Testing the Martingale Difference Hypothesis Using Integrated Regression Functions

This paper proposes an omnibus test for testing a generalized version of the martingale difference hypothesis (MDH). This generalized hypothesis includes the usual MDH, testing for conditional moments constancy such as conditional homoscedasticity (ARCH effects) or testing for directional predictability. Here we propose a unified approach for dealing with all of these testing problems. These hypotheses are long standing problems in econometric time series analysis, and typically have been tested using the sample autocorrelations or in the spectral domain using the periodogram. Since these hypotheses cover also nonlinear predictability, tests based on those second order statistics are inconsistent against uncorrelated processes in the alternative hypothesis. To circumvent this problem we introduce the pairwise integrated regression functions as measures of linear and nonlinear dependence. With our test there is no need to choose a lag order depending on sample size, to smooth the data or to formulate a parametric alternative model. Moreover, our test is robust to higher order dependence, in particular to conditional heteroskedasticity. Under general dependence the asymptotic null distribution depends on the data generating process, so a bootstrap procedure is considered and a Monte Carlo study examines its finite sample performance. Then we investigate the martingale and conditional heteroskedasticity properties of the Pound/Dollar exchange rate.