Model specification testing of time series regressions

Given the specification of the lag length and functional form of a (non)linear time series regression we shall propose a test of the null hypothesis that the expectation of the error conditional on the exogenous variables, all lagged exogenous variables and all lagged dependent variables equals zero with probability 1. In the case that the data-generating process is strictly stationary this test is consistent with respect to the alternative hypothesis that the null is false. The test is also applicable for a particular class of non-stationary time series regressions, although in that case consistency with respect to all possible alternatives is no longer guaranteed. The test involved is a generalization of a test proposed in Bierens (1982b). Moreover, we also present a similar but simpler test of the hypothesis that the errors are martingale differences.