Garch-mPowerPoint PresentationSlide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24K. Ensor, STAT 4211Spring 2004Garch-m•The process or return is dependent on the volatility211211022)()()()(tttttattatactr, c are constantsC is the “risk premium parameter”; c>0 indicates the return is positively related to its volatility.K. Ensor, STAT 4212Spring 2004Time0 200 400 600 800-0.2 0.2Time0 200 400 600 800-0.2 0.2S&P 500-0.2 0.0 0.2 0.40 100 200 300HistogramS&P 500 LagACF0 5 10 15 20 25-1.0 -0.5 0.0 0.5 1.0ACFLagACF0 5 10 15 20 25-1.0 -0.5 0.0 0.5 1.0PACFfrequencyspectrum0.0 0.1 0.2 0.3 0.4 0.5-28 -26 -24 -22Series: x AR ( 21 ) Spectrum using yule-walkerK. Ensor, STAT 4213Spring 2004Estimated Coefficients:--------------------------------------------------------------Value Std.Error t value Pr(>|t|) C 0.00548675 0.00226173 2.426 7.747e-003ARCH-IN-MEAN 1.08783589 0.81822755 1.330 9.203e-002A 0.00008764 0.00002507 3.496 2.494e-004ARCH(1) 0.12268468 0.02047268 5.993 1.571e-009GARCH(1) 0.84939373 0.01957565 43.390 0.000e+000--------------------------------------------------------------Output from Splus m-garch fitgarch(x~1+var.in.mean,~garch(1,1))Differs from Tsay’s fit slightly.K. Ensor, STAT 4214Spring 2004-0.2 0.0 0.2 0.40 200 400 600 800Conditional SD0.05 0.10 0.15 0.20Original SeriesValuesSeries and Conditional SDS&P500 IndexSquare rootOf volatilityK. Ensor, STAT 4215Spring 20040.00.20.40.60.81.00 5 10 15 20 25 30ACFLagsACF of Squared Observations0.00.20.40.60.81.00 5 10 15 20 25 30ACFLagsACF of Observations-4-2020 200 400 600 800residualsStandardized ResidualsGARCH Standardized Residuals0.00.20.40.60.81.00 5 10 15 20 25 30ACFLagsACF of Std. Residuals0.00.20.40.60.81.00 5 10 15 20 25 30ACFLagsACF of Squared Std. Residuals-4-202-3 -2 -1 0 1 2 3QQ-Plot147173742Quantiles of gaussian distributionStandardized ResidualsQQ-Plot of Standardized ResidualsSummary GraphsK. Ensor, STAT 4216Spring 2004-10 -5 0 5 10 150 100 200 300 400 500Conditional SD2 4 6 8Original SeriesValuesSeries and Conditional SDHong Kong stock market index return (bottom graph) and estimated volatility.K. Ensor, STAT 4217Spring 2004Estimated 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 0.001871GARCH(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)K. Ensor, STAT 4218Spring 2004-100100 100 200 300 400 500garchfitValuesSeries with 2 Conditional SD SuperimposedHK - Garch fit +/- 2SDK. Ensor, STAT 4219Spring 2004-0.20.00.20.40.60.81.00 5 10 15 20 25ACFLagsACF of Observations0.00.20.40.60.81.00 5 10 15 20 25ACFLagsACF of Squared Observations-6-4-2020 100 200 300 400 500residualsStandardized ResidualsGARCH Standardized Residuals0.00.20.40.60.81.00 5 10 15 20 25ACFLagsACF of Std. Residuals0.00.20.40.60.81.00 5 10 15 20 25ACFLagsACF of Squared Std. Residuals-6-4-20246-5 0 5QQ-Plot321Quantiles of t distributionStandardized ResidualsQQ-Plot of Standardized ResidualsK. Ensor, STAT 42110Spring 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-007GARCH(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)K. Ensor, STAT 42111Spring 2004-10 -5 0 5 10 150 100 200 300 400 500Conditional SD2 4 6 8 10Original SeriesValuesSeries and Conditional SDK. Ensor, STAT 42112Spring 2004-20-10010200 100 200 300 400 500garchfitValuesSeries with 2 Conditional SD SuperimposedK. Ensor, STAT 42113Spring 2004-6-4-2020 100 200 300 400 500residualsStandardized ResidualsGARCH Standardized Residuals-6-4-202-3 -2 -1 0 1 2 3QQ-Plot47143451Quantiles of gaussian distributionStandardized ResidualsQQ-Plot of Standardized ResidualsK. Ensor, STAT 42114Spring 2004-10 -5 0 5 10 150 100 200 300 400 500Conditional SD2 4 6 8Original SeriesValuesSeries and Conditional SDJapanese stock market index and volatility based on Gaussian GARCH(1,1) modelK. Ensor, STAT 42115Spring 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-007GARCH(1) 0.7708 0.04609 16.722 0.000e+000--------------------------------------------------------------garchfit<-garch(JI~-1,~garch(1,1),cond.dist="gaussian",dist.est=T)K. Ensor, STAT 42116Spring 2004-20-10010200 100 200 300 400 500garchfitValuesSeries with 2 Conditional SD SuperimposedJIK. Ensor, STAT 42117Spring 2004-0.20.00.20.40.60.81.00 5 10 15 20 25ACFLagsACF of Observations0.00.20.40.60.81.00 5 10 15 20 25ACFLagsACF of Squared Observations-6-4-2020 100 200 300 400 500residualsStandardized ResidualsGARCH Standardized Residuals0.00.20.40.60.81.00 5 10 15 20 25ACFLagsACF of Std. Residuals0.00.20.40.60.81.00 5 10 15 20 25ACFLagsACF of Squared Std. Residuals-6-4-202-3 -2 -1 0 1 2 3QQ-Plot47143451Quantiles of gaussian distributionStandardized ResidualsQQ-Plot of Standardized ResidualsJIK. Ensor, STAT 42118Spring 2004-10 -5 0 5 10 150 100 200 300 400 500Series 1-4 -2 0 2 4 6 8Series 2ValuesOriginal ObservationsLet’s trying looking at the multivariate GARCH.K. Ensor, STAT 42119Spring 2004Series 1: Hong Kong Stock IndexSeries 2: Japanese Stock Index Series 1 ACF0 5 10 15 20 25-0.2 0.0 0.2 0.4 0.6 0.8 1.0 Series 1 and Series 20 5 10 15 20 25-0.1 0.0 0.1 0.2 0.3 Series 2 and Series 1LagACF-25 -20 -15 -10 -5 0-0.1 0.0 0.1 0.2 0.3 Series 2 Lag0 5 10 15 20 250.0 0.2 0.4 0.6 0.8 1.0ACF of ObservationsK. Ensor, STAT 42120Spring 2004 Series 1 ACF0 5 10 15 200.0 0.2 0.4 0.6 0.8 1.0 Series 1 and Series 20 5 10 15 20-0.05 0.0 0.05 0.10 0.15 Series 2 and Series 1LagACF-20 -15 -10 -5 00.0 0.2 0.4 0.6 Series 2 Lag0 5 10 15 200.0 0.2 0.4 0.6