The quantile spectral density and comparison based tests for nonlinear time series

In this paper we consider tests for nonlinear time series, which are motivated by the notion of serial dependence. The proposed tests are based on comparisons with the quantile spectral density, which can be considered as a quantile version of the usual spectral density function. The quantile spectral density `measures' the sequential dependence structure of a time series, and is well defined under relatively weak mixing conditions. We propose an estimator for the quantile spectral density and derive its asympototic sampling properties. We use the quantile spectral density to construct a goodness of fit test for time series and explain how this test can also be used for comparing the sequential dependence structure of two time series. The asymptotic sampling properties of the test statistic is derived under the null and an alternative. Furthermore, a bootstrap procedure it proposed to obtain a finite sample approximation. The method is illustrated with simulations and some real data examples.