O'Reilly logo

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

Multiple Time Series Modeling Using the SAS VARMAX Procedure

Book Description

Aimed at econometricians who have completed at least one course in time series modeling, Multiple Time Series Modeling Using the SAS VARMAX Procedure will teach you the time series analytical possibilities that SAS offers today. Estimations of model parameters are now performed in a split second. For this reason, working through the identifications phase to find the correct model is unnecessary. Instead, several competing models can be estimated, and their fit can be compared instantaneously. Consequently, for time series analysis, most of the Box and Jenkins analysis process for univariate series is now obsolete. The former days of looking at cross-correlations and pre-whitening are over, because distributed lag models are easily fitted by an automatic lag identification method. The same goes for bivariate and even multivariate models, for which PROC VARMAX models are automatically fitted. For these models, other interesting variations arise: Subjects like Granger causality testing, feedback, equilibrium, cointegration, and error correction are easily addressed by PROC VARMAX. One problem with multivariate modeling is that it includes many parameters, making parameterizations unstable. This instability can be compensated for by application of Bayesian methods, which are also incorporated in PROC VARMAX. Volatility modeling has now become a standard part of time series modeling, because of the popularity of GARCH models. Both univariate and multivariate GARCH models are supported by PROC VARMAX. This feature is especially interesting for financial analytics in which risk is a focus. This book teaches with examples. Readers who are analyzing a time series for the first time will find PROC VARMAX easy to use; readers who know more advanced theoretical time series models will discover that PROC VARMAX is a useful tool for advanced model building.

