Bootstrap-assisted Goodness-of-fit Tests in the Frequency Domain

The validity of a stationary time series model may be measured by the goodness of fit of the spectral distribution function. Anderson (Technical Report 27, 1991; Technical Report 309, 1995; Stanford University) has worked out the closed‐form characteristic functions for the Cramer–von Mises criterion for general linear processes, under the condition that all values of the parameters are specified. The asymptotic approach is not easily implemented and usually requires a case by case analysis. In this paper we propose a bootstrap goodness‐of‐fit test in the frequency domain. By properly resampling the residuals, we can consistently estimate the p values for many weakly dependent semiparametric models with unspecified parameter values. This is the content of the main theorem that we try to explain. A group of simulations is conducted, showing consistent significance level and good power. The special tests are applied to the lynx data and reveal structure unexplained by the AR(1) model fitted by Tong (J. R. Stat. Soc. A 140 (1977), 432–36). A possible generalization with application to financial data analysis is also discussed.