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## Über dieses Buch

This book contains new aspects of model diagnostics in time series analysis, including variable selection problems and higher-order asymptotics of tests. This is the first book to cover systematic approaches and widely applicable results for nonstandard models including infinite variance processes. The book begins by introducing a unified view of a portmanteau-type test based on a likelihood ratio test, useful to test general parametric hypotheses inherent in statistical models. The conditions for the limit distribution of portmanteau-type tests to be asymptotically pivotal are given under general settings, and very clear implications for the relationships between the parameter of interest and the nuisance parameter are elucidated in terms of Fisher-information matrices. A robust testing procedure against heavy-tailed time series models is also constructed in the context of variable selection problems. The setting is very reasonable in the context of financial data analysis and econometrics, and the result is applicable to causality tests of heavy-tailed time series models. In the last two sections, Bartlett-type adjustments for a class of test statistics are discussed when the parameter of interest is on the boundary of the parameter space. A nonlinear adjustment procedure is proposed for a broad range of test statistics including the likelihood ratio, Wald and score statistics.

## Inhaltsverzeichnis

### Chapter 1. Elements of Stochastic Processes

Abstract
A great deal of data in economics, finance, engineering, and the natural sciences occur in the form of time series where observations are dependent and where the nature of this dependence is of interest. A model which describes the probability structure of time series observations $$X_t$$, $$t=1, 2,\ldots , n$$, is called a stochastic process. An $$X_t$$ might be the value of a stock return at time t, the water level in a river at time t, and so on. In this chapter, we will introduce a variety of stochastic processes, e.g., stationary processes, linear processes, heavy-tailed processes, etc., and explain some of their elements, e.g., stationarity, spectral analysis, prediction, causality, etc. The primary purpose of this book is to provide statistical inference, Whittle estimation, model diagnostics, portmanteau tests, nonstandard testing problems for the boundary parameter, and causality tests for heavy-tailed stochastic processes. This relies on the asymptotic theory for higher order asymptotics, Bartlett adjustment, empirical likelihood method, asymptotic efficiency, robustness, etc. For these problems, some useful fundamentals and tools will be given. Throughout this book, we denote by $$\mathbb {N}$$, $$\mathbb {Z}$$, and $${\mathbb {R}}^m$$, the set of all positive integers, the set of all integers, and the m-dimensional Euclidean space, respectively. Stochastic processes were born as a mathematical model describing random quantities which vary together with time. For each time $$t\in \mathbb {Z}$$, there exists a random variable $$X_t$$ defined on a probability space $$(\Omega , \mathscr {A}, P)$$, then the family of random variables $$\{X_t: t\in \mathbb {Z}\}$$ is called a stochastic process. From the definition, $$\{X_t\}$$ may be a family of any random variables. But, if we want to do mathematical or statistical analysis, a sort of regularity or invariance for $$\{X_t\}$$ is needed. The most fundamental one is stationarity.
Fumiya Akashi, Masanobu Taniguchi, Anna Clara Monti, Tomoyuki Amano

### Chapter 2. Systematic Approach for Portmanteau Tests

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 (pq). $$T_{BP}$$ is called the classical portmanteau test. Under the null hypothesis that the autoregressive–moving-average model of order (pq) 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.
Fumiya Akashi, Masanobu Taniguchi, Anna Clara Monti, Tomoyuki Amano

### Chapter 3. A New Look at Portmanteau Test

Abstract
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.
Fumiya Akashi, Masanobu Taniguchi, Anna Clara Monti, Tomoyuki Amano

### Chapter 4. Adjustments for a Class of Tests Under Nonstandard Conditions

Abstract
Generally, the likelihood ratio statistic $$\varLambda$$ for standard hypotheses is asymptotically $$\chi ^2$$ distributed, and the Bartlett adjustment improves the $$\chi ^2$$ approximation to its asymptotic distribution in the sense of third-order asymptotics. However, if the parameter of interest is on the boundary of the parameter space, Self and Liang (1987) show that the limiting distribution of $$\varLambda$$ is a mixture of $$\chi ^2$$ distributions. For such “nonstandard setting of hypotheses”, the current chapter develops the third-order asymptotic theory for a class $$\mathcal {S}$$ of test statistics, which includes the Likelihood Ratio, the Wald and the Score statistic, in the case of observations generated from a general stochastic process, providing widely applicable results. In particular, it is shown that $$\varLambda$$ is Bartlett adjustable despite its nonstandard asymptotic distribution. Although the other statistics are not Bartlett adjustable, a nonlinear adjustment is provided for them which greatly improves the $$\chi ^2$$ approximation to their distribution and allows a subsequent Bartlett type adjustment. Numerical studies confirm the benefits of the adjustments on the accuracy and on the power of tests whose statistics belong to $$\mathcal {S}$$.
Fumiya Akashi, Masanobu Taniguchi, Anna Clara Monti, Tomoyuki Amano

### Chapter 5. Adjustments for Variance Component Tests in ANOVA Models

Abstract
This chapter 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 the Wald statistic fail to have the usual asymptotic $$\chi ^2$$ distribution. The asymptotic distribution is instead a mixture of a $$\chi ^2_1$$ and 0 with mixing probability $$\pi _0$$ different from 1/2. The chapter develops the Bartlett correction for the Likelihood Ratio statistic under nonstandard conditions. Furthermore, it proposes an adjustment of the Wald and modified Wald statistic which makes these statistics Bartlett correctable. The accuracy of the asymptotic approximation in finite samples is investigated through numerical experiments which provide clear evidence of their quality.
Fumiya Akashi, Masanobu Taniguchi, Anna Clara Monti, Tomoyuki Amano

### Chapter 6. Robust Causality Test of Infinite Variance Processes

Abstract
This chapter develops a robust causality test for time series models with infinite variance innovation processes. The testing problem of linear dependence, feedback and causality between two processes is as important as diagnostics for a time series model itself, which are discussed in Chaps. 25. First, we introduce a measure of dependence for vector nonparametric linear processes, and derive the asymptotic distribution of the test statistic by Taniguchi et al. (1996) in the infinite variance case. Second, we construct a weighted version of the Generalized Empirical Likelihood (GEL) test statistic, called the self-weighted GEL statistic in the time domain. The limiting distribution of the self-weighted GEL statistic is shown to be the usual chi-squared one regardless of whether the model has finite variance or not.
Fumiya Akashi, Masanobu Taniguchi, Anna Clara Monti, Tomoyuki Amano

### Backmatter

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