotic Theory

This section provides asymptotic results of γτ and βτ (u0) defined in Section 2.2.3. All proofs

are relegated to the appendices.

Firstly, define µj =∫∞−∞ u

jK(u)du and νj =∫∞−∞ u

jK2(u)du with j ≥ 0. Let h1 and h2 be

the bandwidths used at the first and third stages, respectively. Let e′1 = (IK∗1 ,0K∗1×K∗2 ) and

e′2 = (IK2 ,0K2×KT ), where K∗ = K∗1 + K∗2 , K∗1 = K1 and K∗2 = K2 + KT . Then, denote

fU (·) by the marginal density of U .

Secondly, we give the following notations and definitions which will be used in the rest of the

paper. Let gτ (a, σ) = ∂ ln(λτ (a, σ))/∂a 8, bit,1(u0) = Yit−Z ′it,1γτ−Z ′it,2θτ (u0) and bit,2(u0) =

Y ∗it − Z ′it,2θτ (u0), then denote mg(u0,Zit, σ) = E[gτ (bit,1(u0), σ)|u0,Zit], m∗g(u0,Zit,2, σ) =

E[gτ (bit,2(u0), σ)|u0,Zit,2], mg2(u0,Zit, σ) = E[g2τ (bit,1(u0), σ)|u0,Zit], m∗g2(u0,Zit,2, σ) =

E[g2τ (bit,2(u0), σ)|u0,Zit,2], mg1t(u0,Zi1,Zit, σ) = E[gτ (bi1,1(u0), σ)gτ (bit,1(u0), σ)|u0,Zi1,Zit],

m∗g1t(u0,Zi1,2,Zit,2, σ) = E[gτ (bi1,2(u0), σ)gτ (bit,2(u0), σ)|u0,Zi1,2,Zit,2], mg(u0,Zit, σ) =

E[gτ (bit,1(u0), σ)|u0,Zit], mg(u0,Zit, σ) = E[gτ (bit,1(u0), σ)|u0,Zit], and m∗g(u0,Zit,2, σ) =

E[gτ (bit,2(u0), σ)|u0,Zit,2].

Moreover, let

Ωτ,z(u0, σ) = τ2Ωz(u0)− 2τΩzg(u0, σ) + Ωzg2(u0, σ)

with Ωz(u0) = E(ZitZ′it|u0), Ωzg(u0, σ) = E(ZitZ

′itmg(u0,Zit, σ)|u0) and Ωzg2(u0, σ) =

E(ZitZ′itmg2(u0,Zit, σ)|u0),

Ωτ,z1t(u0, σ) = τ2Ωz1t(u0)− 2τΩz1tg(u0, σ) + Ωz1tg1t(u0, σ)

with Ωz1t(u0) = E(Zi1Z′it|u0), Ωz1tg(u0, σ) = E(Zi1Z

′itmg(u0,Zit, σ)|u0) and Ωz1tg1t(u0, σ) =

E(Zi1Z′itmg1t(u0,Zi1,Zit, σ)|u0),

Ωzg(u0, σ) = E[ZitZ′itmg(u0,Zit, σ)|u0],

Ωτ,z2(u0, σ) = τ2Ωz2(u0)− 2τΩz2g(u0, σ) + Ωz2g2(u0, σ)

8 ∂ ln(λτ (a,σ))∂a

= ea+(1−2τ)σ2

2 Φ(− a

σ+ (τ − 1)σ

)/[Φ(aσ− τσ

)+ ea+

(1−2τ)σ2

2 Φ(− a

σ+ (τ − 1)σ

)].

14

with Ωz2(u0) = E(Zit,2Z′it,2|u0), Ωz2g(u0, σ) = E(Zit,2Z

′it,2m

∗g(u0,Zit,2, σ)|u0) and Ωz2g2(u0, σ) =

E(Zit,2Z′it,2m

∗g2(u0,Zit,2, σ)|u0),

Ωτ,z1t,2(u0, σ) = τ2Ωz1t,2(u0)− 2τΩz1t,2g(u0, σ) + Ωz1t,2g1t(u0, σ)

with Ωz1t,2(u0) = E(Zi1,2Z′it,2|u0), Ωz1t,2g(u0, σ) = E(Zi1,2Z

′it,2m

∗g(u0,Zit,2, σ)|u0) and

Ωz1t,2g1t(u0, σ) = E(Zi1,2Z′it,2m

∗g1t(u0,Zi1,2,Zit,2, σ)|u0),

Ωz2g(u0, σ) = E[Zit,2Z′it,2m

∗g(u0,Zit,2, σ)|u0].

Consequently, we define the asymptotic biases and variances in the following theorems as

Bγ,τ (σ) = −µ2h21

2e′1E

Ω−1zg (Ui, σ)[2Ωzg(Ui, σ)

(0

θτ (Ui)

)+ Θ(Ui, σ)]

with Θ(Ui, σ) = Emg(Ui,Zit, σ)Zit[Z

′it,2θτ (Ui)]

2|Ui,

Bβ,τ (u0, σ) ≡ Bβ,τ (u0) = −µ2h22

2βτ (u0),

Σγ,τ (σ) = e′1EΩ−1zg (Ui, σ)[1

TΩτ,z(Ui, σ) +

T∑t=2

2(T − t+ 1)

T 2Ωτ,z1t(Ui, σ)]Ω−1zg (Ui, σ)e1,

and

Σβ,τ (u0, σ) =ν0e′2

fU (u0)Ω−1z2g(u0, σ)[

1

TΩτ,z2(u0, σ) +

T∑t=2

2(T − t+ 1)

T 2Ωτ,z1t,2(u0, σ)]Ω−1z2g(u0, σ)e2.

Remark 4. Note that g(a, σ) = Ia<0 when σ = 0, then mg(u0,Zit, σ), m∗g(u0,Zit,2, σ),

mg2(u0,Zit, σ), m∗g2(u0,Zit,2, σ), mg1t(u0,Zi1,Zit, σ) and m∗g1t(u0,Zi1,2,Zit,2, σ) all equal to

τ , moreover, mg(u0,Zit, σ), m∗g(u0,Zit,2, σ) and mg(u0,Zit, σ) equal to fY |U,Z(Z ′it,1γτ +

Z ′it,2θτ (u0)), fY ∗|U,Z2(Z ′it,2θτ (u0)) and fY |U,Z(Z ′it,1γτ + Z ′it,2θτ (u0)), respectively. Accord-

ingly, Ωτ,z(u0, σ) = τ(1− τ)Ωz(u0), Ωzg(u0, σ) = E[ZitZ′itfY |U,Z(Z ′it,1γτ +Z ′it,2θτ (u0))|u0] ≡

Ωzf (u0), Ωτ,z1t(u0, σ) = τ(1 − τ)Ωz1t(u0), Ωτ,z2(u0, σ) = τ(1 − τ)Ωz2(u0), Ωz2g(u0, σ) =

E[Zit,2Z′it,2fY ∗|U,Z2

(Z ′it,2θτ (u0))|u0] ≡ Ωz2f (u0), Ωτ,z1t,2(u0, σ) = τ(1− τ)Ωz1t,2(u0).

Therefore, the asymptotic bias Bγ,τ (σ) reduce to

Bγ,τ = −µ2h21e′1E

Ω−1zf (Ui)[2Ωzf (Ui)

(0

θτ (Ui)

)+ Θ(Ui)]

15

with Θ(Ui) = EZit[Z ′it,2θτ (Ui)]2fY |U,Z(Z ′it,1γτ +Z ′it,2θτ (Ui))|Ui, and the asymptotic vari-

ances Σγ,τ (σ) and Σβ,τ (u0, σ) reduce to

Σγ,τ = τ(1− τ)e′1EΩ−1zf (Ui)[1

TΩz(Ui) +

T∑t=2

2(T − t+ 1)

T 2Ωz1t(Ui)]Ω

−1zf (Ui)e1,

and

Σβ,τ (u0) =τ(1− τ)ν0e

′2

fU (u0)Ω−1z2f (u0)[

1

TΩz2(u0) +

T∑t=2

2(T − t+ 1)

T 2Ωz1t,2(u0)]Ω

−1z2f

(u0)e2.

The asymptotic bias of the varying coefficients is exactly same as the one of the case without

vi no matter σ is zero or not.

