Testing against stochastic trend and seasonality in the presence of unattended breaks and unit roots

摘要

This paper considers the problem of testing against stochastic trend and seasonality in the presence of structural breaks and unit roots at frequencies other than those directly under test, which we term unattended breaks and unattended unit roots,respectively. We show that under unattended breaks the true size of the Kwiatkowski et al. (J. Econom. 54 (1992) 159) (KPSS) test at frequency zero and the Canova and Hansen (J. Business Econom. Stat. 13 (1995) 237) (CH) test at the seasonal frequenciesfall well below the nominal level under the null with an associated, often very dramatic, loss of power under the alternative. We demonstrate that a simple modification of the statistics can recover the usual limiting distribution appropriate to the casewhere there are no breaks, provided unit roots do not exist at any of the unattended frequencies. Where unattended unit roots occur we show that the above statistics converge in probability to zero under the null. However, computing the KPSS and CH statistics after pre-filtering the data is simultaneously efficacious against both unattended breaks and unattended unit roots, in the sense that the statistics retain their usual pivotal limiting null distributions appropriate to the case where neither occurs. The case where breaks may potentially occur at all frequencies is also discussed. The practical relevance of the theoretical contribution of the paper is demonstrated through empirical applications of the tests to data on UK marriages and UK consumers' expenditure on tobacco.