New Introduction to Multiple Time Series Analysis

Description

When I worked on my Introduction to Multiple Time Series Analysis (Lutk ] ]- pohl (1991)), a suitable textbook for this ?eld was not available. Given the great importance these methods have gained in applied econometric work, it is perhaps not surprising in retrospect that the book was quite successful. Now, almost one and a half decades later the ?eld has undergone substantial development and, therefore, the book does not cover all topics of my own courses on the subject anymore. Therefore, I started to think about a serious revision of the book when I moved to the European University Institute in Florence in 2002. Here in the lovely hills of ToscanyIhadthetimetothink about bigger projects again and decided to prepare a substantial revision of my previous book. Because the label Second Edition was already used for a previous reprint of the book, I decided to modify the title and thereby hope to signal to potential readers that signi?cant changes have been made relative to my previous multiple time series book." Title Page Copyright Page Preface Table of contents 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 Part 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 Autocovariances of a VAR(1) Process Autocovariances of a Stable VAR(p) Process Autocorrelations of a Stable VAR(p) Process 2.2 Forecasting 2.2.1 The Loss Function 2.2.2 Point Forecasts Conditional Expectation Linear Minimum MSE Predictor 2.2.3 Interval Forecasts and Forecast Regions 2.3 Structural Analysis with VAR Models 2.3.1 Granger-Causality, Instantaneous Causality, and Multi-Step Causality Definitions of Causality Characterization of Granger-Causality Characterization of Instantaneous Causality Discussion of Instantaneous and Granger-Causality Changing the Information Set 2.3.2 Impulse Response Analysis Responses to Forecast Errors Responses to Orthogonal Impulses 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 Causality 3.6.1 A Wald Test for Granger-Causality 3.6.2 An Example 3.6.3 Testing for Instantaneous Causality 3.6.4 Testing for Multi-Step 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 Theoretical Results An Illustrative Example 4.4.3 Portmanteau Tests 4.4.4 Lagrange Multiplier 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 Chow Tests 4.6.2 Forecast Tests for Structural Change A Test Statistic Based on one Forecast Period A Test Based on Several Forecast Periods 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 Asymptotic Properties Comparison of LS and Restricted EGLS Estimators 5.2.3 Maximum Likelihood Estimation 5.2.4 Constraints for Individual Equations 5.2.5 Restrictions for 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 Elimination of Complete Matrices Top-Down Strategy Bottom-Up Strategy Sequential Elimination of Regressors 5.2.9 Model Checking Residual Autocovariances and Autocorrelations Portmanteau Tests LM Test for Residual Autocorrelation Other Checks of Restricted Models 5.2.10 An Example 5.3 VAR Processes with Nonlinear Parameter Restrictions 5.4 Bayesian Estimation 5.4.1 Basic Terms and Notation 5.4.2 Normal Priors for the Parameters of a Gaussian VAR Process 5.4.3 The Minnesota or Litterman Prior 5.4.4 Practical Considerations 5.4.5 An Example 5.4.6 Classical versus Bayesian Interpretation of α in Forecasting and Structural Analysis 5.5 Exercises 5.5.1 Algebraic Exercises 5.5.2 Numerical Problems Part II Cointegrated Processes 6 Vector Error Correction Models 6.1 Integrated Processes 6.2 VAR Processes with Integrated Variables 6.3 Cointegrated Processes, Common Stochastic Trends, and Vector Error Correction Models 6.4 Deterministic Terms in Cointegrated Processes 6.5 Forecasting Integrated and Cointegrated Variables 6.6 Causality Analysis 6.7 Impulse Response Analysis 6.8 Exercises 7 Estimation of Vector Error Correction Models 7.1 Estimation of a Simple Special Case VECM 7.2 Estimation of General VECMs 7.2.1 LS Estimation Discussion of the Proof of Proposition 7.1 7.2.2 EGLS Estimation of the Cointegration Parameters 7.2.3 ML Estimation The Estimator Asymptotic Properties of the ML Estimator 7.2.4 Including Deterministic Terms 7.2.5 Other Estimation Methods for Cointegrated Systems Two-Stage Estimation 7.2.6 An Example 7.3 Estimating VECMs with Parameter Restrictions 7.3.1 Linear Restrictions for the Cointegration Matrix 7.3.2 Linear