New Introduction to Multiple Time Series Analysis

Transcription

1 Helmut Lütkepohl New Introduction to Multiple Time Series Analysis With 49 Figures and 36 Tables Springer

2 Contents 1 Introduction Objectives of Analyzing Multiple Time Series Some Basics Vector Autoregressive Processes Outline of the Following Chapters 5 Part I Finite Order Vector Autoregressive Processes 2 Stable Vector Autoregressive Processes Basic Assumptions and Properties of VAR Processes Stable VAR(p) Processes The Moving Average Representation of a VAR Process Stationary Processes Computation of Autocovariances and Autocorrelations of Stable VAR Processes Forecasting The Loss Function Point Forecasts Interval Forecasts and Forecast Regions Structural Analysis with VAR Models Granger-Causality, Instantaneous Causality, and Multi-Step Causality Impulse Response Analysis Forecast Error Variance Decomposition Remarks on the Interpretation of VAR Models Exercises 66 3 Estimation of Vector Autoregressive Processes Introduction Multivariate Least Squares Estimation 69

3 XII Contents The Estimator Asymptotic Properties of the Least Squares Estimator An Example Small Sample Properties of the LS Estimator Least Squares Estimation with Mean-Adjusted Data and Yule-Walker Estimation Estimation when the Process Mean Is Known Estimation of the Process Mean Estimation with Unknown Process Mean The Yule-Walker Estimator An Example Maximum Likelihood Estimation The Likelihood Function The ML Estimators Properties of the ML Estimators Forecasting with Estimated Processes General Assumptions and Results The Approximate MSE Matrix An Example A Small Sample Investigation Testing for Causality A Wald Test for Granger-Causality An Example Testing for Instantaneous Causality Testing for Multi-Step Causality The Asymptotic Distributions of Impulse Responses and Forecast Error Variance Decompositions The Main Results Proof of Proposition An Example Investigating the Distributions of the Impulse Responses by Simulation Techniques Exercises Algebraic Problems Numerical Problems VAR Order Selection and Checking the Model Adequacy Introduction A Sequence of Tests for Determining the VAR Order The Impact of the Fitted VAR Order on the Forecast MSE The Likelihood Ratio Test Statistic A Testing Scheme for VAR Order Determination An Example Criteria for VAR Order Selection 146

4 Contents XIII Minimizing the Forecast MSE Consistent Order Selection Comparison of Order Selection Criteria Some Small Sample Simulation Results Checking the Whiteness of the Residuals The Asymptotic Distributions of the Autocovariances and Autocorrelations of a White Noise Process The Asymptotic Distributions of the Residual Autocovariances and Autocorrelations of an Estimated VAR Process Portmanteau Tests Lagrange Multiplier Tests Testing for Nonnormality Tests for Nonnormality of a Vector White Noise Process Tests for Nonnormality of a VAR Process Tests for Structural Change Chow Tests Forecast Tests for Structural Change Exercises Algebraic Problems Numerical Problems 191 VAR Processes with Parameter Constraints Introduction Linear Constraints The Model and the Constraints LS, GLS, and EGLS Estimation Maximum Likelihood Estimation Constraints for Individual Equations Restrictions for the White Noise Covariance Matrix Forecasting Impulse Response Analysis and Forecast Error Variance Decomposition Specification of Subset VAR Models Model Checking An Example VAR Processes with Nonlinear Parameter Restrictions Bayesian Estimation Basic Terms and Notation Normal Priors for the Parameters of a Gaussian VAR Process The Minnesota or Litterman Prior Practical Considerations An Example 227

5 XIV Contents Classical versus Bayesian Interpretation of α in Forecasting and Structural Analysis Exercises Algebraic Exercises Numerical Problems 231 Part II Cointegrated Processes 6 Vector Error Correction Models Integrated Processes VAR Processes with Integrated Variables Cointegrated Processes, Common Stochastic Trends, and Vector Error Correction Models Deterministic Terms in Cointegrated Processes Forecasting Integrated and Cointegrated Variables Causality Analysis Impulse Response Analysis Exercises Estimation of Vector Error Correction Models Estimation of a Simple Special Case VECM Estimation of General VECMs LS Estimation EGLS Estimation of the Cointegration Parameters ML Estimation Including Deterministic Terms Other Estimation Methods for Cointegrated Systems An Example Estimating VECMs with Parameter Restrictions Linear Restrictions for the Cointegration Matrix Linear Restrictions for the Short-Run and Loading Parameters An Example Bayesian Estimation of Integrated Systems The Model Setup The Minnesota or Litterman Prior An Example _ Forecasting Estimated Integrated and Cointegrated Systems Testing for Granger-Causality The Noncausality Restrictions Problems Related to Standard Wald Tests A Wald Test Based on a Lag Augmented VAR An Example 32o 7.7 Impulse Response Analysis 3 21

