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). The generalized hypothesis includes the usual MDH or testing for conditional moments constancy such as conditional homoscedasticity (ARCH effects). Here we propose a unified approach for dealing with all of them. 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 are not only about linear predictability, tests based on these statistics are inconsistent against uncorrelated processes in the alternative hypothesis. To circumvent this problem we use the pairwise integrated regression functions as measures of linear and nonlinear dependence. Our test is consistent against general pairwise Pitman’s local alternatives converging at the parametric rate and presents optimal power properties. There is no need to choose a lag order depending on sample size, to smooth the data or 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 proposed and theoretically justified. A Monte Carlo study examines its finite sample performance and a final section investigates the martingale and conditional heteroskedasticity properties of the Pound/Dollar exchange rate.