面板数据分位数回归的一个简单方法

Econometrics Journal (2011),volume 14,pp.368–386.

doi:10.1111/j.1368-423X.2011.00349.x

A simple approach to quantile regression for panel data

I VAN A.C ANAY ?

?

Department of Economics,Northwestern University,2001Sheridan Rd,Evanston,

IL 60208,USA.

E-mail:iacanay@https://www.360docs.net/doc/7d9869692.html,

First version received:May 2010;?nal version accepted:April 2011

Summary This paper provides a set of suf?cient conditions that point identify a quantile regression model with ?xed effects.It also proposes a simple transformation of the data that gets rid of the ?xed effects under the assumption that these effects are location shifters.The new estimator is consistent and asymptotically normal as both n and T grow.

Keywords:Deconvolution ,Panel data models ,Quantile regression ,Two-step estimator .

1.INTRODUCTION

Panel data models and quantile regression models are both widely used in applied econometrics and popular topics of research in theoretical papers.Quantile regression models allow the researcher to account for unobserved heterogeneity and heterogeneous covariates effects,while the availability of panel data potentially allows the researcher to include ?xed effects to control for some unobserved covariates.There has been little but growing work at the intersection of these two methodologies (e.g.Koenker,2004,Geraci and Bottai,2007,Abrevaya and Dahl,2008,Galvao,2008,Rosen,2009,and Lamarche,2010).This initial lack of attention is possibly due to a fundamental issue associated with conditional quantiles.This is,as it is the case with non-linear panel data models,standard demeaning (or differencing)techniques do not result in feasible approaches.These techniques rely on the fact that expectations are linear operators,which is not the case for conditional quantiles.This paper provides suf?cient conditions under which the parameter of interest is identi?ed for ?xed T and shows that there is a simple transformation of the data that eliminates the ?xed effects as T →∞,when the ?xed effects are viewed as location shift variables (i.e.variables that affect all quantiles in the same way).The resulting two-step estimator is consistent and asymptotically normal when both n and T go to in?nity.Also,the new estimator is extremely simple to compute and can be implemented in standard econometrics packages.The paper is organized as follows.Section 2presents the model.Section 3provides an identi?cation result based on deconvolution arguments.Section 4introduces a two-step estimator for panel data quantile regression models.Asymptotic properties of the new estimator are presented in the same section.Section 5includes a small Monte Carlo experiment to study the ?nite sample properties of the two-step estimator.Finally,Section 6concludes.Appendix A provides proofs of results.An estimator of the covariance kernel and the bootstrap method are given in Appendix B.

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.Published by Blackwell Publishing Ltd,9600

Garsington Road,Oxford OX42DQ,UK and 350Main Street,Malden,MA,02148,USA.

Journal

The

Econometrics

A simple approach to quantile regression for panel data369

2.THE MODEL

Consider the following model

Y it=X itθ(U it)+αi,t=1,...,T,i=1,...,n,(2.1) where(Y it,X it)∈R×R k are observable variables and(U it,αi)∈R×R are unobservable.

Throughout the paper the vector X it is assumed to include a constant term,i.e.X

it =(1,X s it)

with X s

it ∈R k?1.The functionτ→X θ(τ)is assumed to be strictly increasing inτ∈(0,1)and

the parameter of interest is assumed to beθ(τ).Ifαi were observable it would follow that

P

Y it≤X itθ(τ)+αi|X i,αi

=τ,(2.2)

under the assumption that U it～U[0,1]conditional on X i=(X i1,...,X iT) andαi.This type of representation has been extensively used in the literature(e.g.Chernozhukov and Hansen, 2006,2008).The difference with the model in equation(2.1)and the standard quantile regression model introduced by Koenker and Bassett(1978)lies in the presence of the unobservedαi.This random variable could be arbitrarily related to the rest of the random variables in equation(2.1) (i.e.αi=αi(U it,X i,ηi)for some i.i.d.sequenceηi)rendering condition(2.2)as not particularly useful in terms of identi?cation.The question is under what additional conditions on(U it,αi)the parameterθ(τ)can be identi?ed and consistently estimated from the data.

Rosen(2009)recently showed that conditional on covariates quantile restriction alone does not identifyθ(τ).That is,let Q Z(τ|A)denote theτ-quantile of a random variable Z

conditional on another random variable A,let e it(τ)≡X

it [θ(U it)?θ(τ)],and write the model

in equation(2.1)as

Y it=X itθ(τ)+αi+e it(τ),Q e it(τ)(τ|X i)=0.(2.3)

Then,the conditional quantile restriction Q e

it (τ)

(τ|X i)=0does not have suf?cient identi?cation

power.1Rosen(2009)then provides different assumptions,i.e.support conditions and some form of conditional independence of e it(τ)across time,that(point and partially)identifyθ(τ).

Abrevaya and Dahl(2008)use the correlated random-effects model of Chamberlain(1982, 1984)as a way to get an estimator ofθ(τ).This model views the unobservableαi as a linear projection onto the observables plus a disturbance,i.e.

αi(τ,X i,ηi)=X i T(τ)+ηi.

The authors view the model as an approximation to the true conditional quantile and proceed to get estimates ofθ(τ)and T(τ)by running a quantile regression of Y it on X it and X i.In cases where there is no disturbanceηi,such a regression identi?esθ(τ).However,it is immediate to see that a quantile restriction alone does not identifyθ(τ)wheneverηi is present non-trivially since

the conditional behaviour of X

it θ(U it)+X

i

T(τ)+ηi depends on the joint distribution of the

unobservables U it andηi.This is problematic since not even a correctly speci?ed function for αi(τ,X i,ηi)helps in identifyingθ(τ),meaning that the correlated random-effects model might work poorly in many contexts.The simulations of Section5illustrate this point.

1Note that the distribution of e it(τ)need not be identical across t even when U it is i.i.d.,but that e it(τ)has the same τ-quantile for all t.

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

370I.A.Canay

Koenker(2004)takes a different approach and treats{αi}n i=1as parameters to be jointly estimated withθ(τ)for q different quantiles.He proposes the penalized estimator

?θ,{?α

i}n i=1

≡argmin

(θ,{αi}n i=1)

q

k=1

n

i=1

T

t=1

ρτ

k

Y it?X itθ(τk)?αi

+λ

n

i=1

|αi|,(2.4)

whereρτ(u)=u[τ?I(u<0)],I(·)denotes the indicator function,andλ≥0is a penalization parameter that shrinks the?αs towards a common value.Solving equation(2.4) can be computationally demanding when n is large(even forλ=0)and has the additional complication involved in the choice ofλ.2

Finally,there is a related literature on non-separable panel data models.These type of models are?exible enough to provide quantile treatment effects(see,e.g.Chernozhukov et al.,2010, and Graham and Powell,2010).For example,Chernozhukov et al.(2010)show that the quantile treatment effect of interest is partially identi?ed(for?xed T)and provide bounds for those effects in the model

Y it=g0(X it,αi,U it),U it|X i,αi=d U it |X i,αi,(2.5) where X it is assumed discrete.They also derive rates of shrinkage of the identi?ed set to a point as T goes to in?nity.The model in Chernozhukov et al.(2010)is more general than the one in equation(2.1)as it is non-separable inαi and it involves weaker assumptions on the unobservable U it.However,it leads to less powerful identi?cation results and more complicated estimators.

