3. A New Look at Portmanteau Test

The present chapter proposes a portmanteau-type test, based on a sort of likelihood ratio statistic, useful to test general parametric hypotheses inherent to statistical models, which includes the classical portmanteau tests and Whittle-type portmanteau test provided in Chap. 2 as special cases. Sufficient conditions for the statistic to be asymptotically chi-square distributed are elucidated in terms of the Fisher information matrix, and the results have very clear implications for the relationships between the parameter of interest and nuisance parameter. In addition, the power of the test is investigated when local alternative hypotheses are considered. The bias adjustment procedure of the portmanteau-type test is also proposed when the sufficient conditions for asymptotic chi-squared distribution fail to hold, and we relax the conditions in Akashi et al. (2018). Some interesting applications of the proposed test to various problems are illustrated. Since portmanteau tests are widely used in many fields, it appears essential to elucidate the fundamental mechanism in a unified view.