A Consistent Nonparametric Test for Causality in Quantile

1 TT

^r_τ ≡-------∑ ∑ Kεε_s

Tm          tsts

τ(τ- 1)h _{t=1 s≠t}

τhen

Thm/2 J_τ → N(0, σ0 ) in distribution,

where σ₀²=2E{σ_ε⁴(z_t)f_z(z_t)}{∫K²(u)du}
and f_z (∙) is the marginal density function of

zt .

Technical conditions required to derive the asymptotic distribution of J_τ are given in
Appendix, which are adopted from Li (1999) and Franke and Mwita (2003). In the
assumptions we use the definitions of Robinson (1988) for the class of kernel functions Y_v
and the class of functions A _v^∞ , defined in Appendix.

Conditions (A1)(i)-(iv) and (A2)(i)-(ii) are adopted from condition (A1) and (A2) of Li
(1999), which are used to derive the asymptotic normal distribution of a second-order
degenerate U-statistic. Conditions (A1)(v)-(vii) and (A2)(iii) are conditions (A1), (A2), (B1),
(B2), (C1) and (C2) of Franke and Mwita (2003), which are required to get the uniform
convergence rate of nonparametric kernel estimator of conditional quantile with mixing data.
Finally Conditions (A2)(iv)-(v) are adopted from conditions of Lemma 2 of Yoshihara (1976),
which are required to get the asymptotic equivalence of nondegenerate U-statistic and its
projection under the β -mixing process.

We consider testing for local departures from the null that converge to the null at the rate
T¹² h^- m/4. More precisely we consider the sequence of local alternatives:

H,T :   Fyiz {Qθ(X) + dτ>(z,)|z,} = θ,                                         (17)

where d_T = T^-1/2h^-m/4 and the function l(∙) and its first-derivatives are bounded.

Theorem 1. Assume the conditions (A1) and (A2). Then

(i) Under the null hypothesis (3), Thm/2J_τ → N(0,σ2) in distribution, where

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