Systematic Approach for Portmanteau Tests in View of Whittle Likelihood Ratio

Box and Pierce (1970) proposed a test statistic TBP which is the squared sum of m sample autocorrelations of the estimated residual process of autoregressive-moving 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 chi-square distribution with (m-p-q) degrees of freedom, ”if m is moderately large”. This paper shows that TBP is understood as a special form of Whittle likelihood ratio test TPW for autoregressive-moving average spectral density with m-dependent residual process. Then, it is shown that, for any finite m, TPW does not converge to chi-square distribution with (m-p-q) degrees of freedom in distribution, and that, if we assume Bloomfield’s exponential spectral density TPW 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 likelihood ratio, we also mention the asymptotics of a natural Whittle likelihood ratio test TWLR which is always asymptotically chi-square distributed. Its local power is also evaluated. Numerical studies illuminate interesting features of TPW , T † PW , and TWLR. Because many versions of the portmanteau test have been proposed, and been used in variety of fields, our systematic approach for portmanteau tests and proposal of tests will give another view and useful applications.