Quantile Regression with Clustered Data

Appendix

Throughout the Appendix c_r, CS, M, and T denote the c_r, 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^= argminβ∈ℝkSG(β)−SG(β0)=argminβ∈ℝk[1G∑g=1G∑i=1nρθ(ygi−x′giβ)−1G∑g=1G∑i=1nρθ(ugi)].

We have to show that S_G(β)–S_G(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)=1G∑g=1G∑i=1n[ρθ(ygi−x′giβ)−ρθ(ugi)]=1G∑g=1G∑i=1n[ρθ(ugi−x′giδ)−ρθ(ugi)],

where δ=β–β₀. Note that Knight’s identity (Koenker 2005, 121) tells us that

ρθ(u−v)−ρθ(v)=−vψθ(u)+∫0v{I[u≤s]−I[u≤0]}ds,

where ψ_θ(u)=θ–I(u<0). Thus

SG(β)−SG(β0)=1G∑g=1G∑i=1n−x′giδψθ(ugi)+1G∑g=1G∑i=1n∫0x′giδ{I[ugi≤s]−I[ugi≤0]}ds.

Now by a LLN 1G∑g=1G∑i=1n−x′gi δψθ(ugi)=op(1) and the second term of the rhs converges to

S¯(δ)=∑i=1nE[∫0x′giδ{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 x′_giδ>0 for some i F[s|x_gi]–θ>0, thus ∫0x′giδ{F[s|xgi]−θ}ds>0. If x′_giδ<0 for some i F[s|x_gi]–θ<0, thus ∫0x′giδ{F[s|xgi]−θ}ds>0. Since δ=β–β₀=0 is a unique local minimizer and the limiting function is convex, δ=β–β₀=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=1G∑i=1n[ρθ(ugi−x′giδ/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)=1G∑g=1G∑i=1n−x′giδψθ(ugi)Z2G(δ)=∑g=1G∑i=1n∫0G−1/2x′giδ{I[ugi≤s]−I[ugi≤0]}ds.

Now Z_1G(δ)=–δ′W, where W=G−1/2∑g=1G∑i=1nxgiψθ(ugi). Also, by a CLT, W→DN(0,C) where

C=Var(∑i=1nxgiψθ(ugi))=E[∑i=1nxgiψθ(ugi)(∑j=1nxgjψθ(ugj))′]=E[∑i=1n∑j=1nxgix′gjψθ(ugi)ψθ(ugj)].

Now write

Z2Ggi(δ)=∫0G−1/2x′giδ{I[ugi≤s]−I[ugi≤0]}ds,

and Z2G(δ)=∑g=1G∑i=1nZ2Ggi(δ). Note that

Z2G(δ)=∑g=1G∑i=1nE[Z2Ggi(δ)|xgi]+RG(δ),

where

RG(δ)=∑g=1G∑i=1n{Z2Ggi(δ)−E[Z2Ggi(δ)|xgi]}.

Note also that

∑g=1G∑i=1nE[Z2Ggi(δ)|xgi]=∑g=1G∑i=1n∫0G−1/2x′giδ{F[s|xgi]−θ}ds=1G∑g=1G∑i=1n∫0x′giδ{F[tG1/2|xgi]−θ}dt=1G∑g=1G∑i=1n∫0x′giδ{F[tG1/2|xgi]−θt/G}tdt=1G∑g=1G∑i=1n∫0x′giδ{f[0|xgi]} tdt+op(1)=12G∑g=1G∑i=1nδ′f[0|xgi]xgix′giδ+op(1).

Now by CS

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

Var(RG(δ))≤∑g=1GE[(∑i=1nZ2Ggi(δ))2]≤n∑g=1G∑i=1nE[Z2Ggi(δ)2]≤n||δ||G1/2∑g=1G∑i=1nE[Z2Ggi(δ)||xgi||]=n||δ||G1/2∑g=1G∑i=1nE[E[Z2Ggi(δ)|xgi]||xgi||]=n||δ||G3/2∑g=1G∑i=1nE[||xgi||∫0x′giδ{F[tG1/2|xgi]−θt/G}tdt]≤n||δ||2G3/2∑g=1G∑i=1nE[||xgi||δ′f[0|xgi]xgix′giδ]+o(1)≤n||δ||32G1/2f1∑i=1nE[||xgi||3]+o(1)=o(1).

Thus R_G(δ)=o_p(1). Hence by a LLN

Z2G(δ)=12G∑g=1G∑i=1nδ′f[0|xgi]xgix′giδ+op(1)=δ′Bδ/2+op(1).

Therefore

ZG(δ)=−δ′W+δ′Bδ/2+op(1).

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

G(β^−β0)=argminZG(δ)→Dδ^0=argmin−δ′W+δ′Bδ/2.

Now note that δ^0=B−1W (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)−1∑g=1G∑i=1nI(|ugi|≤cG)xgix′gi. Using the mean value theorem we have E[BG]=E[∑i=1nf(c˜G|xgi)xgix′gi], where |c˜G|≤cG and therefore c˜G=o(1). Hence, by the Lebesgue dominated convergence theorem E[B_G]=B. It follows from the law of large numbers for double arrays (Davidson 1994, Corollary 19.9, 301, and Theorem 12.10, 190) that BG→pB. We now show (i) |B˜G−BG|→p0 where B˜G=(2cGG)−1∑g=1G∑i=1nI(|u^gi|≤c^G)xgix′gi, and (ii) |B^−B˜G|→p0. The conclusion follows from T.

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

|(2cGG)−1∑g=1G∑i=1n[I(|u^gi| ≤c^G)−I(|ugi|≤cG)]xgihx′gij|.

