  Using cumulative residual processes, we propose joint goodness-of-fit tests
for conditional means and variances functions in the context of nonlinear time
series with martingale difference innovations. The main challenge comes from
the fact the cumulative residual process no longer admits, under the null
hypothesis, a distribution-free limit. To obtain a practical solution one
either transforms the process in order to achieve a distribution-free limit or
approximates the non-distribution free limit using a numerical or a re-sampling
technique. Here the three solutions will be considered.It is shown that the
proposed tests have nontrivial power against a class of root-n local
alternatives, and are suitable when the conditioning information set is
infinite-dimensional, which allows including models like autoregressive
conditional heteroscedastic stochastic models with dependent innovations. The
approach presented assumes only certain conditions on the first- and
second-order conditional moments, without imposing any autoregression model.
The test procedures introduced are compared with each other and with other
competitors in terms of their power using a simulation study and a real data
application. These simulations have shown that the statistical powers of tests
based on re-sampling or numerical approximation of the original statistics are
in general slightly better than those based on a martingale transformation of
the original process.
