The Limit Distribution of the CUSUM of Square Test Under Genreal MIxing Conditions*

We consider the CUSUM of squares test in a linear regression model with general mixing assumptions on the regressors and the errors. We derive its limit distribution and show how it depends on the nature of the error process. We suggest a corrected version that has a limit distribution free of nuisance parameters. We also discuss how it provides an improvement over the standard approach to testing for a change in the variance in a univariate times series. Simulation evidence is presented to support this. We illustrate the usefulness of our method by analyzing changes in the variance of stock returns and a variety of macroeconomic time series, as well as by testing for change in the variance of the residuals in a typical four-variable VAR model. Our results show the widespread prevalence of changes in the variance of such series and the fact that the variability of shocks affecting the U.S. economy has decreased.