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E-raamat: State-Space Methods for Time Series Analysis: Theory, Applications and Software

(Universidad Complutense de Madrid, Spain), (Universidad Complutense de Madrid, Spain), , (Texas Tech University, Lubbock, USA), (Universidad Complutense de Madrid, Spain)
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State-Space Methods for Time Series Analysis: Theory, Applications and SoftwareSuurem pilt
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The state-space approach provides a formal framework where any result or procedure developed for a basic model can be seamlessly applied to a standard formulation written in state-space form. Moreover, it can accommodate with a reasonable effort nonstandard situations, such as observation errors, aggregation constraints, or missing in-sample values.

Exploring the advantages of this approach, State-Space Methods for Time Series Analysis: Theory, Applications and Software presents many computational procedures that can be applied to a previously specified linear model in state-space form.

After discussing the formulation of the state-space model, the book illustrates the flexibility of the state-space representation and covers the main state estimation algorithms: filtering and smoothing. It then shows how to compute the Gaussian likelihood for unknown coefficients in the state-space matrices of a given model before introducing subspace methods and their application. It also discusses signal extraction, describes two algorithms to obtain the VARMAX matrices corresponding to any linear state-space model, and addresses several issues relating to the aggregation and disaggregation of time series. The book concludes with a cross-sectional extension to the classical state-space formulation in order to accommodate longitudinal or panel data. Missing data is a common occurrence here, and the book explains imputation procedures necessary to treat missingness in both exogenous and endogenous variables.

Web Resource The authors E4 MATLAB® toolbox offers all the computational procedures, administrative and analytical functions, and related materials for time series analysis. This flexible, powerful, and free software tool enables readers to replicate the practical examples in the text and apply the procedures to their own work.

Arvustused

"The way the authors of describe their book, it is the fruit of a long-lasting love affair with state space models, which started in the 1980s, inspired by the work of Box and Jenkins. Judging from the density of equations and symbols, it must be the theory of the subject that attracts them most. This book is not for the fainthearted. It explains a lotabout state space models. To use them, you have to accept the philosophy of detailed modelling of time series. In summary, if you are a specialist, or want to become one, you will like this book." Paul Eilers, ISCB News, May 2017

"This book synthesizes and presents the computational advantages of the statespace approach over the traditional time domain approaches to linear time series analysis. The explicit connection between the mainstream ARIMA time series models and the statespace representation, one of the main features of the book, is achieved by presenting many examples and procedures to combine, decompose, aggregate, and disaggregate an economic time series into the statespace form. More specifically, it provides a bridge for going back and forth between statespace models and the broad class of VARMAX modelsOverall, this is a useful book on satespace methods for time series analysis and covers substantial amount of material lucidly with a focus on computational aspects and software. It is an excellent reference book for self-study and can also be used as a companion for teaching time series analysis along with a standard time series text." Mohsen Pourahmadi, Texas A&M University, in the Journal of Time Series Analysis, June 2017

