Testing Conditional Independence Via Empirical Likelihood

Let f(y|x,z) (resp. f(y|x) be the conditional density of Y given (X,Z) (resp. X). We construct a class of `smoothed` empirical likelihood-based tests for the conditional independence hypothesis: Pr[f(Y|X,Z)=f(Y|X)]=1. We show that the test statistics are asymptotically normal under the null hypothesis and derive their asymptotic distributions under a sequence of local alternatives. The tests are shown to possess a weak optimality property in large samples. Simulation results suggest that the tests behave well in finite samples. Applications to some economic and financial time series indicate that our tests reveal some interesting nonlinear causal relations which the traditional linear Granger causality test fails to detect.