The following assumptions are necessary to establish the consistency and asymptotic nor-

mality of our estimators, although they might not be the weakest ones.9

Assumptions:

A1. The series Ui is iid. The series Zit is iid across individual i, but can be correlated

around t for fixed i. The series vi is iid N(0, σ2).

A2. The distribution of Y given U and Z has an everywhere positive Lebesgue density

fY |U,Z(·), which is bounded and satisfies the Lipschitz continuity condition.

A3. The kernel function K(·) is a symmetric bounded density with a bounded support

region.

A4. Assume that the functional coefficients θ(u0) are 2 times continuously differentiable

in a small neighborhood of u0. The marginal density smoothing variable U , fU (·), is contin-

uous with fU (u0) > 0. Assume all the variance-covariance matrix are positive-definite and

continuously differentiable in a neighborhood of u0.

A5. Assume that E(||Z||δ∗) <∞ with δ∗ > 4.

A6. Assume that bandwidth h1 → 0, h2 → 0, Nh1 → ∞ and Nh2 → ∞ as N → ∞.

Furthermore, Nh41 → 0.

A7. Assume that T ≥ 2.

Remark 5. Assumption A1 specifies that the data is iid across individual i, but can be

correlated around t for fixed i. This is a standard assumption in the panel data literature.

9The assumptions are imposed for a fixed u0 and a fixed τ , and the same assumptions are imposed on the

finite points of interest.

16

The normality assumption of vi can be relaxed by using some approximation approaches such

as Laplace approximation but the quasi-likelihood would be complex since there is no close

form of the quasi-likelihood like (8). We leave the generalization of this normally distributed

restriction to other types of distribution as future work. Assumption A6 requires that N go

to infinity, but T can be fixed. If the model is extended to the case of large T , an appropriate

mixing condition should be included. Assumption A7 excludes the case of T = 1 in order to

identify the coefficients of interest. This is not a severe assumption because this paper focus

on panel data. Assumption A2 does not exclude the dependence between error and covariates.

This kind of dependence will cause heteroscedasticity, which is indicated by the changing of

coefficients under different quantile levels. Assumption A3 is a standard assumption in the

nonparametric literature. Assumption A4 includes some smoothness conditions on functionals

involved.

As mentioned above, a√N consistent estimator of γτ at the second stage is constructed

by using the average method defined in (15). The following Theorem 1 states its asymptotic

normality result which can be obtained by following the U-statistic technique in Powell, Stock

and Stoker (1989).

Theorem 1. Suppose that Assumptions I1, I2 and A1-A7 hold, we have

√N(γτ − γτ +Bγ,τ (σ))

D→ N(0,Σγ,τ (σ)).

Results of Theorem 1 and Remark 4 show that the asymptotic normality of the estimate for

constant coefficients reduce to the one of the case without vi when σ = 0, because Bγ,τ (σ) and

Σγ,τ (σ) are exactly same as Bγ,τ and Σγ,τ , the corresponding asymptotic bias and variance of

the estimate for constant coefficients relating to the case without vi, when σ = 0. Therefore,

the estimator γτ is√N consistent and is asymptotically unbiased when the bandwidth h1

satisfies√Nh21 → 0, which implies that it requires under-smooth at the first stage. Clearly,

compare the above theorem with Theorem 1 in Cai and Xiao (2012), the asymptotic bias

term, Bγ,τ , is the same but the asymptotic variance, Σγ,τ , depends on T . However, when T

goes to infinite, some conditions on the observations (Yit,Xit, Ui), say mixing conditions,

are needed; see Cai and Xiao (2012) for details.

17

At the last stage, the partial residuals Y ∗it are used to estimate θτ (u0). The following

Theorem 2 states the asymptotic normality result of βτ (u0) which can be easily extracted

from the asymptotic normality result of θ0,τ (u0) because of βτ (u0) = e′2θ0,τ (u0).

Theorem 2. Suppose that Assumptions I1, I2 and A1-A7 hold, given the√N consistent

estimator of γτ , we have√Nh2[βτ (u0)− βτ (u0) +Bβ,τ (u0)]→ N(0,Σβ,τ (u0, σ)).

Results of Theorem 2 and Remark 4 show that the asymptotic normality of the estimate for

varying coefficients reduce to the one of the case without vi when σ = 0, because Σβ,τ (u0, σ)

is exactly same as Σβ,τ (u0), the corresponding asymptotic variance of the estimate for varying

coefficients relating to the case without vi, when σ = 0. Compared with Theorem 1 in Cai

and Xu (2008) and Theorem 2 in Cai and Xiao (2012), the asymptotic bias term in the

above theorem is the same but the asymptotic variance, Σβ,τ (u0), depends on T . Clearly,

the optimal bandwidth is h2,opt = O(N−1/5) by minimizing the mean squared error and the

asymptotic mean squared error is in the order of O(N−4/5). This means that the regular

bandwidth selection procedures can be applied here.

2.4 A Specification Test

After deriving the asymptotic results, we now turn to discussing statistical inferences such as

constructing confidence intervals and testing hypotheses. To make statistical inferences for

γτ and βτ (·) in practice, first one needs some consistent covariance estimators of Σγ,τ (σ) and

Σβ,τ (u0, σ). To this end, we need to estimate Ωzg(u0, σ), Ωτ,z(u0, σ), Ωτ,z1t(u0, σ), Ωz2g(u0, σ),

Ωτ,z2(u0, σ) and Ωτ,z1t,2(u0, σ) consistently. The estimation of Ωz2g(u0, σ), Ωτ,z2(u0, σ) and

Ωτ,z1t,2(u0, σ) is similar to Ωzg(u0, σ), Ωτ,z(u0, σ) and Ωτ,z1t(u0, σ). Hence, we here only focus

on the latter to save notations. Following Cai and Xu (2008), we define

Ωz(u0) = (NT )−1N∑i=1

T∑t=1

ZitZ′itKh(Ui − u0),

Ωz1t(u0) = (N(T − t))−1N∑i=1

T−t∑s=1

ZisZ′i(s+t)Kh(Ui − u0),

Ωzg(u0, σ) = (NT )−1N∑i=1

T∑t=1

ZitZ′itmg(u0,Zit, σ)Kh(Ui − u0)

18

and

Ωz1tg(u0, σ) = (N(T − t))−1N∑i=1

T−t∑s=1

ZisZ′i(s+t)mg(u0,Zi(s+t), σ)Kh(Ui − u0),

where

mg(u, z′1γτ − z′2θτ (u), σ) =

N∑i=1

T∑t=1

Kh(Ui − u,Zit − z)g(Yit − z′1γτ − z′2θτ (u), σ)

N∑i=1

T∑t=1

Kh(Ui − u,Zit − z)

.

It can be easily shown that Ωz(u0) = fU (u0)Ωz(u0) + op(1), Ωzg(u0, σ) = fU (u0)Ωzg(u0, σ) +

op(1), Ωz1t(u0) = fU (u0)Ωz1t(u0) + op(1) and Ωz1tg(u0, σ) = fU (u0)Ωz1tg(u0, σ) + op(1).

Similarly, we can get the consistent estimators of Ωzg2(u0, σ), Ωz1tg1t(u0, σ) and Ωzg(u0, σ).

Then, Ωτ,z(u0, σ) = τ2Ωz(u0) − 2τ Ωzg(u0, σ) + Ωzg2(u0, σ) and Ωτ,z1t(u0, σ) = τ2Ωz1t(u0) −

2τ Ωz1tg(u0, σ) + Ωz1tg1t(u0, σ). Finally, the consistent covariance estimators of Σγ,τ (σ) can

be respectively given by

e′1N−1

N∑i=1

Ω−1zg (Ui, σ)[1

TΩτ,z(Ui, σ) +

T∑t=2

2(T − t+ 1)

T 2Ωτ,z1t(Ui, σ)]Ω−1zg (Ui, σ)e1.

Therefore, the consistent estimate of Σβ,τ (u0, σ) can be constructed accordingly in an obvious

manner.

In empirical studies, it is of importance to test the constancy of the varying coefficients.

The null hypothesis H0 : βτ (uj) = βτ for ujqj=1 where ujqj=1 is a set of distinct points

within the domain of Ui. Cai and Xiao (2012) provided some comments on the choice of

ujqj=1 and q in practice. Under the null hypothesis, a simple and easily implemented test

statistics can be constructed as follows

TN =∑

1≤j≤q||√Nh2Σβ,τ (uj , σ)−1/2(βτ (uj)− βτ )||2 → χ2

qK2(17)

where χ2qK2

is a chi-squared random variable with qK2 degrees of freedom. Thus, the null is

rejected if TN is too large. Note that the proposed test statistic TN in (17) is different from

that in Cai and Xiao (2012). Of course, other types of test statistics may be constructed and

it would be warranted as future research topics.

