Garch-m

1. 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. K. Ensor, STAT 421

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3. Output from Splus m-garch fit • garch(x~1+var.in.mean,~garch(1,1)) 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 -------------------------------------------------------------- Differs from Tsay’s fit slightly. K. Ensor, STAT 421

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

5. Summary Graphs K. Ensor, STAT 421

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

7. garchfit<-garch(HK~-1+arma(1,0),~garch(1,1),cond.dist="t",dist.est=T)garchfit<-garch(HK~-1+arma(1,0),~garch(1,1),cond.dist="t",dist.est=T) • 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 • -------------------------------------------------------------- K. Ensor, STAT 421

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

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10. garchfit<-garch(HK~-1+arma(1,0),~garch(1,1),cond.dist="gaussian",dist.est=T)garchfit<-garch(HK~-1+arma(1,0),~garch(1,1),cond.dist="gaussian",dist.est=T) • -------------------------------------------------------------- • 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 • -------------------------------------------------------------- K. Ensor, STAT 421

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

15. garchfit<-garch(JI~-1,~garch(1,1),cond.dist="gaussian",dist.est=T)garchfit<-garch(JI~-1,~garch(1,1),cond.dist="gaussian",dist.est=T) • -------------------------------------------------------------- • 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 • -------------------------------------------------------------- K. Ensor, STAT 421

16. JI K. Ensor, STAT 421

17. JI K. Ensor, STAT 421

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

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

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

21. ---------------------------------------------------------------------------------------------------------------------------- • 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 K. Ensor, STAT 421

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