## The log of the determinant of the autocorrelation matrix for testing goodness of fit in time series.(English)Zbl 1332.62302

Summary: A finite sample modification of a test by the authors [J. Am. Stat. Assoc. 97, No. 458, 601–610 (2002; Zbl 1073.62554)] is proposed. The new modified test is asymptotically equivalent but it has a more intuitive explanation and it can be 25% more powerful for small sample size than the previous one. The test statistic is the log of the determinant of the $$m$$th autocorrelation matrix. We propose two approximations by using the Gamma and the normal distributions to the asymptotic distribution of the test statistic. It is shown that, depending on the model and sample size, the proposed test can be up to 50% more powerful than the ones by G. M. Ljung and G. E. P. Box [Biometrika 65, 297–303 (1978; Zbl 0386.62079)], A. C. Monti [Biometrika 81, No. 4, 776–780 (1994; Zbl 0810.62082)] and Y. Hong [Econometrica 64, No. 4, 837–864 (1996; Zbl 0960.62559)], and for finite sample size is always better than the previous Peña–Rodríguez test. This modified test is applied to the detection of several types of nonlinearity by using either the autocorrelation matrix of the squared or the absolute values of the residuals. It is shown that, in general, the new test is more powerful than the one by A. I. McLeod and W. K. Li [J. Time Ser. Anal. 4, 269–273 (1983; Zbl 0536.62067)].

### MSC:

 62M07 Non-Markovian processes: hypothesis testing 62H20 Measures of association (correlation, canonical correlation, etc.) 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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### References:

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