19

3 Monte Carlo Simulations

In this section, we conduct Monte Carlo simulations to demonstrate the finite sample perfor-

mance of the proposed estimators for both constant and functional coefficients. We consider

the following data generating process

Qτ (Yit|Ui,Xi, vi) = γ0,τ +Xit,1γ1,τ +Xit,2βτ (Ui) +T∑t=1

Xit,2δt(Ui) + vi (18)

with T = 2, where the smoothing variable Ui is generated from iid U(−2.5, 2.5), Xit,1 andXit,2

are respectively generated from iid U(0, 3) and U(0, 2), vi is generated from iid N(0, σ2) with

σ = 0.1, and Yit is generated base on the Skorohod representation. The constant coefficients

are set by γ0,τ = 2 + τ and γ1,τ = −1.5 + τ , respectively. The functional coefficients are

defined as βτ (u) = 0.5 cos(2u) + u/3 + τ , δ1(u) = sin(1.5u), and δ2(u) = 1.5e−u2 − 0.75.

To measure the performance of γj,τ for 0 ≤ j ≤ 2 and βτ (·), we use the mean absolute

deviation errors (MADE) of the estimator, defined by

MADE(βτ (·)) =1

n0

n0∑l=1

|βτ (ul)− βτ (ul)|,

where uln0l=1 are the grid points within the domain of Ui, and

MADE(γj,τ ) = |γj,τ − γj,τ |

for 0 ≤ j ≤ 2.

We consider three sample sizes as N = 200, 500 and 1000. For a given sample size, we

repeat simulations 500 times to calculate the MADE values. We compare the estimation

results using different bandwidths, such as h1 = 5N−2/5 (under-smooth) and h2 = cN−1/5,

where c is chosen from 1.5, 1.7, 2, 2.2, 2.5, 2.7, 3.0, · · · . From the simulation results we

can conclude that the estimation of constant coefficients is not sensitive to the choice of the

bandwidth when the first step is under-smoothed, and the estimation of βτ (·) is quite stable

when the bandwidth selection is chosen within a reasonable range. The optional bandwidth

for estimating functional coefficient βτ (·) is about h2 = 2.7N−1/5.

The boxplots of 500 MADE values of γ0,τ , γ1,τ and βτ (·) are given in Figures 1 for γ0,τ ,

2 for γ1,τ , and 3 for βτ (·). From the boxplots, we can observe that the median and stan-

dard deviation of 500 MADE values are decreasing for all settings when the sample size is

20

increasing. This concludes that the proposed estimators for both constant and functional

coefficients are consistent.

0.0

0.1

0.2

0.3

0.4

0.5

MADE of γ0,0.15

− N=200 − − N=500 − − N=1000 −

0.0

0.1

0.2

0.3

0.4

0.5

0.6

MADE of γ0,0.5

− N=200 − − N=500 − − N=1000 −

0.0

0.1

0.2

0.3

0.4

0.5

MADE of γ0,0.75

− N=200 − − N=500 − − N=1000 −

Figure 1: Boxplot of MADE values of γ0,τ .

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

MADE of γ1,0.15

− N=200 − − N=500 − − N=1000 −

0.0

0.1

0.2

0.3

0.4

MADE of γ1,0.5

− N=200 − − N=500 − − N=1000 −

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

MADE of γ1,0.75

− N=200 − − N=500 − − N=1000 −

Figure 2: Boxplot of MADE values of γ1,τ .

21

0.0

0.1

0.2

0.3

0.4

MADE of β0.15(u)− N=200 − − N=500 − − N=1000 −

0.0

0.1

0.2

0.3

0.4

MADE of β0.5(u)− N=200 − − N=500 − − N=1000 −

0.0

0.1

0.2

0.3

0.4

MADE of β0.75(u)− N=200 − − N=500 − − N=1000 −

Figure 3: Boxplot of MADE values of βτ (u).

Also, the simulation results (the median and standard deviation, denoted by SD, in paren-

theses of 500 MADE values) for the estimators of both constant and functional coefficient are

summarized in Table 1. From Table 1, one can see clearly that the medians of 500 MADE

values for all settings decrease significantly as N increases. For example, when the sample

size increases from 200 to 1000, the medians of MADE values for γ0,0.15, γ1,0.15 and β0.15(·)

shrink quickly from 0.1089 to 0.0439, from 0.0683 to 0.0289, and from 0.2193 to 0.1214, re-

spectively. The standard deviations also shrink quickly when the sample size is enlarged. For

example, for γ0,0.15, the standard deviations shrink from 0.1080 to 0.0384, and for γ1,0.15 and

β0.15(·), they decrease from 0.0721 to 0.0285 and from 0.0577 to 0.0225, respectively. Similar

results can be observed at the median, τ = 0.5, and the upper quantile, τ = 0.75. This

is in line with our asymptotic theory and implies that our proposed estimators are indeed

consistent. Compared with the estimation of βτ (·), the shrinkage speed of the estimation

of γ1,τ is relatively fast, which is also consistent with the theoretical results in the previous

sections.

22

Table 1: The median and SD of the MADE values for γ0,τ , γ1,τ and βτ (·)

τ = 0.15 τ = 0.5 τ = 0.75

γ0,τ γ1,τ βτ (·) γ0,τ γ1,τ βτ (·) γ0,τ γ1,τ βτ (·)200 0.1089 0.0683 0.2193 0.1399 0.0848 0.2498 0.1238 0.0814 0.2320

(0.1080) (0.0721) (0.0577) (0.1146) (0.0757) (0.0555) (0.1078) (0.0765) (0.0559)

500 0.0629 0.0445 0.1550 0.0823 0.0544 0.1968 0.0782 0.0507 0.1751

(0.0594) (0.0411) (0.0313) (0.0732) (0.0518) (0.0397) (0.0677) (0.0440) (0.0356)

1000 0.0439 0.0289 0.1214 0.0613 0.0398 0.1603 0.0514 0.0337 0.1411

(0.0384) (0.0285) (0.0225) (0.0495) (0.0354) (0.0273) (0.0465) (0.0297) (0.0241)

4 Modeling the Effect of FDI on Economic Growth

4.1 Empirical Models

The existing literature presented contradictory empirical evidences on whether or not FDI can

promote the economic growth in host countries. Recent studies in the literature tried to find

the sources of the mixed empirical conclusions and some studies concluded that the reason

may be due to nonlinearities in FDI effects on the economic growth and the heterogeneity

among countries. For example, in the growth literature like Kottaridi and Stengos (2010),

the following is a typical model in empirical studies

yit = αi + β1(FDI/Y )it + β2 log(DI/Y )it + β3nit + β4hit + εit, (19)

where yit denotes the growth rate of GDP per capita in the country or region i during the

period t, αi is the individual effect used to control the unobserved country-specific hetero-

geneity, nit is the logarithm of population growth rate, hit is the human capital, and εit is

random error. Moreover, the FDI and DI in (19) refer to foreign direct investment and do-

mestic investment respectively and Y represents the total output. Hence, (FDI/Y )it denotes

the average ratio between the FDI and the total output during the period t in country i and

(DI/Y )it is defined in the same fashion for the domestic investment. To allow the possible

joint effect of FDI and human capital, some literatures considered to add an interacted term

between FDI and human capital into the empirical growth model, then (19) becomes

yit = αi + β1(FDI/Y )it + β2 log(DI/Y )it + β3nit + β4hit + β5((FDI/Y )it × hit) + εit, (20)

23

which is indeed a nonlinear parametric model; see Kottaridi and Stengos (2010) and the

references therein.

Since the majority of the literature realized that the effect of FDI on the economic growth

depends on the absorptive capacity in host countries and the initial GDP per capita is one

of the most important indicators to reflect the initial conditions and the absorptive capacity

in the host country; see Hansen (2000), Nunnemkamp (2004), and among others, we hereby

propose a partially varying-coefficient model which allows the effect of FDI on the economic

growth to depend on the initial GDP per capita in the host country. Hence, our first empirical

econometric model is the conditional mean model, given by

yit = αi+β1(Ui)(FDI/Y )it+β2 log(DI/Y )it+β3nit+β4hit+β5((FDI/Y )it×hit) + εit, (21)

where Ui is the logarithm of initial GDP per capita in country i and β1(Ui) is the varying

coefficient over the logarithm of initial GDP per capita Ui. Therefore, model (21) has an

ability to characterize how the FDI may have different effects on the economic growth under

the different initial conditions.

