Testing the covariance stationarity of heavy-tailed time series: An overview of the theory with applications to several financial datasets

Abstract

We investigate methods of testing the proposition that the unconditional variance of a time series is constant over time. Motivated by the observation that many financial datasets are “heavy-tailed,” we focus on the properties of statistical tests of covariance stationarity when unconditional fourth and second moments of the data are not finite. We find that sample split prediction tests and cusum of squares tests have nonstandard limiting distributions when fourth unconditional moments are infinite. These tests are consistent provided that variances are finite. However, the rate of divergence under the alternative hypothesis and hence the power of these tests is sensitive to the index of tail thickness in the data. We estimate the maximal moment exponent (which measures tail thickness) for a number of stock market return and exchange rate return series, and conclude that fourth unconditional moments of these series do not appear to be finite. In our formal tests of covariance stationarity, we reject the null hypothesis of constancy of the unconditional variance of these series. This raises questions about the nature of the observed volatility in economic time series, and about appropriate methods of statistically modeling this volatility.