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## Über dieses Buch

In this book, the author adopts a state space approach to time series modeling to provide a new, computer-oriented method for building models for vector-valued time series. This second edition has been completely reorganized and rewritten. Background material leading up to the two types of estimators of the state space models is collected and presented coherently in four consecutive chapters. New, fuller descriptions are given of state space models for autoregressive models commonly used in the econometric and statistical literature. Backward innovation models are newly introduced in this edition in addition to the forward innovation models, and both are used to construct instrumental variable estimators for the model matrices. Further new items in this edition include statistical properties of the two types of estimators, more details on multiplier analysis and identification of structural models using estimated models, incorporation of exogenous signals and choice of model size. A whole new chapter is devoted to modeling of integrated, nearly integrated and co-integrated time series.

## Inhaltsverzeichnis

### 1. Introduction

Abstract
Study of time series has a history much older than modern system theory. Probability theorists, statisticians and econometricians have all contributed to our understanding of time series over the past several decades, as is evidenced by the publication of numerous scholarly books. One may wonder what system theory can add to this well established field and doubt if any new perspective or insight can be gained by this relative newcomer to the field. The history of science shows us, however, that a given problem can fruitfully be examined by different disciplines, partly because when it is viewed from new perspectives, implications of alternative assumptions are explored by researchers with different backgrounds or interests, and partly because new techniques developed elsewhere are imported to explore areas left untouched by the discipline in which the problem originated. Although a latecomer to the field of time series analysis, system theory has introduced a set of viewpoints, concepts, and tools sufficiently different from the traditional ones, and these have proved effective in dealing with vector-valued time-indexed data.
Masanao Aoki

### 2. The Notion of State

Abstract
System theory has shown that the behavior of dynamic systems can be conveniently and succinctly described by introducing the notions of state space and state vectors. Some time series behavior may be so complex that it can’t be described by a finite number of parameters, i.e., its description may go beyond the framework of finite dimensional dynamic models. When a finite dimensional state space model does not suffice to capture time series behavior, we attempt to approximate the series by another of lesser complexity which admits a finite dimensional characterization. In the transform domain description, spectral density functions of finite dimensional dynamics are rational functions of frequencies. They are used to approximate irrational spectral density functions of infinite dimensional dynamics.
Masanao Aoki

### 3. Data Generating Processes

Abstract
Before we attempt to build models for some data series, we need to describe the scope of coverage, i.e., the class of data generating processes for which we design model-building algorithms. First, we assume that data are generated by linear processes. Second, throughout most of the book, we further assume that the processes are at least covariance (or weakly) stationary. We assume, that is, that the mean of the process is a constant not varying with time, and that the covariance matrices are functions of the differences of the two time instants and not of these two instants themselves. Later, when we discuss series with apparent trends, we drop this constant-mean assumption. Third, the spectral density matrix of the processes are rational—i.e., the elements of the matrix are ratios of finite order polynomials. This last assumption means that we can use a finite number of parameters to describe the process, such as finite-order ARMA processes or finite-dimensional state space models which are discussed in Chapter 4.
Masanao Aoki

### 4. State Space and ARMA Models

Abstract
This chapter establishes the theoretical equivalence of time series model descriptions in terms of well-known ARMA models and less familiar Markovian (state space) models, introduces the notion of minimal dimensional models and the associated minimal dimensional state vectors, and presents several methods for putting models into state space forms in general and into the observable or observability canonical form in particular. Although econometricians and statisticians are perhaps less accustomed to the state space representation of time series, this representation is quite useful in building models of times series for the purposes of either forecasting or analyzing dynamic interdependence of components of the series.
Masanao Aoki

### 5. Properties of State Space Models

Abstract
This chapter briefly discusses three properties of dynamic systems in state space form that are important in our model-building procedure. They are stability, observability, and reachability (controllability). The notion of stability is well known. A dynamic system is asymptotically stable if the effects of initial conditions vanish asymptotically over time. This property holds if and only if all the eigenvalues of the transition matrix A of a state space model are less than one in magnitude. The other two properties are less familiar to statisticians and econometricians. They may be motivated by requiring that state space model representations be parsimonious—i.e., by the “minimality” of state vector dimension. We briefly mentioned observability in connection with the observability canonical form in Chapter 4. The pair of matrices A and C in a state space model determines this property. More on these properties, as well as economic examples of state space models, can be found in Aoki [1976]. The pair of matrices A and B jointly determines the reachability property. When the model possesses both of these properties, then the minimality of the state vector dimension is assured (see Kalman [1960], or Lindquist and Picci [1979]).
Masanao Aoki

