2. Systematic Approach for Portmanteau Tests

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-squared distribution with \((m-p-q)\) degrees of freedom, “if m is moderately large”. This chapter 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-squared 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. 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 compare \(T_{WLR}\) with other famous portmanteau tests Ljung–Box’s \(T_{LB}\) and Li–McLeod’s \(T_{LM}\) and prove its accuracy. 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.