Stationarity and Goodness‐of‐Fit Tests for Locally Stationary Time Series

ABSTRACT This paper considers the trajectory of a time series with time‐varying coefficients and proposes to test the adequacy of these parameters at a finite and fixed number of instants of the trajectory. For this purpose, a Wald test is constructed from point estimates of the parameters obtained by minimization of a kernel contrast. This can take the form of a localized near‐maximum likelihood estimator for ARMA or GARCH processes, or a localized least squares estimator for a GLARCH process, but many other time‐varying time series such as AR, ARCH, ARMA‐GARCH, APARCH,…, could be considered. Above all, this allows the introduction of a new stationarity test for these processes, whose very good numerical performance has been demonstrated by numerical experiments.