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Time Series Analysis Nonstationary and Noninvertible Distribution Theory von Tanaka, Katsuto (eBook)

  • Erscheinungsdatum: 27.03.2017
  • Verlag: Wiley-Interscience
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Time Series Analysis

Reflects the developments and new directions in the field since the publication of the first successful edition and contains a complete set of problems and solutions

This revised and expanded edition reflects the developments and new directions in the field since the publication of the first edition. In particular, sections on nonstationary panel data analysis and a discussion on the distinction between deterministic and stochastic trends have been added. Three new chapters on long-memory discrete-time and continuous-time processes have also been created, whereas some chapters have been merged and some sections deleted. The first eleven chapters of the first edition have been compressed into ten chapters, with a chapter on nonstationary panel added and located under Part I: Analysis of Non-fractional Time Series. Chapters 12 to 14 have been newly written under Part II: Analysis of Fractional Time Series. Chapter 12 discusses the basic theory of long-memory processes by introducing ARFIMA models and the fractional Brownian motion (fBm). Chapter 13 is concerned with the computation of distributions of quadratic functionals of the fBm and its ratio. Next, Chapter 14 introduces the fractional Ornstein-Uhlenbeck process, on which the statistical inference is discussed. Finally, Chapter 15 gives a complete set of solutions to problems posed at the end of most sections. This new edition features:

- Sections to discuss nonstationary panel data analysis, the problem of differentiating between deterministic and stochastic trends, and nonstationary processes of local deviations from a unit root

- Consideration of the maximum likelihood estimator of the drift parameter, as well as asymptotics as the sampling span increases

- Discussions on not only nonstationary but also noninvertible time series from a theoretical viewpoint

- New topics such as the computation of limiting local powers of panel unit root tests, the derivation of the fractional unit root distribution, and unit root tests under the fBm error

Time Series Analysis: Nonstationary and Noninvertible Distribution Theory, Second Edition , is a reference for graduate students in econometrics or time series analysis.

Katsuto Tanaka, PhD, is a professor in the Faculty of Economics at Gakushuin University and was previously a professor at Hitotsubashi University. He is a recipient of the Tjalling C. Koopmans Econometric Theory Prize (1996), the Japan Statistical Society Prize (1998), and the Econometric Theory Award (1999). Aside from the first edition of Time Series Analysis (Wiley, 1996), Dr. Tanaka had published five econometrics and statistics books in Japanese.

Katsuto Tanaka, PhD, is a professor in the Faculty of Economics at Gakushuin University and was previously a professor at Hitotsubashi University. He is a recipient of the Tjalling C. Koopmans Econometric Theory Prize (1996), the Japan Statistical Society Prize (1998), and the Econometric Theory Award (1999). Aside from the first edition of Time Series Analysis (Wiley, 1996), Dr. Tanaka had published five econometrics and statistics books in Japanese.

Produktinformationen

    Format: ePUB
    Kopierschutz: AdobeDRM
    Seitenzahl: 904
    Erscheinungsdatum: 27.03.2017
    Sprache: Englisch
    ISBN: 9781119132134
    Verlag: Wiley-Interscience
    Größe: 55609 kBytes
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Time Series Analysis

Chapter 1
Models for Nonstationarity and Noninvertibility

We deal with linear time series models on which stationarity or invertibility is not imposed. Using simple examples arising from estimation and testing problems, we indicate nonstandard aspects of the departure from stationarity or invertibility. In particular, asymptotic distributions of various statistics are derived by the eigenvalue approach under the normality assumption on the underlying processes. As a prelude to discussions in later chapters, we also present equivalent expressions for limiting random variables based on the other two approaches, which I call the stochastic process approach and the Fredholm approach.
1.1 Statistics from the One-Dimensional Random Walk

Let us consider the following simple nonstationary model:
1.1
where are independent and identically distributed with common mean 0 and variance 1, which is abbreviated as i.i.d.. The model ( 1.1 ) is usually referred to as the random walk . It is also called the unit root process in the econometrics literature.

Let us deal with the following two statistics arising from the model ( 1.1 ):
1.2
where . Each second moment statistic has a normalizer T 2, which is different from the stationary case, and is necessary to discuss the limiting distribution as T . In fact, noting that , we have

It holds [Fuller (1996, p. 220)] that

where means that, for every 0, there exists a positive number T such that for all T . It is anticipated that and have different nondegenerate limiting distributions.

We now attempt to derive the limiting distributions of and . There are three approaches for this purpose, which I call the eigenvalue approach , the stochastic process approach , and the Fredholm approach . The first approach is described here in detail, whereas the second and third are only briefly described and the details are discussed in later chapters.
1.1.1 Eigenvalue Approach

The eigenvalue approach requires a distributional assumption on . We assume that are independent and identically normally distributed with common mean 0 and variance 1, which is abbreviated as NID.

We also need to compute the eigenvalues of the matrices appearing in quadratic forms. To see this the observation vector may be expressed as
1.3
where the matrix C and its inverse are given by
1.4
The matrix C may be called the random walk generating matrix and play an important role in subsequent discussions.

We can now rewrite and as
1.5 1.6
where

Let us compute the eigenvalues and eigenvectors of and . The eigenvalues of were obtained by Rutherford (1946) (see also Problem 1.1 in this chapter) by computing those of

The j th largest eigenvalue j of is found to be
1.7
There exists an orthogonal matrix P such that , where the k th column of P is an eigenvector corresponding to k . It can be shown [Dickey and Fuller (1979)] that the th component of P is given by

On the other hand, is evidently singular because the vector e is the first column of C and so that the first column of is a zero vector. In fact, it holds that
1.8
where the matrix G is given by
1.9
Here C and are the last and submatrices of C and e , respectively, whereas . The eigenvalues of

can be easily obtained (Prob

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