A Consistent Nonparametric Test for Causality in Quantile

Thm/2 [ J2(Qθ) - J2(Qθ - Ct)]

₌ O_p₍C_Th-m/2₎

=op(1) .

(A.25)

[3] Thm/² [J3(Qθ)- J3(Qθ -Ct)] = Op(1):

Noting that H₃_t (5, t, Q_θ ) = 0 because of F(Q_θ (x_j )| Zj ) - θ = 0 for j = t, s, we have

J3(Qθ)-J3(Qθ-CT)

₁ ʃ. ʃ. ₁ (_z -_z A

=--¹--∑ ∑ ɪ K ^z-÷- I

T(T -1) ⅛ ⅛ hm L h )

× {F(Qθ(x_t)-Ct |z,)-θ}{F(Qθ(x_s)-Ct∣Zs)-θ}

1^TT1(zzI

= ^~∑ ∑ ∑ mK l^^tv^ I CT S (Qθ ( Xt )) S (Qθ ( Xs ))

T(T- 1) _t₌₁_s_≠_th Lh )

= CT¹ ∑ S(Q(X_t ))S(Q( X ))f Z(z, ) (A.26)

T t=1

Thus, we have

EJ3( Qθ ) - J3( Qθ - ' )∣

1^T

≤ ΛCT-∑ E∖fZ(z■,)

T t=1

≤ ΛCT¹ ∑ Ef (z-,)∣ + ΛCT¹ ∑ E∖f■ (z,)-f,(z,)∣

Tt=1 Tt=1

1T1T 2

≤ ΛCT-∑ Efz(Zt) + ΛCT-∑ E{f■ (Z_t)-f_z(Z_t)}
Tt=1Tt=1

= O(C_T² ) (A.27)

Finally, we have

Th""-[J3(Qθ)- J3(Qθ -Ct)]

= O_p(Th^m^/2C_T²)