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Quantile Regression with Clustered Data

Paulo M.D.C. Parente and João M.C. Santos Silva

Abstract

We study the properties of the quantile regression estimator when data are sampled from independent and identically distributed clusters, and show that the estimator is consistent and asymptotically normal even when there is intra-cluster correlation. A consistent estimator of the covariance matrix of the asymptotic distribution is provided, and we propose a specification test capable of detecting the presence of intra-cluster correlation. A small simulation study illustrates the finite sample performance of the test and of the covariance matrix estimator.

JEL Classifications:: C12; C21; C23

Corresponding author: João M.C. Santos Silva, Department of Economics, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK; and CEMAPRE, Rua do Quelhas 6, 1200-781 Lisboa, Portugal, E-mail:

Acknowledgments

We thank the Editor Jason Abrevaya and an anonymous referee for many useful comments and suggestions; the usual disclaimer applies. Santos Silva also gratefully acknowledges partial financial support from Fundação para a Ciência e Tecnologia (Programme PEst-OE/EGE/UI0491/2013).

Appendix

Throughout the Appendix cr, CS, M, and T denote the cr, Cauchy-Schwarz, Markov, and triangle inequalities, respectively. LLN denotes the Khintchine’s Weak Law of Large Numbers, UWL denotes a uniform weak law of large numbers such as Lemma 2.4 of Newey and McFadden (1994), and CLT is the Lindeberg-Lévy central limit theorem.

Proof of Theorem 1: We use Theorem 2.7 of Newey and McFadden (1994). Note that

β^=argminβkSG(β)SG(β0)=argminβk[1Gg=1Gi=1nρθ(ygixgiβ)1Gg=1Gi=1nρθ(ugi)].

We have to show that SG(β)–SG(β0) converges uniformly to a function. In this case pointwise convergence suffices as pointwise convergence of convex functions implies uniform convergence on compact subsets. Note that

SG(β)SG(β0)=1Gg=1Gi=1n[ρθ(ygixgiβ)ρθ(ugi)]=1Gg=1Gi=1n[ρθ(ugixgiδ)ρθ(ugi)],

where δ=ββ0. Note that Knight’s identity (Koenker 2005, 121) tells us that

ρθ(uv)ρθ(v)=vψθ(u)+0v{I[us]I[u0]}ds,

where ψθ(u)=θ–I(u<0). Thus

SG(β)SG(β0)=1Gg=1Gi=1nxgiδψθ(ugi)+1Gg=1Gi=1n0xgiδ{I[ugis]I[ugi0]}ds.

Now by a LLN 1Gg=1Gi=1nxgiδψθ(ugi)=op(1) and the second term of the rhs converges to

S¯(δ)=i=1nE[0xgiδ{F[s|xgi]θ}ds].

Note that S¯(δ)=0 if and only if δ=0 and S¯(δ)>0 if δ≠0. To see this note that if xgiδ>0 for some i F[s|xgi]–θ>0, thus 0xgiδ{F[s|xgi]θ}ds>0. If xgiδ<0 for some i F[s|xgi]–θ<0, thus 0xgiδ{F[s|xgi]θ}ds>0. Since δ=ββ0=0 is a unique local minimizer and the limiting function is convex, δ=ββ0=0 is also a global minimizer and the function is convex and consequently β^=β0+op(1).    ■

Proof of Theorem 2: We adapt the proof of Koenker (2005, 121). Consider the objective function

ZG(δ)=g=1Gi=1n[ρθ(ugixgiδ/G)ρθ(ugi)].

This function is convex and minimized at δ^G=G(β^β0). Using Knight’s identity we have

ZG(δ)=Z1G(δ)+Z2G(δ)Z1G(δ)=1Gg=1Gi=1nxgiδψθ(ugi)Z2G(δ)=g=1Gi=1n0G1/2xgiδ{I[ugis]I[ugi0]}ds.

Now Z1G(δ)=–δW, where W=G1/2g=1Gi=1nxgiψθ(ugi). Also, by a CLT, WDN(0,C) where

C=Var(i=1nxgiψθ(ugi))=E[i=1nxgiψθ(ugi)(j=1nxgjψθ(ugj))]=E[i=1nj=1nxgixgjψθ(ugi)ψθ(ugj)].

Now write

Z2Ggi(δ)=0G1/2xgiδ{I[ugis]I[ugi0]}ds,

and Z2G(δ)=g=1Gi=1nZ2Ggi(δ). Note that

Z2G(δ)=g=1Gi=1nE[Z2Ggi(δ)|xgi]+RG(δ),

where

RG(δ)=g=1Gi=1n{Z2Ggi(δ)E[Z2Ggi(δ)|xgi]}.