In this context this paper contributes to the literature in two ways.The next section shows that when the model in equation(2.1)is viewed as a deconvolution model,a result from Neumann(2007)can be applied to show thatθ(τ)is identi?ed when there are at least two time periods available andαi has a pure location shift effect.3This identi?cation result could be potentially used to construct estimators ofθ(τ)based on non-parametric estimators of conditional distribution functions.Such non-parametric estimators would rarely satisfy the end-goal of this paper,that is,to provide an easy-to-use estimator that can be implemented in standard econometric packages,would suffer from the common curse of dimensionality,and would typically involve a delicate choice of tuning parameters for their implementation.4Thus,when moving from identi?cation to estimation,this paper takes a different approach and shows that there exists a simple transformation of the data that eliminates the?xed effectsαi as T→∞. The transformation leads to an extremely simple asymptotically normal estimator forθ(τ)that can be easily computed even for very large values of n.Standard errors for this new estimator can be computed from the asymptotically normal representation.

3.IDENTIFICATION

In this section,I prove that the parameter of interestθ(τ)is identi?ed for T≥2under independence restrictions and existence of moments.The intuition behind the result is quite 2Lamarche(2010)proposes a method to choseλunder the additional assumption thatαi and X i are independent. Galvao(2008)further extends this idea to dynamic panels.

3This means that ifαi captures unobserved covariates Z

i β(τ)that enter the model and are constant over time,such

variables must have coef?cients that are constant acrossτ,that is,β=β(τ)for allτ.

4The advantage of such non-parametric estimators would be their consistency for asymptotics in which n→∞and T remains?xed.

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

A simple approach to quantile regression for panel data 371

simple.Letting S t ≡X t θ(U t )(the dependence on i is omitted for convenience here),it follows from equation (2.1)that Y t =S t +αis a convolution of S t and αconditional on X ,provided αand U t are independent conditional on X .It then follows that the conditional distributions of S t and αcan be identi?ed from the conditional distribution of Y t by using a deconvolution argument similar to that in Neumann (2007).This in turn results in identi?cation of θ(τ)after exploiting the fact that U t is conditionally U [0,1]together with some regularity conditions.

For ease of exposition let T =2and consider the following assumption where the lower case x =(x 1,x 2)denotes a realization of the random variable X =(X 1,X 2).

A SSUMPTION 3.1.Denote by φS t |x and φα|x the conditional on X =x characteristic functions of the distributions P S t |x and P α|x ,respectively.Then (a)conditional on X =x the random

variables S 1,S 2and αare independent for all x ∈ˉX

,where ˉX denotes the support of X =(X 1,X 2);(b)the set ≡{ω:φS t |x (ω2?k )=0for t ∈{1,2}and φα|x (ω2?k )=0,k =0,1,...}

is dense in R for all x ∈ˉX

.Assumption 3.1follows Neumann (2007),but it does not impose P S 1|x =P S 2|x for all x ∈

ˉX

,which does not hold in equation (2.1).Assumption 3.1(a)implies that αi does not change across quantiles as αi is independent of (U 1,U 2).Assumption 3.1(b)excludes characteristic functions that vanish on non-empty open subsets of R but allows the characteristic function to have countably many zeros.This includes cases ruled out in the deconvolution analysis of Li and Vuong (1998)and Evdokimov (2010),among others.More generally,any deconvolution analysis based on Kotlarski’s Lemma (which has been widely applied to identify and estimate a variety of models in economics)will fail to include such cases.5For example,if α～U [?1,1],or if αis discrete uniform,φαhas countably many zeros and so Kotlarski’s Lemma would not apply while Assumption 3.1(b)would be satis?ed.Having this extra dimension of generality is important here as the distribution of αi is left unspeci?ed in this paper.

Assumption 3.1implies that Y t is a convolution of S t and αconditional on X =x and so

φ(Y 1,Y 2)|x (ω1,ω2)=E [exp(iω Y )]=E [exp(iω1S 1+iω2S 2+i (ω1+ω2)α)|X =x ]

(3.1)=φS 1|x (ω1)φS 2|x (ω2)φα|x (ω1+ω2).

(3.2)

L EMMA 3.1.Suppose that P S t |x and P α|x are distributions with characteristic functions φS t |x

and φα|x satisfying Assumption 3.1.Let ?P

α|x and ?P S t |x be further distributions with respective characteristic functions ?φ

α|x and ?φS t |x .If now φS 1|x (ω1)φS 2|x (ω2)φα|x (ω1+ω2)=?φ

S 1|x (ω1)?φS 2|x (ω2)?φα|x (ω1+ω2),?ω1,ω2∈R ,

(3.3)

and all x ∈ˉX

,then there exist constants c x ∈R such that ?P

S t |x =P (S t ?c x )|x and

?P

α|x =P (α+c x )|x .(3.4)

5

An extension of the result by Kotlarski for the case of characteristic functions with zeros was recently proposed by Evdokimov and White (2010)by using conditions on the derivatives of the characteristic functions.

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

372I.A.Canay

That is,?P

S t |x and P S t |x for t ∈{1,2}as well as ?P α|x and P α|x are equal up to a location shift.

R EMARK 3.1.Lemma 3.1extends immediately to models that are slightly more general than the one in equation (2.1).For example,consider the case where Y it =q (U it ,X it )+αi and q (τ,

x )is strictly increasing in τfor all x ∈ˉX

.This model reduces to equation (2.1)if q (U it ,X it )=X

it

θ(U it ).Lemma 3.1holds in this case by letting S t =q (U t ,X t ).Another example would be a random coef?cients model where Y it =X

it

θi (U it )and θi (U it )=θ(U it )+?αi .This model can be written as Y it =X

it θ(U it )+αit and so the additional generality relative to equation (2.1)comes

from αit ≡X it

?αi varying across i and t .Lemma 3.1also applies to this model conditional on the event X 1=X 2.

R EMARK 3.2.It is worth noting that the identi?cation result in Lemma 3.1also applies to the correlated random effects model in Abrevaya and Dahl (2008)where

Y t =θ(U t )X t +α(U t ,X,η),

α(U t ,X,η)= 1(U t )X 1+ 2(U t )X 2+η.

(3.5)

Here Assumption 3.1must hold for φS t |x and φη|x where S t =(θ(U t )+ t (U t ))X t + ?t (U t )X ?t .Lemma 3.1identi?es the distributions up to location.Typically,to be able to identify the entire distribution one would need to add a location assumption.In the case of quantile regression the standard cross-section speci?cation assumes that U ～U [0,1]independent of X .The extension of this assumption to the panel case together with some additional regularity conditions allows identi?cation of the location of S t conditional on X =x as well as the parameter θ(τ).This is the role of Assumption 3.2.