List of Figures xvii
List of Tables xix
Preface xxiii
About the Authors xxvii
1 Introduction 1(4)
2 Linear State-Space models 5(14)
2.1 The multiple error model
6(6)
2.1.1 Model formulation
6(2)
2.1.2 Similar transformations
8(1)
2.1.3 Properties of the State-Space model
8(4)
2.2 Single error models
12(7)
2.2.1 Model formulation
12(1)
2.2.2 State estimation in the SEM
12(1)
2.2.3 Obtaining the SEM equivalent to a MEM
13(2)
2.2.4 Obtaining the SEM equivalent to a general linear process
15(4)
3 Model transformations 19(14)
3.1 Model decomposition
19(5)
3.1.1 Deterministic and stochastic subsystems
19(2)
3.1.2 Implied univariate models
21(1)
3.1.3 Block-diagonal forms
22(2)
3.2 Model combination
24(4)
3.2.1 Endogeneization of stochastic inputs
24(2)
3.2.2 Colored errors
26(2)
3.3 Change of variables in the output
28(3)
3.3.1 Observation errors
28(1)
3.3.2 Missing values and aggregated data
29(1)
3.3.3 Temporal aggregation
30(1)
3.4 Uses of these transformations
31(2)
4 Filtering and Smoothing 33(12)
4.1 The conditional moments of a State-Space model
33(1)
4.2 The Kalman Filter
34(2)
4.3 Decomposition of the smoothed moments
36(1)
4.4 Smoothing for a general State-Space model
37(2)
4.5 Smoothing for fixed-coefficients and single-error models
39(1)
4.6 Uncertainty of the smoothed estimates in a SEM
40(2)
4.7 Examples
42(3)
4.7.1 Recursive Least Squares
42(1)
4.7.2 Cleaning the Wolf sunspot series
42(2)
4.7.3 Extracting the Hodrick-Prescott trend
44(1)
5 Likelihood computation for fixed-coefficients models 45(20)
5.1 Maximum likelihood estimation
45(3)
5.1.1 Problem statement
45(1)
5.1.2 Prediction error decomposition
46(1)
5.1.3 Initialization of the Kalman filter in the stationary case
47(1)
5.2 The likelihood for a nonstationary model
48(6)
5.2.1 Diffuse likelihood
49(1)
5.2.2 Minimally conditioned likelihood
50(2)
5.2.3 Likelihood computation for a fixed-parameter SEM
52(1)
5.2.4 Initialization of the Kalman filter in the nonstationary case
52(2)
5.3 The likelihood for a model with inputs
54(3)
5.3.1 Models with deterministic inputs
54(1)
5.3.2 Models with stochastic inputs
55(2)
5.4 Examples
57(8)
5.4.1 Models for the Airline Passenger series
57(3)
5.4.2 Modeling the series of Housing Starts and Sales
60(5)
6 The likelihood of models with varying parameters 65(18)
6.1 Regression with time-varying parameters
66(2)
6.1.1 SS formulation
66(1)
6.1.2 Maximum likelihood estimation
67(1)
6.2 Periodic models
68(6)
6.2.1 All the seasons have the same dynamic structure
69(1)
6.2.2 The s models do not have the same dynamic structure
70(1)
6.2.3 Stationarity and invertibility
71(1)
6.2.4 Maximum likelihood estimation
72(2)
6.3 The likelihood of models with GARCH errors
74(2)
6.4 Examples
76(7)
6.4.1 A time-varying CAPM regression
76(2)
6.4.2 A periodic model for West German Consumption
78(1)
6.4.3 A model with vector-GARCH errors for two exchange rate series
79(4)
7 Subspace methods 83(24)
7.1 Theoretical foundations
83(6)
7.1.1 Subspace structure and notation
84(2)
7.1.2 Assumptions, projections and model reduction
86(1)
7.1.3 Estimating the system matrices
87(2)
7.2 System order estimation
89(3)
7.2.1 Preliminary data analysis methods
89(2)
7.2.2 Model comparison methods
91(1)
7.3 Constrained estimation
92(2)
7.3.1 State sequence structure
92(1)
7.3.2 Subspace-based likelihood
93(1)
7.4 Multiplicative seasonal models
94(1)
7.5 Examples
95(12)
7.5.1 Univariate models
95(4)
7.5.2 A multivariate model for the interest rates
99(8)
8 Signal extraction 107(22)
8.1 Input and error-related components
108(2)
8.1.1 The deterministic and stochastic subsystems
108(1)
8.1.2 Enforcing minimality
109(1)
8.2 Estimation of the deterministic components
110(4)
8.2.1 Estimating the total effect of the inputs
111(2)
8.2.2 Estimating the individual effect of each input
113(1)
8.3 Decomposition of the stochastic component
114(2)
8.3.1 Characterization of the structural components
114(2)
8.3.2 Estimation of the structural components
116(1)
8.4 Structure of the method
116(1)
8.5 Examples
116(13)
8.5.1 Comparing different methods with simulated data
116(4)
8.5.2 Common features in wheat prices
120(3)
8.5.3 The effect of advertising on sales
123(6)
9 The VARMAX representation of a State-Space model 129(16)
9.1 Notation and previous results
130(1)
9.2 Obtaining the VARMAX form of a State-Space model
131(4)
9.2.1 From State-Space to standard VARMAX
132(1)
9.2.2 From State-Space to canonical VARMAX
133(2)
9.3 Practical applications and examples
135(10)
9.3.1 The VARMAX form of some common State-Space models
135(1)
9.3.2 Identifiability and conditioning of the estimates
135(4)
9.3.3 Fitting an errors-in-variables model to Wolf's sunspot series
139(1)
9.3.4 "Bottom-up" modeling of quarterly US GDP trend
140(5)
