Contents

About This Book

About The Authors

Acknowledgment

Chapter 1: Introduction

Introduction

Ordinary Regression Models

Regression Models in Time Series Analysis

Time Series Models

Which Time Series Features to Model

Parameterized Models for Time Series

Chapter 2: Regression Analysis for Time Series Data

Introduction

The Data Series

Durbin-Watson Test Using PROC REG

Definition of the Durbin-Watson Test Statistic

Procedure Output

Cochrane-Orcutt Estimation

Conclusion

Chapter 3: Regression Analysis with Autocorrelated Errors

Introduction

Correction of Standard Errors with PROC AUTOREG

Adjustment of Standard Deviations by the Newey-West Method

Cochrane-Orcutt Estimation Using PROC AUTOREG

Simultaneous Estimation Using PROC AUTOREG

Conclusion

Chapter 4: Regression Models for Differenced Series

Introduction

Regression Model for the Differenced Series

Regression Results

Inclusion of the Lagged Independent Variable

Reverted Regression

Inclusion of the Lagged Independent Variable in the Model

Two Lags of the Independent Variables

Inclusion of the Lagged Dependent Variable in the Regression

How to Interpret a Model with a Lagged Dependent Variable

Conclusions about the Models in Chapters 2, 3, and 4

Chapter 5: Tests for Differencing Time Series

Introduction

Stationarity

Unit Roots

Dickey-Fuller Tests for Unit Roots

Simple Applications of the Dickey-Fuller Test

Augmented Dickey-Fuller Tests for Milk Production

KPSS Unit Root Tests

An Application of the KPSS Unit Root Test

Seasonal Differencing

Conclusion

Chapter 6: Models for Univariate Time Series

Introduction

Autocorrelations

Autoregressive Models

Moving Average Models

ARIMA Models

Infinite-Order Representations

Multiplicative Seasonal ARIMA Models

Information Criteria

Use of SAS to Estimate Univariate ARIMA Models

Conclusion

Chapter 7: Use of the VARMAX Procedure to Model Univariate Series

Introduction

Wage-Price Time Series

PROC VARMAX Applied to the Wage Series

PROC VARMAX Applied to the Differenced Wage Series

Estimation of the AR(2) Model

Check of the Fit of the AR(2) Model

PROC VARMAX Applied to the Price Series

PROC VARMAX Applied to the Number of Cows Series

PROC VARMAX Applied to the Series of Milk Production

A Simple Moving Average Model of Order 1

Conclusion

Chapter 8: Models for Multivariate Time Series

Introduction

Multivariate Time Series

VARMAX Models

Infinite-Order Representations

Correlation Matrix at Lag 0

VARMAX Models

VARMAX Building in Practice

Conclusion

Chapter 9: Use of the VARMAX Procedure to Model Multivariate Series

Introduction

Use of PROC VARMAX to Model Multivariate Time Series

Dickey-Fuller Tests for Differenced Series

Selection of Model Orders

Fit of a Fourth-Order Autoregressive Model

Estimation for the Parameters

Restriction of Insignificant Model Parameters

Residual Autocorrelation in a VARMA(2,0) Model

Cross-Correlation Significance

Portmanteau Tests

Distribution of the Residuals in a VARMA(2,0) Model

Identification of Outliers

Use of a VARMA Model for Milk Production and the Number of Cows

Analysis of the Standardized Series

Correlation Matrix of the Error Terms

The Model Fit

Properties of the Fitted Model

Conclusion

Chapter 10: Exploration of the Output

Introduction

Roots of the Fitted Second-Order Autoregressive Model

Forecasts

Lag 0 Correlation of the Error Terms

The Infinite-Order Representations

Plots of the Impulse Response

Accumulated Effects

Effects of Orthogonal Shocks

Conclusion

Chapter 11: Causality Tests for the Danish Egg Market

Introduction

The Danish Egg Market

Formulation of the VARMA Model for the Egg Market Data

Estimation Results

Model Fit

Causality Tests of the Total Market Series

Granger Causality Tests in the VARMAX Procedure

Causality Tests of the Production Series

Causality Tests That Use Extended Information Sets

Estimation of a Final Causality Model

Fit of the Final Model

Conclusion

Chapter 12: Bayesian Vector Autoregressive Models

Introduction

The Prior Covariance of the Autoregressive Parameter Matrices

The Prior Distribution for the Diagonal Elements

The Prior Distribution for the Off-Diagonal Elements

The BVAR Model in PROC VARMAX

Specific Parameters in the Prior Distribution

Further Shrinkage toward Zero

Application of the BVAR(1) Model

BVAR Models for the Egg Market

Conclusion

Chapter 13: Vector Error Correction Models

Introduction

The Error Correction Model

The Matrix Formulation of the Error Correction Model

The Long-Run Relation

A Simple Example: The Price of Potatoes in Ohio and Pennsylvania

A Simple Regression

Estimation of an Error Correction Model by PROC VARMAX

Dickey-Fuller Test Results

Estimated Error Correction Parameters

The αβT Matrix

Properties of the Estimated Model

The Autoregressive Terms in the Model

Theory for Testing Hypotheses on β Parameters

Tests of Hypotheses on the β Parameters Using PROC VARMAX

Tests for Two Restrictions on the β Parameters

Estimated α Parameters under the Restrictions

Tests of Hypotheses on the α Parameters by PROC VARMAX

The TEST Statement for Hypotheses on the α Parameters

The RESTRICT Statement for the β Parameters

Restrictions on Both α Parameters and β Parameters

Properties of the Final Model

Conclusion

Chapter 14: Cointegration

Introduction

Test for a Cointegration Relation in the Bivariate Case

Cointegration Test Using PROC VARMAX for Two Price Series

Cointegration Tests in a Five-Dimensional Series

Initial Estimates for the β Values

A Model with Rank 2

Use of the RESTRICT Statement to Determine the Form of the Model

Stock-Watson Test for Common Trends for Five Series

A Rank 4 Model for Five Series Specified with Restrictions

An Alternative Form of the Restrictions

Estimation of the Model Parameters by a RESTRICT Statement

Estimation with Restrictions on Both the α and β Parameters

Conclusion

Chapter 15: Univariate GARCH Models

Introduction

The GARCH Model

GARCH Models for a Univariate Financial Time Series

Use of PROC VARMAX to Fit a GARCH(1,1) Model

The Fitted Model

Use of PROC VARMAX to Fit an IGARCH Model

The Wage Series

Use of PROC VARMAX to Fit an AR(2)-GARCH(1,1) Model

The Conditional Variance Series

Other Forms of GARCH Models

The QGARCH Model

The TGARCH Model

The PGARCH Model

The EGARCH Model

Conclusion

Chapter 16: Multivariate GARCH Models

Introduction

Multivariate GARCH Models

The CCC Parameterization

The DCC Parameterization

The BEKK Parameterization

A Bivariate Example Using Two Quotations for Danish Stocks

Using the CCC Parameterization

Using the DCC Parameterization

Using the BEKK Parameterization

Using the CCC Bivariate Combination of Univariate TGARCH Models

Conclusion

Chapter 17: Multivariate VARMA-GARCH Models

Introduction

Multivariate VARMA-GARCH Models

The Wage-Price Time Series

A VARMA Model with a CCC-GARCH Model for the Residuals

A VARMA Model with a DCC-GARCH Model for the Residuals

Refinement of the Estimation Algorithm

The Final VARMA Model with DCC-GARCH Residuals

Conclusion

References

Index