A SSUMPTION 3.2.(a)U it ⊥(X i ,αi )and U it ～U [0,1];(b) UU ≡E [(θ(U it )?θμ)(θ(U it )?θμ) ],where θμ≡E [θ(U it )],is non-singular with ?nite norm;(c)letting X t =(1,X s t )for t =1,

2,there exists no A ?R k ?1such that A has probability 1under the distribution of X s 2

?X s 1and A is a proper linear subspace of R k ?1

;(d)(Y t ,X t )have ?nite ?rst moments for t ={1,2}.Assumption 3.2(a)is standard in quantile regression models except that here U it is also assumed independent of αi .Assumption 3.2(b)implies that θμ∈R k exists and this implies that the location of S t is well de?ned.The restriction on UU is not used in Lemma 3.2but it is important for the derivation of the asymptotic variance of the two-step estimator of the next section.Assumption 3.2(c)is a standard rank-type condition on the subvector of regressors that excludes the constant term.Assumption 3.2(d)is implied by Assumption 4.1in the next section.

It is immediate to see that Assumption 3.2(a)(b)implies E [Y 2?Y 1|X ]=(X s 2

?X s 1) θs μand θ0μ=E [Y 1]?E [X s 1] θs μ,where θ μ=(θ0μ,θs

μ).Assumption 3.2(c)(d)then implies that θμis

identi?ed.6Since E (S t |X )=X

t θμ,the location of S t conditional on X =x is identi?ed.Also,note that Assumption 3.2implies

Q S t |X =x (t |x )=θ[Q U t |X =x (t |x )]x =θ(τ)x.

(3.6)

The following Lemma then follows immediately.

6

This follows from A ={x ∈R k ?1:x (θ?μ?θ??μ)=0}being a proper linear subspace of R k ?1if θ?μ=θ??μ

.In addition,if X t does not include a constant then Assumption 3.2(c)must hold for X 2?X 1.C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

A simple approach to quantile regression for panel data 373

L EMMA 3.2.Under Assumptions 3.1and 3.2the location θμand the function θ(τ)for τ∈(0,1)are identi?ed.

R EMARK 3.3.The extension to the case T >2is straightforward under the same assumptions.The exception is Assumption 3.1(ii),which should be replaced by ≡{ω:φS t |x (ω(T ?1)?k )=

0,for t =1,...,T ,k =0,1,...}is dense in R for all x ∈ˉX

;see Neumann (2007).The results of this section show that the parameter of interest θ(τ)is point identi?ed from the

distribution of the observed data and Assumptions 3.1and 3.2.The next step then is to derive a consistent estimator for this parameter and study its asymptotic properties.Section 4presents a two-step estimator that follows a different intuition relative to the one behind the identi?cation result,but has the virtue of being extremely simple to compute and has an asymptotically normal distribution as both n and T go to in?nity.

4.TWO-STEP ESTIMATOR

The two-step estimator that I introduce in this section exploits two direct consequences of Assumption 3.2and the fact that αi is a location shift (Assumption 3.1(a)).The ?rst implication is in equation (2.3),where only θ(τ)and e it (τ)depend on τ.The second implication arises by

letting u it ≡X

it

[θ(U it )?θμ]and writing a conditional mean equation for Y it as follows 7Y it =X

it θμ+αi +u it ,

E (u it |X i ,αi )=0.

(4.1)

Equation (4.1)implies that αi is also present in the conditional mean of Y it .Therefore,from

equation (4.1)I can compute a √T -consistent estimator of αi given a √nT -consistent estimator of θμ.This includes for example the standard within estimator of θμgiven in equation (A.17)in Appendix A.Then,using equation (2.3)I estimate θ(τ)by a quantile regression of the random

variable ?Y

it ≡Y it ??αi on X it .To be precise,the two steps are described below where I use the notation E T (·)≡T ?1

T t =1(·)and E nT (·)≡(nT )?1 T t =1 n i =1(·).

Step 1.Let ?θμbe a √nT -consistent estimator of θμ.De?ne ?αi ≡E T [Y it ?X it ?θμ].Step 2.Let ?Y it ≡Y it ??α

i and de?ne the two-step estimator ?θ(τ)as:?θ(τ)≡argmin θ∈

E nT ρτ ?Y it ?X it θ .(4.2)

Intuitively,the two-step estimator in equation (4.2)works because ?Y it Y ?it

≡Y it ?αi as T →∞,where denotes weak convergence.This is so because ?Y it ≡Y ?it +?r

i ,where ?r i ≡(αi ??αi )=E T (X it ) (?θμ?θμ)?E T [Y ?it ?X it θμ]→p 0,as T →∞.

(4.3)

Then,the random variable ?Y it converges in probability,as T →∞,to the variable Y ?it

which implies weak convergence,?Y it Y ?it .The next proposition shows that the two-step

estimator de?ned in equation (4.2)is consistent and asymptotically normal under the following assumptions.

7

Note that E (e it (τ)|X i )=X it [θμ?θ(τ)]=0unless θμ=θ(τ).Also,Pr(u it ≤0|X i )=Pr(X it θ(U it )≤

X it θμ|X i ) τ,depending on whether θ(U it ) θμ.C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

374I.A.Canay

A SSUMPTION 4.1.Let ?τ(u )=τ?I (u <0),I (·)denote the indicator function,W =

(Y ?,X ),and g τ(W ,θ,r )≡?τ(Y ??X θ+r )X .(a)(Y ?

it

,X it ,αi )are i.i.d.de?ned on the probability space (W ,F ,P ),take values in a compact set Y ×X ×A ,and E (αi )=0;(b)for all τ∈T ,θ(τ)∈int ,where is compact and convex and T is a closed subinterval of (0,

1);(c)Y ?∈Y has bounded conditional on X density a.s.,sup ?y ∈Y f (?y

)

?θ (θ,τ,r )that is continuous and has full

rank uniformly over ×T ×R ,and J 2(θ,τ,r )=?

?r (θ,τ,r )is uniformly continuous over ×T ×R .

Assumption 4.1imposes similar conditions to those in Chernozhukov and Hansen (2006).

Note that 4.1(a)imposes i.i.d.on the unobserved variable Y ?

it and not on Y it .Condition 4.1(c)is used for asymptotic normality.Finally,in order to derive the expression for the covariance kernel of the limiting Gaussian process provided in equation (4.7),I use the following assumption on

the preliminary estimator ?θ

μ.A SSUMPTION 4.2.The preliminary estimator ?θμadmits the expansion √nT (?θμ?θμ)=√

nT E nT (ψit )+o p (1),

(4.4)

where ψit is an i.i.d.sequence of random variables with E [ψit ]=0and ?nite ψψ=E [ψit ψ

it ].

