Elements of Multivariate Time Series Analysis (Google eBook)

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Springer Science & Business Media, Dec 2, 2003 - Mathematics - 358 pages
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In this revised edition, some additional topics have been added to the original version, and certain existing materials have been expanded, in an attempt to pro vide a more complete coverage of the topics of time-domain multivariate time series modeling and analysis. The most notable new addition is an entirely new chapter that gives accounts on various topics that arise when exogenous vari ables are involved in the model structures, generally through consideration of the so-called ARMAX models; this includes some consideration of multivariate linear regression models with ARMA noise structure for the errors. Some other new material consists of the inclusion of a new Section 2. 6, which introduces state-space forms of the vector ARMA model at an earlier stage so that readers have some exposure to this important concept much sooner than in the first edi tion; a new Appendix A2, which provides explicit details concerning the rela tionships between the autoregressive (AR) and moving average (MA) parameter coefficient matrices and the corresponding covariance matrices of a vector ARMA process, with descriptions of methods to compute the covariance matrices in terms of the AR and MA parameter matrices; a new Section 5.
  

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Contents

Vector Time Series and Model Representations
1
11 Stationary Multivariate Time Series and Their Properties
2
112 Some Spectral Characteristics for a Stationary Vector Process
4
113 Some Relations for Linear Filtering of a Stationary Vector Process
5
12 Linear Model Representations for a Stationary Vector Process
7
Review of Multivariate Normal Distribution and Related Topics
12
A12 Vec Operator and Kronecker Product of Matrices
13
A13 Expected Values and Covariance Matrices of Random Vectors
14
522 LR Testing of the Hypothesis of the Linear Constraints
132
53 Exact Likelihood Function for Vector ARMA Models
134
531 Expressions for the Exact Likelihood Function and Exact Backcasts
135
532 Special Cases of the Exact Likelihood Results
138
533 Finite Sample Forecast Results Based on the Exact Likelihood Approach
140
54 Innovations Form of the Exact Likelihood Function for ARMA Models
145
542 Prediction of Vector ARMA Processes Using the Innovations Approach
147
55 Overall Checking for Model Adequacy
149

