Factor instrumental variable quantile regression

Appendix

Proof of Theorem 2. For simplicity, we assume W is known, replace F˜ with the unobservable F, and consider a model with endogenous regressors only. It can be shown that the same result holds with W being estimated and using the estimated factor, F˜, as instruments. Let y_t=d_^′tα(ψ)+ε_t, φ(ψ)=ψ−I(y≤Dα(ψ)), a T×1 vector, g^(θ(ψ), F)=(1/T)∑t=1T(ψ−I(εt(ψ)<0))Ft=(1/T)F′φ(ψ) containing r moment restrictions, g^(θ(ψ), Z)=(1/T)∑t=1T(ψ−I(εt(ψ)<0))zt=(1/T)Z′φ(ψ) containing N moment restrictions,

W(ψ, Z)=E [ztz′t(ψ−I(εt(ψ)<0))2]=ψ(1−ψ) E [ztz′t]=ψ(1−ψ) E [(λ′iFt+eit)(λ′iFt+eit)]=ψ(1−ψ)[ΛF′FTΛ′+E],

where E:=ee′ is assumed to be diagonal for ease of analysis, and finally denote

W(ψ, F)=ψ(1−ψ)F′FT.

For the time being, letting N be a fixed number and under Assumptions F and G, we have,

T(θ^gmm, Z(ψ)−θ0(ψ))→d𝒩(0, Ωgmm, Z(ψ)),

and

T(θ^gmm, F(ψ)−θ0(ψ))→d𝒩(0, Ωgmm, F(ψ)),

where

Ωgmm, Z(ψ)=plim (f2(0)D′ZTW(ψ, Z)−1Z′DT)−1

and

Ωgmm, F(ψ)=plim (f2(0)D′FTW(ψ, F)−1F′DT)−1.

We first show that Ω_{gmm, Z}=Ω_{gmm, F} if N/T→0. From Z=FΛ′+e with e=(e₁, e₂, …, e_T), we can write

D′ZTW(ψ, Z)−1Z′DT=D′FTΛ′W(ψ, Z)−1ΛF′DT+D′eTW(ψ, Z)−1e′DT     +D′FTΛ′W(ψ, Z)−1e′DT+D′eTW(ψ, Z)−1ΛF′DT=:w1+w2+w3+w4.

We will show that the first term has a limit that is the inverse of Ω_{gmm, F}. Using the fact that

W(ψ, Z)−1=1ψ(1−ψ)[E−1−E−1Λ[(F′FT)−1+Λ′E−1Λ]−1Λ′E−1],

and the result in Bai and Ng (2010, Appendix II), we can show that

Λ′W(ψ, Z)−1Λ=1ψ(1−ψ)[(F′FT)−1+(Λ′E−1Λ)−1]−1=1ψ(1−ψ)(F′FT)−1+O(1N),

since Λ′E^–1Λ=O(N). It is straightforward to verify, also by using the results in Bai and Ng (2010, Appendix II), that the sum of the last three terms is negligible in the sense that

w2+w3+w4=[OP (NT)+OP (1T)]+OP (1NT)+OP (1NT).

as N/T→0. Thus, if N/T→0,

Ωgmm, Z(ψ)−1= plim (f2(0)D′ZTW(ψ, Z)−1Z′DT)−1=f2(0)D′FT[1ψ(1−ψ)(F′FT)−1+O(1N)]F′DT+oP (1)=f2(0)D′FTW(ψ, F)−1F′DT+o P (1)=Ωgmm, F(ψ)−1.

we next show that θ^gmm, Z(ψ) is inconsistent if N/T→c>0. By stochastic equicontinuity,

θ^gmm, Z(ψ)−θ0(ψ)=(f2(0)D′ZTW(ψ, Z)−1Z′DT)−1⋅f(0)D′ZTW(ψ, Z)−1g^(θ0(ψ), Z)

