Joint parametric specification checking of conditional mean and volatility in time series models with martingale difference innovations

Using cumulative residual processes, we introduce powerful joint specification tests for conditional mean and variance functions in the context of nonlinear time series with martingale difference innovations. The main challenge comes from the fact that, the cumulative residual process no longer admits a distribution-free limit. To obtain a practical solution one either transforms the process to achieve a distribution-free limit or approximates the non-distribution free limit using numerical or re-sampling techniques. In this paper, the three solutions are considered and compared. The proposed tests have nontrivial power against a class of root-n local alternatives and are suitable when the conditioning set is infinite-dimensional, which allows including more general models such as ARMAX-GARCH with dependent innovations. Numerical results based on simulated and real data show that the powers of tests based on re-sampling or numerical approximation are in general slightly better than those based on martingale transformation.