As we discussed in the introduction, the conditional mean model in (21) is usually not

sufficiently enough to control the heterogeneity among countries. The existing literature

dealt with the aforementioned issue by simply looking at sub-samples. Instead, in this paper,

we propose to adopt quantile regression approach to investigate the impact of FDI on the

economic growth. Our method is capable of dealing with heterogeneity among countries by

allowing different quantiles to have different empirical growth equations, and at the same

time, we can avoid splitting the sample. Different from the mean model, another advance

of considering the quantile model is that one can see how the FDI effects differently on the

economic growth in the different groups of countries, say the economy fast growing countries

(upper quantile) and the economy slowly growing countries (lower quantile) conditional on

controlled covariates.

By assuming that the εit in (21) takes a linearly heteroscedastic form as εit = (X ′itϕ)ε∗it; see

Koenker and Bassett (1978), where Xit includes all regressors in (21) and ε∗it is independent

of all covariates, but given i, ε∗it is allowed to be correlated around t, then we can obtain the

24

following conditional quantile model,

Qτ (yit | Ui,Xit, αi) = αi + β1,τ (Ui)(FDI/Y )it + β2,τ log(DI/Y )it

+β3,τnit + β4,τhit + β5,τ ((FDI/Y )it × hit), (22)

which can be regarded as a special case of model (1). Imposing the correlated random effect

assumption in (3), we can derive the conditional quantile regression model in (5) for studying

this empirical example. Therefore, the three-stage estimation procedures described in Section

2 can be applied here.

4.2 The Data and Empirical Results

Our data set includes 95 countries or regions from 1970 to 1999. To smooth the yearly

fluctuations in aggregate economic variables, we take five-year averages by following the

convention in the empirical growth literature; see Maasoumi, Racine and Stengos (2007),

Durlauf, Kourtellos and Tan (2008), and Kottaridi and Stengos (2010). Hence, there are six

periods totally. The population growth is computed by the average annual growth rate in

each period, the human capital is measured as mean years of schooling in each period, and

the domestic investment refers to the average of the domestic gross fixed capital formation

measured by the US dollars in 2000 constant values. We measure the initial GDP by the GDP

per capita of each country in the beginning year of each decade in constant 2000 US dollars.10

All the above data are available to be downloaded from World Development Indicators (WDI).

The FDI flows, in constant 2000 US dollars, are taken from United Nations Conference on

Trade and Development (UNCTAD). The full list of countries and regions can be found in

Table 5 in Appendix A.

Firstly, we consider the classical linear regression model in (20). Table 2 presents corre-

sponding estimation results, including coefficient estimates, standard deviations, t-values and

p-values from Column 2 to Column 5. The estimate of the FDI effect, denoted by β1, is about

0.56, which is positive and significant with a p-value of 0.027. On average, the linear condi-

tional mean model reports a mild positive effect of FDI on promoting the economic growth.

Compared to the growth effect of FDI, Table 2 reports a larger effect of domestic investments

on the economic growth, which is about 2.72 and highly significant with the p-value of 0.009.

10We combine three decades, from 1970 to 1979 (69 countries), from 1980 to 1989 (93 countries), and from

1990 to 1999 (95 countries), and then obtain a panel of 514 observations with N = 257 and T = 2.

25

The effect of population growth (β3) is also positive and significant, with an estimate of 0.65.

However, other estimates (the effect of human capital and the effect of the interacted term

between human capital and FDI) are not significant.

Table 2: Empirical results of a linear conditional mean model in (20)

Mean Model Coefficient Standard Deviation t-value p-value

β1 0.5589 0.2513 2.2248 0.0270 ∗

β2 2.7180 1.0374 2.6206 0.0093 ∗∗

β3 0.6483 0.2575 2.5183 0.0124 ∗

β4 -0.0290 0.4269 -0.0682 0.9458

β5 -0.0359 0.0375 -0.9561 0.3401

Next, we move to the partially varying-coefficient conditional mean model in (21). Com-

pared to the linear model in (20), we now allow the effect of FDI to depend on the initial

conditions. Figure 4 and Table 3 present the corresponding estimation results. The solid

line in Figure 4 represents the nonparametric estimate of the varying coefficient β1(·) along

various values of initial GDP, and the shaded area is the corresponding 90% pointwise con-

fidence intervals with the bias ignored. The nonparametric estimate shows a mild but clear

pattern that the growth effect of FDI increases as the initial GDP improves, which is in line

with the hypothesis of the absorptive capacity. The range of the estimated values of the

varying coefficient is between 0.9 and 1.6 for different initial GDPs, much larger than 0.56,

the estimated value of the linear model. Table 3 reports the estimates of constant coefficients

in (21), which are quite different from the corresponding estimation results in Table 2. For

example, in Table 3, the estimate of β2 is now 3.81 in stead of 2.72. The impact of popula-

tion growth rate on the economic growth now becomes to be significantly negative with an

estimate of −1.18. Moreover, both the coefficients of human capital and the interacted term

between FDI and human capital become significant in Table 3. The estimate of the impact

of human capital is positive with a value of 0.17 and the estimate of the interacted term is

−0.18. We attribute the different estimation results to the existence of nonlinearity in the

regression model.

26

Table 3: Constant coefficients of a partial linear conditional mean model in (21)

Mean Model Coefficient Standard Deviation t-value p-value

β2 3.8100 0.1321 28.8326 0.0000 ∗∗∗

β3 -1.1838 0.3357 -3.5268 0.0004 ∗∗∗

β4 0.1728 0.0268 6.4519 0.0000 ∗∗∗

β5 -0.1753 0.0112 -15.5394 0.0000 ∗∗∗

7.0 7.5 8.0 8.5 9.0

0.8

1.0

1.2

1.4

1.6

1.8

log(Initial GDP)

Figure 4: Estimated curve of functional coefficient β1(·) in model (21) together with the

pointless 90% confidence interval with the bias ignored.

Finally, we consider the partially varying-coefficient quantile model in (22). We conduct a

constancy test as in Section 3.3 to testing whether the coefficient β1,τ (·) does not vary with

the initial GDP at different quantiles and whether the coefficients βj,τ where j = 2, · · · , 5

do not vary with the initial GDP at different quantiles. The testing results are summarized

in Table 4. It turns out that the p-values are 0.0000, 0.0000, and 0.0000 for 0.15, 0.50, and

0.75 quantiles, respectively. The test results strongly reject the null of constancy of β1,τ (·).

All these results verify the existence of the heterogeneity among countries and regions with

different development stages.

27

Table 4: p-values of testing constancyτ β1,τ β2,τ β3,τ β4,τ β5,τ

0.15 0.0000 0.9999 0.9999 0.9999 0.9999

0.50 0.0000 0.9999 0.9999 0.9999 0.9999

0.75 0.0000 0.9999 0.9999 0.9999 0.9999

Figure 5 presents the estimates of all four constant coefficients βj,τ for 2 ≤ j ≤ 5 under

different quantile levels as τ = 0.15, 0.25, 0.35, 0.45, · · · , 0.75, 0.85. The horizontal axis

represents different quantiles and the vertical axis measures the estimated value of βj,τ . The

curves in solid line denote the estimates under different quantiles and the areas in dark

gray color are corresponding 90% confidence intervals. The horizontal solid lines denote the

estimates under conditional mean model and the areas in light gray color are corresponding

90% confidence intervals. Except the estimated values of β3,τ in the upper left panel in Figure

5, most estimated values are outside the 90% confidence intervals of the conditional mean

estimates, implying that indeed, βj,τ changes over τ and the conditional mean model might

be inadequate to catch the heterogeneity effect. We observe that the estimated values of β2,τ

increase with τ when τ is bigger than 0.35 but the estimated values of β5,τ decrease with

τ when τ is in the range from 0.25 to 0.75. The estimated values of β4,τ decrease with τ

when τ < 0.5 and increase with τ when τ > 0.5. Moreover, the estimated values of β2,τ

and β4,τ are all positive but the estimated values of β3,τ and β5,τ are all negative. Hence,

generally speaking, we find a clear evidence that domestic investments and human capitals

have positive effects on the economic growth, while the effects of domestic investments are

larger and increase more significantly in countries or regions with better economic growth

performance than those with poor growth performance and the effects of human capitals show

a U-shape across the countries or regions from poor growth performance to better economic

growth performance.