Note also that

g=1Gi=1nE[Z2Ggi(δ)|xgi]=g=1Gi=1n0G1/2xgiδ{F[s|xgi]θ}ds=1Gg=1Gi=1n0xgiδ{F[tG1/2|xgi]θ}dt=1Gg=1Gi=1n0xgiδ{F[tG1/2|xgi]θt/G}tdt=1Gg=1Gi=1n0xgiδ{f[0|xgi]}tdt+op(1)=12Gg=1Gi=1nδf[0|xgi]xgixgiδ+op(1).

Now by CS

(8)Z2Ggi(δ)=0G1/2xgiδ{I[ugis]I[ugi0]}ds|xgiδ|G1/2||δ||||xgi||G1/2. (8)

Note that E[RG(δ)]=0 and that by cr, (8), and CS we have

Var(RG(δ))g=1GE[(i=1nZ2Ggi(δ))2]ng=1Gi=1nE[Z2Ggi(δ)2]n||δ||G1/2g=1Gi=1nE[Z2Ggi(δ)||xgi||]=n||δ||G1/2g=1Gi=1nE[E[Z2Ggi(δ)|xgi]||xgi||]=n||δ||G3/2g=1Gi=1nE[||xgi||0xgiδ{F[tG1/2|xgi]θt/G}tdt]n||δ||2G3/2g=1Gi=1nE[||xgi||δf[0|xgi]xgixgiδ]+o(1)n||δ||32G1/2f1i=1nE[||xgi||3]+o(1)=o(1).

Thus RG(δ)=op(1). Hence by a LLN

Z2G(δ)=12Gg=1Gi=1nδf[0|xgi]xgixgiδ+op(1)=δBδ/2+op(1).

Therefore

ZG(δ)=δW+δBδ/2+op(1).

The convexity of –δW+δ/2 assures that the minimizer is unique and therefore

G(β^β0)=argminZG(δ)Dδ^0=argminδW+δBδ/2.

Now note that δ^0=B1W (see Koenker 2005, 122, and the references therein).    ■

Proof of Theorem 3: The proof is similar to that of Lemma 5 of Kim and White (2003). Let BG=(2cGG)1g=1Gi=1nI(|ugi|cG)xgixgi. Using the mean value theorem we have E[BG]=E[i=1nf(c˜G|xgi)xgixgi], where |c˜G|cG and therefore c˜G=o(1). Hence, by the Lebesgue dominated convergence theorem E[BG]=B. It follows from the law of large numbers for double arrays (Davidson 1994, Corollary 19.9, 301, and Theorem 12.10, 190) that BGpB. We now show (i) |B˜GBG|p0 where B˜G=(2cGG)1g=1Gi=1nI(|u^gi|c^G)xgixgi, and (ii) |B^B˜G|p0. The conclusion follows from T.

To prove (i) consider the (h, j)th element of |B˜GBG|, which is given by

|(2cGG)1g=1Gi=1n[I(|u^gi|c^G)I(|ugi|cG)]xgihxgij|.

Now using the facts that u^gi=ugi(β^β0)xgi, I(|a|≤b)=I(ab)–I(a<–b), |I(x≤0)–I(y≤0)|≤I(|x|≤|xy|), |I(x<0)–I(y<0)|≤I(|x|≤|xy|), T, and CS we have

|(2cGG)1g=1Gi=1n[I(|u^gi|c^G)I(|ugi|cG)]xgihxgij|U1G+U2GU1G=(2cGG)1g=1Gi=1n[I(|ugicG|dG)]|xgih||xgij|U2G=(2cGG)1g=1Gi=1n[I(|ugi+cG|dG)]|xgih||xgij|,

where dG=|cGc^G|+β^β0||xgi||. We prove that U1Gp0, the proof U2Gp0 is similar.

Let 𝒟1G={U1G}, D2G={cG1β^β0Δ}, and D3G={cG1|cGc^G|Δ} for a constant Δ>0. Thus

Pr(U1G>η)=Pr(D1G)Pr(D1GD2GD3G)+Pr(D2Gc)+Pr(D3Gc).

Now as G(β^β0)=Op(1) and cG1=o(G) it follows that limGPr(D2Gc)=0. Also as c^G/cGp1, we have limGPr(D3Gc)=0. Additionally if cG1β^β0Δ and cG1|cGc^G|Δ we have |dG|≤cGΔ+cGΔ||xgi||. Hence by M

Pr(D1GD2GD3G)(2ηcGG)1g=1Gi=1nE[cGΔcGΔ||xgi||cGΔ+cGΔ||xgi||f(s|xgi)ds|xgih||xgij|](2ηcGG)1g=1Gi=1nE[cGΔcGΔ||xgi||cGΔ+cGΔ||xgi||f1ds|xgih||xgij|]=(η)1Δi=1nE[(||xgi||+1)|xgih||xgij|]<

under Assumptions 3. Now take Δ arbitrarily small and consequently U1Gp0.

