On the power transformation of kernel-based tests for serial correlation in vector time series: Some finite sample results and a comparison with the bootstrap
Portmanteau test statistics represent useful diagnostic tools for checking the adequacy of multivariate time series models. For stationary and partially non-stationary vector time series models, Duchesne and Roy [Duchesne,Â P., Roy,Â R., 2004. On consistent testing for serial correlation of unknown form in vector time series models. Journal of Multivariate Analysis 89, 148-180] and Duchesne [Duchesne,Â P., 2005a. Testing for serial correlation of unknown form in cointegrated time series models. Annals of the Institute of Statistical Mathematics 57, 575-595] have proposed kernel-based test statistics, obtained by comparing the spectral density of the errors under the null hypothesis of non-correlation with a kernel-based spectral density estimator; these test statistics are asymptotically standard normal under the null hypothesis of non-correlation in the error term of the model. Following the method of Chen and Deo [Chen,Â W.W., Deo,Â R.S., 2004a. Power transformations to induce normality and their applications. Journal of the Royal Statistical Society, Ser. B 66, 117-130], we determine an appropriate power transformation to improve the normal approximation in small samples. Additional corrections for the mean and variance of the distance measures intervening in these test statistics are obtained. An alternative procedure to estimate the finite distribution of the test statistics is to use the bootstrap method; we introduce bootstrap-based versions of the original spectral test statistics. In a Monte Carlo study, comparisons are made under various alternatives between: the original spectral test statistics, the new corrected test statistics, the bootstrap-based versions, and finally the classical Hosking portmanteau test statistic.
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