A Generalization of Some Classical Time Series Tools

In classical time series analysis the sample autocorrelation function (SACF) and the sample partial autocorrelation function (SPACF) has gained wide application for structural identification of linear time series models. We suggest generalizations, founded on smoothing techniques, applicable for structural identification of non-linear time series models. A similar generalization of the sample cross correlation function is discussed. Furthermore, a measure of the departure from linearity is suggested. It is shown how bootstrapping can be applied to construct confidence intervals under independence or linearity. The generalizations do not prescribe a particular smoothing technique. In fact, when the smoother is replaced by a linear regression the generalizations reduce to close approximations of SACF and SPACF. For this reason a smooth transition from the linear to the non-linear case can be obtained by varying the bandwidth of a local linear smoother. By adjusting the flexibility of the smoother the power of the tests for independence and linearity against specific alternatives can be adjusted. The generalizations allow for graphical presentations, very similar to those used for SACF and SPACF. In this paper the generalizations are applied to some simulated data sets and to the Canadian lynx data. The generalizations seem to perform well and the measure of the departure from linearity proves to be an important additional tool.