A Consistent Nonparametric Test for Causality in Quantile

= -J2(Qθ -CT)

1 ʃ. ʃ.   1 (_z -_z A

=--¹— ∑ ∑ ɪ κ ∖z¹-^ I

T(T -1) ⅛ ⅛ h^m L h )

× {1( y_t ≤ Qθ ( x_t ) - Ct ) - Fyz ( Qθ ( x_t ) - Cτ∣z_t )}{Fyz ( Qθ ( x_s ) - C ∖z, ) - θ} (A.22)

Denote S(g) ≡ ∂F[g] / ∂g . By taking a Taylor expansion of F_y∣_z (Q_θ (x_s ) - C_τ ∣ z_s) around
Q_θ(x_s), we have

J2(Qθ)-J2(Qθ-Cτ)

-zI

_τ^s- I {1(y_t ≤ Qθ(x_t) - Ct) - Fyz(Qθ(x_t) - Ct | z_t)}
h)

× (-Ct )S(Qθ (Xs ))

1^τ

= Ct -∑ {1(y_t ≤ Qθ(x_t) - Ct ) -Fyz(Qθ(x_t) - Ct)}S(Qθ(Xs)) f (Zt)
τ t=1

1^τ

≡ Ct V∑ UtS(Qθ(x,))f,(zà.                                    (A.23)

τ t=1

where Q_θ is between Q_θ and Q_θ - C_τ . Thus we have

EJ2(Qθ ) - J2(Qθ - Ct )∣

1^τ

≤Λ Ct - ∑ E∖ufz (z■ )

τ t=1

1^τ

≤ΛCt -∑ E {ut f z²(Z_t)}
τ t=1

= O(C_τ(τh^m)^-1).                                                       (A.24)

where the first inequality holds due to Assumption (1)(v) and the last equality is derived by
using Lemma C.3(iii) of Li (1999) that is proved in the proof of Lemma A.4(i) of Fan and Li
(1996c).

Thus. we have

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