Restrictions for the Short-Run and Loading Parameters 7.3.3 An Example 7.4 Bayesian Estimation of Integrated Systems 7.4.1 The Model Setup 7.4.2 The Minnesota or Litterman Prior 7.4.3 An Example 7.5 Forecasting Estimated Integrated and Cointegrated Systems 7.6 Testing for Granger-Causality 7.6.1 The Noncausality Restrictions 7.6.2 Problems Related to Standard Wald Tests 7.6.3 A Wald Test Based on a Lag Augmented VAR 7.6.4 An Example 7.7 Impulse Response Analysis 7.8 Exercises 7.8.1 Algebraic Exercises 7.8.2 Numerical Exercises 8 Specification of VECMs 8.1 Lag Order Selection 8.2 Testing for the Rank of Cointegration 8.2.1 A VECM without Deterministic Terms 8.2.2 A Nonzero Mean Term 8.2.3 A Linear Trend 8.2.4 A Linear Trend in the Variables and Not in the Cointegration Relations 8.2.5 Summary of Results and Other Deterministic Terms 8.2.6 An Example 8.2.7 Prior Adjustment for Deterministic Terms 8.2.8 Choice of Deterministic Terms 8.2.9 Other Approaches to Testing for the Cointegrating Rank 8.3 Subset VECMs 8.4 Model Diagnostics 8.4.1 Checking for Residual Autocorrelation Asymptotic Properties of Residual Autocovariances and Autocorrelations Portmanteau and LM Tests for Residual Autocorrelation 8.4.2 Testing for Nonnormality 8.4.3 Tests for Structural Change Chow Tests Prediction Tests for Structural Change 8.5 Exercises 8.5.1 Algebraic Exercises 8.5.2 Numerical Exercises Part III Structural and Conditional Models 9 Structural VARs and VECMs 9.1 Structural Vector Autoregressions 9.1.1 The A-Model 9.1.2 The B-Model 9.1.3 The AB-Model 9.1.4 Long-Run Restrictions a la Blanchard-Quah 9.2 Structural Vector Error Correction Models 9.3 Estimation of Structural Parameters 9.3.1 Estimating SVAR Models Computation of ML Estimates Estimation with Long-Run Restrictions a la Blanchard-Quah 9.3.2 Estimating Structural VECMs 9.4 Impulse Response Analysis and Forecast Error Variance Decomposition 9.5 Further Issues 9.6 Exercises 9.6.1 Algebraic Problems 9.6.2 Numerical Problems 10 Systems of Dynamic Simultaneous Equations 10.1 Background 10.2 Systems with Unmodelled Variables 10.2.1 Types of Variables 10.2.2 Structural Form, Reduced Form, Final Form 10.2.3 Models with Rational Expectations 10.2.4 Cointegrated Variables 10.3 Estimation 10.3.1 Stationary Variables 10.3.2 Estimation of Models with I(1) Variables 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 Part IV Infinite Order Vector Autoregressive Processes 11 Vector Autoregressive Moving Average Processes 11.1 Introduction 11.2 Finite Order Moving Average Processes 11.3 VARMA Processes 11.3.1 The Pure MA and Pure VAR Representations of a VARMA Process 11.3.2 A VAR(1) Representation of a VARMA Process 11.4 The Autocovariances and Autocorrelations of a VARMA(p, q) Process 11.5 Forecasting VARMA Processes 11.6 Transforming and Aggregating VARMA Processes 11.6.1 Linear Transformations of VARMA Processes 11.6.2 Aggregation of VARMA Processes 11.7 Interpretation of VARMA Models 11.7.1 Granger-Causality 11.7.2 Impulse Response Analysis 11.8 Exercises 12 Estimation of VARMA Models 12.1 The Identification Problem 12.1.1 Nonuniqueness of VARMA Representations 12.1.2 Final Equations Form and Echelon Form 12.1.3 Illustrations 12.2 The Gaussian Likelihood Function 12.2.1 The Likelihood Function of an MA(1) Process 12.2.2 The MA(q) Case 12.2.3 The VARMA(1, 1) Case 12.2.4 The General VARMA(p, q) Case 12.3 Computation of the ML Estimates 12.3.1 The Normal Equations 12.3.2 Optimization Algorithms 12.3.3 The Information Matrix 12.3.4 Preliminary Estimation 12.3.5 An Illustration 12.4 Asymptotic Properties of the ML Estimators 12.4.1 Theoretical Results 12.4.2 A Real Data Example 12.5 Forecasting Estimated VARMA Processes 12.6 Estimated Impulse Responses 12.7 Exercises 13 Specification and Checking the Adequacy of VARMA Models 13.1 Introduction 13.2 Specification of the Final Equations Form 13.2.1 A Specification Procedure 13.2.2 An Example 13.3 Specification of Echelon Forms 13.3.1 A Procedure for Small Systems 13.3.2 A Full Search Procedure Based on Linear Least Squares Computations An Example 13.3.3 Hannan-Kavalieris Procedure 13.3.4 Poskitt’s Procedure 13.4 Remarks on Other Specification Strategies for VARMA Models 13.5 Model Checking 13.5.1 LM Tests 13.5.2 Residual Autocorrelations and Portmanteau Tests 13.5.3 Prediction Tests for Structural Change 13.6 Critique of VARMA Model Fitting 13.7 Exercises 