Introduction to Multiple Time Series Analysis

1. Introduction.- 1.1 Objectives of Analyzing Multiple Time Series.- 1.2 Some Basics.- 1.3 Vector Autoregressive Processes.- 1.4 Outline of the Following Chapters.- I. Finite Order Vector Autoregressive Processes.- 2. Stable Vector Autoregressive Processes.- 2.1 Basic Assumptions and Properties of VAR Processes.- 2.1.1 Stable VAR(p) Processes.- 2.1.2 The Moving Average Representation of a VAR Process.- 2.1.3 Stationary Processes.- 2.1.4 Computation of Autocovariances and Autocorrelations of Stable VAR Processes.- 2.1.4a Autocovariances of a VAR(1) Process.- 2.1.4b Autocovariances of a Stable VAR(p) Process.- 2.1.4c Autocorrelations of a Stable VAR(p) Process.- 2.2 Forecasting.- 2.2.1 The Loss Function.- 2.2.2 Point Forecasts.- 2.2.2a Conditional Expectation.- 2.2.2b Linear Minimum MSE Predictor.- 2.2.3 Interval Forecasts and Forecast Regions.- 2.3 Structural Analysis with VAR Models.- 2.3.1 Granger-Causality and Instantaneous Causality.- 2.3.1a Definitions of Causality.- 2.3.1b Characterization of Granger-Causality.- 2.3.1c Characterization of Instantaneous Causality.- 2.3.1d Interpretation and Critique of Instantaneous and Granger-Causality.- 2.3.2 Impulse Response Analysis.- 2.3.2a Responses to Forecast Errors.- 2.3.2b Responses to Orthogonal Impulses.- 2.3.2c Critique of Impulse Response Analysis.- 2.3.3 Forecast Error Variance Decomposition.- 2.3.4 Remarks on the Interpretation of VAR Models.- 2.4 Exercises.- 3. Estimation of Vector Autoregressive Processes.- 3.1 Introduction.- 3.2 Multivariate Least Squares Estimation.- 3.2.1 The Estimator.- 3.2.2 Asymptotic Properties of the Least Squares Estimator.- 3.2.3 An Example.- 3.2.4 Small Sample Properties of the LS Estimator.- 3.3 Least Squares Estimation with Mean-Adjusted Data and Yule-Walker Estimation.- 3.3.1 Estimation when the Process Mean Is Known.- 3.3.2 Estimation of the Process Mean.- 3.3.3 Estimation with Unknown Process Mean.- 3.3.4 The Yule-Walker Estimator.- 3.3.5 An Example.- 3.4 Maximum Likelihood Estimation.- 3.4.1 The Likelihood Function.- 3.4.2 The ML Estimators.- 3.4.3 Properties of the ML Estimators.- 3.5 Forecasting with Estimated Processes.- 3.5.1 General Assumptions and Results.- 3.5.2 The Approximate MSE Matrix.- 3.5.3 An Example.- 3.5.4 A Small Sample Investigation.- 3.6 Testing for Granger-Causality and Instantaneous Causality.- 3.6.1 A Wald Test for Granger-Causality.- 3.6.2 An Example.- 3.6.3 Testing for Instantaneous Causality.- 3.7 The Asymptotic Distributions of Impulse Responses and Forecast Error Variance Decompositions.- 3.7.1 The Main Results.- 3.7.2 Proof of Proposition 3.6.- 3.7.3 An Example.- 3.7.4 Investigating the Distributions of the Impulse Responses by Simulation Techniques.- 3.8 Exercises.- 3.8.1 Algebraic Problems.- 3.8.2 Numerical Problems.- 4. VAR Order Selection and Checking the Model Adequacy.- 4.1 Introduction.- 4.2 A Sequence of Tests for Determining the VAR Order.- 4.2.1 The Impact of the Fitted VAR Order on the Forecast MSE.- 4.2.2 The Likelihood Ratio Test Statistic.- 4.2.3 A Testing Scheme for VAR Order Determination.- 4.2.4 An Example.- 4.3 Criteria for VAR Order Selection.- 4.3.1 Minimizing the Forecast MSE.- 4.3.2 Consistent Order Selection.- 4.3.3 Comparison of Order Selection Criteria.- 4.3.4 Some Small Sample Simulation Results.- 4.4 Checking the Whiteness of the Residuals.- 4.4.1 The Asymptotic Distributions of the Autocovariances and Autocorrelations of a White Noise Process.