T HEOREM 4.1.Let n /T s →0for some s ∈(1,∞).Under Assumptions 3.2,4.1and 4.2

sup τ∈T

?θ

(τ)?θ(τ) →p 0,and

√

nT (?θ

(·)?θ(·))=[?J 1(·)]?1√nT E nT {?τ(εit (τ))X it +J 2(·)ξit }+o p (1),(4.5) G (·)

in ∞(T ),

(4.6)

where εit (τ)≡Y ?it ?X it θ(τ),ξit ≡μ X ψit ?u it ,u it ≡Y ?it ?X

it θμ,μX ≡E [X it ],J 1(τ)≡J 1(θ(τ),τ,0),J 2(τ)≡J 2(θ(τ),τ,0),G (·)is a mean-zero Gaussian process with covariance function E G (τ)G (τ ) =J 1(τ)?1 (τ,τ )[J 1(τ )?1] , (τ,τ )is de?ned in equation (4.7)below,and ∞(T )in the set of all uniformly bounded functions on T .The matrix (τ,τ )is given by

(τ,τ )=S (τ,τ )+J 2(τ) ξg (τ )+ gξ(τ)J 2(τ )+J 2(τ) ξξJ 2(τ ) ,

(4.7)

where S (τ,τ )≡(min {τ,τ }?ττ )E (XX ), g ξ(τ)≡E [g τ(W ,θ(τ))ξ],and ξξ≡E [ξ2].

The asymptotic expansion for √

nT (?θ

(·)?θ(·))presented in Theorem 4.1has two terms.The ?rst term is the standard term in quantile regressions while the second term captures the

fact αi is being estimated by ?α

i .Assumption 4.2is not necessary for convergence to a limit Gaussian process but it is used to derive the expression of (τ,τ )in equation (4.7).This is a mild assumption as,for example,the usual within estimator of θμsatis?es Assumption 4.2under Assumptions 3.2and 4.1(see Lemma A.4in Appendix A).

Appendix B contains expressions for estimating each element entering the covariance

function J 1(τ)?1 (τ,τ )[J 1(τ )?1] that can be used to make inference on ?θ

(τ).Alternatively,I conjecture that the standard i.i.d.bootstrap on (Y i ,X i )would work in this context and therefore provide a different approach to inference (Appendix B).The proof of this conjecture is however

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

A simple approach to quantile regression for panel data375 beyond the scope of this paper and the supporting evidence is limited to the simulations of the Section5.

5.SIMULATIONS

To illustrate the performance of the two-step estimator I conduct a small simulation study. Tables1and2summarize the results by reporting percentage bias(%Bias)and mean squared error(MSE)for each estimator considered.The simulated model is

Y it=( it?1)+ it X it+αi,

(5.1)

αi=γ(X i1+···+X iT+ηi)?E(αi),

Table1.Bias and MSE for Model5.1:γ=2and0.25.

n=100n=5000

θ(·)～N(2,1),θ[1](τ)=1.3255θ(·)～N(2,1),θ[1](τ)=1.3255 Estimator QR CRE INFE2-STEP QR CRE INFE2-STEP T=5%Bias1.75690.21570.00680.14941.74440.20740.00040.1496 MSE5.82200.27940.10320.14735.35430.07980.00200.0414 T=10%Bias1.76600.21270.00250.07931.76870.2063?0.00200.0757 MSE5.75570.16390.04940.06055.50210.07650.00100.0111 T=20%Bias1.77500.19030.00320.03771.80840.2070?0.00010.0397 MSE5.76780.10440.02280.02645.75060.07620.00050.0032θ(·)～exp(1)+2,θ[1](τ)=2.2877θ(·)～exp(1)+2,θ[1](τ)=2.2877 Estimator QR CRE INFE2-STEP QR CRE INFE2-STEP T=5%Bias1.02020.1291?0.00030.09761.02850.1367?0.00000.0961 MSE5.81060.20660.01740.10035.54340.10020.00030.0493 T=10%Bias1.03100.1184?0.00020.04871.03930.1368?0.00010.0489 MSE5.83060.11980.00870.02865.65930.09900.00020.0128 T=20%Bias1.05240.09190.00080.02151.05930.1361?0.00010.0209 MSE6.04320.06410.00450.00875.87770.09750.00010.0024

θ(·)～Mixture,θ[1](τ)=1.3097θ(·)～Mixture,θ[1](τ)=1.3097 Estimator QR CRE INFE2-STEP QR CRE INFE2-STEP T=5%Bias2.05650.53690.01900.25612.10610.53700.00120.1897 MSE7.68370.78550.51150.49637.56250.49650.00360.0682 T=10%Bias2.12210.56830.00260.04242.14330.54130.00230.0280 MSE7.97630.66790.14500.20887.82950.50160.00180.0046 T=20%Bias2.17700.6010?0.0039?0.00082.17500.5412?0.0013?0.0032 MSE8.32000.67440.05360.07638.06160.50010.00090.0013 Note:1000MC replications.QR:standard quantile regression estimator;CRE:correlated random effect estimator;INFE: infeasible estimator,2-STEP:two-step estimator from4.2.All regressions include an intercept term.

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

376I.A.Canay

Table2.Bias and MSE for Model5.1:γ=2and0.90.

n=100n=5000

θ(·)～N(2,1),θ[1](τ)=3.2815θ(·)～N(2,1),θ[1](τ)=3.2815 Estimator QR CRE INFE2-STEP QR CRE INFE2-STEP T=5%Bias0.4069?0.1697?0.0065?0.12230.4321?0.1575?0.0002?0.1144 MSE2.40480.64550.15090.31622.02200.27430.00320.1439 T=10%Bias0.4039?0.1547?0.0069?0.06450.4079?0.1594?0.0006?0.0603 MSE2.18940.39950.07630.12281.80010.27680.00150.0406 T=20%Bias0.3777?0.13070.0013?0.02800.3833?0.15790.0001?0.0301 MSE1.84100.24660.03930.04791.58840.27000.00080.0105θ(·)～exp(1)+2,θ[1](τ)=4.3026θ(·)～exp(1)+2,θ[1](τ)=4.3026 Estimator QR CRE INFE2-STEP QR CRE INFE2-STEP T=5%Bias0.3120?0.1517?0.0015?0.08140.3134?0.1366?0.0002?0.0804 MSE2.57810.95980.47300.45831.83330.35660.00950.1264 T=10%Bias0.2911?0.14420.0012?0.04070.2984?0.1361?0.0005?0.0417 MSE2.04930.61000.23370.24441.65850.34790.00490.0363 T=20%Bias0.2748?0.12610.0008?0.02240.2823?0.1371?0.0003?0.0216 MSE1.81360.40230.11660.11811.48340.35010.00230.0107

θ(·)～Mixture,θ[1](τ)=3.335θ(·)～Mixture,θ[1](τ)=3.335 Estimator QR CRE INFE2-STEP QR CRE INFE2-STEP T=5%Bias0.4921?0.0843?0.0040?0.09840.4922?0.09480.0004?0.0958 MSE3.25730.24990.01960.17512.70500.10470.00040.1034 T=10%Bias0.4779?0.0808?0.0005?0.05840.4784?0.09490.0009?0.0568 MSE2.96310.12750.01010.05902.55420.10210.00020.0363 T=20%Bias0.4556?0.0732?0.0002?0.03190.4585?0.09530.0006?0.0325 MSE2.65370.07840.00480.01912.34480.10180.00010.0119 Note:1000MC replications.QR:standard quantile regression estimator;CRE:correlated random effect estimator;INFE: infeasible estimator,2-STEP:two-step estimator from4.2.All regressions include an intercept term.

where X it～Beta(1,1),ηi～N(0,1)and the distribution of it changes with the model speci?cation.In Model1, it～N(2,1);in Model2, it～exp(1)+2;while in Model3, it～B it N(1,.1)+(1?B it)N(3,.1)with B it～Bernoulli(p=0.3).The conditionalτth quantiles are given byθ[0](τ)+θ[1](τ)X+α.In Model1,for example,θ[0](τ)= ?1(τ)+1and θ[1](τ)= ?1(τ)+2for ?1(τ)the inverse of the normal CDF evaluated atτ.For all models I set n={100,5000},T={5,10,20},τ={0.25,0.90},γ=2and consider four estimators. These are a standard quantile regression estimator of Y it on X it(QR),a correlated random effect

estimator(CRE)from Abrevaya and Dahl(2008),an infeasible estimator of Y?

it ≡Y it?αi on

X it(INFE)and the two-step estimator(2-STEP)from Section4.8Note that QR and CRE are inconsistent estimators.Finally,I set the number of Monte Carlo simulations to1000.