14 Cointegrated VARMA Processes 14.1 Introduction 14.2 The VARMA Framework for I(1) Variables 14.2.1 Levels VARMA Models 14.2.2 The Reverse Echelon Form 14.2.3 The Error Correction Echelon Form 14.3 Estimation 14.3.1 Estimation of ARMARE Models 14.3.2 Estimation of EC-ARMARE Models 14.4 Specification of EC-ARMARE Models 14.4.1 Specification of Kronecker Indices 14.4.2 Specification of the Cointegrating Rank 14.5 Forecasting Cointegrated VARMA Processes 14.6 An Example 14.7 Exercises 14.7.1 Algebraic Exercises 14.7.2 Numerical Exercises 15 Fitting Finite Order VAR Models to Infinite Order Processes 15.1 Background 15.2 Multivariate Least Squares Estimation 15.3 Forecasting 15.3.1 Theoretical Results 15.3.2 An Example 15.4 Impulse Response Analysis and Forecast Error Variance Decompositions 15.4.1 Asymptotic Theory 15.4.2 An Example 15.5 Cointegrated Infinite Order VARs 15.5.1 The Model Setup 15.5.2 Estimation 15.5.3 Testing for the Cointegrating Rank 15.6 Exercises Part V Time Series Topics 16 Multivariate ARCH and GARCH Models 16.1 Background 16.2 Univariate GARCH Models 16.2.1 Definitions 16.2.2 Forecasting 16.3 Multivariate GARCH Models 16.3.1 Multivariate ARCH 16.3.2 MGARCH 16.3.3 Other Multivariate ARCH and GARCH Models 16.4 Estimation 16.4.1 Theory 16.4.2 An Example 16.5 Checking MGARCH Models 16.5.1 ARCH-LM and ARCH-Portmanteau Tests 16.5.2 LM and Portmanteau Tests for Remaining ARCH 16.5.3 Other Diagnostic Tests 16.5.4 An Example 16.6 Interpreting GARCH Models 16.6.1 Causality in Variance 16.6.2 Conditional Moment Profiles and Generalized Impulse Responses 16.7 Problems and Extensions 16.8 Exercises 17 Periodic VAR Processes and Intervention Models 17.1 Introduction 17.2 The VAR(p) Model with Time Varying Coefficients 17.2.1 General Properties 17.2.2 ML Estimation 17.3 Periodic Processes 17.3.1 A VAR Representation with Time Invariant Coefficients 17.3.2 ML Estimation and Testing for Time Varying Coefficients All Coefficients Time Varying All Coefficients Time Invariant Time Invariant White Noise Time Invariant Covariance Structure LR Tests Testing a Model with Time Varying Error Covariance Matrix Only Against One Where All Coefficients Are Time Varying Testing a Time Invariant Model Against One with Time Varying Error Covariance 17.3.3 An Example 17.3.4 Bibliographical Notes and Extensions 17.4 Intervention Models 17.4.1 Interventions in the Intercept Model 17.4.2 A Discrete Change in the Mean 17.4.3 An Illustrative Example 17.4.4 Extensions and References 17.5 Exercises 18 State Space Models 18.1 Background 18.2 State Space Models 18.2.1 The Model Setup A Finite Order VAR Process A VARMA(p, q) Process The VARX Model Systematic Sampling and Aggregation Structural Time Series Models Factor Analytic Models VARX Models with Systematically Varying Coefficients Random Coefficients VARX M 18.2.2 More General State Space Models 18.3 The Kalman Filter 18.3.1 The Kalman Filter Recursions Assumptions for the State Space Model The Recursions Computational Aspects and Extensions 18.3.2 Proof of the Kalman Filter Recursions 18.4 Maximum Likelihood Estimation of State Space Models 18.4.1 The Log-Likelihood Function 18.4.2 The Identification Problem 18.4.3 Maximization of the Log-Likelihood Function The Gradient of the Log-Likelihood The Information Matrix Discussion of the Scoring Algorithm 18.4.4 Asymptotic Properties of the ML Estimator 18.5 A Real Data Example 18.6 Exercises 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 and Orthogonal Complements 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 Elimination, Duplication, and Commutation Matrices A.13 Vector and Matrix Differentiation A.14 Optimization of Vector Functions A.15 Problems B Multivariate Normal and Related Distributions B.1 Multivariate Normal Distributions B.2 Related Distributions C Stochastic Convergence and Asymptotic Distributions C.1 Concepts of Stochastic Convergence C.2 Order in Probability C.3 Infinite Sums of Random Variables C.4 Laws of Large Numbers and Central Limit Theorems C.5 Standard Asymptotic Properties of Estimators and Test Statistics C.6 Maximum Likelihood Estimation C.7 Likelihood Ratio, Lagrange Multiplier, and Wald Tests C.8 Unit Root Asymptotics C.8.1 Univariate Processes C.8.2 Multivariate Processes 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 Resampling Methods References Index of Notation Author Index Subject Index