A Consistent Nonparametric Test for Causality in Quantile



εt I {yt Q (xt)}- θ.

(11)


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

J ----1---

TT(T- 1)hm


TT

∑ ∑ Ktsεtεs

t=1   st


1 TT

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

T(T- 1)h t=1 st

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= Tap(sTlogT)-1 →∞ for
some
sT →∞. Let ST=a2+S%T-1/2. Then for the nonparametric kernel estimator of
conditional quantile of
Qθ(xt)of equation (9), we have

sup Q (x)- qθ (x)=O (Sτ )+OI ɪ I as

(13)


||x|P                  i                к Ta )

Lemma 2 (Li / Fan and Li) Let Lt= (εt,zt)T be a strictly stationary process that

satisfies the condition (A1)(i)-(iv) of Appendix, εt R and zt Rm, K() be the kernel

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

σl(z) = E[εt2 | zt = z] and                                                (14)



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