A Consistent Nonparametric Test for Causality in Quantile

1^T

= -∑ S(Qθ (^xt, Zt))CtS(Qθ(x_s))fz (z_t)

T t=1

1^T

+-∑ C_rS(Qθ(x_t))S(Qθ(x_s, Zs))f(Zt)

T _t=1

1^r

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

r t=1

where Q_θ(x_s,Z_s) is between Q_θ(x_s) and Q_θ(Z_s). 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 J_T=J+o_p(1) under the alternative hypothesis.

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

process, we have

1 TT

T =------∑ ∑ Kεε_s

Tm          tsts

T(T- 1)h _{t=1 s≠t}

1

= -∑ ^E(st | z)^fz⁽z^ε

T t=1

= ⅛ ∑ ^e(ε | z ) ^fz⁽z )ε

T t=1

⁺1 ∑ {^EE:- | ^zt )^{f (z}t )^{- e}(ε | z_t ) ^fz⁽z_t )} ^εt

T t=1

= ⅛ ∑ ^e(ε | ^zt ) ^fz⁽z_t )^ε_t⁺op⁽¹⁾

T t=1

=^e [^e(ε | ^z_t ) ^fz^(z_t ^ε ]+op⁽¹⁾

=J+op (1)

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