8I use the within estimator of equation(A.17)as?rst step estimator.

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

A simple approach to quantile regression for panel data377

Table3.Bias and MSE for Model5.1:γ=2and n=100.

Model1Model2Model3

τ=0.25Estimator%Bias MSE Time%Bias MSE Time%Bias MSE Time

T=52-STEP0.14940.14690.0090.09760.10030.0090.25610.49630.009 KOEN0.10840.14730.133?0.00140.02190.1360.50380.55650.135 T=102-STEP0.07220.05790.0170.04870.02860.0180.04240.20880.019 KOEN0.05040.05450.258?0.00610.00860.2540.34880.28410.284 T=202-STEP0.03770.02640.0160.02150.00870.020?0.00080.07630.015 KOEN0.02230.02680.240?0.00240.00480.2410.19240.11290.233

τ=0.90Estimator%Bias MSE Time%Bias MSE Time%Bias MSE Time

T=52-STEP?0.12230.31620.009?0.08140.45830.013?0.09840.17510.011 KOEN?0.18600.53770.137?0.23821.25740.187?0.06170.06910.158 T=102-STEP?0.06450.12280.019?0.04070.24440.018?0.05840.05900.022 KOEN?0.05550.13710.262?0.08500.37080.239?0.01770.01800.270 T=202-STEP?0.02800.04790.021?0.02240.11810.024?0.03190.01910.026 KOEN?0.02800.05430.241?0.04840.15940.253?0.00810.00680.275 Note:1000MC replications.KOEN:Koenker’s estimator for panel data quantile regression,2-STEP:two-step estimator from4.2.All regressions include an intercept term.Time is reported in seconds.

Tables1and2show equivalent patterns.CRE has a larger bias than2-STEP and its bias does not improve as T grows.92-STEP does show a bias that decreases as T grows.This is consistent with the analysis presented in Sections2and4.Also,it is worth noticing that the bias of 2-STEP is not affected as N increases and T remains?https://www.360docs.net/doc/7d9869692.html,parisons based on MSE arrive to similar conclusions.Finally,additional simulations not reported show that these results also hold for other quantiles,and that the two-step estimator performs similarly whenαi is speci?ed as a non-linear function of X i andηi.

Table3reports bias,MSE and computational time for2-STEP and KOEN,the estimator proposed by Koenker(2004),see equation(2.4).Only the case n=100is shown since KOEN had problems handling the big matrices for the case n=5000.The results are quite mixed.2-STEP performs better than KOEN in about half of the cases.However,the worst performance of 2-STEP(bias:25%,MSE:0.50)is better than that of KOEN(bias:50%,MSE:1.26).Note also that2-STEP is about15times faster than KOEN.

Finally,Table4reports standard errors and95%con?dence intervals computed using the formulas provided in Appendix B.The coverage is very close to the nominal level when T= 20and below the nominal level for T=5.It is worth noting that the coverage is expected to deteriorate in two circumstances.Given a value of n,a smaller T implies a larger?nite sample bias and so a?nite sample distribution centred further away from the truth.In addition,given a value of T,a larger value of n keeps the?nite sample bias unaffected but implies a?nite sample distribution that is more concentrated about the wrong place.However,even for a case with12% of bias(model1,T=5),the actual coverage levels are about85%,which look very decent for such small values of T and bias above10%.

9The performance of CRE depends on the parameterγ,so asγgrows its performance deteriorates.

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

378I.A.Canay

Table4.Standard Errors and95%Con?dence Intervals for Model5.1:γ=2and

n=100.

τ=0.25τ=0.90

T=5T=10T=20T=5T=10T=20

Estimator 1.5329 1.4194 1.3719Estimator 2.9057 3.0995 3.1822

Asy SE0.29340.21620.1555Asy SE0.35030.26200.1901 Model1Boot SE0.28680.21310.1537Boot SE0.36890.27130.1944 Asy Cov0.85400.92300.9420Asy Cov0.76000.89200.9240

Boot Cov0.83200.90300.9340Boot Cov0.79300.88700.9240

Estimator 2.5028 2.4046 2.3369Estimator 3.9575 4.1257 4.2009

Asy SE0.19610.12310.0773Asy SE0.53360.42900.3254 Model2Boot SE0.19040.11810.0738Boot SE0.56970.44470.3335 Asy Cov0.77600.82400.8910Asy Cov0.86200.90900.9220

Boot Cov0.72600.79800.8910Boot Cov0.87200.90900.9220

Estimator 1.6305 1.3778 1.3110Estimator 3.0058 3.1488 3.2298

Asy SE0.47510.39430.2630Asy SE0.22180.13200.0822 Model3Boot SE0.53640.44940.2918Boot SE0.23330.13680.0842 Asy Cov0.86300.92900.9460Asy Cov0.65200.70500.7400

Boot Cov0.87200.93100.9460Boot Cov0.68000.70500.7400 Note:1000MC replications.Asy SE:asymptotic standard errors.Boot SE:bootstrap standard errors.Asy Cov:Coverage of the asymptotic con?dence interval.Boot Cov:Coverage of the Bootstrap percentile interval.

6.DISCUSSION

This paper provided an identi?cation result for quantile regression in panel data models and introduced a two-step estimator that is attractive for its computational simplicity.There are many issues that remain to be investigated.First,several panels available have a short time span and therefore approximations taking T to in?nity might result in poor approximations for those cases. However,a computationally simple estimator that works for?xed T and large N is extremely challenging since,even under the assumption thatαi is independent of the rest of the variables of the model,we would still have to face similar problems to those discussed in Sections2and3. Second,the assumption thatαi does not depend on the quantiles restricts the type of unobserved heterogeneity that the model can handle.Improvements in any of these directions are important for future research.

ACKNOWLEDGMENTS

This paper was previously circulated under the title‘A Note on Quantile Regression for Panel Data Models.’I am deeply grateful to Jack Porter and Bruce Hansen for thoughtful discussions.I would also like to thank the Editor and two anonymous referees whose comments have led to an improved version of this paper.