10 Aggregation and disaggregation of time series 145(22)
10.1 The effect of aggregation on an SS model
146(4)
10.1.1 The high-frequency model in stacked form
146(2)
10.1.2 Aggregation relationships
148(1)
10.1.3 Relationships between the models for high, low, and mixed-frequency data
149(1)
10.1.4 The effect of aggregation on predictive accuracy
150(1)
10.2 Observability in the aggregated model
150(4)
10.2.1 Unobservable modes
150(2)
10.2.2 Observability and fixed-interval smoothing
152(1)
10.2.3 An algorithm to aggregate a linear model: theory and examples
153(1)
10.3 Specification of the high-frequency model
154(4)
10.3.1 Enforcing approximate consistency
154(4)
10.3.2 "Bottom-up" determination of the quarterly model
158(1)
10.4 Empirical example
158(9)
10.4.1 Annual model
159(1)
10.4.2 Decomposition of the quarterly indicator
159(1)
10.4.3 Specification and estimation of the quarterly model
160(1)
10.4.4 Diagnostics
160(2)
10.4.5 Forecast accuracy and non-conformable samples
162(1)
10.4.6 Comparison with alternative methods
163(4)
11 Cross-sectional extension: longitudinal and panel data 167(24)
11.1 Model formulation
168(1)
11.2 The Kalman Filter
169(2)
11.2.1 Case of uncorrelated state and observational errors
170(1)
11.2.2 Case of correlated state and observational errors
171(1)
11.3 The linear mixed model in SS form
171(3)
11.4 Maximum likelihood estimation
174(2)
11.5 Missing data modifications
176(6)
11.5.1 Missingness in responses only
176(2)
11.5.2 Missingness in both responses and covariates: method 1
178(2)
11.5.3 Missingness in both responses and covariates: method 2
180(2)
11.6 Real data examples
182(9)
11.6.1 A LMM for the mare ovarian follicles data
182(1)
11.6.2 Smoothing and prediction of missing values for the beluga whales data
183(8)
Appendices 191(56)
A Some results in numerical algebra and linear systems
193(18)
A.1 QR Decomposition
193(2)
A.2 Schur decomposition
195(1)
A.3 The Hessenberg Form
196(1)
A.4 SVD Decomposition
197(1)
A.5 Canonical Correlations
198(1)
A.6 Algebraic Lyapunov and Sylvester equations
198(2)
A.7 Numerical solution of a Sylvester equation
200(1)
A.8 Block-diagonalization of a matrix
201(1)
A.9 Reduced rank least squares
202(1)
A.10 Riccati equations
203(5)
A.10.1 Definition
204(1)
A.10.2 Solving the ARE in the general case
205(2)
A.10.3 Solving the ARE for GARCH models
207(1)
A.11 Kalman filter
208(3)
B Asymptotic properties of maximum likelihood estimates
211(12)
B.1 Preliminaries
211(3)
B.2 Basic likelihood results for the State-Space model
214(4)
B.2.1 The information matrix
214(2)
B.2.2 Regularity conditions
216(1)
B.2.3 Choice of estimation method
217(1)
B.3 The State-Space model with cross-sectional extension
218(5)
C Software (E4)
223(20)
C.1 Models supported in E4
224(9)
C.1.1 State-Space model
224(1)
C.1.2 The THD format
224(2)
C.1.3 Basic models
226(2)
C.1.3.1 Mathematical definition
226(1)
C.1.3.2 Definition in THD format
227(1)
C.1.4 Models with GARCH errors
228(2)
C.1.4.1 Mathematical definition of the GARCH process
228(1)
C.1.4.2 Defining a model with GARCH errors in THD format
229(1)
C.1.5 Nested models
230(3)
C.2 Overview of computational procedures
233(8)
C.2.1 Standard procedures for time series analysis
233(2)
C.2.2 Signal extraction methods
235(3)
C.2.3 Likelihood and model estimation
238(3)
C.3 Who can benefit from E4?
241(2)
D Downloading E4 and the examples in this book
243(4)
D.1 The e website
243(1)
D.2 Downloading and installing E4
243(2)
D.3 Downloading the code for the examples in this book
245(2)
Bibliography 247(14)
Author Index 261(4)
Subject Index 265
Jose Casals is head of global risk management at Bankia. He is also an associate professor of econometrics at Universidad Complutense de Madrid.

Alfredo Garcia-Hiernaux is an associate professor of econometrics at Universidad Complutense de Madrid and a freelance consultant.

Miguel Jerez is an associate professor of econometrics at Universidad Complutense de Madrid and a freelance consultant. He was previously executive vice-president at Caja de Madrid for six years.

Sonia Sotoca is an associate professor of econometrics at Universidad Complutense de Madrid.

Drs. Casals, Garcia-Hiernaux, Jerez, and Sotoca are all engaged in a long-term research project to apply state-space techniques to standard econometric problems. Their common research interests include state-space methods and time series econometrics.

A. Alexandre (Alex) Trindade is a professor of statistics in the Department of Mathematics and Statistics at Texas Tech University and an adjunct professor in the Graduate School of Biomedical Sciences at Texas Tech University Health Sciences Center. His research spans a broad swath of theoretical and computational statistics.
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