### 6. Hankel Matrix and Singular Value Decomposition

Abstract
This chapter constructs a Hankel matrix from auto-covariance matrices of time series which are used in Chapter 9 to estimate system matrices in the state space models for the data-generating processes. This chapter describes several concepts and relations needed in the later chapters.
Masanao Aoki

### 7. Innovation Models, Riccati Equations, and Multiplier Analysis

Abstract
Suppose that a representation in the form of (2) of Chapter 4 is adopted for time series models. The orthogonally-projected image of the state vector, given in (8) of Chapter 6, showes that the filtered estimate of the state vector evolves dynamically with the same dynamic matrix A as the original model. This vector is a vector of statistics which summarizes the information carried by the stacked data vector. Now, we show by a different reasoning that this vector is updated by exactly the same dynamics as the original model. To be explicit about the timing, and to show that the derivation is valid also for time-varying processes, we add time subscripts to the matrices Ω and R. Advance t by one unit in (8) of Chapter 6:
$$Z_{t + 1} = \Omega _{t + 1} R_t ^{ - 1} y_t^ - .$$
(1)
Masanao Aoki

### 8. State Vectors and Optimality Measures

Abstract
The state vector introduced in Chapter 5 is by no means the only way of summarizing information in data sets. Two other choices, one used by Akaike and the other by Arun et al., are discussed in this chapter.
Masanao Aoki

### 9. Estimation of System Matrices

Abstract
This chapter is central to this book. It describes two types of estimators for system matrices of state space models from given sets of time series data, and examines statistical properties of the proposed estimators. The estimators of the first type are based on the stochastic realization theory, and are suggested by the systems literature on the deterministic realization theory. The estimators of the second type are instrumental variable estimators often found in the econometric literature. We describe an important relation between these two estimators. The first type of estimators is computationally faster than the second, but the second is asymptotically more efficient than the first. There is an important class of VAR data-generating processes for which the two types of estimators are equally efficient asymptotically.
Masanao Aoki

### 10. Approximate Models and Error Analysis

Abstract
Exogenous disturbances affect time series variables in complex and varied ways. Their relationships are usually only approximately captured by models. In the frequency domain, rational transfer functions of the models are best viewed as approximations to more complex rational, or possibly irrational, transfer functions. In the time domain, finite-dimensional state space (innovation) models merely approximate dynamic phenomena of greater complexity that cannot be conveniently captured. Model builders can only hope to reproduce some salient features of actual dynamics by judicious choice of the dimension and the values of model system matrices, and by analyzing the consequences of adding (deleting) a time series from the data vector, or of retaining (dropping) correlations between some of the data components.
Masanao Aoki

### 11. Integrated Time Series

Abstract
Earlier in Chapter 3 we touched on random trends and described two aggregation schemes to decompose series into trends and other, weakly stationary, components. Series with random walk components are called integrated (of order one) because they are the sums (integrals) of weakly-stationary components. When some linear combinations of components of an integrated vector-valued series become weakly stationary rather than being integrated, we say that these components are cointegrated. This notion is proposed by Granger—see Granger [1981], Granger and Weiss [1983], and Engle and Granger [1987], for example. In state space modeling of cointegrated series, some components of time series at different time points may enter as state variables. We therefore interpret the notion of cointegration broadly and allow for the possibility of linear combination of not-necessarily-contemporaneous components of time series being weakly stationary, as in Aoki [1990], for example. We devote this chapter to modeling of integrated time series and related series such as cointegrated or nearly integrated series.
Masanao Aoki

### 12. Numerical Examples

Abstract
This chapter presents examples of models estimated from post World War II macroeconomic time series data for the United States of America, the United Kingdom, West Germany, and Japan. Actual macroeconomic time series used in model construction are described in the data appendix. Models of dimension n constructed using K sub-blocks of Hankel matrices are denoted by mK.n, m2.3, for example. In these examples J is taken to be equal to K. Recall the constraint that nKp where p is the number of a series in the data set. These models thus utilize information contained in the first 2K — 1 covariance matrices of the data vectors. Estimated models are then used to obtain dynamic multipliers in some cases, utilizing the procedure described in Section 7.4.
Masanao Aoki

### Backmatter

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