Diagnostic Methods in Time Series

Model diagnostics is a central issue in statistical modeling and inference. It embeds checking the adequacy of fitted autoregressive–moving-average models, verifying the absence of serial correlation of the error term, performing variable selection in linear regression models, testing causality in multivariate time series models, and so on. In this field, a variety of test statistics have been suggested, and it is worth elucidating their theoretical background and fundamental mechanisms in a unified view. The primary aim of this book is providing a systematic approach and widely applicable results in the context ofmodel diagnostics including classical portmanteau tests and their generalized versions. Moreover, the focus of this book extends also to higher order asymptotics and bias adjustment methods for test statistics applied in time series models. The methods in this book are widely applicable to tests of causality in the infinite variance processes and tests for boundary parameters. The book is designed not only for researchers who specialize in but also are interested in time series analysis and model diagnostics. Chapter 1 reviews the basic elements of stochastic processes to make the book accessible to readers without specific expertise in this field. Chapter 2 shows that the asymptotic χ2 distribution of the Box and Pierce’s portmanteau statistic may fail to hold when a finite number of lags are considered and proposes a Whittle-type portmanteau test statistic, whose local power is evaluated. Numerical studies compare theWhittle-type portmanteau test with other famous portmanteau tests and prove its accuracy. Chapter 3 provides a general framework for hypothesis testing based on a portmanteau-type test statistic. Sufficient conditions for the proposed test statistic to have an asymptotic chi-squared distribution in terms of the Fisher information matrices are provided, and an adjustment procedure for the test statistic is implemented. The delicate limit behavior of the proposed test is investigated with respect to the local asymptotic power. Finally, the fundamental mechanism of portmanteautype tests is discussed in a unified view. Chapter 4 deals with the asymptotics related to the likelihood ratio test and the Wald test when the parameter of interest is on the boundary of the parameter space. In this context, the likelihood ratio statistic asymptotically has a mixed χ2 distribution.We introduce a class S of test statistics which includes the likelihood ratio and theWald and the Rao statistic, in the case of observations generated from a general stochastic process.We develop the third-order vii viii Preface asymptotic theory for S, prove that Λ is Bartlett adjustable, and derive nonlinear adjustments for the other statistics. Numerical studies confirm the benefits of the adjustments on the accuracy and on the power of tests whose statistics belong to S. Chapter 5 focuses on the test for variance components in ANOVA models. Under the null hypothesis, the parameter of interest occurs on the boundary of the parameter space; hence, the likelihood ratio and theWald statistic fail to have the usual asymptotic χ2 distribution. The asymptotic distribution is instead a mixture of a χ2 1 and 0 with mixing probability π0 different from 1/2. The chapter develops the Bartlett adjustment for the likelihood ratio statistic under nonstandard conditions. Furthermore, it proposes an adjustment of the Wald and the modified Wald statistic which makes these statistics Bartlett adjustable. The accuracy of the asymptotic approximation in finite samples is illustrated through numerical experiments. Chapter 6 develops testing procedures of causality between two time series whose innovation processes do not necessarily have finite variance. We introduce frequency and time domain approaches to the testing problem. The limit distribution of the frequency domain statistic, based on Taniguchi et al. (1996), is derived in the infinite variance case, and a robust generalized empirical likelihood test statistic is proposed in the time domain.