To prove (ii), note that B^B˜G=(cGc^G1)B˜G. Note also that by (i) B˜G=Op(1) and since (cGc^G1)=op(1) by assumption, the result follows.    ■

Proof of Theorem 4: For simplicity of notation we write gi:=g=1Gi=1n and j:=j=1n. Note that

T=1Ggi(jzgizgjψθ(u^gi)ψθ(u^gj)ψθ(u^gi)2zgi2)=1Ggij,ijzgizgjψθ(u^gi)ψθ(u^gj)=1Ggi(j,ijzgizgjψθ(ugi)ψθ(ugj))+G,

where

G=1Ggi(j,ijzgizgj[ψθ(u^gi)ψθ(u^gj)ψθ(ugi)ψθ(ugj)]).

Now by Lemma 1

G=1G1/2gij,ijzgizgjhgij(β^)+op(1),

where

hgij(β^)=θ2θF(xgj(β^β0)|xgj)θF(xgi(β^β0)|xgi)+F(xgi(β^β0)|xgi)×F(xgj(β^β0)|xgj),ij.

By a Taylor expansion around β0 we have

G=1Ggij,ijzgizgjHgij(β˜)G(β^β0)+op(1),

where β˜ is on the line segment joining β^ and β0 and

Hgij(β˜)=θf(xgj(β˜β0)|xgj)xgjθf(xgi(β˜β0)|xgi)xgi+f(xgi(β˜β0)|xgi)×F(xgj(β˜β0)|xgj)xgi+f(xgj(β˜β0)|xgj)F(xgi(β˜β0)|xgi)xgj,

where ij. Now notice that

1Ggij,ijzgizgjHgij(β˜)=op(1)

by a UWL. Since G(β^β0)=Op(1) we have RG=op(1). Thus

T=1Ggi(j,ijzgizgjψθ(ugi)ψθ(ugj))+op(1)

and consequently TDN(0,D) as D>0 and

D=E[(i=1n[j=1,ijnzgizgjψθ(ugi)ψθ(ugj)])2]n2i=1nj=1,ijnE[zgi4]1/2E[zgj4]1/2(θ+1)2<

by two applications of cr and one of CS.    ■

Let

mG(β)=1Ggi(j,ijzgizgj[θI(ugixig(ββ0))][θI(ugjxgi(ββ0))]).

Lemma 1Suppose that Assumption 4 holds. Then, under H0, for any δG=o(1) we have

sup||ββ0||δG|G(mG(β)mG(β0))1G1/2gij,ijzgizgjhgij(β)|=op(1),

where

hgij(β)=θ2θF(xgj(ββ0)|xgj)θF(xgi(ββ0)|xgi)+F(xgi(ββ0)|xgi)×F(xgj(ββ0)|xgj).

Proof: Note that

mG(β)=1Ggij,ijzgizgjhgij(β),

where

hgij(β)=[θ2θI(ugjxgj(ββ0))θI(ugixgi(ββ0))+I(ugixig(ββ0))I(ugjxgj(ββ0))],ij.

Now taking the expected value of hgij(β) conditional on xg we have

E[hgij(β)|xg]=θ2θFj(xgj(ββ0)|xg)+θFi(xgi(ββ0)|xg)+Fi,j(xig(ββ0),xgj(ββ0)|xg)=θ2θF(xgj(ββ0)|xgj)+θF(xgi(ββ0)|xgi)+F(xgi(ββ0)|xgi)×F(xgj(ββ0)|xgj),

where the last line follows from H0 and i≠j.

Note now that

G(mG(β)mG(β0))1G1/2gij,ijzgizgjhgij(β)=1G1/2gij,ijzgizgj[hgij(β)hgij(β0)hgij(β)].

Since the indicator functions I(ugixig(ββ0)) and I(ugjxgi(ββ0)) and the conditional distribution functions F(xgi(ββ0)|xgi) and F(xgj(ββ0)|xgj) are functions of bounded variation (and hence type I class of functions in the sense of Andrews 1994) and as Assumptions 1 (a) and 4 (a) hold, it follows that

1G1/2gij,ijzgizgj[hgij(β)hgij(β0)hgij(β)]

is stochastic equicontinuous by Theorems 1, 2 and 3 of Andrews (1994).    ■

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The online version of this article (DOI: 10.1515/jem-2014-0011) offers supplementary material, available to authorized users.


Published Online: 2015-2-27
Published in Print: 2016-1-1

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