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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
2.2 Forecasting
2.2.1 The Loss Function
2.2.2 Point Forecasts
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
2.3.2 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
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
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
5.2.3 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
5.2.9 Model Checking
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
7.2.2 EGLS Estimation of the Cointegration Parameters
7.2.3 ML Estimation
7.2.4 Including Deterministic Terms
7.2.5 Other Estimation Methods for Cointegrated Systems
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
8.4.2 Testing for Nonnormality
8.4.3 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
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
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
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
18.2.2 More General State Space Models
18.3 The Kalman Filter
18.3.1 The Kalman Filter Recursions
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
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