A Consistent Nonparametric Test for Causality in Quantile



1T

= -∑ S(Qθ (xt, Zt))CtS(Qθ(xs))fz (zt)

T t=1

1T

+-∑ CrS(Qθ(xt))S(Qθ(xs, Zs))f(Zt)

T t=1

1r

- - ∑ CTS(Qθ (xt ))S(Qθ (Xs )) f z (Zt ),                                   (A.36)

r t=1

where Qθ(xs,Zs) is between Qθ(xs) and Qθ(Zs). Then by using the same procedures as

in (A.27), we have

J3(Qθ)-J3(Qθ-CT)=O(CT).                                           (A.37)

Now we have the result of Step 1 for the proof of Theorem (iii).                          □

Step 2: Show that JT=J+op(1) under the alternative hypothesis.

Using (7) and uniform convergence rate of kernel regression estimator under β-mixing

process, we have

1 TT

T =------∑ ∑ Kεεs

Tm          tsts

T(T- 1)h t=1 st

1

= -∑ E(st | z)fz(zε

T t=1

= ⅛ ∑ e(ε | z ) fz(z )ε

T t=1

+1 {EE:- | zt )f (zt )- e | zt ) fz(zt )} εt

T t=1

= ⅛ ∑ e(ε | zt ) fz(zt )εt+op(1)

T t=1

=e [e(ε | zt ) fz(zt ε ]+op(1)

=J+op (1)

(A.38)


21



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