Elements of Multivariate Time Series Analysis

1. Vector Time Series and Model Representations.- 1.1 Stationary Multivariate Time Series and Their Properties.- 1.1.1 Covariance and Correlation Matrices for a Stationary Vector Process.- 1.1.2 Some Spectral Characteristics for a Stationary Vector Process.- 1.1.3 Some Relations for Linear Filtering of a Stationary Vector Process.- 1.2 Linear Model Representations for a Stationary Vector Process.- 1.2.1 Infinite Moving Average (Wold) Representation of a Stationary Vector Process.- 1.2.2 Vector Autoregressive Moving Average (ARMA) Model Representations.- 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.- 2.1.1 Invertibility of the Vector Moving Average Model.- 2.1.2 Covariance Matrices of the Vector Moving Average Model.- 2.1.3 Features of the Vector MA(1) Model.- 2.1.4 Model Structure for Subset of Components in the Vector MA Model.- 2.2 Vector Autoregressive Models.- 2.2.1 Stationarity of the Vector Autoregressive Model.- 2.2.2 Yule-Walker Relations for Covariance Matrices of a Vector AR Process.- 2.2.3 Covariance Features of the Vector AR(1) Model.- 2.2.4 Univariate Model Structure Implied by Vector AR Model.- 2.3 Vector Mixed Autoregressive Moving Average Models.- 2.3.1 Stationarity and Invertibility of the Vector ARMA Model.- 2.3.2 Relations for the Covariance Matrices of the Vector ARMA Model.- 2.3.3 Some Features of the Vector ARMA(1,1) Model.- 2.3.4 Consideration of Parameter Identifiability for Vector ARMA Models.- 2.3.5 Further Aspects of Nonuniqueness of Vector ARMA Model Representations.- 2.4 Nonstationary Vector ARMA Models.- 2.4.1 Vector ARIMA Models for Nonstationary Processes.- 2.4.2 Cointegration in Nonstationary Vector Processes.- 2.4.3 The Vector IMA(1,1) Process or Exponential Smoothing Model.- 2.5 Prediction for Vector ARMA Models.- 2.5.1 Minimum Mean Squared Error Prediction.- 2.5.2 Forecasting for Vector ARMA Processes and Covariance Matrices of Forecast Errors.- 2.5.3 Computation of Forecasts for Vector ARMA Processes.- 2.5.4 Some Examples of Forecast Functions 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.- 3.1.1 Kronecker Indices and McMillan Degree of Vector ARMA Process.- 3.1.2 Echelon Form Structure of Vector ARMA Model Implied by Kronecker Indices.- 3.1.3 Reduced-Rank Form of Vector ARMA Model Implied by Kronecker Indices.- 3.2 Canonical Correlation Structure for ARMA Time Series.- 3.2.1 Review of Canonical Correlations in Multivariate Analysis.- 3.2.2 Canonical Correlations for Vector ARMA Processes.- 3.2.3 Relation to Scalar Component Model Structure.- 3.3 Partial Autoregressive and Partial Correlation Matrices.- 3.3.1 Vector Autoregressive Model Approximations and Partial Autoregression Matrices.- 3.3.2 Recursive Fitting of Vector AR Model Approximations.- 3.3.3 Partial Cross-Correlation Matrices for a Stationary Vector Process.- 3.3.4 Partial Canonical Correlations for a Stationary Vector Process.- 4. Initial Model Building and Least Squares Estimation for Vector AR Models.- 4.1 Sample Cross-Covariance and Correlation Matrices and Their Properties.- 4.1.1 Sample Estimates of Mean Vector and of Covariance and Correlation Matrices.- 4.1.2 Asymptotic Properties of Sample Correlations.- 4.2 Sample Partial AR and Partial Correlation Matrices and Their Properties.- 4.2.1 Test for Order of AR Model Based on Sample Partial Autoregression Matrices.- 4.2.2 Equivalent Test Statistics Based on Sample Partial Correlation Matrices.- 4.3 Conditional Least Squares Estimation of Vector AR Models.- 4.3.1 Least Squares Estimation for the Vector AR(1) Model.- 4.3.2 Least Squares Estimation for the Vector AR Model of General Order.- 4.3.3 Likelihood Ratio Testing for the Order of the AR Model.- 4.3.4 Derivation of the Wald Statistic for Testing the Order of the AR Model.- 4.4 Relation of LSE to Yule-Walker Estimate for Vector AR Models.- 4.5 Additional Techniques for Specification of Vector ARMA Models.- 4.5.1 Use of Order Selection Criteria for Model Specification.- 4.5.2 Sample Canonical Correlation Analysis Methods.- 4.5.3 Order Determination Using Linear LSE Methods for the Vector ARMA Model.- 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.- 5.1.1 Conditional Likelihood Function for the Vector ARMA Model.- 5.1.2 Likelihood Equations for Conditional ML Estimation.- 5.1.3 Iterative Computation of the Conditional MLE by GLS Estimation.- 5.1.4 Asymptotic Distribution for the MLE in the Vector ARMA Model.- 5.2 ML Estimation and LR Testing of ARMA Models Under Linear Restrictions.- 5.2.1 ML Estimation of Vector ARMA Models with Linear Constraints on the Parameters.- 5.2.2 LR Testing of the Hypothesis of the Linear Constraints.