Elements of Multivariate Time Series Analysis

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.

Table of Contents:
1. Vector Time Series and Model Representations.- 1.1 Stationary Multivariate Time Series and Their Properties.- 1.2 Linear Model Representations for a Stationary Vector Process.- A1 Appendix: Review of Multivariate Normal Distribution and Related Topics.- A l. l Review of Some Basic Matrix Theory Results.- A l. 2 Vec Operator and Kronecker Products of Matrices.- A l. 3 Expected Values and Covariance Matrices of Random Vectors.- A1.4 The Multivariate Normal Distribution.- A1.5 Some Basic Results on Stochastic Convergence.- 2. Vector ARMA Time Series Models and Forecasting.- 2.1 Vector Moving Average Models.- 2.2 Vector Autoregressive Models.- 2.3 Vector Mixed Autoregressive Moving Average Models.- 2.4 Nonstationary Vector ARMA Models.- 2.5 Prediction for Vector ARMA Models.- 2.6 State-Space Form of the Vector ARMA Model.- A2 Appendix: Methods for Obtaining Autoregressive and Moving Average Parameters from Covariance Matrices.- A2.1 Iterative Algorithm for Factorization of Moving Average Spectral Density Matrix in Terms of Covariance Matrices.- A2.2 Autoregressive and Moving Average Parameter Matrices in Terms of Covariance Matrices for the Vector ARMA Model.- A2.3 Evaluation of Covariance Matrices in Terms of the AR and MA Parameters for the Vector ARMA Model.- 3. Canonical Structure of Vector ARMA Models.- 3.1 Consideration of Kronecker Structure for Vector ARMA Models.- 3.2 Canonical Correlation Structure for ARMA Time Series.- 3.3 Partial Autoregressive and Partial Correlation Matrices.- 4. Initial Model Building and Least Squares Estimation for Vector AR Models.- 4.1 Sample Cross-Covariance and Correlation Matrices and Their Properties.- 4.2 Sample Partial AR and Partial Correlation Matrices and Their Properties.- 4.3 Conditional Least Squares Estimation of Vector AR Models.- 4.4 Relation of LSE to Yule-Walker Estimate for Vector AR Models.- 4.5 Additional Techniques for Specification of Vector ARMA Models.- A4 Appendix: Review of the General Multivariate Linear Regression Model.- A4.1 Properties of the Maximum Likelihood Estimator of the Regression Matrix.- A4.2 Likelihood Ratio Test of Linear Hypothesis About Regression Coefficients.- A4.3 Asymptotically Equivalent Forms of the Test of Linear Hypothesis.- A4.4 Multivariate Linear Model with Reduced-Rank Structure.- A4.5 Generalization to Seemingly Unrelated Regressions Model.- 5. Maximum Likelihood Estimation and Model Checking for Vector ARMA Models.- 5.1 Conditional Maximum Likelihood Estimation for Vector ARMA Models.- 5.2 ML Estimation and LR Testing of ARMA Models Under Linear Restrictions.- 5.3 Exact Likelihood Function for Vector ARMA Models.- 5.4 Innovations Form of the Exact Likelihood Function for ARMA Models.- 5.5 Overall Checking for Model Adequacy.- 5.6 Effects of Parameter Estimation Errors on Prediction Properties.- 5.7 Motivation for AIC as Criterion for Model Selection, and Corrected Versions of AIC.- 5.8 Numerical Examples.- 6. Reduced-Rank and Nonstationary Cointegrated Models.- 6.1 Nested Reduced-Rank AR Models and Partial Canonical Correlation Analysis.- 6.2 Review of Estimation and Testing for Nonstationarity (Unit Roots) in Univariate ARIMA Models.- 6.3 Nonstationary (Unit-Root) Multivariate AR Models, Estimation, and Testing.- 6.4 A Canonical Analysis for Vector Autoregressive Time Series.- 6.5 Multiplicative Seasonal Vector ARMA Models.- 7. State-Space Models, Kaiman Filtering, and Related Topics.- 7.1 State-Variable Models and Kaiman Filtering.- 7.2 State-Variable Representations of the Vector ARMA Model.- 7.3 Exact Likelihood Estimation for Vector ARMAProcesses with Missing Values.- 7.4 Classical Approach to Smoothing and Filtering of Time Series.- 8. Linear Models with Exogenous Variables.- 8.1 Representations of Linear Models with Exogenous Variables.- 8.2 Forecasting in ARMAX Models.- 8.3 Optimal Feedback Control in ARMAX Models.- 8.4 Model Specification, ML Estimation, and Model Checking for ARMAX Models.- 8.5 Numerical Example.- Appendix: Time Series Data Sets.- Exercises and Problems.- References.- Author Index.