Multivariate Modelling of Non-Stationary Economic Time Series

Preface

Contents

1 Introduction

References

2 Multivariate Time Series

2.1 Introduction

2.2 Stationarity

2.2.1 Strict Stationarity

2.2.2 Strict (Joint Distribution) Stationarity

2.2.3 Describing Covariance Non-Stationarity: Parametric Models

2.2.4 The White Noise Process

2.2.4.1 White Noise

2.2.5 The Moving Average Process

2.2.6 Wold's Representation Theorem

2.2.7 The Autoregressive Process

2.2.8 Lag Polynomials and Their Roots

2.2.8.1 The Lag Operator and Lag Polynomials

2.2.9 Non-Stationarity and the Autoregressive Process

2.2.9.1 Stationarity of an Autoregressive Process

2.2.10 The Random Walk and the Unit Root

2.2.10.1 The Random Walk Process

2.2.10.2 Differencing and Stationarity

2.2.10.3 The Random Walk as a Stochastic Trend

2.2.10.4 The Random Walk with Drift

2.2.11 The Autoregressive Moving Average Process and Operator Inversion

2.2.11.1 Illustration of Operator Inversion

2.2.12 Testing Stationarity in Single Series

2.2.12.1 Reparameterizing the Autoregressive Model

2.2.12.2 Semi-parametric Methods

2.3 Multivariate Time Series Models

2.3.1 The VAR and VECM Models

2.3.2 The VMA Model

2.3.3 Estimation

2.3.4 The Procedure

2.4 Persistence

2.4.1 Reparameterizing the VAR

2.4.2 Long-Run Growth Models

2.5 Impulse Responses

2.5.1 Impulse Responses and VAR Models

2.5.2 Orthogonality and the IRF

2.5.3 The Choleski Decomposition

2.5.4 IRFs in the General VAR Case

2.5.4.1 IRFs and Time Series Identification

2.6 Variance Decomposition

2.6.1 Prediction Errors and Forecasts

2.7 Conclusion

References

3 Cointegration

3.1 Cointegration of the VMA, VAR and VECM

3.1.1 The Granger Representation Theorem: Systems Representation of Cointegrated Variables

3.1.1.1 Cointegration Starting from a VMA and Deriving VAR and VECM Forms

3.1.2 VARMA Representation of CI(1,1) Variables

3.2 The Smith-McMillan-Yoo Form

3.2.1 Using the Smith Form to Reparameterize a Finite Order VMA

3.2.1.1 Reparameterizing a VMA in Differences

3.2.2 The SM Form in General Applied to a Rational VMA: The SMY Form

3.2.2.1 The SMY Form and Cointegration of Order (1,1)

3.2.3 Cointegrating Vectors in the VMA and VAR Representations of CI(1,1)

3.2.3.1 A(L) as Partial Inverse of C(L) in the CI(1,1) Case

3.2.4 Equivalence of VAR and VMA Representations in the CI(1,1) Case

3.3 Johansen's VAR Representation of Cointegration

3.3.1 Cointegration Assuming Integration of Order 1

3.3.1.1 Cointegrated VARs with I(1) Processes

3.3.2 Conditions for the VAR Process to be I(1) and Cointegrated

3.3.2.1 Discussion

3.3.3 The MA Representation

3.4 Cointegration with Intercept and Trend

3.4.1 Levels Process for the VECM with Intercept

3.4.2 Levels Process for the VECM with Higher Order Trends and Other Deterministic Terms

3.5 Alternative Representations of the Cointegrating VAR, VMA and VARMA

3.5.1 The Sargan-Bézout Factorization

3.5.2 A VAR(1) Representation of a VMA(1) Model Under Cointegration

3.6 Single Equation Implications and Examples

3.6.1 Cointegration: Static Equilibrium with I(1) Variables

3.6.2 ADL Models, Cointegration and Equilibrium

3.6.2.1 ADL Models, Cointegration and Equilibrium

3.6.2.2 Example

3.7 Conclusion

References

