A Consistent Nonparametric Test for Causality in Quantile

variance σ₁²(z) is o(1) with i.i.d data. We have

E^% [H1T(s,t,Qθ)-H1T(s,t,Qθ-CT)]²

≤ Λ E^% { [1t(Qθ) - Ft(Qθ)][1s(Qθ) - Fs(Qθ)]

-[1t(Qθ-CT)-Ft(Qθ-CT)][1s(Qθ-CT)-Fs(Qθ-CT)] }²

≤Λ E^%{Ft(Qθ)[1-Ft(Qθ)]Fs(Qθ)[1-Fs(Qθ)]}

+E^%{Ft(Qθ-CT)[1-Ft(Qθ-CT)]Fs(Qθ-CT)[1-Fs(Qθ-CT)]}

-2E{ [Ft(min(Qθ,Qθ-CT)-Ft(Qθ)Ft(Qθ-CT)]

×[Fs(min(Qθ,Qθ-CT)-Fs(Qθ)Fs(Qθ-CT)] }

=Λ E^%{[Ft(Qθ)-Ft(Qθ)Ft(Qθ)][Fs(Qθ)-Fs(Qθ)Fs(Qθ)]}

-Λ E^%{ [Ft(min(Qθ,Qθ-CT)-Ft(Qθ)Ft(Qθ-CT)]

×[Fs(min(Qθ,Qθ-CT)-Fs(Qθ)Fs(Qθ-CT)] }

+Λ E^%{ [Ft(Qθ-CT)-Ft(Qθ-CT)Ft(Qθ-CT)]

×[Fs(Qθ-CT)-Fs(Qθ-CT)Fs(Qθ-CT)] }

-Λ E^%{ [Ft(min(Qθ,Qθ-CT)-Ft(Qθ)Ft(Qθ-CT)]

×[Fs(min(Qθ,Qθ-CT)-Fs(Qθ)Fs(Qθ-CT)] }

≤Λ ∣C_t∣ = o (1). (A.20)

where the last equality holds by the smoothness of conditional distribution function and its

bounded first derivative due to Assumption (A.8). Thus we have

Th^m,[ J,( Qθ ) - Jɪ( Qθ - C )] = Op (1) (A.21)
[2] Th^m'2[J₂(Q_β)- J₂(Q_β -C)] = Op(1):

Noting that H₂_t (5, t, Q_θ ) = 0 because of Fyz (Q_θ (x_s ) | z_s ) - θ = 0, we have

J2(Qθ)-J2(Qθ-CT)