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

A simple approach to quantile regression for panel data379

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380I.A.Canay

APPENDIX A:PROOF OF THE LEMMAS AND THEOREMS Throughout the Appendix I use the following notation.For W=(Y?,X),

g→E nT g(W)≡

1

nT

T

t=1

n

i=1

g(W it),g→G nT g(W)≡

1

√

nT

T

t=1

n

i=1

[g(W it)?Eg(W it)].

In addition, denotes weak convergence,CMT refers to the Continuous Mapping Theorem and LLN refers to the Law of Large Numbers.The symbols o,O,o p and O p denote the usual order of magnitudes for non-random and random sequences.

Proof of Lemma3.1:The proof is a simple extension of the result Lemma2.1from Neumann(2007).I

write it here for completeness.SinceφS

t|x andφα|x are characteristic functions there exists anω0>0such

thatφS

t|x (ω)=0for t∈{1,2}andφα|x(2ω)=0if|ω|≤ω0(In this case,ω0might depend on x but we

omit this dependence for simplicity).For|ω|≤2ω0de?ne,

gα|x(ω)=?φα|x(ω)/φα|x(ω),

a continuous complex function which equals1at0.It follows from equation(3.3),forω1,ω2∈[?ω0,ω0], that

φα|x(ω1+ω2)φα|x(ω1)φα|x(ω2)=φα|x(ω1

+ω2)φS

1|x

(ω1)φS

2|x

(ω2)

φα|x(ω1+0)φS1|x(ω1)φS2|x(0)φα|x(0+ω2)φS1|x(0)φS2|x(ω2)

=

?φ

α|x

(ω1+ω2)?φS1|x(ω1)?φS2|x(ω2)

?φ

α|x

(ω1?φS

1|x

(ω1)?φS

2|x

(0)?φα|x(0+ω2)?φS

1|x

(0)?φS

2|x

(ω2)

=

?φ

α|x

(ω1+ω2)

?φ

α|x

(ω1)?φα|x(ω2)

,

(A.1)

which implies

gα|x(ω1+ω2)=gα|x(ω1)gα|x(ω2)?ω1,ω2∈[?ω0,ω0].

The unique solution to this equation satisfying gα|x(0)=1and gα|x(?ω)=gα|x(ω)(a Hermitian function) is gα|x(ω)=e icω,from some real c.Therefore we conclude that,

?φ

α|x

(ω)=e icωφα|x(ω)?ω∈[?2ω0,2ω0].(A.2) Furthermore,equation(3.3)yields that forω2=0,

φS

1|x (ω1)φα|x(ω1)=e icω1φα|x(ω1)?φS

1|x

(ω1),?ω1∈[?2ω0,2ω0],

so that,

?φ

S1|x (ω)=e?icωφS

1|x

(ω)?ω∈[?2ω0,2ω0].(A.3)

Settingω1=0it follows that equation(A.3)holds for S2as well.Now it remains to extend these results to the whole real line.Letω∈ be arbitrary.We obtain,analogously to equation(A.1),that

φα|x(ω) (φα|x(ω/2))2=

?φ

α|x

(ω)

(?φα|x(ω/2))2

and

φα|x(ω)

(φα|x(ω2?k))2k

=

?φ

α|x

(ω)

(?φα|x(ω2?k))2k

,

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

A simple approach to quantile regression for panel data 381

after https://www.360docs.net/doc/7d9869692.html,ing this equation with a k large enough such that |ω2?k |≤2ω0we conclude from equation (A.2)that

?φ

α|x (ω)=φα|x (ω) ?φα|x (ω2?k

)φα|x (ω2?k )

2k =e icωφα|x (ω).Since is dense in R and {ω:?φ

α|x (ω)=e icωφα|x (ω)}is a closed set,we conclude that ?φα|x (ω)=e icωφα|x (ω),?ω∈R ,this is,?P

α|x =P (α+c )|x .This implies,again by equation (3.3)that ?φS t |x (ω)=e

?icωφS t |x (ω),?ω∈R ,which yields ?P S t |x =P (S t ?c )|x for t ∈{1,2}. L EMMA A.1.Under Assumptions 3.2and 4.1,the following statements are true.(a)G nT g ·(W ,θ(·))

G ?1(·)in ∞(T ),where G ?1is a Gaussian process with covariance function E G ?1(τ)G ?1(τ ) =(min {τ,τ

}?

ττ )E (XX ).(b)If sup τ∈T

θ(τ)?θ(τ) =o p (1)and max i ≤n | r i |=o p (1),then sup τ∈T

G nT g τ(W , θ(τ), r i )?G nT g τ(W ,θ(τ),0) =o p (1).

Proof:The proof follows by similar arguments to those in Lemma B.2of Chernozhukov and Hansen

(2006)after noticing that the class of functions,H ={h =(θ,τ,r )→?τ(Y ??X θ+r )X,θ∈ ,τ∈T ,r ∈R }is Donsker (it is formed by taking products and sums of bounded Donsker classes)by Theorem 2.10.6in van der Vaart and Wellner (1996). L EMMA A.2.Under Assumptions 3.2and 4.1,max i ≤n |?r i |≡max i ≤n |αi ??αi |→0provided n

T s

→0

for some s ∈(1,∞).

Proof:Let u it ≡Y ?it

?X it θμand note that by the triangle inequality,|αi ??αi |≤|E T (X it )| |?θ

μ?θμ|+|E T (u it )|.X it and Y it have compact support so that max i ≤n |X it |≤C x <∞and max i ≤n |u it |≤C z <∞.It is immediate

then that

max i ≤n

|E T (X it )| |?θ

μ?θμ|=o p (1).Since E (u it )=0and E (|u it |2s )≤C 2s

z

<∞for all s ∈(1,∞),it follows from the Markov inequality that for any η>0,Pr(|E T (u it )|>η)=O (T ?s )and then

Pr

max i ≤n

|E T (u it )|>η ≤n Pr(|E T (u it )|>η)=O (n/T s )=o (1).

Proof of Theorem 4.1:

Consistency .De?ne the following two criterion functions,

Q nT (θ,τ)=E nT ρτ ?Y it ?X it θ and Q (θ,τ)=E ρτ Y ?it ?X it θ .

The ?rst step shows that Q nT (θ,τ)converge uniformly to Q (θ,τ).To this end,note that since ?Y

it Y ?it ,it follows from van der Vaart (1998,Lemma 2.2)that

E ρτ ?Y it ?X it θ ?E ρτ Y ?it ?X it θ →0,as T →∞,

since ρτ(·)is a bounded Lipschitz function (it is bounded due to 4.1(a)and 4.1(b)).Due to the compactness

of ×T and the continuity of E [ρτ(?Y

it ?X it θ)]and E [ρτ(Y ?it ?X it θ)]implied by 4.1(c),the above convergence is also uniform,

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

382

I.A.Canay

sup

(θ,τ)∈ ×T

|E [ρτ(?Y

?X θ)]?E [ρτ(Y ??X θ)]|→0,as T →∞.(A.4)

Next note that functions in the class F =[(θ,τ)?→ρτ(Y ?X θ)]are bounded,uniformly Lipschitz

over ×T and form a Donsker class.This also means that F is Glivenko-Cantelli so that,

sup

(θ,τ)∈ ×T

sup |E nT [ρτ(?Y

?X θ)]?E [ρτ(?Y ?X θ)]|→p 0,as n,T →∞.(A.5)

It follows from equations (A.4)and (A.5)that Q nT (θ,τ)converges uniformly to Q (θ,τ)as both n

and T go to in?nity.Under Assumption 4.1(c)θ(τ)uniquely solves θ(τ)≡arg inf θ∈ E [ρτ(Y ??X θ)]and Q (θ,τ)is continuous over ×T so that

sup τ∈T

?θ

(τ)?θ(τ) →p 0,after invoking Chernozhukov and Hansen (2006,Lemma B.1).