A15 Some Basic Results on Stochastic Convergence
19
Vector ARMA Time Series Models and Forecasting
22
212 Covariance Matrices of the Vector Moving Average Model
23
213 Features of the Vector MA1 Model
24
214 Model Structure for Subset of Components in the Vector MA Model
25
22 Vector Autoregressive Models
27
222 YuleWalker Relations for Covariance Matrices of a Vector AR Process
29
224 Univariate Model Structure Implied by Vector AR Model
30
23 Vector Mixed Autoregressive Moving Average Models
34
232 Relations for the Covariance Matrices of the Vector ARMA Model
35
233 Some Features of the Vector ARMA11 Model
36
234 Consideration of Parameter Identifiability for Vector ARM A Models
37
235 Further Aspects of Nonuniqueness of Vector ARMA Model Representations
40
24 Nonstationary Vector ARMA Models
41
241 Vector ARIMA Models for Nonstationary Processes
42
242 Cointegration in Nonstationary Vector Processes
43
243 The Vector IMA1 1 Process or Exponential Smoothing Model
44
25 Prediction for Vector ARMA Models
46
251 Minimum Mean Squared Error Prediction
47
253 Computation of Forecasts for Vector ARMA Processes
49
254 Some Examples of Forecast Functions for Vector ARMA Models
50
26 StateSpace Form of the Vector ARMA Model
52
Methods for Obtaining Autoregressive and Moving Average Parameters from Covariance Matrices
56
A22 Autoregressive and Moving Average Parameter Matrices in Terms of Covariance Matrices for the Vector ARMA Model
58
A23 Evaluation of Covariance Matrices in Terms of the AR and MA Parameters for the Vector ARM A Model
59
Canonical Structure of Vector ARMA Models
61
311 Kronecker Indices and McMillan Degree of Vector ARMA Process
62
312 Echelon Form Structure of Vector ARMA Model Implied by Kronecker Indices
63
313 ReducedRank Form of Vector ARMA Model Implied by Kronecker Indices
65
32 Canonical Correlation Structure for ARMA Time Series
68
322 Canonical Correlations for Vector ARMA Processes
70
323 Relation to Scalar Component Model Structure
71
33 Partial Autoregressive and Partial Correlation Matrices
74
332 Recursive Fitting of Vector AR Model Approximations
76
333 Partial CrossCorrelation Matrices for a Stationary Vector Process
79
334 Partial Canonical Correlations for a Stationary Vector Process
81
Initial Model Building and Least Squares Estimation for Vector AR Models
84
412 Asymptotic Properties of Sample Correlations
86
42 Sample Partial AR and Partial Correlation Matrices and Their Properties
88
421 Test for Order of AR Model Based on Sample Partial Autoregression Matrices
89
43 Conditional Least Squares Estimation of Vector AR Models
91
432 Least Squares Estimation for the Vector AR Model of General Order
93
433 Likelihood Ratio Testing for the Order of the AR Model
95
44 Relation of LSE to YuleWalker Estimate for Vector AR Models
99
45 Additional Techniques for Specification of Vector ARMA Models
101
451 Use of Order Selection Criteria for Model Specification
102
452 Sample Canonical Correlation Analysis Methods
103
453 Order Determination Using Linear LSE Methods for the Vector ARMA Model
106
Review of the General Multivariate Linear Regression Model
115
A42 Likelihood Ratio Test of Linear Hypothesis About Regression Coefficients
116
A4 3 Asymptotically Equivalent Forms of the Test of Linear Hypothesis
118
A44 Multivariate Linear Model with ReducedRank Structure
119
A45 Generalization to Seemingly Unrelated Regressions Model
120
Maximum Likelihood Estimation and Model Checking for Vector ARMA Models
122
511 Conditional Likelihood Function for the Vector ARMA Model
123
512 Likelihood Equations for Conditional ML Estimation
124
513 Iterative Computation of the Conditional MLE by GLS Estimation
125
514 Asymptotic Distribution for the MLE in the Vector ARMA Model
129
52 ML Estimation and LR Testing of ARM A Models Under Linear Restrictions
130
552 Asymptotic Distribution of Residual Covariances and GoodnessofFit Statistic
150
553 Use of the Score Test Statistic for Model Diagnostic Checking
151
56 Effects of Parameter Estimation Errors on Prediction Properties
155
561 Effects of Parameter Estimation Errors on Forecasting in the Vector ARp Model
156
562 Prediction Through Approximation by Autoregressive Model Fitting
158
57 Motivation for AIC as Criterion for Model Selection and Corrected Versions of AIC
160
58 Numerical Examples
163
ReducedRank and Nonstationary Cointegrated Models
175
611 Specification of Ranks Through Partial Canonical Correlation Analysis
176
612 Canonical Form for the ReducedRank Model
178
613 Maximum Likelihood Estimation of Parameters in the Model
179
614 Relation of ReducedRank AR Model with Scalar Component Models and Kronecker Indices
181
62 Review of Estimation and Testing for Nonstationarity Unit Roots in Univariate ARIMA Models
183
622 UnitRoot Distribution Results for General Order AR Models
185
63 Nonstationary UnitRoot Multivariate AR Models Estimation and Testing
189
632 Asymptotic Properties of the Least Squares Estimator
192
633 ReducedRank Estimation of the ErrorCorrection Form of the Model
194
634 Likelihood Ratio Test for the Number of Unit Roots
199
635 ReducedRank Estimation Through Partial Canonical Correlation Analysis
202
636 Extension to Account for a Constant Term in the Estimation
203
637 Forecast Properties for the Cointegrated Model
209
638 Explicit UnitRoot Structure of the Nonstationary AR Model and Implications
210
639 Further Numerical Examples
212
64 A Canonical Analysis for Vector Autoregressive Time Series
215
641 Canonical Analysis Based on Measure of Predictability
216
642 Application to the Analysis of Nonstationary Series for Cointegration
218
65 Multiplicative Seasonal Vector ARMA Models
219
651 Some Special Seasonal ARM A Models for Vector Time Series
220
StateSpace Models Kalman Filtering and Related Topics
226
711 The Kalman Filtering Relations
227
712 Smoothing Relations in the StateVariable Model
230
713 Innovations Form of StateSpace Model and Steady State for TimeInvariant Models
231
714 Controllability Observability and Minimality for TimeInvariant Models
232
72 StateVariable Representations of the Vector ARMA Model
236
722 Exact Likelihood Function Through the StateVariable Approach
237
723 Alternate StateSpace Forms for the Vector ARMA Model
242
724 Minimal Dimension State Variable Representation and Kronecker Indices
247
73 Exact Likelihood Estimation for Vector ARMA Processes with Missing Values
255
732 Estimation of Missing Values in ARMA Models
257
733 Initialization for Kalman Filtering Smoothing and Likelihood Evaluation in Nonstationary Models
260
74 Classical Approach to Smoothing and Filtering of Time Series
265
741 Smoothing for Univariate Time Series
266
742 Smoothing Relations for the Signal Plus Noise or Structural Components Model
269
743 A Simple Vector Structural Component Model for Trend
272
Linear Models with Exogenous Variables
274
82 Forecasting in ARMAX Models
276
822 MSB Matrix of Optimal Forecasts
278
823 Forecasting When Future Exogenous Variables Are Specified
279
83 Optimal Feedback Control in ARMAX Models
280
84 Model Specification ML Estimation and Model Checking for ARMAX Models
285
842 ML Estimation for ARMAX Models
286
843 Asymptotic Distribution Theory of Estimators in ARMAX Models
289
85 Numerical Example
292
Appendix Time Series Data Sets
299
Exercises and Problems
315
References
332
Subject Index
345
Author Index
354
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