28

0.2 0.3 0.4 0.5 0.6 0.7 0.8

3.0

3.5

4.0

Tau

beta_2

0.2 0.3 0.4 0.5 0.6 0.7 0.8

−1.

5−

1.0

−0.

5

Tau

beta_3

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.15

0.25

0.35

Tau

beta_4

0.2 0.3 0.4 0.5 0.6 0.7 0.8

−0.

24−

0.20

−0.

16−

0.12

Tau

beta_5

Figure 5: Estimated values of constant coefficients βj,τ in model (22) for 2 ≤ j ≤ 5 and βj in

model (21) and their corresponding 90% confidence intervals.

The nonparametric estimates of functional coefficient β1,τ (·) with the upper (τ = 0.75, in

the dashed line) and lower (τ = 0.15, in the solid line) quantiles are demonstrated in Figure

6. The horizontal axis measures different values of log of initial GDP, Ui, and the vertical

axis measures the nonparametric estimated values of β1,τ (·). The shaded areas represent the

90% confidence intervals of β1,τ (·), where the higher order bias is ignored. We observe that

the estimated values of β1,τ (·) at the upper quantile significantly higher than those at the

lower quantile uniformly over the values of initial GDPs, which is on line with the testing

results in Table 4. In general, our empirical findings support the hypothesis of absorptive

capacity. The initial conditions really matter for host countries to benefit from adopting

foreign direct investments. At the upper quantile, the estimated values of β1,τ (·) generally

increase with the value of initial GDPs, and furthermore, the tendency of increase speeds up

when Ui > 8.2. However, at the lower quantile, although the estimated curve has an overall

positive slope, it becomes almost flat when Ui is larger than 8.2 for host countries.

29

7.0 7.5 8.0 8.5 9.0

0.5

1.0

1.5

2.0

log(Initial GDP)

Tau=0.75Tau=0.15

Figure 6: Estimated curves of functional coefficient β1,τ (·) in model (22) for τ = 0.15 (solid

line) and τ = 0.75 (dashed line) and their concresponding 90% pointwise confidence intervals

with the bias ignored.

5 Conclusion

Quantile panel data models have gained a lot of attentions in the literature during recent

years. In this paper, we propose a partially varying-coefficient quantile panel data model

with correlated random effects. Compared to quantile panel data models with fixed effect,

our estimation assumes only large N and short (fixed) T , while the latter requires that both

N and T go to infinity. In our semiparametric model, we allow some coefficients to vary with

other economic variables while others keep constant. This novel semiparametric quantile

panel data model is applied to estimating the impact of FDI on the economic growth.

There are several issues still worth of further studies. First, it is reasonable to allow for

cross sectional dependence in the current model. In the literature of conditional mean models,

some methods have been developed to deal with cross sectional dependence, for example,

using the factor structure or the interactive effect. However, due to the nature of conditional

quantile model, it is not obvious to extend these under the quantile setup. Second, it is

also interesting to address a dynamic structure and endogeneity issue in conditional quantile

30

panel data models. We leave these important issues as future research topics.

Appendix A: Table of Countries and Regions

Table 5: Countries and regions in the empirical data setAlgeria Australia Austria Bahrain

Bangladesh Barbados Belgium Benin

Bolivia Botswana Brazil Cameroon

Canada Central African Republic Chile China

Colombia Congo, Rep. Costa Rica Cyprus

Denmark Dominican Republic Ecuador Egypt, Arab Rep.

El Salvador Fiji Finland France

Gambia Germany Ghana Greece

Guatemala Guyana Honduras Hong Kong SAR, China

Hungary Iceland India Indonesia

Iran, Islamic Rep. Ireland Israel Italy

Jamaica Japan Jordan Kenya

Korea, Rep. Lesotho Malawi Malaysia

Mali Malta Mauritius Mexico

Mozambique Nepal Netherlands New Zealand

Nicaragua Niger Norway Pakistan

Panama Papua New Guinea Paraguay Peru

Philippines Poland Portugal Rwanda

Senegal Sierra Leone Singapore South Africa

Spain Sri Lanka Sudan Swaziland

Sweden Switzerland Syrian Arab Republic Thailand

Togo Trinidad and Tobago Tunisia Turkey

Uganda United Kingdom United States Uruguay

Venezuela, RB Zambia Zimbabwe

Appendix B: Proof of Theorem 1

In order to establish the asymptotic theory of γτ in Theorem 1, the local Bahadur representa-

tion for the estimators obtained from the first stage should be derived. At first, we introduce

the following additional notations and definitions: H = diag(1K∗ , h11K∗2 )(K∗+K∗2 )×(K∗+K∗2 )

and G =

(IK∗

0K∗2×K∗1 , Uih1IK∗2

)(K∗+K∗2 )×K∗

, where Uih1 = (Ui − u0)/h1. Following Cai and

31

Xu (2008) and Cai and Xiao (2012), we have

√Nh1H

γτ (u0)− γτ (u0)

θ0,τ (u0)− θτ (u0)

θ1,τ (u0)− θτ (u0)

=Ω−1(u0, σ)√Nh1TfU (u0)

N∑j=1

T∑t=1

GZjtψτ (ajt,1, σ)K(Ujh1) + op(1),

where Ω(u0, σ) = diag(Ωzg(u0, σ), µ2e′0Ωzg(u0, σ)e0), e

′0 = (0K∗2×K∗1 , IK∗2 ), and ψτ (a, σ) =

τ − gτ (a, σ). In particular, we can obtain

√Nh1

(γτ (u0)− γτ (u0)

θ0,τ (u0)− θτ (u0)

)=

Ω−1zg (u0, σ)√Nh1TfU (u0)

N∑j=1

T∑t=1

Zjtψτ (ajt,1, σ)K(Ujh1) + op(1),

(23)

which is useful for establishing the asymptotic results for our estimators.

For any u0, (23) can be rewritten as

√Nh1

(γτ (u0)− γτ (u0)

θ0,τ (u0)− θτ (u0)

)' h1√

Nh1T

N∑j=1

T∑t=1

B−1(u0, σ)Z(u0,Zjt, σ)

+h1√Nh1T

N∑j=1

T∑t=1

B−1(u0, σ)Zjt[ψτ (ajt,1, σ)− ψτ (bjt,1(u0), σ)]Kh(Uj − u0),

where B(u0, σ) = fU (u0)Ωzg(u0, σ), Z(u0,Zjt, σ) = Zjtψτ (bjt,1(u0), σ)Kh(Uj − u0) and

bjt,1(u0) = Yjt − Z ′jt,1γτ − Z ′jt,2θτ (u0) for Uj in a small neighborhood of u0. In particu-

lar,

γτ (u0)− γτ (u0) '1

NT

N∑j=1

T∑t=1

e′1B−1(u0, σ)Z(u0,Zjt, σ)

+1

NT

N∑j=1

T∑t=1

e′1B−1(u0, σ)Zjt[ψτ (ajt,1, σ)− ψτ (bjt,1(u0), σ)]Kh(Uj − u0)

≡ 1

NT

N∑j=1

T∑t=1

e′1B−1(u0, σ)Z(u0,Zjt, σ) +BN (u0, σ)

32

holds uniformly for all u0 under Assumption A1-A4. Thus,

γτ − γτ =1

N

N∑i=1

[γτ (Ui)− γτ (Ui)]

=1

N2

N∑i=1

N∑j=1

1

T

T∑t=1

e′1B−1(Ui, σ)Z(Ui,Zjt, σ) +

1

N

N∑i=1

BN (Ui, σ)

=2

N2

∑1≤i<j≤N

e′1B−1(Ui, σ)

1

T

T∑t=1

Z(Ui,Zjt, σ) +1

N

N∑i=1

BN (Ui, σ)

=1

N2

∑1≤i<j≤N

[e′1B−1(Ui, σ)

1

T

T∑t=1

Z(Ui,Zjt, σ) + e′1B−1(Uj , σ)

1

T

T∑t=1

Z(Uj ,Zit, σ)]

+1

N

N∑i=1

BN (Ui, σ)

=N − 1

2NUN + BN ,

where BN = 1N

N∑i=1

BN (Ui, σ) and UN = 2N(N−1)

∑1≤i<j≤N

pN (ξi, ξj , σ) with

pN (ξi, ξj , σ) = e′1B−1(Ui, σ)

1

T

T∑t=1

Z(Ui,Zjt, σ) + e′1B−1(Uj , σ)

1

T

T∑t=1

Z(Uj ,Zit, σ),

and ξi = (Ui,ZZZi) indicates all the information for i. Define rN (ξi, σ) = E[pN (ξi, ξj , σ)|ξi],

θN (σ) = E[rN (ξi, σ)] = E[pN (ξi, ξj , σ)], and UN = θN (σ) + 2N

N∑i=1

[rN (ξi, σ) − θN (σ)]. The

following two lemmas are useful to prove Theorem 1 and their detailed proofs are relegated

to Appendix D.

