1. K. Ensor, STAT 421 1 Spring 2004 Garch-m The process or return is dependent on the volatility , c are constants C is the “risk premium parameter”; c>0 indicates the return is positively related to its volatility.

2. K. Ensor, STAT 421 2 Spring 2004

3. K. Ensor, STAT 421 3 Spring 2004 Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) C 0.00548675 0.00226173 2.426 7.747e-003 ARCH-IN-MEAN 1.08783589 0.81822755 1.330 9.203e-002 A 0.00008764 0.00002507 3.496 2.494e-004 ARCH(1) 0.12268468 0.02047268 5.993 1.571e-009 GARCH(1) 0.84939373 0.01957565 43.390 0.000e+000 -------------------------------------------------------------- Output from Splus m-garch fit garch(x~1+var.in.mean,~garch(1,1)) Differs from Tsay’s fit slightly.

4. K. Ensor, STAT 421 4 Spring 2004 S&P500 Index Square root Of volatility

5. K. Ensor, STAT 421 5 Spring 2004 Summary Graphs

6. K. Ensor, STAT 421 6 Spring 2004 Hong Kong stock market index return (bottom graph) and estimated volatility.

7. K. Ensor, STAT 421 7 Spring 2004 Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) AR(1) 0.0450 0.04578 0.983 0.163052 A 0.1688 0.08404 2.009 0.022568 ARCH(1) 0.1700 0.05835 2.913 0.001871 GARCH(1) 0.7732 0.06454 11.980 0.000000 -------------------------------------------------------------- garchfit<-garch(HK~- 1+arma(1,0),~garch(1,1),cond.dist="t",dist.est=T)

8. K. Ensor, STAT 421 8 Spring 2004 HK - Garch fit +/- 2SD

9. K. Ensor, STAT 421 9 Spring 2004

10. K. Ensor, STAT 421 10 Spring 2004 -------------------------------------------------------------- Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) AR(1) 0.1199 0.05709 2.100 1.811e-002 A 0.1424 0.04834 2.946 1.687e-003 ARCH(1) 0.1782 0.03693 4.827 9.287e-007 GARCH(1) 0.7592 0.04913 15.452 0.000e+000 -------------------------------------------------------------- garchfit<-garch(HK~- 1+arma(1,0),~garch(1,1),cond.dist="gaussian",dist.est=T)

11. K. Ensor, STAT 421 11 Spring 2004

12. K. Ensor, STAT 421 12 Spring 2004

13. K. Ensor, STAT 421 13 Spring 2004

14. K. Ensor, STAT 421 14 Spring 2004 Japanese stock market index and volatility based on Gaussian GARCH(1,1) model

15. K. Ensor, STAT 421 15 Spring 2004 -------------------------------------------------------------- Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) A 0.1352 0.04517 2.993 1.452e-003 ARCH(1) 0.1713 0.03409 5.024 3.552e-007 GARCH(1) 0.7708 0.04609 16.722 0.000e+000 -------------------------------------------------------------- garchfit <-garch(JI~- 1,~garch(1,1),cond.dist="gaussian",dist.est= T)

16. K. Ensor, STAT 421 16 Spring 2004 JI

17. K. Ensor, STAT 421 17 Spring 2004 JI

18. K. Ensor, STAT 421 18 Spring 2004 Let’s trying looking at the multivariate GARCH.

19. K. Ensor, STAT 421 19 Spring 2004 Series 1: Hong Kong Stock Index Series 2: Japanese Stock Index

20. K. Ensor, STAT 421 20 Spring 2004 Series 1: Hong Kong Stock Index Squared Series 2: Japanese Stock Index Squared

21. K. Ensor, STAT 421 21 Spring 2004 -------------------------------------------------------------- Estimated Coefficients: -------------------------------------------------------------- Value Std.Error t value Pr(>|t|) AR(1; 1, 1) 0.124329 0.058850 2.1126 1.757e-002 AR(1; 2, 2) 0.017088 0.047872 0.3569 3.606e-001 A(1, 1) 0.144756 0.050129 2.8877 2.027e-003 A(2, 2) 0.003265 0.006921 0.4718 3.187e-001 ARCH(1; 1, 1) 0.186976 0.039732 4.7059 1.649e-006 ARCH(1; 2, 2) 0.069114 0.016141 4.2818 1.117e-005 GARCH(1; 1, 1) 0.755876 0.050284 15.0320 0.000e+000 GARCH(1; 2, 2) 0.937297 0.017199 54.4981 0.000e+000 -------------------------------------------------------------- mgarchfit=mgarch(X~- 1+arma(1,0),~garch(1, 1)) Page 367 of text

22. K. Ensor, STAT 421 22 Spring 2004

23. K. Ensor, STAT 421 23 Spring 2004

24. K. Ensor, STAT 421 24 Spring 2004