A Consistent Nonparametric Test for Causality in Quantile

^εt ≡ I {y_t ≤ Q (x_t)}^- θ.

Replacing ε by ε, we have a kernel-based feasible test statistic of J,

1 ^TT

_τ._τ .... ∑∑ KtsI{y, ≤Qθ(x,)}-θI{y₁ ≤Q(x,)}-θ (12)

T(T- 1)h _{t=1 s≠t}

3. The Limiting Distributions of the Test Statistic

Two existing works are useful in deriving the limiting distribution of the test statistic; one
is Theorem 2.3 of Franke and Mwita (2003) on the uniform convergence rate of the
nonparametric kernel estimator of conditional quantile; another is Lemma 2.1 of Li (1999) on
the asymptotic distribution of a second-order degenerate U-statistic, which is derived from
Theorem 2.1 of Fan and Li (1999). We restate these results in lemmas below for ease of
reference.

Lemma 1 (Franke and Mwita) Suppose Conditions (A1)(v)-(vii) and (A2)(iii) of Appendix
hold. The bandwidth sequence is such that a= o(1) and S^%_T= Ta^p(s_TlogT)^-1 →∞ for
some s_T →∞. Let S_T=a²+S^%_T^-1/2. Then for the nonparametric kernel estimator of
conditional quantile of Q_θ(x_t)of equation (9), we have

sup Q (x)^{- q}θ (x)∣=O (Sτ )+OI ɪ I as

||x|P                  ⁱ                к Ta )

Lemma 2 (Li / Fan and Li) Let L_t= (ε_t,z_t)^T be a strictly stationary process that

satisfies the condition (A1)(i)-(iv) of Appendix, ε_t ∈ R and z_t ∈ R^m, K(∙) be the kernel

function with h being the smoothing parameter that satisfies the condition (A2)(i)-(ii) of
Appendix. Define

σl(z) = E[ε_t² | z_t = z] and                                                (14)

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