## Elements of Multivariate Time Series AnalysisIn 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 |

332 | |

345 | |

354 | |

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### Common terms and phrases

approximately AR(p ARMAX asymptotic distribution autocovariance autoregressive bivariate canonical correlation analysis coefficient matrices cointegration computation conditional consider convergent correlation matrices corresponding covariance matrix denotes determined Diag diagonal discussed in Section echelon canonical eigenvalues elements equal equations exact likelihood function example expressed forecast error future values given Hankel matrix Hannan Hence infinite MA representation Kalman filtering Kronecker indices least squares estimates likelihood function linear combination lower triangular LR test mean squared error mean vector methods ML estimation model representation moving average MSE matrix multivariate linear nonstationary Note observations obtained one-step parameters partial canonical correlations predictor procedure recursive Reinsel relation rows sample canonical correlation specification state-space form state-space model state-space representation stationary process stationary vector test statistic tion transfer function unit roots univariate models vector ARMA model vector ARMA process vector ARMA(p,q vector process white noise Y,_j

### Popular passages

Page 344 - The Estimation of Parameters in Multivariate Time Series Models, Journal of the Royal Statistical Society B, Vol.

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