Testing Conditional Independence: Martingale Transforms and Bootstrap 1

Testing Conditional Independence: Martingale Transforms and Bootstrap 1

This paper investigates the problem of testing conditional independence of Y and Z given (X) for some 2 R d ; for a parametric function ( ): First, this paper proposes several classes of asymptotic pivotal tests that are based on Rosenblatt transforms and establishes that against certain alternatives of generic nature, a martingale transform can be employed to improve the asymptotic power of the tests in terms of limiting Pitman e¢ ciency. The martingale transform applied here is extremely simple, causing almost zero additional computational cost. Second, this paper proposes wild bootstrap procedures applied to Rosenblatt transform-based tests and Rosenblatt-martingale transform-based tests. Third, this paper …nds that even when one knows exactly the conditional distribution functions used in the test statistics, using their nonparametric estimators improves the asymptotic power in terms of limiting Pitman e¢ ciency. Lastly, the …nite sample performances of the proposed tests are compared via a Monte Carlo simulation study.