Moment tests for window length selection in singular spectrum analysis of short– and long–memory processes

In this paper, we propose a new methodology for selecting the window length in Singular Spectral Analysis in which the window length is determined from the data prior to the commencement of modelling. The selection procedure is based on statistical tests designed to test the convergence of the autocovariance function. A classical time series portmanteau type statistic and two test statistics derived using a conditional moment principle are considered. The first two are applicable to short–memory processes, and the third is applicable to both short– and long–memory processes. We derive the asymptotic null and alternative distributions of the statistics under fairly general regularity conditions. Consistency of the tests implies that the selection criteria will identify true convergence with a finite window length with probability arbitrarily close to one as the sample size increases. Results obtained using Monte Carlo simulation point to the relevance of the asymptotic theory and show that the conditional moment tests will choose a window length consistent with the Whitney embedding theorem. Application to observations on the Southern Oscillation Index shows how observed experimental behaviour can be reflected in features seen with real world data sets.