Asymptotic Normality .From the properties of standard quantile regression it follows that

√nT E nT g ·(W ,?θ

(·),?r i )is o p (1),and then the following expansion is valid,o p (1)=G nT g ·(W ,?θ

(·),?r i )+1√nT T t =1n i =1

Eg ·(W ,?θ(·),?r i )(1)

=G nT g ·(W ,θ(·))+o p (1)+1√nT T

t =1n i =1Eg ·(W ,?θ(·),?r i )in ∞(T ).

Here =(1)follows from Lemma A.1.Now,expand Eg ·(W ,?θ

(·),?r i ).1√

nT T t =1n i =1Eg ·(W ,?θ(·),?r i )=1√nT T t =1n i =1

J 1 θ?(·),·,r ?i (?θ(·)?θ(·))+J 2 θ?(·),·,r ?i ?r i =J 1(·)√

?θ

(·)?θ(·))+J 2(·)1√nT T t =1n i =1

?r i +o p (1),where θ?is on the line connecting ?θ

(τ)and θ(τ)for each τand r ?i is on the line connecting 0and ?r i .The second equality follows from

sup i ≤n,τ∈T

J k θ?(·),τ,r ?i

?J k (θ(τ),τ,0) =o p (1),for k =1,2,

(A.6)which in turn follows from the uniform continuity assumption,and the fact that max i ≤n |?r

i |=o p (1)by Lemma A.2.Solving for √nT (?θ(·)?θ(·)),

√

nT (?θ

(·)?θ(·))=[?J 1(·)]?1

G nT g τ(W ,θ(·))+[?J 1(·)]

?1

J 2(·)1√n n

i =1

√

T ?r

i +o p (1) G (·),

in ∞(T ),

where G (·)is a gaussian process with covariance kernel J 1(τ)?1 (τ,τ )[J 1(τ )?1] ,where (τ,τ )is de?ned in equation (A.10).This follows from the ?rst term converging to a Gaussian process by Lemma A.1,and the second term

1√

n n i =1√?r i =1√n n

i =1

√ E T (X it ) (?θμ?θμ)?E T Y ?it ?X it θμ ,=E nT (X it ) √nT (?θ

μ?θμ)?√nT E nT Y ?it ?X it θμ ,C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

A simple approach to quantile regression for panel data 383

being asymptotically normal due to √nT (?θμ?θμ),and 1√nT

T t =1 n

i =1(Y ?it ?X it θμ)both being asymptotically normal.

Covariance Kernel .We ?rst need to derive the expression for (τ,τ

).Under Assumption 4.2we can

write the expansion √nT (?θ

(·)?θ(·))as √nT (?θ(·)?θ(·))=[?J 1(·)]?1 G nT g τ(W ,θ(·))+J 2(·)1√n n i =1

√T ?r

i +o p (1),=[?J 1(·)]?1√nT {g ·(W it ,θ(·))+J 2(·)ξit }+o p (1),

(A.7)where ξit ≡μ X ψit ?u it ,μX ≡E (X ),and u it ≡Y ?it ?X

it θμ.In addition,

J 1(τ)=E [f ε(τ)(0|X )XX ],

where f ε(τ)(0|X )denotes the conditional on X density of ε(τ)≡Y ??X θ(τ)at 0and

J 2(τ)=E [f ε(τ)(0|X )X ].

Under Assumptions 3.2and 4.1it follows that

√nT E nT g ·(W it ,θ(·))ξit

G ?(·),

in ∞(T ),

(A.8)

where G ?(·)is a zero-mean gaussian process with covariance kernel

(τ,τ )≡ S (τ,τ

) gξ

(τ) ξg (τ ) ξξ

,

(A.9)

and S (τ,τ )≡(min {τ,τ }?ττ )E (XX ), g ξ(τ)≡E [g τ(W ,θ(τ))ξ],and ξξ≡E [ξ2].The above result

implies that

(τ,τ )≡E [(g τ(W ,θ(τ))+J 2(τ)ξit )(g τ (W ,θ(τ ))+J 2(τ )ξit ) ]

=S (τ,τ )+J 2(τ) ξg (τ )+ gξ(τ)J 2(τ )+J 2(τ) ξξJ 2(τ ) .

(A.10)

We can conclude from equation (A.7)that

√

nT (?θ(·)?θ(·)) G (·),in ∞(T ),

(A.11)

where G (·)is a gaussian process with covariance kernel

E G (τ)G (τ ) =J 1(τ)?1 (τ,τ )[J 1(τ )?1] .

(A.12)

L EMMA A.3.If ||v i ,T ?v ||→0a.s.uniformly in i as T →∞,and there exists a function q i ≥0such

that ||v i ,T ||≤q i for all i and T with E [sup i q i ]<∞,then sup i E v i,T ?v →0,as T →∞.Proof:Let h i ,T =||v i ,T ?v ||and note that 0≤sup i h i,T ≤2sup i q i .By Fatou’s Lemma

2E

sup i

q i =E lim inf T →∞

2sup i

q i ?sup i

h i,T

≤lim inf T →∞

E 2sup i

q i ?sup i

h i,T ,(A.13)

≤2E sup i

q i ?lim sup T →∞

E

sup i

h i,T ,

(A.14)

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

384I.A.Canay

meaning that lim sup T→∞E[sup i h i,T]≤0.Then,the result directly follows from

0≤lim sup

T→∞sup

i

E[h i,T]≤lim sup

T→∞

E

sup

i

h i,T

≤0.(A.15)

L EMMA A.4.Assume XX≡E[(X s it?μs X)(X s it?μs X) ]is non-singular with?nite norm,n T a→0for some a∈(0,∞)and let Assumptions3.2and4.1hold.The within estimator ofθμsatis?es Assumption4.2

with the in?uence function

ψit=

ψ0

it

ψs

it

≡

Y it?μY?μs X ?1XX

X s

it

?μs X

u it

?1

XX

X s

it

?μs X

u it

,(A.16)

where X

it =(1,X s it),μs X≡E(X s it),μY≡E(Y it),u it is i.i.d.with E[u it|X i]=0and E[u2it|X i]=

X

it

UU X it,and UU non-singular with?nite norm.

