Data-driven portmanteau tests for time series

Portmanteau tests and information criteria are widely used for checking the hypothesis of independence in time series. More recently, data-driven versions were proposed, where the tests are calibrated based on the largest estimated autocorrelation. It seems natural to introduce a double test statistic (M, Q) where Q is the portmanteau and M is the largest squared autocorrelation. Both statistics have been investigated at length in the past decades. We computed under reasonable assumptions the bivariate probability distribution of this double statistic, conditional, in addition, to the lag at which the largest autocorrelation is found. Tests of the null hypothesis of independence based on rejection regions in the plane (M, Q) are proposed, and some methods to select the rejection region in order to maximize power when the alternative hypothesis is unknown are suggested. A simulation study and a thorough comparison with some popular tests have been performed to show the advantages of our proposal. Notice that this latter includes some well-known univariate tests, so we could expect not only an optimal choice but also additional information which may turn useful for a better understanding of the time series for both model building and forecasting.