Some tools for identification of nonlinear time series

Henrik Aa. Nielsen ( and Henrik Madsen (

For a copy of this paper, either


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. For non-linear time series these tools are not applicable since they only address variation which can be explained by linear models, and for this reason they may completely fail to detect non-linear dependencies. 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 test for independence and for linearity. The generalizations do not prescribe a particular smoothing technique. In fact, when the smoother are replaced by a linear regression the generalizations of SACF and SPACF reduce to a close approximation of their linear counterparts. For this reason a smooth transition form 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 report the generalizations are tested on some simulated data sets and on 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.

KEYWORDS: Lagged scatter plot, Non-linear time series, Smoothing, Non-parametric, Independence, Bootstrap.

Last modified Nov 11, 1998

IMM HomePage