Table of Contents

  1. About This Book
  2. About The Authors
  3. Acknowledgment
  4. Chapter 1: Introduction
    1. Introduction
    2. Ordinary Regression Models
    3. Regression Models in Time Series Analysis
    4. Time Series Models
      1. Which Time Series Features to Model
      2. Parameterized Models for Time Series
  5. Chapter 2: Regression Analysis for Time Series Data
    1. Introduction
    2. The Data Series
    3. Durbin-Watson Test Using PROC REG
      1. Definition of the Durbin-Watson Test Statistic
      2. Procedure Output
    4. Cochrane-Orcutt Estimation
    5. Conclusion
  6. Chapter 3: Regression Analysis with Autocorrelated Errors
    1. Introduction
    2. Correction of Standard Errors with PROC AUTOREG
    3. Adjustment of Standard Deviations by the Newey-West Method
    4. Cochrane-Orcutt Estimation Using PROC AUTOREG
    5. Simultaneous Estimation Using PROC AUTOREG
    6. Conclusion
  7. Chapter 4: Regression Models for Differenced Series
    1. Introduction
    2. Regression Model for the Differenced Series
      1. Regression Results
      2. Inclusion of the Lagged Independent Variable
    3. Reverted Regression
    4. Inclusion of the Lagged Independent Variable in the Model
    5. Two Lags of the Independent Variables
    6. Inclusion of the Lagged Dependent Variable in the Regression
    7. How to Interpret a Model with a Lagged Dependent Variable
    8. Conclusions about the Models in Chapters 2, 3, and 4
  8. Chapter 5: Tests for Differencing Time Series
    1. Introduction
    2. Stationarity
    3. Unit Roots
    4. Dickey-Fuller Tests for Unit Roots
    5. Simple Applications of the Dickey-Fuller Test
    6. Augmented Dickey-Fuller Tests for Milk Production
    7. KPSS Unit Root Tests
    8. An Application of the KPSS Unit Root Test
    9. Seasonal Differencing
    10. Conclusion
  9. Chapter 6: Models for Univariate Time Series
    1. Introduction
    2. Autocorrelations
    3. Autoregressive Models
    4. Moving Average Models
    5. ARIMA Models
      1. Infinite-Order Representations
      2. Multiplicative Seasonal ARIMA Models
    6. Information Criteria
    7. Use of SAS to Estimate Univariate ARIMA Models
    8. Conclusion
  10. Chapter 7: Use of the VARMAX Procedure to Model Univariate Series
    1. Introduction
    2. Wage-Price Time Series
    3. PROC VARMAX Applied to the Wage Series
    4. PROC VARMAX Applied to the Differenced Wage Series
    5. Estimation of the AR(2) Model
    6. Check of the Fit of the AR(2) Model
    7. PROC VARMAX Applied to the Price Series
    8. PROC VARMAX Applied to the Number of Cows Series
    9. PROC VARMAX Applied to the Series of Milk Production
    10. A Simple Moving Average Model of Order 1
    11. Conclusion
  11. Chapter 8: Models for Multivariate Time Series
    1. Introduction
    2. Multivariate Time Series
      1. VARMAX Models
      2. Infinite-Order Representations
    3. Correlation Matrix at Lag 0
    4. VARMAX Models
    5. VARMAX Building in Practice
    6. Conclusion
  12. Chapter 9: Use of the VARMAX Procedure to Model Multivariate Series
    1. Introduction
    2. Use of PROC VARMAX to Model Multivariate Time Series
      1. Dickey-Fuller Tests for Differenced Series
      2. Selection of Model Orders
    3. Fit of a Fourth-Order Autoregressive Model
      1. Estimation for the Parameters
      2. Restriction of Insignificant Model Parameters
    4. Residual Autocorrelation in a VARMA(2,0) Model
      1. Cross-Correlation Significance
      2. Portmanteau Tests
    5. Distribution of the Residuals in a VARMA(2,0) Model
    6. Identification of Outliers
    7. Use of a VARMA Model for Milk Production and the Number of Cows
      1. Analysis of the Standardized Series
      2. Correlation Matrix of the Error Terms
      3. The Model Fit
      4. Properties of the Fitted Model
    8. Conclusion
  13. Chapter 10: Exploration of the Output
    1. Introduction
    2. Roots of the Fitted Second-Order Autoregressive Model
    3. Forecasts
    4. Lag 0 Correlation of the Error Terms
    5. The Infinite-Order Representations
      1. Plots of the Impulse Response
      2. Accumulated Effects
      3. Effects of Orthogonal Shocks
    6. Conclusion
  14. Chapter 11: Causality Tests for the Danish Egg Market
    1. Introduction
    2. The Danish Egg Market
    3. Formulation of the VARMA Model for the Egg Market Data
      1. Estimation Results
      2. Model Fit
    4. Causality Tests of the Total Market Series
    5. Granger Causality Tests in the VARMAX Procedure
    6. Causality Tests of the Production Series
    7. Causality Tests That Use Extended Information Sets
    8. Estimation of a Final Causality Model
    9. Fit of the Final Model
    10. Conclusion
  15. Chapter 12: Bayesian Vector Autoregressive Models
    1. Introduction
    2. The Prior Covariance of the Autoregressive Parameter Matrices
      1. The Prior Distribution for the Diagonal Elements
      2. The Prior Distribution for the Off-Diagonal Elements
    3. The BVAR Model in PROC VARMAX
    4. Specific Parameters in the Prior Distribution
      1. Further Shrinkage toward Zero
      2. Application of the BVAR(1) Model
    5. BVAR Models for the Egg Market
    6. Conclusion
  16. Chapter 13: Vector Error Correction Models
    1. Introduction
    2. The Error Correction Model
      1. The Matrix Formulation of the Error Correction Model
      2. The Long-Run Relation
    3. A Simple Example: The Price of Potatoes in Ohio and Pennsylvania
      1. A Simple Regression
      2. Estimation of an Error Correction Model by PROC VARMAX
      3. Dickey-Fuller Test Results
    4. Estimated Error Correction Parameters
      1. The αβT Matrix
      2. Properties of the Estimated Model
      3. The Autoregressive Terms in the Model
    5. Theory for Testing Hypotheses on β Parameters
    6. Tests of Hypotheses on the β Parameters Using PROC VARMAX
      1. Tests for Two Restrictions on the β Parameters
      2. Estimated α Parameters under the Restrictions
    7. Tests of Hypotheses on the α Parameters by PROC VARMAX
    8. The TEST Statement for Hypotheses on the α Parameters
    9. The RESTRICT Statement for the β Parameters
    10. Restrictions on Both α Parameters and β Parameters
    11. Properties of the Final Model
    12. Conclusion
  17. Chapter 14: Cointegration
    1. Introduction
    2. Test for a Cointegration Relation in the Bivariate Case
    3. Cointegration Test Using PROC VARMAX for Two Price Series
    4. Cointegration Tests in a Five-Dimensional Series
      1. Initial Estimates for the β Values
      2. A Model with Rank 2
    5. Use of the RESTRICT Statement to Determine the Form of the Model
    6. Stock-Watson Test for Common Trends for Five Series
    7. A Rank 4 Model for Five Series Specified with Restrictions
      1. An Alternative Form of the Restrictions
      2. Estimation of the Model Parameters by a RESTRICT Statement
      3. Estimation with Restrictions on Both the α and β Parameters
    8. Conclusion
  18. Chapter 15: Univariate GARCH Models
    1. Introduction
    2. The GARCH Model
    3. GARCH Models for a Univariate Financial Time Series
      1. Use of PROC VARMAX to Fit a GARCH(1,1) Model
      2. The Fitted Model
      3. Use of PROC VARMAX to Fit an IGARCH Model
    4. The Wage Series
      1. Use of PROC VARMAX to Fit an AR(2)-GARCH(1,1) Model
      2. The Conditional Variance Series
    5. Other Forms of GARCH Models
      1. The QGARCH Model
      2. The TGARCH Model
      3. The PGARCH Model
      4. The EGARCH Model
    6. Conclusion
  19. Chapter 16: Multivariate GARCH Models
    1. Introduction
    2. Multivariate GARCH Models
      1. The CCC Parameterization
      2. The DCC Parameterization
      3. The BEKK Parameterization
    3. A Bivariate Example Using Two Quotations for Danish Stocks
      1. Using the CCC Parameterization
      2. Using the DCC Parameterization
      3. Using the BEKK Parameterization
      4. Using the CCC Bivariate Combination of Univariate TGARCH Models
    4. Conclusion
  20. Chapter 17: Multivariate VARMA-GARCH Models
    1. Introduction
    2. Multivariate VARMA-GARCH Models
    3. The Wage-Price Time Series
    4. A VARMA Model with a CCC-GARCH Model for the Residuals
    5. A VARMA Model with a DCC-GARCH Model for the Residuals
    6. Refinement of the Estimation Algorithm
    7. The Final VARMA Model with DCC-GARCH Residuals
    8. Conclusion
  21. References
  22. Index