- 5.2.3 ML Estimation of Vector ARMA Models in the Echelon Canonical Form.- 5.3 Exact Likelihood Function for Vector ARMA Models.- 5.3.1 Expressions for the Exact Likelihood Function and Exact Backcasts.- 5.3.2 Special Cases of the Exact Likelihood Results.- 5.3.3 Finite Sample Forecast Results Based on the Exact Likelihood Approach.- 5.4 Innovations Form of the Exact Likelihood Function for ARMA Models.- 5.4.1 Use of Innovations Algorithm Approach for the Exact Likelihood.- 5.4.2 Prediction of Vector ARMA Processes Using the Innovations Approach.- 5.5 Overall Checking for Model Adequacy.- 5.5.1 Residual Correlation Matrices and Overall Goodness-of-Fit Test.- 5.5.2 Asymptotic Distribution of Residual Covariances and Goodness-of-Fit Statistic.- 5.5.3 Use of the Score Test Statistic for Model Diagnostic Checking.- 5.6 Effects of Parameter Estimation Errors on Prediction Properties.- 5.6.1 Effects of Parameter Estimation Errors on Forecasting in the Vector AR(p) Model.- 5.6.2 Prediction Through Approximation by Autoregressive Model Fitting.- 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.- 6.1.1 Specification of Ranks Through Partial Canonical Correlation Analysis.- 6.1.2 Canonical Form for the Reduced-Rank Model.- 6.1.3 Maximum Likelihood Estimation of Parameters in the Model.- 6.1.4 Relation of Reduced-Rank AR Model with Scalar Component Models and Kronecker Indices.- 6.2 Review of Estimation and Testing for Nonstationarity (Unit Roots) in Univariate ARIMA Models.- 6.2.1 Limiting Distribution Results in the AR(1) Model with a Unit Root.- 6.2.2 Unit-Root Distribution Results for General Order AR Models.- 6.3 Nonstationary (Unit-Root) Multivariate AR Models, Estimation, and Testing.- 6.3.1 Unit-Root Nonstationary Vector AR Model, the Error-Correction Form, and Cointegration.- 6.3.2 Asymptotic Properties of the Least Squares Estimator.- 6.3.3 Reduced-Rank Estimation of the Error-Correction Form of the Model.- 6.3.4 Likelihood Ratio Test for the Number of Unit Roots 199 6.3.5 Reduced-Rank Estimation Through Partial Canonical Correlation Analysis.- 6.3.6 Extension to Account for a Constant Term in the Estimation.- 6.3.7 Forecast Properties for the Cointegrated Model.- 6.3.8 Explicit Unit-Root Structure of the Nonstationary AR Model and Implications.- 6.3.9 Further Numerical Examples.- 6.4 A Canonical Analysis for Vector Autoregressive Time Series.- 6.4.1 Canonical Analysis Based on Measure of Predictability.- 6.4.2 Application to the Analysis of Nonstationary Series for Cointegration.- 6.5 Multiplicative Seasonal Vector ARMA Models.- 6.5.1 Some Special Seasonal ARMA Models for Vector Time Series.- 7. State-Space Models, Kaiman Filtering, and Related Topics.- 7.1 State-Variable Models and Kaiman Filtering.- 7.1.1 The Kaiman Filtering Relations.- 7.1.2 Smoothing Relations in the State-Variable Model.- 7.1.3 Innovations Form of State-Space Model and Steady State for Time-Invariant Models.- 7.1.4 Controllability, Observability, and Minimality for Time-Invariant Models.- 7.2 State-Variable Representations of the Vector ARMA Model.- 7.2.1 A State-Space Form Based on the Prediction Space of Future Values.- 7.2.2 Exact Likelihood Function Through the State-Variable Approach.- 7.2.3 Alternate State-Space Forms for the Vector ARMA Model.- 7.2.4 Minimal Dimension State-Variable Representation and Kronecker Indices.- 7.2.5 (Minimal Dimension) Echelon Canonical State-Space Representation.- 7.3 Exact Likelihood Estimation for Vector ARMA Processes with Missing Values.- 7.3.1 State-Space Model and Kaiman Filtering with Missing Values.- 7.3.2 Estimation of Missing Values in ARMA Models.- 7.3.3 Initialization for Kaiman Filtering, Smoothing, and Likelihood Evaluation in Nonstationary Models.- 7.4 Classical Approach to Smoothing and Filtering of Time Series.- 7.4.1 Smoothing for Univariate Time Series.- 7.4.2 Smoothing Relations for the Signal Plus Noise or Structural Components Model.- 7.4.3 A Simple Vector Structural Component Model for Trend.- 8. Linear Models with Exogenous Variables.- 8.1 Representations of Linear Models with Exogenous Variables.- 8.2 Forecasting in ARMAX Models.- 8.2.1 Forecasts When Future Exogenous Variables Must Be Forecasted.- 8.2.2 MSE Matrix of Optimal Forecasts.- 8.2.3 Forecasting When Future Exogenous Variables Are Specified.- 8.3 Optimal Feedback Control in ARMAX Models.- 8.4 Model Specification, ML Estimation, and Model Checking for ARMAX Models.- 8.4.1 Some Comments on Specification and Checking of ARMAX Models.- 8.4.2 ML Estimation for ARMAX Models.- 8.4.3 Asymptotic Distribution Theory of Estimators in ARMAX Models.- 8.5 Numerical Example.- Appendix: Time Series Data Sets.- Exercises and Problems.- References.- Author Index.