Riassunto
This graduate level textbook deals with analyzing and forecasting multiple time series. It considers a wide range of multiple time series models and methods. The models include vector autoregressive, vector autoregressive moving average, cointegrated, and periodic processes as well as state space and dynamic simultaneous equations models. Least squares, maximum likelihood, and Bayesian methods are considered for estimating these models. Different procedures for model selection or specification are treated and a range of tests and criteria for evaluating the adequacy of a chosen model are introduced. The choice of point and interval forecasts is considered and impulse response analysis, dynamic multipliers as well as innovation accounting are presented as tools for structural analysis within the multiple time series context. This book is accessible to graduate students in business and economics. In addition, multiple time series courses in other fields such as statistics and engineering may be based on this book. Applied researchers involved in analyzing multiple time series may benefit from the book as it provides the background and tools for their task. It enables the reader to perform his or her analyses in a gap to the difficult technical literature on the topic.
Contenuti
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.
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