- 4.4.2 The Asymptotic Distributions of the Residual Autocovariances and Autocorrelations of an Estimated VAR Process.- 4.4.2a Theoretical Results.- 4.4.2b An Illustrative Example.- 4.4.3 Portmanteau Tests.- 4.5 Testing for Nonnormality.- 4.5.1 Tests for Nonnormality of a Vector White Noise Process.- 4.5.2 Tests for Nonnormality of a VAR Process.- 4.6 Tests for Structural Change.- 4.6.1 A Test Statistic Based on one Forecast Period.- 4.6.2 A Test Based on Several Forecast Periods.- 4.6.3 An Example.- 4.7 Exercises.- 4.7.1 Algebraic Problems.- 4.7.2 Numerical Problems.- 5. VAR Processes with Parameter Constraints.- 5.1 Introduction.- 5.2 Linear Constraints.- 5.2.1 The Model and the Constraints.- 5.2.2 LS, GLS, and EGLS Estimation.- 5.2.2a Asymptotic Properties.- 5.2.2b Comparison of LS and Restricted EGLS Estimators.- 5.2.3 Maximum Likelihood Estimation.- 5.2.4 Constraints for Individual Equations.- 5.2.5 Restrictions on the White Noise Covariance Matrix.- 5.2.6 Forecasting.- 5.2.7 Impulse Response Analysis and Forecast Error Variance Decomposition.- 5.2.8 Specification of Subset VAR Models.- 5.2.8a Elimination of Complete Matrices.- 5.2.8b Top-down Strategy.- 5.2.8c Bottom-up Strategy.- 5.2.8d Monte Carlo Comparison of Strategies for Subset VAR Modeling.- 5.2.9 Model Checking.- 5.2.9a Residual Autocovariances and Autocorrelations.- 5.2.9b Portmanteau Tests.- 5.2.9c Other Checks of Restricted Models.- 5.2.10 An Example.- 5.3 VAR Processes with Nonlinear Parameter Restrictions.- 5.3.1 Some Types of Nonlinear Constraints.- 5.3.2 Reduced Rank VAR Models.- 5.3.3 Multivariate LS Estimation of Reduced Rank VAR Models.- 5.3.4 Asymptotic Properties of Reduced Rank LS Estimators.- 5.3.5 Specification and Checking of Reduced Rank VAR Models.- 5.3.6 An Illustrative Example.- 5.4 Bayesian Estimation.- 5.4.1 Basic Terms and Notations.- 5.4.2 Normal Priors for the Parameters of a Gaussian VAR Process.- 5.4.3 The Minnesota or Litterman Priors.- 5.4.4 Practical Considerations.- 5.4.5 An Example.- 5.4.6 Classical versus Bayesian Interpretation of ?? in Forecasting and Structural Analyses.- 5.5 Exercises.- 5.5.1 Algebraic Exercises.- 5.5.2 Numerical Problems.- II. Infinite Order Vector Autoregressive Processes.- 6. Vector Autoregressive Moving Average Processes.- 6.1 Introduction.- 6.2 Finite Order Moving Average Processes.- 6.3 VARMA Processes.- 6.3.1 The Pure MA and Pure VAR Representations of a VARMA Process.- 6.3.2 A VAR(1) Representation of a VARMA Process.- 6.4 The Autocovariances and Autocorrelations of a VARMA(p, q) Process.- 6.5 Forecasting VARMA Processes.- 6.6 Transforming and Aggregating VARMA Processes.- 6.6.1 Linear Transformations of VARMA Processes.- 6.6.2 Aggregation of VARMA Processes.- 6.7 Interpretation of VARMA Models.- 6.7.1 Granger-Causality.- 6.7.2 Impulse Response Analysis.- 6.8 Exercises.- 7. Estimation of VARMA Models.- 7.1 The Identification Problem.- 7.1.1 Nonuniqueness of VARMA Representations.- 7.1.2 Final Equations Form and Echelon Form.- 7.1.3 Illustrations.- 7.2 The Gaussian Likelihood Function.- 7.2.1 The Likelihood Function of an MA(1) Process.- 7.2.2 The MA(q) Case.- 7.2.3 The VARMA(1, 1) Case.- 7.2.4 The General VARMA(p, q) Case.- 7.3 Computation of the ML Estimates.- 7.3.1 The Normal Equations.- 7.3.2 Optimization Algorithms.- 7.3.3 The Information Matrix.- 7.3.4 Preliminary Estimation.- 7.3.5 An Illustration.- 7.4 Asymptotic Properties of the ML Estimators.- 7.4.1 Theoretical Results.