Improved Peňa–Rodriguez portmanteau test

Several problems with the diagnostic check suggested by Peňa and Rodriguez [2002. A powerful portmanteau test of lack of fit for time series. J. Amer. Statist. Assoc. 97, 601–610.] are noted and an improved Monte-Carlo version of this test is suggested. It is shown that quite often the test statistic recommended by Peňa and Rodriguez [2002. A powerful portmanteau test of lack of fit for time series. J. Amer. Statist. Assoc. 97, 601–610.] may not exist and their asymptotic distribution of the test does not agree with the suggested gamma approximation very well if the number of lags used by the test is small. It is shown that the convergence of this test statistic to its asymptotic distribution may be quite slow when the series length is less than 1000 and so a Monte-Carlo test is recommended. Simulation experiments suggest the Monte-Carlo test is usually more powerful than the test given by Peňa and Rodriguez [2002. A powerful portmanteau test of lack of fit for time series. J. Amer. Statist. Assoc. 97, 601–610.] and often much more powerful than the Ljung–Box portmanteau test. Two illustrative examples of enhanced diagnostic checking with the Monte-Carlo test are given.