Abstract
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.
Highlights
Singular spectrum analysis (SSA) is a non-parametric technique that has gained popularity in the analysis of meteorological (Ghil et al 2002), bio-mechanical (Alonso et al 2005) and hydrological time series (Marques et al 2006), and following its successful application in the physical sciences, applications in economics and finance are finding favour (Hassani & Zhigljavsky 2009)
The techniques that we develop are related to the work of Tzagkarakis et al (2009), who selected the window length as the point of first crossing of a confidence interval (CI) for the sample autocorrelation function (SACF)
In this paper we presented a new methodology for selecting the window length in SSA based on the use of statistical tests designed to ascertain convergence of the autocovariance function of the observed process
Summary
Singular spectrum analysis (SSA) is a non-parametric technique that has gained popularity in the analysis of meteorological (Ghil et al 2002), bio-mechanical (Alonso et al 2005) and hydrological time series (Marques et al 2006), and following its successful application in the physical sciences, applications in economics and finance are finding favour (Hassani & Zhigljavsky 2009). In this paper we propose a methodology in which m is determined from the data prior to the construction of the trajectory matrix and commencement of the SSA modeling In this we are motivated by the fact that Khan & Poskitt (2010) have developed a description length principle that enables the user to consistently extract signal components (both theoretically and in practice) given a preassigned window length compatible with the Whitney embedding theorem. The techniques that we develop are related to the work of Tzagkarakis et al (2009), who selected the window length as the point of first crossing of a confidence interval (CI) for the sample autocorrelation function (SACF) This might be appropriate for the type of data examined in Tzagkarakis et al (2009), in general the point of first crossing of a (”white noise” 95%) CI by the SACF does not represent a time interval beyond which there is no memory left in the process.
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