Systematic Approach for Portmanteau Tests in View of the Whittle Likelihood Ratio

Abstract

Box and Pierce (1970) proposed a test statistic T_BP which is the squared sum of m sample autocorrelations of the estimated residual process of an autoregressive-moving average model of order (p,q). T_BP 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 T_BP is approximated by a chi-square distribution with (m-p-q) degrees of freedom, ``if m is moderately large". This paper shows that T_BP 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 chi-square distributed for any finite m. From 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_WLR 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_WLR. 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.