testing procedures, see Li and Wang (1998) and Hsiao and Li (2001). A feasible test statistic
based on the measure J has a second order degenerate U-statistics as the leading term under
the null hypothesis. Generalizing Hall’s (1984) result for independent data, Fan and Li (1999)
establish the asymptotic normal distribution for a general second order degenerate U-statistics
with dependent data.
All the results stated above on testing mean restrictions are however irrelevant when
testing quantile restrictions. Zheng (1998) proposed an idea to transform quantile restrictions
to mean restrictions in independent data. Following his idea, one can use the existing
technical results on testing mean restrictions in testing quantile restrictions. In this paper, by
combining the Zheng’s idea and the results of Fan and Li (1999) and Li (1999), we derive a
test statistic for Granger causality in quantile and establish the asymptotic normal distribution
of the proposed test statistic under the beta-mixing process. Our testing procedure can be
extended to several hypotheses testing problems with conditional quantile in dependent data;
for example, testing a parametric regression functional form, testing the insignificance of a
subset of regressors, and testing semiparametric versus nonparametric regression models.
The paper is organized as follows. Section 2 presents the test statistic. Section 3
establishes the asymptotic normal distribution under the null hypothesis of no causalty in
quantile. Technical proofs are given in Appendix.
2. Nonparametric Test for Granger-Causality in Quantile
To simplify the exposition, we assume a bivariate case, or only{yt,wt}are observable.
Denote Ut-1 = {yt-1,L,yt-p,wt-1,L,wt-q} and Wt-1={wt-1,L,wt-q}. Granger causality
in mean (Granger, 1988) is defined as
(i) wt does not cause yt in mean with respect to Ut-1 if
E(yt|Ut-1) = E(yt|Ut-1-Wt-1) and
(ii) wt is a prima facie cause in mean of yt with respect to Ut-1 if
E(yt∖Ut-ι) ≠ E(yt∖Ut—1 - Wt-ι),
Motivated by the definition of Granger-causality in mean, we define Granger causality in
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