A Consistent Nonparametric Test for Causality in Quantile

=op(1) .

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{f_z(z_t)}∫K²(u)du and

σ^02 ≡ 2θ² (i - θ)² —¹— ∑ кI,

⁰                 T(T- 1)h^m s≠t ^ts

it is enough to show that

=E{fz(zt)}∫K²(u)du+op(1)

Note that σ_T² is a nondegenerate U-statistic of order 2 with kernel

H_t ( z_t, z ) = ɪ K²1 ^z-^z-
^{t t s}    h^m    I h

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

=∫∫HT(z1,z2)dFz(z1)dFz(z2)

T

+2T-¹ ∑Γ∫ Ht ( z_t, z 2) dFz ( z 2)-∫∫ H ( z_p z 2) dFz ( z_γ) dF_z ( z 2)] + Op (T^-1^-γ )
t=1

=∫∫HT(z1,z2)dFz(z1)dFz(z2)+op(1)

= ∫∫-^K² [^h-p-)dF,(zɪ)dF,(Z2)+o_p(1)

18

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