A Consistent Nonparametric Test for Causality in Quantile



1. Introduction

Whether movements in one economic variable cause reactions in another variable is an
important issue in economic policy and also for the financial investment decisions. A
framework for investigating causality has been developed by Granger (1969). Testing for
Granger causality between economic time series has been studied intensively in empirical
macroeconomics and empirical finance. The majority of research results have been obtained
in the context of Granger causality in the conditional mean. The conditional mean, though, is
a questionable element of analysis if the distributions of the variables involved are
non-elliptic or fat tailed as to be expected with financial returns. The fixation of causality
analysis on the mean might result in many unclear results on Granger causality. Also, the
conditional mean targets on an overall summary for the conditional distribution. A tail area
causal relation may be quite different to that of the center of the distribution. Lee and Yang
(2007) explore money-income Granger causality in the conditional quantile by using
parametric quantile regression and find that Granger causality is significant in tail quantiles,
while it is not significant in the center of the distribution.

This paper investigates Granger causality in the conditional quantile. It is well known that
the conditional quantile is insensitive to outlying observations and a collection of conditional
quantiles can characterize the entire conditional distribution. Based on the kernel method, we
propose a nonparametric test for Granger causality in quantile. Testing conditional quantile
restrictions by nonparametric estimation techniques in dependent data situations has not been
considered in the literature before. This paper therefore intends to fill this literature gap.

Recently, the problem of testing the conditional mean restrictions using nonparametric
estimation techniques has been actively extended from independent data to dependent data.
Among the related work, only the testing procedures of Fan and Li (1999) and Li (1999) are
consistent and have the standard asymptotic distributions of the test statistics. For the general
hypothesis testing problem of the form
E(ε |z)= 0 a.e., where ε and z are the
regression error term and the vector of regressors respectively, Fan and Li (1999) and Li
(1999) all consider the distance measure of
J= E[εE(ε |z)f(z)] to construct kernel-based
consistent test procedures. For the advantages of using distance measure
J in kernel-based



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