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Multiple Time Series Modeling Using the SAS® VARMAX Procedure
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Multiple Time Series Modeling Using the SAS® VARMAX Procedure
by Anders Milhoj
Multiple Time Series Modeling Using the SAS VARMAX Procedure
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
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