Costationarity of Locally Stationary Time Series Usingcostat

Abstract

This article describes the R package costat. This package enables a user to (i) perform a test for time series stationarity; (ii) compute and plot time-localized autocovariances, and (iii) to determine and explore any costationary relationship between two locally stationary time series. Two locally stationary time series are said to be costationary if there exists two time-varying combination functions such that the linear combination of the two series with the functions produces another time series which is stationary. Costationarity existing between two time series indicates a relationship between the series that might be usefully exploited in a number of ways. Sometimes the relationship itself is of interest, sometimes the derived stationary series is of interest and useful as a substitute for either of the original stationary series in some applications.Keywords: local stationarity, costationary time series, local autocovariance, costat. SummaryThe costat package, available from the Comprehensive R Archive Network at http://CRAN. R-project.org/package=costat, is designed for the analysis of locally stationary time series, particularly locally stationary wavelet time series. The package includes functionality for computing tests of stationarity, BootTOS; computing localized autocovariances, lacv; discovering costationarity between two time series, findstysols Several method functions exist that print, summarize or plot the various outputs of these functions. 2 Costationarity of Locally Stationary Time Series Using costat Introduction: Locally stationary time seriesThis article concerns the analysis of discrete time series, X t . That is, a sequence of observations taken through time where t could be an integer or some other regular spacing. Many are familiar with the concept of a stationary time series: that is, loosely speaking, a series whose statistical properties do not change over time. In much of classical time series analysis stationary means second-order stationary where the mean and variance of a series, X t , do not depend on time and the autocovariance:only depends on the lag τ between two different variates X t and X t+τ , but not the time point t = 1, . . . , T . There are several excellent books dealing with (stationary) time series analysis, for example, Brockwell and Davis (1991), Chatfield (2003), Hamilton (1994), Hannan (1960, Priestley (1983) or Shumway and Stoffer (2006).Mathematical theory demands that a stationary time series X t possesses the following representation:where A(ω) is an amplitude function, {e itω } is the usual system of harmonic complex exponentials and dz(ω) is an orthonormal increments process. The amplitude function controls the second-order properties of the time series. The usual spectrum f (ω) = |A(ω)| 2 and the spectrum and autocovariance are a Fourier transform pair. The amplitude, spectrum and autocovariance here reflect the time-invariant nature of the time series X t : none of them depend on time, t.However, many actual time series, arising in many disc...

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