Variance Shifts, Structural Breaks, and Stationarity Tests

Abstract

This article considers the problem of testing the null hypothesis of stochastic stationarity in time series characterized by variance shifts at some (known or unknown) point in the sample. It is shown that existing stationarity tests can be severely biased in the presence of such shifts, either oversized or undersized, with associated spurious power gains or losses, depending on the values of the breakpoint parameter and on the ratio of the prebreak to postbreak variance. Under the assumption of a serially independent Gaussian error term with known break date and known variance ratio, a locally best invariant (LBI) test of the null hypothesis of stationarity in the presence of variance shifts is then derived. Both the test statistic and its asymptotic null distribution depend on the breakpoint parameter and also, in general, on the variance ratio. Modi cations of the LBI test statistic are proposed for which the limiting distribution is independent of such nuisance parameters and belongs to the family of Cramér-von Mises distributions. One such modi cation is particularly appealing in that it is simultaneously exact invariant to variance shifts and to structural breaks in the slope and/or level of the series. Monte Carlo simulations demonstrate that the power loss from using our modi ed statistics in place of the LBI statistic is not large, even in the neighborhood of the null hypothesis, and particularly for series with shifts in the slope and/or level.The tests are extended to cover the cases of weakly dependent error processes and unknown breakpoints. The implementation of the tests are illustrated using output, in ation, and exchange rate data series.

New variance ratio tests to identify random walk from the general mean reversion model

“…In addition, Wright [53] proposes using variance ratio tests based on ranks and signs of a time series to test the null hypothesis that the series is a martingale difference sequence. Recently, Busetti and Taylor [6] develop a new statistic to test the hypothesis of stochastic stationarity in time series characterized by variance shifts at some point in the sample. Further research includes applying our approach to the above studies to test the GMR model in place of the non-random walk model or the fads model.…”

Wild bootstrap tests for unit root in ESTAR models

“…Many authors have examined the empirical size of the standard unit root tests in the presence of heteroskedastic variances. Examples of such studies are Hamori and Tokihisa (1997), Kim et al (2002), Busetti and Taylor (2003), Cavaliere (2004a), and Maki (2008). While performances of standard unit root tests in the presence of heteroskedastic variances have been investigated, properties of unit root tests in ESTAR models have not been sufficiently examined for heteroskedastic variances.…”

“…1 In contrast, ESTAR models tend to yield spurious nonlinearity under heteroskedastic variances as shown by Dijk et al (1999a), Dijk et al (1999b) and Pavlidis et al (2010). Monetary and financial variables often have heteroskedastic variances modeled by generalized autoregressive conditional heteroskedasticity (GARCH), stochastic volatility, and variance breaks (e.g., Chou 1988;Harvey et al 1994;Kim et al 1998;Busetti and Taylor 2003;Sensier and Dijk 2004). Heteroskedastic variances influence unit root test performance.…”

The Seasonal KPSS Test When Neglecting Seasonal Dummies: A Monte Carlo Analysis

“…Specifically, we focus, through a Monte Carlo analysis, on the impacts of neglecting seasonal dummies on the seasonal KPSS test which is but the seasonal version of stationarity tests as defined by Busetti and Taylor (2003). Remember that other most commonly used tests of seasonal stationarity adopt also the KPSS framework, either in the specification of the basic regression equation or in the construction of the test statistic.…”