- 7.4.2 A Real Data Example.- 7.5 Forecasting Estimated VARMA Processes.- 7.6 Estimated Impulse Responses.- 7.7 Exercises.- 8. Specification and Checking the Adequacy of VARMA Models.- 8.1 Introduction.- 8.2 Specification of the Final Equations Form.- 8.2.1 A Specification Procedure.- 8.2.2 An Example.- 8.3 Specification of Echelon Forms.- 8.3.1 A Procedure for Small Systems.- 8.3.2 A Full Search Procedure Based on Linear Least Squares Computations.- 8.3.2a The Procedure.- 8.3.2b An Example.- 8.3.3 Hannan-Kavalieris Procedure.- 8.3.4 Poskitt’s Procedure.- 8.4 Remarks on other Specification Strategies for VARMA Models.- 8.5 Model Checking.- 8.5.1 LM Tests.- 8.5.2 Residual Autocorrelations and Portmanteau Tests.- 8.5.3 Prediction Tests for Structural Change.- 8.6 Critique of VARMA Model Fitting.- 8.7 Exercises.- 9. Fitting Finite Order VAR Models to Infinite Order Processes.- 9.1 Background.- 9.2 Multivariate Least Squares Estimation.- 9.3 Forecasting.- 9.3.1 Theoretical Results.- 9.3.2 An Example.- 9.4 Impulse Response Analysis and Forecast Error Variance Decompositions.- 9.4.1 Asymptotic Theory.- 9.4.2 An Example.- 9.5 Exercises.- III. Systems with Exogenous Variables and Nonstationary Processes.- 10. Systems of Dynamic Simultaneous Equations.- 10.1 Background.- 10.2 Systems with Exogenous Variables.- 10.2.1 Types of Variables.- 10.2.2 Structural Form, Reduced Form, Final Form.- 10.2.3 Models with Rational Expectations.- 10.3 Estimation.- 10.4 Remarks on Model Specification and Model Checking.- 10.5 Forecasting.- 10.5.1 Unconditional and Conditional Forecasts.- 10.5.2 Forecasting Estimated Dynamic SEMs.- 10.6 Multiplier Analysis.- 10.7 Optimal Control.- 10.8 Concluding Remarks on Dynamic SEMs.- 10.9 Exercises.- 11. Nonstationary Systems with Integrated and Cointegrated Variables.- 11.1 Introduction.- 11.1.1 Integrated Processes.- 11.1.2 Cointegrated Processes.- 11.2 Estimation of Integrated and Cointegrated VAR(p) Processes.- 11.2.1 ML Estimation of a Gaussian Cointegrated VAR(p) Process.- 11.2.1a The ML Estimators and their Properties.- 11.2.1b An Example.- 11.2.1c Discussion of the Proof of Proposition 11.2.- 11.2.2 Other Estimation Methods for Cointegrated Systems.- 11.2.2a Unconstrained LS Estimation.- 11.2.2b A Two-Stage Procedure.- 11.2.3 Bayesian Estimation of Integrated Systems.- 11.2.3a Generalities.- 11.2.3b The Minnesota or Litterman Prior.- 11.2.3c An Example.- 11.3 Forecasting and Structural Analysis.- 11.3.1 Forecasting Integrated and Cointegrated Systems.- 11.3.2 Testing for Granger-Causality.- 11.3.2a The Noncausality Restrictions.- 11.3.2b A Wald Test for Linear Constraints.- 11.3.3 Impulse Response Analysis.- 11.3.3a Theoretical Considerations.- 11.3.3b An Example.- 11.4 Model Selection and Model Checking.- 11.4.1 VAR Order Selection.- 11.4.2 Testing for the Rank of Cointegration.- 11.4.3 Prediction Tests for Structural Change.- 11.5 Exercises.- 11.5.1 Algebraic Exercises.- 11.5.2 Numerical Exercises.- 12. Periodic VAR Processes and Intervention Models.- 12.1 Introduction.- 12.2 The VAR(p) Model with Time Varying Coefficients.- 12.2.1 General Properties.- 12.2.2 ML Estimation.- 12.3 Periodic Processes.- 12.3.1 A VAR Representation with Time Invariant Coefficients.- 12.3.2 ML Estimation and Testing for Varying Parameters.- 12.3.2a All Coefficients Time Varying.- 12.3.2b All Coefficients Time Invariant.- 12.3.2c Time Invariant White Noise.- 12.3.2d Time Invariant Covariance Structure.- 12.3.2e LR Tests.