It was shown that f2(0)(1/T)D′ZW(ψ, Z)−1(1/T)Z′D→P Ωgmm, Z(ψ)−1 if N/T→c=0. If c>0 but is bounded, its limit becomes Ω_{gmm, Z}(ψ)^–1+Ξ, where Ξ is the limit of (1/T)D′eW(ψ, Z)^–1(1/T)e′D which is O_IP(N/T). Now we will show that f(0)(1/T)D′ZW(ψ, Z)−1g^(θ0(ψ), Z)=OP (N/T). Using Z=FΛ′+e,

f(0)D′ZTW(ψ, Z)−1g^(θ0(ψ), Z)=f(0)D′ZTW(ψ, Z)−11TZ′φ(ψ)=f(0)D′FTΛ′W(ψ, Z)−1ΛF′φ(ψ)T+f(0)D′eTW(ψ, Z)−1e′φ(ψ)T   +f(0)D′FTΛ′W(ψ, Z)−1e′φ(ψ)T+f(0)D′eTW(ψ, Z)−1ΛF′φ(ψ)T=:I1+I2+I3+I4.

From Assumption F, Λ′W(ψ, Z)^–1Λ→[ψ(1–ψ)]^–1[(1/T)F′F]^–1, and (1/T)F′φ(ψ)→d𝒩(0, plim  ψ(1−ψ)(1/T)F′F), we have

TI1→d𝒩(0, plim f2(0)D′FT1ψ(1−ψ)(F′FT)−1F′DT)=𝒩(0, Ωgmm, F(ψ)−1).

It can be shown that

I2=f(0)X′eTE−1e′φ(ψ)T−OP (1T)

and

f(0)D′eTE−1e′φ(ψ)T=f(0)NT1N∑i=1N[(1T∑t=1T1σi, edteit)(1T∑t=1T1σi, eφt(ψ)eit)]=f(0)NT(1N∑i=1Nξ(d)iξ(ε)i),

where ξ(d)i:=(1/T)∑t=1T(1/σi, e)dteit and ξ(ε)i:=(1/T)∑t=1T(1/σi, e)φt(ψ)eit. Note that  E[ξ(d)iξ(ε)i]≠0 due to endogeneity. Let γ=(1/N)∑i=1NE[ξ(d)iξ(ε)i] and thus E[I2]=f(0)(N/T)γ, i.e., I₂=O_{I P}(N/T). Again, using the results in Bai and Ng (2010, Appendix II), we can verify that I3=O P (1/NT) and I4=O P (1/(TN)). Therefore

θ^gmm, Z(ψ)−θ0(ψ)=(f2(0)D′ZTW(ψ, Z)−1Z′DT)−1⋅      [I1+f(0)NT(1N∑i=1Nξ(d)iξ(ε)i)+OP (1T)+OP (1NT)].

Assume (1/N)∑i=1N(ξ(d)iξ(ε)i−γ)=OP (1), where γ=E[ξ(d)_iξ(ε)u] and then θ^gmm, Z(ψ)−θ0(ψ)=OP (N/T). From the results above,

θ^gmm, Z(ψ)−θ0(ψ)=(f2(0)D′ZTW(ψ, Z)−1Z′DT)−1⋅       [I1+f(0)NT(1N(Nγ+OP (1)))+OP (1T)+OP (1NT)],

and then

θ^gmm, Z(ψ)−θ0(ψ)−NTb=(f2(0)D′ZTW(ψ, Z)−1Z′DT)−1[I1+OP (NT)+OP (1T)+OP (1NT)],

where

b:=(f2(0)D′ZTW(ψ, Z)−1Z′DT)−1f(0)γ.

Recall that TI1→d𝒩(0, Ωgmm, F(ψ)−1) and f²(0)(1/T)D′ZW(ψ, Z)^–1(1/T)Z′D→Ω_{gmm, F}(ψ)^–1 provided that N/T→0; thus

T(θ^gmm, Z(ψ)−θ0(ψ)−NTb)=(f2(0)D′ZTW(ψ, Z)−1Z′DT)−1TI1+oP(1),

which implies that

T(θ^gmm, Z(ψ)−θ0(ψ)−NTb)→d𝒩(0, Ωgmm, F(ψ)).   □