4 Testing for Cointegration: Standard and Non-Standard Conditions

4.1 Introduction

4.2 Maximum Likelihood Estimation

4.3 Johansen's Approach to Testing for Cointegration in Systems

4.3.1 Testing for Reduced Rank and Estimating Cointegrating Vectors

4.3.1.1 Review of Source of Reduced Rank in Cointegrated Systems

4.3.1.2 Using Eigenvalues and Eigenvectors in Cointegration Analysis

4.3.2 The Removal of Nuisance Parameters

4.3.3 Estimating Potentially Cointegrating Relations

4.3.4 Testing Cointegrating Rank

4.4 Performing Tests of Cointegrating Rank in the Presence of Deterministic Components

4.4.1 Intercepts and Trends and the Preliminary Regressions to Remove Nuisance Parameters

4.5 Examples of Tests of Cointegration in VAR Models

4.5.1 Special Cases of the Johansen Test

4.5.2 Empirical Examples of the Johansen Test

4.6 The VMA and VARMA Form

4.6.1 The Removal of Nuisance Parameters

4.6.2 The Impact of the VMA Structure on the Tests of Cointegration

4.6.3 A Simple Multi-Cointegration Extension

4.7 Quasi-Maximum Likelihood Estimator (QMLE) and Non-Gaussianity

4.7.1 Further Evidence on the Performance of the Johansen Test

4.7.2 Breaks in Structure

4.7.3 Outliers in the Mean Equation and the Johansen Trace Test

4.8 Conclusion

References

5 Structure and Evaluation

5.1 An Introduction to Exogeneity

5.1.1 Conditional Models and Testing for Cointegration and Exogeneity

5.1.2 Cointegration and Exogeneity

5.1.3 Tests of Long-Run Exogeneity

5.2 Identification

5.2.1 I(0) Systems and Some Preliminaries

5.2.1.1 The Cointegration Case

5.2.2 A Simple Indirect Procedure for Generic Identification

5.2.3 Johansen Identification Conditions

5.2.4 Boswijk Conditions and Observational Equivalence

5.2.5 Hunter's Conditions for Identification

5.2.6 An Example of Empirical and Generic Identification

5.3 Exogeneity and Identification

5.3.1 Empirical Examples

5.4 Impulse Response Functions

5.4.1 The Cointegration Case

5.4.2 IRF of a Bivariate VAR(1)

5.4.3 Lütkepohl's Method

5.5 Forecasting in Cointegrated Systems

5.5.1 VMA Analysis

5.5.2 Forecasting from the VAR

5.5.3 The Mechanics of Forecasting from a VECM

5.5.4 Forecast Performance

5.5.4.1 Lin and Tsay

5.5.4.2 Forecast Evaluation

5.5.4.3 Other Issues Relevant to Forecasting Performance in Practice

5.6 Conclusion

References

6 Testing in VECMs with Small Samples

6.1 Introduction

6.2 Testing for Cointegrating Rank in Finite Samples

6.2.1 Bartlett Correction Factor for the Trace Test

6.2.2 The Bootstrap p-Value Test

6.3 Testing Linear Restrictions on ?

6.3.1 A Monte Carlo Experiment

6.3.1.1 Some Simulation Results

6.3.1.2 The Probability of a Type II Error

6.4 An Empirical Application

6.5 Conclusion

References

7 Heteroscedasticity and Multivariate Volatility

7.1 Introduction

7.2 VAR Models for Multivariate Heteroscedasticity

7.2.1 The MGARCH-VECM in Systems Form

7.2.1.1 The Model

7.2.1.2 The Disturbance Variance-Covariance Matrix

7.2.2 The VAR-GARCH FIML (Full Information Maximum Likelihood) Approach

7.2.3 The Optimization Problem

7.2.4 An Example Estimating the Variance by BEKK

7.2.4.1 Cointegration Testing and the Mean Specification

7.2.5 BEKK Estimation of the Variance Equation

7.3 Estimation of the Transformed Mean Equation

7.3.1 The Stacked GLS Problem