Lemma 1. Under the assumptions in Theorem 1, we have

(i) rN (ξi, σ) = e′1Ω−1zg (Ui, σ) 1

T

T∑t=1

Zitψτ (bit,1(Ui), σ)(1 + o(1)),

(ii) θN (σ) = −µ2h21e′1EΩ−1zg (Ui, σ)[2Ωzg(Ui, σ)

(0

θτ (Ui)

)+ Θ(Ui, σ)]+ o(h21),

(iii) V ar[rN (ξi, σ)] = Σγ(σ) + o(h1).

Lemma 2. Under the assumptions in Theorem 1, we have

BN = µ2h21e′1EΩ−1zg (Ui, σ)[2Ωzg(Ui, σ)

(0

θτ (Ui)

)+ Θ(Ui, σ)]+ o(h21).

33

Proof of Theorem 1: First, note that E[||pN (ξi, ξj , σ)||2] = O(h−1) = O[N(Nh1)−1] →

o(N) if and only if Nh1 → ∞ as h1 → 0. Lemma 3.1 in Powell, Stock and Stoker (1989)

gives that√N(UN − UN ) = op(1). Then the result follows from Lemma 1, Lemma 2 and the

Lindeberg-Levy central limit theorem.

Appendix C: Proof of Theorem 2

The asymptotic distribution of βτ (u0) can be easily extracted from the asymptotic distri-

bution of θ0,τ (u0) because of βτ (u0) = e′2θ0,τ (u0). Similar to the proof of Theorem 1, the

local Bahadur representation for the estimators obtained from the third stage should be

derived at first. We introduce the following additional notations and definitions: H2 =

diag(1K∗2 , h21K∗2 )2K∗2×2K∗2 and G2 =

(IK∗2

Uih2IK∗2

)2K∗2×K∗2

, where Uih2 = (Ui−u0)/h2. For

a given√N consistent estimator γτ of γτ , we can obtain that

√Nh2H2

(θ0,τ (u0)− θτ (u0)

θ1,τ (u0)− θτ (u0)

)=

Ω−1(u0, σ)√Nh2TfU (u0)

N∑i=1

T∑t=1

G2Zit,2ψτ (ait,2, σ)K(Uih2)+op(1),

where Ω(u0, σ) = diag(Ωz2g(u0, σ), µ2Ωz2g(u0, σ)). Similar to (23), we have

√Nh2

(θ0,τ (u0)− θτ (u0)

)=

Ω−1z2g(u0, σ)√Nh2TfU (u0)

N∑i=1

T∑t=1

Zit,2ψτ (ait,2, σ)K(Uih2) + op(1),

which is useful to establish the asymptotic result for θ0,τ (u0). The above equation can be

rewritten as

√Nh2(θ0,τ − θτ (u0)) '

Ω−1z2g(u0, σ)√Nh2TfU (u0)

N∑i=1

T∑t=1

Zit,2ψτ (ait,2, σ)K(Uih2)

=Ω−1z2g(u0, σ)√Nh2TfU (u0)

N∑i=1

T∑t=1

Zit,2[ψτ (ait,2, σ)− ψτ (bit,2(Ui), σ)]K(Uih2)

+Ω−1z2g(u0, σ)√Nh2TfU (u0)

N∑i=1

T∑t=1

Zit,2ψτ (bit,2(Ui), σ)K(Uih2)

≡ BN + ΨN ,

where bit,2(Ui) = Y ∗it − Z ′it,2θτ (Ui). We will show that the first term BN determines the

asymptotic bias and the second term ΨN gives the asymptotic normality.

34

First, note that the first-order conditional moment of ψτ (bit,2(Ui), σ) is

E(ψτ (bit,2(Ui), σ)|Ui,Zit,2) = E(τ − g(bit,2(Ui), σ)|Ui,Zit,2)

= τ − E[g(bit,2(Ui), σ)|Ui,Zit,2]

≡ τ −m∗g(Ui,Zit,2, σ).

Then, the second-order conditional moments are

E(ψ2τ (bit,2(Ui), σ)|Ui,Zit,2) = E[τ2 − 2τg(bit,2(Ui), σ) + g2(bit,2(Ui), σ)|Ui,Zit,2]

= τ2 − 2τE[g(bit,2(Ui), σ)|Ui,Zit,2] + E[g2(bit,2(Ui), σ)|Ui,Zit,2]

≡ τ2 − 2τm∗g(Ui,Zit,2, σ) +m∗g2(Ui,Zit,2, σ),

and

E(ψτ (bi1,2(Ui), σ)ψτ (bit,2(Ui), σ)|Ui,Zi1,2,Zit,2)

= E[τ2 − τg(bi1,2(Ui), σ)− τg(bit,2(Ui), σ) + g(bi1,2(Ui), σ)g(bit,2(Ui), σ)|Ui,Zi1,2,Zit,2]

= τ2 − τE[g(bi1,2(Ui), σ)|Ui,Zi1,2]− τE[g(bit,2(Ui), σ)|Ui,Zit,2]

+E[g(bi1,2(Ui), σ)g(bit,2(Ui), σ)|Ui,Zi1,2,Zit,2]

≡ τ2 − τm∗g(Ui,Zi1,2, σ)− τm∗g(Ui,Zit,2, σ) +m∗g1t(Ui,Zi1,2,Zit,2, σ).

Thus,

E(ΨN ) =Ω−1z2g(u0, σ)√Nh2fU (u0)

NE[Zit,2ψτ (bit,2(Ui), σ)K(Uih2)]

=Ω−1z2g(u0, σ)√Nh2fU (u0)

NEE[Zit,2ψτ (bit,2(Ui), σ)|Ui]K(Uih2) = 0

with E[Zit,2ψτ (bit,2(Ui), σ)|Ui] = 0, since we know that bit,2(Ui) is the maximizer of the

35

corresponding likelihood function, and

V ar(ΨN )

=Ω−1z2g(u0, σ)

h2T 2f2U (u0)V ar

T∑t=1

Zit,2ψτ (bit,2(Ui), σ)K(Uih2)Ω−1z2g(u0, σ)

=Ω−1z2g(u0, σ)

h2T 2f2U (u0)TV ar[Zit,2ψτ (bit,2(Ui), σ)K(Uih2)]

+

T∑t=2

2(T − t+ 1)Cov((Zi1,2ψτ (bi1,2(Ui), σ)K(Uih2),Zit,2ψτ (bit,2(Ui), σ)K(Uih2))Ω−1z2g(u0, σ)

=ν0

TfU (u0)Ω−1z2g(u0, σ)[Ωτ,z2(u0, σ) +

T∑t=2

2(T − t+ 1)

TΩτ,z1t,2(u0, σ)]Ω−1z2g(u0, σ),

since

E[Zit,2Z′it,2ψ

2τ (bit,2(Ui), σ)K2(Uih2)]

= EZit,2Z ′it,2E[ψ2τ (bit,2(Ui), σ)|Ui,Zit,2]K2(Uih2)

= EZit,2Z ′it,2[τ2 − 2τm∗g(Ui,Zit,2, σ) +m∗g2(Ui,Zit,2, σ)]K2(Uih2)

≡ ν0h2fU (u0)[τ2Ωz2(u0)− 2τΩz2g(u0, σ) + Ωz2g2(u0, σ)](1 + o(1))

≡ ν0h2fU (u0)Ωτ,z2(u0, σ)(1 + o(1)),

and

E[Zi1,2Z′it,2ψτ (bi1,2(Ui), σ)ψτ (bit,2(Ui), σ)K2(Uih2)]

= EZi1,2Z ′it,2E[ψτ (bi1,2(Ui), σ)ψτ (bit,2(Ui), σ)|Ui,Zi1,2,Zit,2]K2(Uih2)

= EZi1,2Z ′it,2[τ2 − τm∗g(Ui,Zi1,2, σ)− τm∗g(Ui,Zit,2, σ) +m∗g1t(Ui,Zi1,2,Zit,2, σ)]K2(Uih2)

≡ ν0h2fU (u0)[τ2Ωz1t,2(u0)− 2τΩz1t,2g(u0, σ) + Ωz1t,2g1t(u0, σ)](1 + o(1))

≡ ν0h2fU (u0)Ωτ,z1t,2(u0, σ)(1 + o(1)).