Proof:Use the partitionθ

μ

=(θ0μ,θs μ),whereθ0μ∈R andθsμ∈R k?1.Then

?θs μ≡

E nT

?X s

it

?X s

it

?1

E nT

?X s

it

Y it

,and?θ0

μ

≡E nT(Y it)?E nT

X s

it

?θs

μ

,(A.17)

where?X s

it ≡X s it?ˉX s i andˉX s i≡E T(X s it).By equations(4.1)and(A.17)it follows that

√

?θs

μ

?θsμ

=

E nT

?X s

it

?X s

it

?1√

E nT

?X s

it

u it

,(A.18)

where u it≡X it(θ(U it)?θμ).By Assumption3.2,E[u it]=0,E[u2it|X it]=X it UU X it,and UU is non-singular and has?nite norm.By Assumptions3.2and4.1

1 nT

n

i=1

T

t=1

?X s

it

?X s

it

= XX+o p(1),as n,T→∞,(A.19)

for XX non-singular and therefore

√nT

?θs

μ

?θsμ

= ?1XX1

√

nT

n

i=1

T

t=1

?X s

it

u it+o p(1).(A.20)

Next note that

√nT

?θs

μ

?θsμ

= ?1XX

1

√

nT

n

i=1

T

t=1

X s

it

?μs X

u it?

1

√

nT

n

i=1

T

t=1

ˉX s

i

?μs X

u it

+o p(1),≡1

√

nT

n

i=1

T

t=1

ψs

it

? ?1XX1

√

nT

n

i=1

T

t=1

ˉX s

i

?μs X

u it+o p(1),(A.21)

meaning that the result for the slope coef?cients would follow provided the second term in equation(A.21) is o p(1).To show this,write this term as

1√nT

n

i=1

T

t=1

ˉX s

i

?μs X

u it=

1

√

n

n

i=1

ˉX s

i

?μs X

1

√

T

T

t=1

u it=

1

√

n

n

i=1

?i,T,(A.22)

where?i,T is i.i.d.across i for all T and satis?es E[?i,T]=0and

E ?i,T 2=E

ˉX s

i

?μs X

1

√

T

T

t=1

u it

2

≤E

ˉ

X s

i

?μs X

E

1

√

T

T

t=1

u it

(A.23)

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

A simple approach to quantile regression for panel data385

≤sup

i E

ˉ

X s

i

?μs X

×O(1)→0,as T→∞,(A.24)

since sup i E ˉX s

i ?μs X →0as T→∞by Lemma A.4(note that by Assumption4.1,the function q i is

just the upper bound of the support X)and the fact thatˉX s

i →pμs X uniformly over i as T→∞by similar

arguments to those in Lemma A.2provided n

T a →0for some a∈(0,∞).

Finally,from equation(A.17),θ0

μ=μY?μs Xθsμ,the expansion in equation(A.21),and a few algebraic

manipulations,it follows that

√

nT(?θ0

μ?θ0μ)=

√

nT

E nT(Y it)?μY+μs Xθsμ?E nT

X s

it

?θs

μ

=

√

nT E nT[Y it?μY]?μs X

√

nT

?θs

μ

?θsμ

+o p(1)

=

√

nT E nT

Y it?μY?μs Xψs it

+o p(1).(A.25)

Lettingψ0

it

≡Y it?μY?μs Xψs it=Y it?μY?μs X ?1XX(X s it?μs X)u it,the result follows.

APPENDIX B:ESTIMATOR OF THE COV ARIANCE KERNEL AND THE

BOOTSTRAP

The components of the asymptotic variance in equation(A.10)can be estimated using sample analogs. The expressions below correspond to the case where?θμis the within estimator and soψit is given by equation(A.16).They can be naturally extended to cover any other preliminary estimator satisfying Assumption4.2.

The matrix S(τ,τ )can be estimated by its sample counterpart

?S(τ,τ )≡(min{τ,τ }?ττ )1

nT

n

i=1

T

t=1

X it X

it

.(B.1)

For the matrices J1(τ)and J2(τ),I follow Powell(1991)and propose

?J 1(τ)≡

1

2nT h n

n

i=1

T

t=1

I(|?εit(τ)|≤h n)X it X

it

,

?J 2(τ)≡

1

2nT h n

n

i=1

T

t=1

I(|?εit(τ)|≤h n)X it,

where?εit(τ)≡?Y it?X it?θ(τ),?Y it≡Y it??αi,and h n is an appropriately chosen bandwidth such that h n→

0and nT h2

n

→∞.Following Koenker(2005,pp.81and140),one possible choice is

h n=κ( ?1(τ+b n)? ?1(τ?b n)),b n=(nT)?1/3z2/3α

1.5φ2( ?1(τ))

2 ?1(τ)2+1

1/3

,(B.2)

whereκis a robust estimate of scale,zα= ?1(1?α/2)andαdenotes the desired size of the test.

For the terms involvingξit and gτ(W,θ(τ))de?ne

?ψ

it≡

Y it??μY??μs X? ?1XX?X s it?u it

? ?1

XX

?X s

it

?u it

,(B.3)

where?X s

it ≡X s it?ˉX s i,?u it≡?Y it?X it?θμ,?μY≡E nt(Y it),?μs X≡E nT(X s it)and? XX≡E nT[?X s it?X s it].This

way,we can de?ne?ξit≡?μ X?ψit??u it,where?μX≡E nT(X it).Finally,letting?gτ,it≡?τ(?εit(τ))X it we have C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.

386I.A.Canay the following sample counterparts for the remaining terms

?

gξ≡

1

nT

n

i=1

T

t=1

?gτ,it?ξit,and? ξξ≡

1

nT

n

i=1

T

t=1

?ξ2

it

.(B.4)

The estimator of the covariance matrix would then be?J1(τ)?1? (τ,τ )[?J1(τ )?1] ,where? (τ,τ )is the matrix in equation(A.10)where all matrices have been replaced by their respective sample analogs.

In the simulations of Section5I use the following bootstrap algorithm to compute standard errors and

con?dence intervals for?θ(τ).Let{Y?

i ,X?

i

}n i=1,j=1,...,B,denote the j th i.i.d.sample of size n distributed

according to?P n,the empirical measure of{Y i,X i}n i=1,where Y i=(Y i1,...,Y iT)and X i=(X i1,...,X iT). For each j=1,...,B compute the two step estimator as described in Section4and denote this estimator by

?θ?j (τ).This involves computing preliminary estimators?θ?

μ,j

and?xed effects?α?

i,j

for each bootstrap sample

j=1,...,B.

The bootstrap estimate of the variance covariance matrix for?θ(τ)is given by

1 B

B

j=1

?θ?

j

(τ)?ˉθ?(τ)

?θ?

j

(τ)?ˉθ?(τ)

,(B.5)

whereˉθ?(τ)≡1 B

j=1

?θ?

j

(τ).In the simulations of Section5I also report the coverage of the1?α

percentile interval

C?n (τ)=

q?

n

(α/2,τ),q?

n

(1?α/2,τ)

,(B.6)

where q?

n (α,τ)is theα-quantile of the empirical distribution of{?θ?

j

(τ)}B

j=1

.This con?dence interval is

translation invariant,which is a good property when working with quantile regressions.Symmetric and equally-tailed intervals can be alternatively computed using the same algorithm.

C 2011The Author(s).The Econometrics Journal C 2011Royal Economic Society.