- 12.3.2f Testing a Model with Time Varying White Noise only Against one with all Coefficients Time Varying.- 12.3.2g Testing a Time Invariant Model Against one with Time Varying White Noise.- 12.3.3 An Example.- 12.3.4 Bibliographical Notes and Extensions.- 12.4 Intervention Models.- 12.4.1 Interventions in the Intercept Model.- 12.4.2 A Discrete Change in the Mean.- 12.4.3 An Illustrative Example.- 12.4.4 Extensions and References.- 12.5 Exercises.- 13. State Space Models.- 13.1 Background.- 13.2 State Space Models.- 13.2.1 The General Linear State Space Model.- 13.2.1a A Finite Order VAR Process.- 13.2.1b A VARMA(p, q) Process.- 13.2.1c The VARX Model.- 13.2.1d Systematic Sampling and Aggregation.- 13.2.1e Structural Time Series Models.- 13.2.1f Factor Analytic Models.- 13.2.1g VARX Models with Systematically Varying Coefficients.- 13.2.1h Random Coefficient VARX Models.- 13.2.2 Nonlinear State Space Models.- 13.3 The Kalman Filter.- 13.3.1 The Kalman Filter Recursions.- 13.3.1a Assumptions for the State Space Model.- 13.3.1b The Recursions.- 13.3.1c Computational Aspects and Extensions.- 13.3.2 Proof of the Kalman Filter Recursions.- 13.4 Maximum Likelihood Estimation of State Space Models.- 13.4.1 The Log-Likelihood Function.- 13.4.2 The Identification Problem.- 13.4.3 Maximization of the Log-Likelihood Function.- 13.4.3a The Gradient of the Log-Likelihood.- 13.4.3b The Information Matrix.- 13.4.3c Discussion of the Scoring Algorithm.- 13.4.4 Asymptotic Properties of the ML Estimators.- 13.5 A Real Data Example.- 13.6 Exercises.- Appendices.- Appendix A. Vectors and Matrices.- A.1 Basic Definitions.- A.2 Basic Matrix Operations.- A.3 The Determinant.- A.4 The Inverse, the Adjoint, and Generalized Inverses.- A.4.1 Inverse and Adjoint of a Square Matrix.- A.4.2 Generalized Inverses.- A.5 The Rank.- A.6 Eigenvalues and -vectors — Characteristic Values and Vectors.- A.7 The Trace.- A.8 Some Special Matrices and Vectors.- A.8.1 Idempotent and Nilpotent Matrices.- A.8.2 Orthogonal Matrices and Vectors.- A.8.3 Definite Matrices and Quadratic Forms.- A.9 Decomposition and Diagonalization of Matrices.- A.9.1 The Jordan Canonical Form.- A.9.2 Decomposition of Symmetric Matrices.- A.9.3 The Choleski Decomposition of a Positive Definite Matrix.- A.10 Partitioned Matrices.- A.11 The Kronecker Product.- A.12 The vec and vech Operators and Related Matrices.- A.12.1 The Operators.- A.12.2 The Elimination, Duplication, and Commutation Matrices.- A.13 Vector and Matrix Differentiation.- A.14 Optimization of Vector Functions.- A.15 Problems.- Appendix B. Multivariate Normal and Related Distributions.- B.1 Multivariate Normal Distributions.- B.2 Related Distributions.- Appendix C. Convergence of Sequences of Random Variables and Asymptotic Distributions.- C.1 Concepts of Stochastic Convergence.- C.2 Asymptotic Properties of Estimators and Test Statistics.- C.3 Infinite Sums of Random Variables.- C.4 Maximum Likelihood Estimation.- C.5 Likelihood Ratio, Lagrange Multiplier, and Wald Tests.- Appendix D. Evaluating Properties of Estimators and Test Statistics by Simulation and Resampling Techniques.- D.1 Simulating a Multiple Time Series with VAR Generation Process.- D.2 Evaluating Distributions of Functions of Multiple Time Series by Simulation.- D.3 Evaluating Distributions of Functions of Multiple Time Series by Resampling.- Appendix E. Data Used for Examples and Exercises.- References.- List of Propositions and Definitions.- Index of Notation.- Author Index.