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Goodness-of-fit tests for linear and nonlinear time series models. (English) Zbl 1119.62359
Summary: We study a general class of goodness-of-fit tests for a parametric conditional mean of a linear or nonlinear time series model. Among the properties of the proposed tests are that they are suitable when the conditioning set is infinite-dimensional; that they are consistent against a broad class of alternatives, including Pitman’s local alternatives converging at the parametric rate $$n^{-1/2}$$, with $$n$$ the sample size; and that they do not need to choose a lag order depending on the sample size or to smooth the data. It turns out that the asymptotic null distributions of the tests depend on the data generating process, so a new bootstrap procedure is proposed and theoretically justified. The proposed bootstrap tests are robust to higher-order dependence, particularly to conditional heteroscedasticity of unknown form. A simulation study compares the finite-sample performance of the proposed and competing tests and shows that our tests can play a valuable role in time series modeling. Finally, an application to an economic price series highlights the merits of our approach.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F03 Parametric hypothesis testing 62F40 Bootstrap, jackknife and other resampling methods 62G10 Nonparametric hypothesis testing
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