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Quantile Regression with Clustered Data

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


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:


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).


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


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


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


where ψθ(u)=θ–I(u<0). Thus


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


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


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


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


Now write


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




Note also that


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


Thus RG(δ)=op(1). Hence by a LLN




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


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


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


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


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


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




Now by Lemma 1




By a Taylor expansion around β0 we have


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


where ij. Now notice that


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


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


by two applications of cr and one of CS.    ■



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




Proof: Note that




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


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

Note now that


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


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


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Supplemental Material

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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