GCov-Based Portmanteau Test

We study nonlinear serial dependence tests for non-Gaussian time series and residuals of dynamic models based on portmanteau statistics involving nonlinear autocovariances. A new test with an asymptotic $\chi^2$ distribution is introduced for testing nonlinear serial dependence (NLSD) in time series. This test is inspired by the Generalized Covariance (GCov) residual-based specification test, recently proposed as a diagnostic tool for semi-parametric dynamic models with i.i.d. non-Gaussian errors. It has a $\chi^2$ distribution when the model is correctly specified and estimated by the GCov estimator. We derive new asymptotic results under local alternatives for testing hypotheses on the parameters of a semi-parametric model. We extend it by introducing a GCov bootstrap test for residual diagnostics,\color{black} which is also available for models estimated by a different method, such as the maximum likelihood estimator under a parametric assumption on the error distribution. \color{black} 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, i.e. bubbles and spikes between 2005 and 2024.