6 Contents 7.8 Exercises Algebraic Exercises Numerical Exercises Specification of VECMs Lag Order Selection Testing for the Rank of Cointegration A VECM without Deterministic Terms A Nonzero Mean Term A Linear Trend A Linear Trend in the Variables and Not in the Cointegration Relations Summary of Results and Other Deterministic Terms An Example Prior Adjustment for Deterministic Terms Choice of Deterministic Terms Other Approaches to Testing for the Cointegrating Rank Subset VECMs Model Diagnostics Checking for Residual Autocorrelation Testing for Nonnormality Tests for Structural Change Exercises Algebraic Exercises Numerical Exercises 352 XV Part III Structural and Conditional Models 9 Structural VARs and VECMs Structural Vector Autoregressions The A-Model The B-Model The AB-Model Long-Run Restrictions à 1a Blanchard-Quah Structural Vector Error Correction Models Estimation of Structural Parameters Estimating SVAR Models Estimating Structural VECMs Impulse Response Analysis and Forecast Error Variance Decomposition Further Issues Exercises Algebraic Problems Numerical Problems 385

7 XVI Contents 10 Systems of Dynamic Simultaneous Equations Background Systems with Unmodelled Variables Types of Variables Structural Form, Reduced Form, Final Form Models with Rational Expectations Cointegrated Variables Estimation Stationary Variables Estimation of Models with I(1) Variables Remarks on Model Specification and Model Checking Forecasting Unconditional and Conditional Forecasts Forecasting Estimated Dynamic SEMs Multiplier Analysis Optimal Control Concluding Remarks on Dynamic SEMs Exercises 412 Part IV Infinite Order Vector Autoregressive Processes 11 Vector Autoregressive Moving Average Processes Introduction Finite Order Moving Average Processes VARMA Processes The Pure MA and Pure VAR Representations of a VARMA Process A VAR(l) Representation of a VARMA Process The Autocovariances and Autocorrelations of a VARMA(p, q) Process Forecasting VARMA Processes Transforming and Aggregating VARMA Processes Linear Transformations of VARMA Processes Aggregation of VARMA Processes Interpretation of VARMA Models Granger-Causality Impulse Response Analysis Exercises Estimation of VARMA Models The Identification Problem Nonuniqueness of VARMA Representations Final Equations Form and Echelon Form Illustrations 455

8 Contents XVII 12.2 The Gaussian Likelihood Function The Likelihood Function of an MA(1) Process The ΜΑ(ρ) Case The VARMA(1,1) Case The General VARMA(p,q) Case Computation of the ML Estimates The Normal Equations Optimization Algorithms The Information Matrix Preliminary Estimation An Illustration Asymptotic Properties of the ML Estimators Theoretical Results A Real Data Example Forecasting Estimated VARMA Processes Estimated Impulse Responses Exercises Specification and Checking the Adequacy of VARMA Models Introduction, Specification of the Final Equations Form A Specification Procedure An Example Specification of Echelon Forms A Procedure for Small Systems A Full Search Procedure Based on Linear Least Squares Computations Hannan-Kavalieris Procedure Poskitt's Procedure Remarks on Other Specification Strategies for VARMA Models Model Checking LM Tests Residual Autocorrelations and Portmanteau Tests Prediction Tests for Structural Change Critique of VARMA Model Fitting Exercises Cointegrated VARMA Processes Introduction The VARMA Framework for 1(1) Variables Levels VARMA Models The Reverse Echelon Form The Error Correction Echelon Form Estimation 521

