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 1 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 Autocovariates 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 Normality

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 Decompositions

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 2 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 Numeric Exercises

8 Specification of VECMs

8.1 Lag Order Specification

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 of Deterministic Terms

8.2.8 Choice of Deterministic Terms

8.2.9 Other Approaches to Testing for teh 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 3 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 à 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

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 4 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 Reponse 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 Echelong 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

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’ Procedure

13.4 Remarks on Other Specification Strategies for VARMA Models

13.5 Model Checking

13.2.1 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 of VARMA Models

14.2.2 The Reverse Echelon Form

14.2.3 The Error Correction Echelong Form

14.3 Estimation

14.3.1 Estimation of ARMA_{RE} Models

14.3.2 Estimation of EC-ARMA_{RE} Models

14.4 Specification of EC-ARMA

_{RE} 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 5 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