Testing the martingale difference hypothesis using martingale difference divergence function

Abstract

This article proposes a novel test for the martingale difference hypothesis based on the martingale difference divergence function, a recently developed dependence measure suitable for measuring the degree of conditional mean dependence of a random variable with respect to another. First, we discuss the use of martingale difference divergence in a time series framework as an alternative to the autocovariance function for detecting the existence of forms of nonlinear serial dependence. In particular, the measure equals zero if and only if the considered time-series components are conditionally mean-independent. This characteristic makes it suitable for studying the behavior of white noise processes characterized by non-null mean conditional on the past. We discuss the asymptotic properties of sample martingale difference divergence in a univariate time series framework, refining some of the results existing in the literature. Doing this allows us to build a Ljung-Box-type test statistic by summing the sample martingale difference divergence function over a finite number of lags. Under suitable conditions, the asymptotic null distribution of our test statistic is also established. The finite sample performance is discussed via a Monte Carlo study as we demonstrate its consistency against uncorrelated non-martingale processes. Finally, we show an empirical application for our methodology in analyzing the properties of the Standard and Poor's 500 stock index.