A Measure of Stationarity in Locally Stationary Processes With Applications to Testing

Abstract

In this article we investigate the problem of measuring deviations from stationarity in locally stationary time series. Our approach is based on a direct estimate of the L 2 -distance between the spectral density of the locally stationary process and its best approximation by a spectral density of a stationary process. An explicit expression of the minimal distance is derived, which depends only on integrals of the spectral density of the locally stationary process and its square. These integrals can be estimated directly without estimating the spectral density, and as a consequence, the estimation of the measure of stationarity does not require the specification of a smoothing bandwidth. We show weak convergence of an appropriately standardized version of the statistic to a standard normal distribution. The results are used to construct confidence intervals for the measure of stationarity and to develop a new test for the hypothesis of stationarity. Finally, we investigate the finite sample properties of the resulting confidence intervals and tests by means of a simulation study and illustrate the methodology in two data examples. Parts of the proofs are available online as supplemental material to this article.