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Using cumulative residual processes, we propose joint goodness-of-fit testsfor conditional means and variances functions in the context of nonlinear timeseries with martingale difference innovations. The main challenge comes fromthe fact the cumulative residual process no longer admits, under the nullhypothesis, a distribution-free limit. To obtain a practical solution oneeither transforms the process in order to achieve a distribution-free limit orapproximates the non-distribution free limit using a numerical or a re-samplingtechnique. Here the three solutions will be considered.It is shown that theproposed tests have nontrivial power against a class of root-n localalternatives, and are suitable when the conditioning information set isinfinite-dimensional, which allows including models like autoregressiveconditional heteroscedastic stochastic models with dependent innovations. Theapproach presented assumes only certain conditions on the first- andsecond-order conditional moments, without imposing any autoregression model.The test procedures introduced are compared with each other and with othercompetitors in terms of their power using a simulation study and a real dataapplication. These simulations have shown that the statistical powers of testsbased on re-sampling or numerical approximation of the original statistics arein general slightly better than those based on a martingale transformation ofthe original process.

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