We study nonlinear serial dependence tests based on portmanteau statistics with nonlinear autocovariances for non-Gaussian time series and residuals of dynamic models. A new test with an asymptotic χ2 distribution is introduced for testing nonlinear serial dependence (NLSD) in time series. It stems from the Generalized Covariance (GCov) residualbased specification test with an asymptotic χ2 distribution for semiparametric dynamic models with i.i.d. non Gaussian errors. We derive new asymptotic results under local alternative hypotheses on the parameters of a dynamic model, and extend the GCov test to an infinite set of nonlinear autocovariance conditions. A GCov bootstrap test is introduced for size adjustments in finite samples, or residual diagnostics in models estimated parametrically by a maximum likelihood method. A simulation study shows that the tests perform well in applications to mixed causal-noncausal autoregressive models. The GCov specification test is used to assess the fit of a mixed causal-noncausal model of aluminum prices with locally explosive patterns, such as bubbles and spikes, between 2005 and 2024.