A Consistent Nonparametric Test for Causality in Quantile



=op(1) .

(A.28)


By combining (A.21), (A.25) and (A.28), we have the result of Step 3

Proof of Theorem 1 (ii)

Since

σ02=2θ2(1-θ)2E{fz(zt)}K2(u)du and

σ^02 2 (i - θ)21кI,

0                 T(T- 1)hm st ts

it is enough to show that

2

σT


1

T (T -1) hm


Kt2s
s
t


=E{fz(zt)}K2(u)du+op(1)

Note that σT2 is a nondegenerate U-statistic of order 2 with kernel

Ht ( zt, z ) = ɪ K21 z-z-
t t s    hm    I h

(A.29)


(A.30)


Since Assumption (A2)(iv)-(v) satisfy the conditions of Lemma 2 of Yoshihara (1976) on the
asymptotic equivalence of U-statistic and its projection under
β-mixing, we have for

γ=2(δ-δ') /δ'(2+δ) >0

2

σT


1

T ( T -1)


HT(zt,zs)

st


=∫∫HT(z1,z2)dFz(z1)dFz(z2)

T

+2T-1ΓHt ( zt, z 2) dFz ( z 2)-∫∫ H ( zp z 2) dFz ( zγ) dFz ( z 2)] + Op (T-1 )
t=1

=∫∫HT(z1,z2)dFz(z1)dFz(z2)+op(1)

= ∫∫-^K2 [h-p-)dF,(zɪ)dF,(Z2)+op(1)

18



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