Lecture 9: Multivariate Time Series Analysis

1. Lecture 9: Multivariate Time Series Analysis • The following topics will be covered: • Modeling Mean • Cross-correlation Matrixes of returns • VAR • VMA • VARMA • Cointegration • Modeling Volatility • VGARCH models L9: Vector Time Series

2. Lag-0 Cross-correlation Matrix L9: Vector Time Series

3. Lag-l Cross-correlation Matrix L9: Vector Time Series

4. Linear Dependence L9: Vector Time Series

5. Sample Cross-Correlation Matrixes (CCM) L9: Vector Time Series

6. Multivariate Portmanteau Test • For a multivariate series, the null hypothesis is H0: ρ1=…=ρm=0 and the alternative hypothesis H0: ρi ne 0 for some i. The statistic is used to test that there are no auto- and cross-correlations in the vector series rt. Portmanteau test is listed on page 308, where T is the sample size, k is the dimension of rt. L9: Vector Time Series

7. VAR (1) L9: Vector Time Series

8. VAR (1): Reduced Form System L9: Vector Time Series

9. Stationarity Conditionof VAR(1) L9: Vector Time Series

10. VAR(p) Models L9: Vector Time Series

11. Building VAR(p) Model L9: Vector Time Series

12. Building VAR(p) Model L9: Vector Time Series

13. VMA and VARMA L9: Vector Time Series

14. Unit Root Nonstationarity and Co-integration L9: Vector Time Series

15. Error-Correction Form L9: Vector Time Series

16. Procedure in Cointegration tests L9: Vector Time Series

17. Conditional Covariance Matrix L9: Vector Time Series

18. Use of Correlations L9: Vector Time Series

19. Cholesky Decomposition L9: Vector Time Series

20. Bivariate GARCH For a k-dimensional return series rt, a multivariate GARCH model uses “exact equations” to describe the evolution of the k(k+1)/2-dimentional vector over time. By exact equation, we mean that the equation does not contain any stochastic shock. However, the exact equation may become complicated. To keep the model simple, some restrictions are often imposed on the equations. • Constant-correlation models: cross-correlation is a constant. – see (9.16) and (9.17) on page 364 proc varmax data=all; model ibm sp / p=1 garch=(q=1); nloptions tech=qn; output out=for lead=5 back=3; run; (all contains two sets of returns) (2) Time-Varying Correlation models L9: Vector Time Series

21. Exercises • Ch8, problem 2 • Replicate Goeij and Marqliering (2004, J. Fin. Econometrics), Modeling the conditional covariance between Stock and Bond Returns: A multivariate GARCH Approach, 2(4), 531-564. L9: Vector Time Series