7.4 The FWL Simplification to the Vectorized System

7.4.1 Purging the Data Equation by Equation

7.4.1.1 Adding Lagged Differences

7.5 Testing for Cointegration Using the GLS Transformed Data

7.6 Dynamic Heteroscedasticity and Market Imperfection

7.7 Conclusion

References

8 Models with Alternative Orders of Integration

8.1 Introduction

8.2 Cointegration Mixing I(0) and I(1) Series

8.2.1 Mixing I(0) and I(1) Variables

8.3 Some Examples

8.4 Inference and Estimation When Series Are Not I(1)

8.4.1 Relations Between I(1) and I(2) Variables

8.4.2 Cointegration When Series Are I(2)

8.4.2.1 The Johansen Procedure for Testing Cointegrating Rank with I(2) Variables

8.4.2.2 An Example of I(2)

8.5 Modified Estimators and Fractional Cointegration

8.5.1 Fractional Integration

8.5.2 Fractional Cointegration

8.5.3 Cointegration Testing and Selection of the Difference Order

8.6 Conclusion

References

9 The Structural Analysis of Time Series

9.1 Introduction

9.2 Cointegration and Models of Expectations

9.2.1 Linear Quadratic Adjustment Cost Models

9.2.2 Cointegration Solutions to Forward Behaviour with n2 Weakly Exogenous Variables

9.2.3 Estimation and Inference

9.2.4 The Effect of Cointegration on Solutions to Rational Expectations Models

9.2.4.1 Cointegration, Exogeneity and the VARMA

9.2.4.2 Rational Expectations and Smith-McMillan Forms

9.3 Singular Spectral Analysis

9.3.1 The Relation Between SSA and TSA

9.3.2 The Singular Spectral Analysis

9.3.3 Multivariate Singular Spectral Analysis

9.3.4 Difference Stationarity, Cointegration and Economic Time Series

9.3.5 Forecasting, Missing Data and Structural Change

9.4 Structural Time Series Models

9.4.1 State Space Form

9.4.2 Structural Time Series

9.4.3 The Multivariate Case

9.4.4 Stochastic Trends and Cointegration

9.4.5 Further Developments

9.5 Further Methods

9.5.1 Factor Models

9.5.2 Non-Linear Error Correction Model

9.5.3 Wavelets

9.6 Conclusion

References

Appendix A Matrix Preliminaries

A.1 Elementary Row Operations and Elementary Matrices

A.2 Unimodular Matrices

Appendix B Matrix Algebra for B:Engle and Granger1987 Representation

B.1 Determinant/Adjoint Representation of a Polynomial Matrix

B.2 Expansions of the Determinant and Adjoint About z[ 0,1 ]

B.3 Drawing Out a Factor of z from a Reduced Rank Matrix Polynomial

Application to Lag Polynomial to Draw Out Unit Root Factor

Appendix C Johansen's Procedure as a Maximum Likelihood Procedure

Appendix D The Maximum Likelihood Procedure in Terms of Canonical Correlations

Appendix E Distribution Theory

E.1 Some Univariate Theory

E.2 Vector Processes and Cointegration

E.3 Testing the Null Hypothesis of Non-Cointegration

E.4 Testing a Null Hypothesis of Non-Zero Rank

E.5 Distribution Theory When There Are Deterministic Trends in the Data

E.5.1 Tables of Approximate Asymptotic and Finite Sample Distributions

E.6 Other Issues

E.6.1 The Maximal Eigenvalue Statistic

E.6.2 Sequential Testing and Model Selection

E.6.3 Partial Systems

Appendix F Estimation Under General Restrictions

Appendix G Proof of Identification Based on an Indirect Solution

Appendix H Generic Identification of Long-Run Parameters in Sect.5.3

Appendix I IRF MA Parameters for the Case in Sect.5.4.3

References

Bibliography

Index