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Testing for residual correlation of any order in the autoregressive process. (English) Zbl 1462.62546

Summary: We are interested in the implications of a linearly autocorrelated driven noise on the asymptotic behavior of the usual least-squares estimator in a stable autoregressive process. We show that the least-squares estimator is not consistent and we suggest a sharp analysis of its almost sure limiting value as well as its asymptotic normality. We also establish the almost sure convergence and the asymptotic normality of the estimated serial correlation parameter of the driven noise. Then, we derive a statistical procedure enabling to test for correlation of any order in the residuals of an autoregressive modelling, giving clearly better results than the commonly used portmanteau tests of G. M. Ljung and G. E. P. Box [Biometrika 65, 297–303 (1978; Zbl 0386.62079)] and G. E. P. Box and D. A. Pierce [J. Am. Stat. Assoc. 65, 1509–1526 (1970; Zbl 0224.62041)], and appearing to outperform the Breusch-Godfrey procedure [T. Breusch, “Testing for autocorrelation in dynamic linear models”, Aust. Econ. Papers 17, No. 31, 334–355 (1978; doi:10.1111/j.1467-8454.1978.tb00635.x); L. G. Godfrey, Econometrica 46, 1293–1301 (1978; Zbl 0395.62062)] on small-sized samples.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62F03 Parametric hypothesis testing
62F12 Asymptotic properties of parametric estimators
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