Let QN = 1NT

N∑i=1

T∑t=1Zit,2ψτ (bit,2(Ui), σ)K(Uih2). Using the Cramer-Wold device, for any

d ∈ RK∗2 , define ZN,it =√

h2T d′Zit,2ψτ (bit,2(Ui), σ)K(Uih2), then we have

√Nh2d

′QN =1√NT

N∑i=1

T∑t=1

ZN,it =1√NT

N∑i=1

Z∗N,i,

where Z∗N,i =T∑t=1

ZN,it, which is iid across i. Hence, it follows by the Lindeberg-Levy central

limit theorem that the asymptotic normality holds.

36

Next, we move to work on the first term BN . Note that

bit,2(Ui)− ait,2 = Z ′it,2θτ (u0) +Z ′it,2θτ (u0)h2Uih2 −Z ′it,2θτ (Ui) = −h22

2Z ′it,2θτ (u)U2

ih2 ,

then

g(bit,2(Ui), σ)− g(ait,2, σ) = g(ait,2 + ζ(bit,2(Ui)− ait,2), σ)(bit,2(Ui)− ait,2)

= −h22

2g(ait,2 − ζ

h222Z ′it,2θτ (u)U2

ih2 , σ)Z ′it,2θτ (u)U2ih2

and

[g(bit,2(Ui), σ)− g(ait,2, σ)]2

= 2[g(ait,2 + ζ(bit,2(Ui)− ait,2), σ)− g(ait,2, σ)]g(ait,2 + ζ(bit,2(Ui)− ait,2), σ)(bit,2(Ui)− ait,2)

= 2[g(ait,2 + η(ait,2 + ζ(bit,2(Ui)− ait,2)− ait,2), σ)(ait,2 + ζ(bit,2(Ui)− ait,2)− ait,2)]

g(ait,2 + ζ(bit,2(Ui)− ait,2), σ)(bit,2(Ui)− ait,2)

= 2ζg(ait,2 + ηζ(bit,2(Ui)− ait,2), σ)g(ait,2 + ζ(bit,2(Ui)− ait,2), σ)(bit,2(Ui)− ait,2)2

= O(h42).

Thus,

E[ψτ (ait,2, σ)− ψτ (bit,2(Ui), σ)|Ui,Zit,2]

= −h22

2E[g(ait,2 − ζ

h222Z ′it,2θτ (u)U2

ih2 , σ)Z ′it,2θτ (u)U2ih2 |Ui,Zit,2]

= −h22

2E[g(bit,2(Ui), σ)|Ui,Zit,2]Z ′it,2θτ (Ui)U

2ih2(1 + o(1))

≡ −h22

2m∗g(Ui,Zit,2, σ)Z ′it,2θτ (Ui)U

2ih2(1 + o(1))

and

E[ψτ (ait,2, σ)− ψτ (bit,2(Ui), σ)]2|Ui,Zit,2 = O(h42).

Let Bit = Zit,2[ψτ (ait,2, σ)− ψτ (bit,2(Ui), σ)]K(Uih2), we have

E(Bit) = −h22

2E[Zit,2m

∗g(Ui,Zit,2, σ)Z ′it,2θτ (Ui)U

2ih2K(Uih2)](1 + o(1))

= −h22

2EE[Zit,2Z

′it,2m

∗g(Ui,Zit,2, σ)|Ui]θτ (Ui)U

2ih2K(Uih2)(1 + o(1))

= −h22

2E[Ωz2g(Ui, σ)θτ (Ui)U

2ih2K(Uih2)](1 + o(1))

= −h32

2fU (u0)µ2Ωz2g(u0, σ)θτ (u0)(1 + o(1))

37

and E[B2it] = o(h42) and similarly, we have E[Bit1Bit2 ] = o(h42). Finally, we show that

E(BN ) =Ω−1z2g(u0, σ)√Nh2fU (u0)

NEZit,2[ψτ (ait,2, σ)− ψτ (bit,2(Ui), σ)]K(Uih2)

= −Ω−1z2g(u0, σ)√Nh2fU (u0)

Nh322fU (u0)µ2Ωz2g(u0, σ)θτ (u0)(1 + o(1))

=µ2h

22

√Nh2

2θτ (u0)(1 + o(1)).

and V ar(BN ) = o(h32). This completes the proof of Theorem 2.

Appendix D: Proofs of Lemmas

Proof of Lemma 1: For any Ui, note that

E[ψτ (bjt,1(Ui), σ)|ξj ] = τ − E[g(bjt,1(Ui), σ)|ξj ]

' τ − Eg(bjt,1(Uj), σ) + g(bjt,1(Uj), σ)Z ′jt,2[θτ (Uj)− θτ (Ui)]

+1

2g(bjt,1(Uj), σ)Z ′jt,2[θτ (Uj)− θτ (Ui)]2|ξj

≡ τ −mg(Uj ,Zjt, σ)−mg(Uj ,Zjt, σ)Z ′jt,2[θτ (Uj)− θτ (Ui)]

−1

2mg(Uj ,Zjt, σ)Z ′jt,2[θτ (Uj)− θτ (Ui)]2

= τ −mg(Uj ,Zjt, σ)−mg(Uj ,Zjt, σ)Z ′jt

(0

θτ (Uj)− θτ (Ui)

)−1

2mg(Uj ,Zjt, σ)Z ′jt,2[θτ (Uj)− θτ (Ui)]2,

since bjt,1(Ui)− bjt,1(Uj) = Z ′jt,2[θτ (Uj)− θτ (Ui)]. Hence, we have

E[Z(Ui,Zjt, σ)|ξi]

= EZjtψτ (bjt,1(Uj), σ)Kh(Uj − Ui)|ξi+ Eψτ bjt,1(Uj), σ)[bjt,1(Uj)− bjt,1(Ui)]Kh(Uj − Ui)|ξi

+Eψτ (bjt,1(Uj), σ)[bjt,1(Uj)− bjt,1(Ui)]2Kh(Uj − Ui)|ξi+ o(h21)

= −E[mg(Uj ,Zjt, σ)ZjtZ′jt

(0

θτ (Uj)− θτ (Ui)

)Kh(Uj − Ui)|ξi]

−1

2Emg(Uj ,Zjt, σ)ZjtZ ′jt,2[θτ (Uj)− θτ (Ui)]2Kh(Uj − Ui)|ξi+ o(h21)

≡ −I1 −I22

+ o(h21),

since EZjtψτ (bjt,1(Uj), σ)Kh(Uj−Ui)|ξi = EE[Zjtψτ (bjt,1(Uj), σ)|Uj ]Kh(Uj−Ui)|ξi = 0.

38

Then, we obtain that

I1 = E[mg(Uj ,Zjt, σ)ZjtZ′jt

(0

θτ (Uj)− θτ (Ui)

)Kh(Uj − Ui)|ξi]

= EE[mg(Uj ,Zjt, σ)ZjtZ′jt|Uj ]

(0

θτ (Ui)(Uj − Ui)− θτ (Ui)(Uj − Ui)2

)Kh(Uj − Ui)|ξi+ o(h21)

≡ E[Ωzg(Uj , σ)

(0

θτ (Ui)(Uj − Ui)− θτ (Ui)(Uj − Ui)2

)Kh(Uj − Ui)|ξi] + o(h21)

=

∫[Ωzg(Uj , σ) + uh1Ωzg(Uj , σ)]

(0

uh1θτ (Ui) + (uh1)2

2 θτ (Ui)

)K(u)[fU (Ui) + uh1fU (Ui)]du+ o(h21)

= µ2h21B(Ui, σ)

(0

12 θτ (Ui) + fU (Ui)

fU (Ui)θτ (Ui)

)+ µ2h

21fU (Ui)Ωzg(Ui, σ)

(0

θτ (Ui)

)+ o(h21),

and

I2 = Emg(Uj ,Zjt, σ)ZjtZ ′jt,2[θτ (Uj)− θτ (Ui)]2Kh(Uj − Ui)|ξi

= Emg(Uj ,Zjt, σ)Zjt[Z′jt,2θτ (Uj)]

2(Uj − Ui)2Kh(Uj − Ui)|ξi+ o(h21)

= EEmg(Uj ,Zjt, σ)Zjt[Z′jt,2θτ (Uj)]

2|Uj(Uj − Ui)2Kh(Uj − Ui)|ξi+ o(h21)

≡ E[Θ(Uj , σ)(Uj − Ui)2Kh(Uj − Ui)|ξi] + o(h21)

= µ2h21fU (Ui)Θ(Ui, σ) + o(h21).