9 XVIII Contents Estimation of ARMA Models Estimation of EC-ARMA Models Specification of EC-ARMA RE Models Specification of Kronecker Indices Specification of the Cointegrating Rank Forecasting Cointegrated VARMA Processes An Example Exercises Algebraic Exercises Numerical Exercises Fitting Finite Order VAR Models to Infinite Order Processes Background Multivariate Least Squares Estimation Forecasting Theoretical Results An Example Impulse Response Analysis and Forecast Error Variance Decompositions Asymptotic Theory An Example Cointegrated Infinite Order VARs The Model Setup Estimation Testing for the Cointegrating Rank Exercises m 552 Part V Time Series Topics 16 Multivariate ARCH and GARCH Models Background Univariate GARCH Models. : Definitions Forecasting Multivariate GARCH Models Multivariate ARCH MGARCH Other Multivariate ARCH and GARCH Models Estimation Theory An Example Checking MGARCH Models ARCH-LM and ARCH-Portmanteau Tests 576

10 Contents XIX LM and Portmanteau Tests for Remaining ARCH Other Diagnostic Tests An Example Interpreting GARCH Models Causality in Variance Conditional Moment Profiles and Generalized Impulse Responses Problems and Extensions Exercises Periodic VAR Processes and Intervention Models Introduction The VAR(p) Model with Time Varying Coefficients General Properties ML Estimation Periodic Processes A VAR Representation with Time Invariant Coefficients ML Estimation and Testing for Time Varying Coefficients An Example Bibliographical Notes and Extensions Intervention Models Interventions in the Intercept Model A Discrete Change in the Mean An Illustrative Example Extensions and References Exercises State Space Models Background State Space Models The Model Setup More General State Space Models The Kalman Filter The Kalman Filter Recursions Proof of the Kalman Filter Recursions Maximum Likelihood Estimation of State Space Models The Log-Likelihood Function The Identification Problem Maximization of the Log-Likelihood Function Asymptotic Properties of the ML Estimator A Real Data Example Exercises 641

11 XX Contents Appendix Vectors and Matrices 645 A.I Basic Definitions 645 A.2 Basic Matrix Operations 646 A.3 The Determinant 647 A.4 The Inverse, the Adjoint, and Generalized Inverses 649 A.4.1 Inverse and Adjoint of a Square Matrix 649 A.4.2 Generalized Inverses 650 A.5 The Rank 651 A.6 Eigenvalues and -vectors - Characteristic Values and Vectors A.7 The Trace 653 A.8 Some Special Matrices and Vectors 653 A.8.1 Idempotent and Nilpotent Matrices 653 A.8.2 Orthogonal Matrices and Vectors and Orthogonal Complements 654 A.8.3 Definite Matrices and Quadratic Forms 655 A.9 Decomposition and Diagonalization of Matrices 656 A.9.1 The Jordan Canonical Form 656 A.9.2 Decomposition of Symmetric Matrices 658 A.9.3 The Choleski Decomposition of a Positive Definite Matrix 658 A.10 Partitioned Matrices 659 A.ll The Kronecker Product 660 A. 12 The vec and vech Operators and Related Matrices 661 A.12.1 The Operators 661 A.12.2 Elimination, Duplication, and Commutation Matrices A.13 Vector and Matrix Differentiation 664 A.14 Optimization of Vector Functions 671 A.15 Problems 675 Multivariate Normal and Related Distributions 677 B.1 Multivariate Normal Distributions 677 B.2 Related Distributions 678 Stochastic Convergence and Asymptotic Distributions 681 C.3 Concepts of Stochastic Convergence 681 C.2 Order in Probability 684 C.3 Infinite Sums of Random Variables 685 C.4 Laws of Large Numbers and Central Limit Theorems 689 C.5 Standard Asymptotic Properties of Estimators and Test Statistics 692 C.6 Maximum Likelihood Estimation 693 C.7 Likelihood Ratio, Lagrange Multiplier, and Wald Tests 694

12 Contents XXI C.8 Unit Root Asymptotics 698 C.8.1 Univariate Processes 698 C.8.2 Multivariate Processes 703 D Evaluating Properties of Estimators and Test Statistics by Simulation and Resampling Techniques 707 D.I Simulating a Multiple Time Series with VAR Generation Process 707 D.2 Evaluating Distributions of Functions of Multiple Time Series by Simulation 708 D.3 Resampling Methods 709 References 713 Index of Notation 733 Author Index 741 Subject Index 747