Tests of the Seasonal Unit-Root Hypothesis Against Heteroscedastic Seasonal Integration

Abstract

This article considers the problem of testing for a nonstochastic seasonal unit root in a seasonally observed time series process against the alternative of a randomized seasonal root with mean unity; that is, the process displays heteroscedastic seasonal integration. The alternative hypothesis allows for potentially frequently occurring changes of regime in the process under investigation, allowing for more volatile forms of seasonal nonstationarity. We discuss a family of models that allow for a potentially smooth transition between the explosive and stationary phases of the seasonal model. To test this hypothesis we consider extensions to existing approaches developed to test against nonseasonal stochastic unit roots. Asymptotic representations of the test statistics are derived. An empirical application to a variety of quarterly measures of U.K. consumer's expenditure is also considered.