On the power of Portmanteau serial correlation tests

This paper studies properties of the portmanteau statistic proposed by Box and Pierce 1 Box, G. and Pierce, D. 1970. Distribution of residual autocorrelations in autoregressive-integrated moving average time series models. Journal of the American Statistical Association, 65: 1509–1526. [Taylor & Francis Online] , [Google Scholar] and its modification of Ljung and Box 2 Ljung, G. and Box, G. 1978. On a measure of lack of fit in time series models. Biometrika, 65: 297–303. [Crossref], [Web of Science ®] , [Google Scholar]. We show that these portmanteau statistics are feasible analogs to optimal tests for the class of statistics which are linear combinations of consistent estimates of serial correlations. We find, however, that for sample sizes commonly encountered in practice, the efficiency loss in power of portmanteau statistics relative to optimal tests can be substantial, although their size properties are broadly comparable. Our results indicate that tests based on some other non-optimal weighting schemes, including tests with optimal weights constructed from moderately misspecified alternatives, deliver tests with better power than the Box–Pierce or Ljung–Box statistics.