It follows that

rN (ξi, σ) = E[e′1B−1(Ui, σ)

1

T

T∑t=1

Z(Ui,Zjt, σ) + e′1B−1(Uj , σ)

1

T

T∑t=1

Z(Uj ,Zit, σ)|ξi]

= E[e′1B−1(Ui, σ)

1

T

T∑t=1

Z(Ui,Zjt, σ)|ξi] + E[e′1B−1(Uj , σ)

1

T

T∑t=1

Z(Uj ,Zit, σ)|ξi]

=1

T

T∑t=1

e′1

∫B−1(u, σ)Zitψτ (bit,1(u), σ)Kh(Ui − u)fU (u)du

=1

T

T∑t=1

e′1B−1(Ui, σ)Zitψτ (bit,1(Ui), σ)fU (Ui)(1 + o(1))

=1

T

T∑t=1

Zitψτ (bit,1(Ui), σ)(1 + o(1))

39

and furthermore, we obtain that

θN (σ) = E[pN (ξi, ξj , σ)] = 2E[e′1B−1(Ui, σ)Z(Ui,Zjt, σ)]

= −µ2h21E[e′1

(0

θτ (Ui) + 2 fU (Ui)fU (Ui)θτ (Ui)

)] + 2E[e′1Ω

−1zg (Ui, σ)Ωzg(Ui, σ)

(0

θτ (Ui)

)]

+E[e′1Ω−1zg (Ui, σ)Θ(Ui, σ)]+ o(h21)

= −µ2h21e′1EΩ−1zg (Ui, σ)[2Ωzg(Ui, σ)

(0

θτ (Ui)

)+ Θ(Ui, σ)]+ o(h21).

Similar to the derivation of V ar(ΨN ), it follows that

V ar[rN (ξi)] = e′1EΩ−1zg (Ui, σ)[1

TΩτ,z(Ui, σ)

+T∑t=2

2(T − t+ 1)

T 2Ωτ,z1t(Ui, σ)]Ω−1zg (Ui, σ)e1 + o(h1)

≡ Σγ(σ) + o(h1).

Therefore, Lemma 1 is established.

Proof of Lemma 2: Similar to Lemma 1 (ii), we show that

BN = µ2h21e′1EΩ−1zg (Ui, σ)[2Ωzg(Ui, σ)

(0

θτ (Ui)

)+ Θ(Ui, σ)]+ o(h21).

The lemma is established due to the fact that V ar(BN ) = O( 1N h

21).

40

Appendix E:

Derivation of (8)

1√2πσ

∞∫−∞

exp[−ρτ (a− v)− v2

2σ2]dv

=1√2πσ

a∫−∞

exp[−τ(a− v)− v2

2σ2]dv +

∞∫a

exp[−(τ − 1)(a− v)− v2

2σ2]dv

=1√2πσe−τa

a∫−∞

exp[− 1

2σ2(v2 − 2τσ2v)]dv + e−(τ−1)a

∞∫a

exp[− 1

2σ2(v2 − 2(τ − 1)σ2v)]dv

= e−τa+τ2σ2

2 Φ(aσ− τσ

)+ e−(τ−1)a+

(τ−1)2σ2

2 [1− Φ(aσ− (τ − 1)σ

)]

= e−τa+τ2σ2

2 Φ(aσ− τσ

)+ e−(τ−1)a+

(τ−1)2σ2

2 Φ(− a

σ+ (τ − 1)σ

)=

e−τa[eτ2σ2

2 Φ(aσ − τσ

)+ ea+

(τ−1)2σ2

2 Φ(− a

σ + (τ − 1)σ)] , when a ≥ 0

e−(τ−1)a[e−a+τ2σ2

2 Φ(aσ − τσ

)+ e

(τ−1)2σ2

2 Φ(− a

σ + (τ − 1)σ)] , when a < 0

= e−ρτ (a)Ia≥0[eτ2σ2

2 Φ(aσ− τσ

)+ ea+

(τ−1)2σ2

2 Φ(− a

σ+ (τ − 1)σ

)]

+e−ρτ (a)Ia<0[e−a+ τ2σ2

2 Φ(aσ− τσ

)+ e

(τ−1)2σ2

2 Φ(− a

σ+ (τ − 1)σ

)]

= e−ρτ (a)[eτ2σ2

2 Φ(aσ− τσ

)+ e

(τ−1)2σ2

2 Φ(− a

σ+ (τ − 1)σ

)ea](Ia≥0 + e−aIa<0)

≡ e−ρτ (a)λτ (a, σ)(Ia≥0 + e−aIa<0)

Derivation of (11)

Differentiation maxσ

N∑i=1

T∑t=1

ln(λτ (ait, σ)) with respect to σ leads to

N∑i=1

T∑t=1

∂λτ (ait,σ)∂σ

λτ (ait, σ)|σ=σ = 0.

By definition of λτ (a, σ) = eτ2σ2

2 Φ(aσ − τσ

)+ e

(τ−1)2σ2

2 Φ(− a

σ + (τ − 1)σ)ea,

∂λτ (a, σ)

∂σ= τ2σe

τ2σ2

2 Φ(aσ− τσ

)+ (τ − 1)2σea+

(τ−1)2σ2

2 Φ(− a

σ+ (τ − 1)σ

)+(− a

σ2− τ)e

τ2σ2

2 φ(aσ− τσ

)+ (

a

σ2+ τ − 1)ea+

(τ−1)2σ2

2 φ(− a

σ+ (τ − 1)σ

)= τ2σλτ (a, σ) + (1− 2τ)σea+

(τ−1)2σ2

2 Φ(− a

σ+ (τ − 1)σ

)− e

τ2σ2

2 φ(aσ− τσ

).

41

Thus,

NTτ2σ +N∑i=1

T∑t=1

−φ(aitσ − τ σ

)+ (1− 2τ)σeait+

(1−2τ)σ2

2 Φ(− ait

σ + (τ − 1)σ)

Φ(aitσ − τ σ

)+ eait+

(1−2τ)σ2

2 Φ(− ait

σ + (τ − 1)σ) = 0.

Derivation of (13)

Assume Yit is generated based on Skorohod representation,

Yit = Z ′it,1γ(Vit) +Z ′it,2θ(Vit, Ui) + vi

where Vit is distributed uniformly from 0 to 1 conditional on all the observarions. Then,

Yit−Z ′it,1γ0.5−Z ′it,2θ0.5(Ui) = Z ′it,1[γ(Vit)−γ0.5]+Z ′it,2[θ(Vit, Ui)−θ0.5(Ui)]+vi. Therefore,

by delta method, we have

V ar

(1

NT

N∑i=1

T∑i=1

(Yit −Z ′it,1γ0.5 −Z ′it,2θ0.5(Ui))

)

= V ar

(1

NT

N∑i=1

T∑i=1

Z ′it,1[γ(Vit)− γ0.5] +Z ′it,2[θ(Vit, Ui)− θ0.5(Ui)]

)+ V ar

(1

NT

N∑i=1

T∑i=1

vi

)

≈ 1

N2T 2

N∑i=1

T∑i=1

Z ′it,1∂γτ∂τ|τ=0.5 +Z ′it,2

∂θτ (Ui)

∂τ|τ=0.52V ar (Vit) +

1

NV ar (vi)

=1

12N2T 2

N∑i=1

T∑i=1

Z ′it,1∂γτ∂τ|τ=0.5 +Z ′it,2

∂θτ (Ui)

∂τ|τ=0.52 +

1

NV ar (vi) .

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