SYSTEMATIC APPROACH FOR PORTMANTEAU TESTS IN VIEW OF THE WHITTLE LIKELIHOOD RATIO

Abstract

Box and Pierce (1970) proposed a test statistic TBP which is the squared sum of m sample autocorrelations of the estimated residual process of an autoregressivemoving average model of order (p,q). TBP is called the classical portmanteau test. Under the null hypothesis that the autoregressive-moving average model of order (p,q) is adequate, they suggested that the distribution of TBP is approximated by a chi-square distribution with (m-p-q) degrees of freedom, “if m is moderately large”. This paper shows that TBP is understood to be a special form of the Whittle likelihood ratio test T PW for autoregressive-moving average spectral density with m-dependent residual processes. Then, it is shown that, for any finite m, T PW does not converge to a chi-square distribution with (m-p-q) degrees of freedom in distribution, and that if we assume Bloomfield’s exponential spectral density, T PW is asymptotically chisquare distributed for any finite m .F rom this observation we propose a modified T † PW which is asymptotically chi-square distributed. In view of the likelihood ratio, we also mention the asymptotics of a natural Whittle likelihood ratio test T WL R which is always asymptotically chi-square distributed. Its local power is also evaluated. Numerical studies illuminate interesting features of T PW , T † PW , and T WL R. Because many versions of the portmanteau test have been proposed and been used in a variety of fields, our systematic approach for portmanteau tests and proposal of tests will give another view and useful applications.