A Consistent Nonparametric Test for Causality in Quantile



+ 2 × Ks{1(yt g(xt ))- F(g(xt)| Z)}{F(g(xs)| zs)- θ}

+ Kts {F (g ( xt ) | zt ) - θ} {F (g ( xs ) | Zs ) - θ}

= H1T(s,t,g)+2H2T(s,t,g)+H3T(s,t,g)

(A.17)


1 TT

Then let   J.[g] =-----------∑ ∑ H.τSs,t,g) , i = 1,2,3 . We will treat

j      T(T- 1)hm            jT

Jj[Qθ]-Jj[Qθ-CT] for j=1,2,3 separately.

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

By simple manipulation, we have

J1(Qθ)-J1(Qθ-CT)

=-----1-----∑ ∑ [H1 t (s, t, Qθ ) - H1T (s, t, Qθ - Ct )]

m              1Tθ1TθT

T(T- 1)h t=1 st

1 TT

= τ,τ nm∑∑ Ks{ [11(Qθ)-F(Qθ)][1 s(Q)-F(Qθ)]

T(T- 1)h t=1 st

-[1t(Qθ-CT)-Ft(Qθ-CT)][1s(Qθ-CT)-Fs(Qθ-CT)] }   (A.18)

To avoid tedious works to get bounds on the second moment of J1(Qθ)-J1(Qθ-CT )
with dependent data, we note that the R.H.S. of (A.18) is a degenerate U-statistic of order 2.
Thus we can apply Lemma 2 and have

Thm/2 [J1(Qθ) - J1(Qθ -Ct)] N(0,c12) in distribution,             (A.19)

where the definition of the asymptotic variance σ12 is based on the i.i.d. sequence having the
same marginal distributions as weakly dependent variables in (A.18). That is,

σ12 =E%[H1T(s,t,Qθ)-H1T(s,t,Qθ-CT)],

where the notation E% is expectation evaluated at an i.i.d. sequence having the same
marginal distribution as the mixing sequences in (A.18) (Fan and Li (1999), p. 248). Now, to
show that
Thm/2 [ J1 (Qθ ) - J1(Qθ - Ct )] = Op (1) , we only need to show that the asymptotic

14



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