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

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

where dG=|cG−c^G|+‖β^−β0‖ ||xgi||. We prove that U1G→p0, the proof U2G→p0 is similar.

Let 𝒟_1G={U_1G>η}, D2G={cG−1‖β^−β0‖≤Δ}, and D3G={cG−1|cG−c^G|≤Δ} for a constant Δ>0. Thus

Pr(U1G>η)=Pr(D1G)≤Pr(D1G∩D2G∩D3G)+Pr(D2Gc)+Pr(D3Gc).

Now as G(β^−β0)=Op(1) and cG−1=o(G) it follows that limG→∞Pr(D2Gc)=0. Also as c^G/cG→p1, we have limG→∞Pr(D3Gc)=0. Additionally if cG−1‖β^−β0‖≤Δ and cG−1|cG−c^G|≤Δ we have |d_G|≤c_GΔ+c_GΔ||x_gi||. Hence by M

Pr(D1G∩D2G∩D3G)≤(2ηcGG)−1∑g=1G∑i=1nE[∫−cGΔ−cGΔ||xgi||cGΔ+cGΔ||xgi||f(s|xgi)ds|xgih| |xgij|]≤(2ηcGG)−1∑g=1G∑i=1nE[∫−cGΔ−cGΔ||xgi||cGΔ+cGΔ||xgi||f1ds|xgih| |xgij|]=(η)−1Δ∑i=1nE[(||xgi||+1)|xgih| |xgij|]<∞

under Assumptions 3. Now take Δ arbitrarily small and consequently U1G→p0.

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

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

T=1G∑gi(∑jzgizgjψθ(u^gi)ψθ(u^gj)−ψθ(u^gi)2zgi2)=1G∑gi∑j,i≠jzgizgjψθ(u^gi)ψθ(u^gj)=1G∑gi(∑j,i≠jzgizgjψθ(ugi)ψθ(ugj))+ℛG,

where

ℛG=1G∑gi(∑j,i≠jzgizgj[ψθ(u^gi)ψθ(u^gj)−ψθ(ugi)ψθ(ugj)]).

Now by Lemma 1

ℛG=1G1/2∑gi∑j,i≠jzgizgjhgij∗(β^)+op(1),

where

hgij∗(β^)=θ2−θ F(x′gj(β^−β0)|xgj)−θ F(x′gi(β^−β0)|xgi)+F(x′gi(β^−β0)|xgi)×F(x′gj(β^−β0)|xgj), i≠j.

By a Taylor expansion around β₀ we have

ℛG=1G∑gi∑j,i≠jzgizgjHgij∗(β˜)G(β^−β0)+op(1),

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

Hgij∗(β˜)=−θf(x′gj(β˜−β0)|xgj)x′gj−θf(x′gi(β˜−β0)|xgi)x′gi+f(x′gi(β˜−β0)|xgi)×F(x′gj(β˜−β0)|xgj)x′gi+f(x′gj(β˜−β0)|xgj)F(x′gi(β˜−β0)|xgi)x′gj,

where i≠j. Now notice that

1G∑gi∑j,i≠jzgizgjHgij∗(β˜)=op(1)

by a UWL. Since G(β^−β0)=Op(1) we have R_G=o_p(1). Thus

T=1G∑gi(∑j,i≠jzgizgjψθ(ugi)ψθ(ugj))+op(1)

and consequently T→DN(0, D) as D>0 and

D=E[(∑i=1n[∑j=1,i≠jnzgizgjψθ(ugi)ψθ(ugj)])2]≤n2∑i=1n∑j=1,i≠jnE[zgi4]1/2E[zgj4]1/2(θ+1)2<∞

by two applications of c_r and one of CS.    ■

Let

mG(β)=1G∑gi(∑j,i≠jzgizgj[θ−I(ugi≤x′ig(β−β0))][θ−I(ugj≤x′gi(β−β0))]).

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

sup||β−β0||≤δG|G(mG(β)−mG(β0))−1G1/2∑gi∑j,i≠jzgizgjhgij∗(β)|=op(1),

where

hgij∗(β)=θ2−θF(x′gj(β−β0)|xgj)−θF(x′gi(β−β0)|xgi)+F(x′gi(β−β0)|xgi)×F(x′gj(β−β0)|xgj).

Proof: Note that

mG(β)=1G∑gi∑j,i≠jzgizgjhgij(β),

where

hgij(β)=[θ2−θ I(ugj≤x′gj(β−β0))−θ I(ugi≤x′gi(β−β0))+I(ugi≤x′ig(β−β0))I(ugj≤x′gj(β−β0))], i≠j.

Now taking the expected value of h_gij(β) conditional on x_g we have

E[hgij(β)|xg]=θ2−θFj(x′gj(β−β0)|xg)+θ Fi(x′gi(β−β0)|xg)+Fi,j(x′ig(β−β0),x′gj(β−β0)|xg)=θ2−θ F(x′gj(β−β0)|xgj)+θ F(x′gi(β−β0)|xgi)+F(x′gi(β−β0)|xgi)×F(x′gj(β−β0)|xgj),

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

Note now that

G(mG(β)−mG(β0))−1G1/2∑gi∑j,i≠jzgizgjhgij∗(β)=1G1/2∑gi∑j,i≠jzgizgj[hgij(β)−hgij(β0)−hgij∗(β)].

Since the indicator functions I(u_gi≤x′_ig(β–β₀)) and I(u_gj≤x′_gi(β–β₀)) and the conditional distribution functions F(x′_gi(β–β₀)|x_gi) and F(x′_gj(β–β₀)|x_gj) 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

1G1/2∑gi∑j,i≠jzgizgj[hgij(β)−hgij(β0)−hgij∗(β)]

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