A Consistent Nonparametric Test for Causality in Quantile

(A.24), we have

J2(Qθ)=O(T^-1h^-m).                                                       (A.33)

Next, we show that the result of Step 3 in the proof of Theorem (i) holds under the
alternative hypothesis. Since F(Q_θ (Xj )∣ zj ) - θ ≠ Ofor j = t, s under the alternative
hypothesis, we have

J3(Qθ)-J3(Qθ-CT)

1 ɪ ɪ 1   ( Z.-Z I

=∑ ∑ mK∖-^lγ~^l×{F(Qθ(x)|zt)-θ}{F(Qθ(x°)|Zs)-θ}

T (T ¹) t=1 s ≠ th L h J

1TT1₍ZZ_A

--¹— ∑ ∑ J- K l^-ʌ I

t(T -1) ⅛ ⅛ h^m L h J

× {F(Qθ(x_t)-Ct∣z₁ )-θ}{F(Qθ(x_s)-Qk)-θ}

1^T

= 7 ∑ ! F(Qθ(X, ) I z, ) - θi!F(Qθ (Xs ) I zs ) - θ}f ■ (Z, )

T t=1

1^T

- 7 ∑ {F ( Q ( Xt ) - Ct∣z, ) - θ}{ F ( Q. ( x ) - Cτ∣z, ) - θ} fz ( z, ).      (A.34)

T t=1

By taking a Taylor expansion of F_yj, (Q_θ (Xj ) - C_t | zj ) around Q_θ (zj ) for j = t, s, we
have

J3(Qθ)-J3(Qθ-CT)

1^T

= -∑ {F(Qe(X_t)∖z_t)-θ}CτS(Qe(X_s))f,(z,)

T t=1

1^T

+7∑ CtS(Qθ(X, )){F(Qθ(Xs ) I z,) -θifz(z■,)

T t=1

-ɪ∑ CTS(Qθ(X. ))S(Qθ(Xs)).f■ (z, ).                                   (A.35)

T _t=1

We further take Taylor expansion of F_y∣_z (Q_θ (X_j ) | z_j) around Q_θ (Zj ) for j = t, s and have

J3(Qθ)-J3(Qθ-CT)

20

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