7. Lab Discrete volatility models Let us rst play around the simulated GARCH series. You need to load the Finmetrics or Garch module rst: Files > Load modules. > garch21 <- simulate.garch(model=list(a=1.5, arch=c(0.4,0.1), garch=0.3),n=1000,n.start=500,rseed=1996126) #simulate a GARCH(2,1) series of length 1000 # component et contains the series and sigma.t contains Cond. SD. > par(mfrow=c(3,2)) > tsplot(garch21$et) # plot of the series > tsplot(garch21$sigma.t) # plot of cond. SD > tsplot(garch21$et/garch21$sigma.t) > acf(garch21$et, lag=30) #acf of the series > acf( (garch21$et)^2, lag=30) #acf of squared series > aqqplot(garch21$et, dist=t, distpar=6.2) #comparing with t_6.2 > qqline(garch21$et) #adding a line to the plot
Now, let us t a GARCH(1,1) model to the daily log-returns of the SP500 index. > > > > > > > > > > return500 <- 100*diff(sp500) par(mfrow=c(3,2)) tsplot(return500) acf(return500) acf(return500^2) #acf of the squared return aqqplot(return500, dist=t, distpar=5) qqline(return500) aqqplot(return500, dist=t, distpar=6) qqline(return500) garchfit <- garch(return500~1,~garch(1,1),cond.dist=t,dist.est=T) #fitting a GARCH(1,1) model ~ garch(1, 1),
Call: garch(formula.mean = return500 ~ 1, formula.var = cond.dist = "t", dist.est = T) Mean Equation: return500 ~ 1 Conditional Variance Equation: Conditional Distribution: t 1 ~ garch(1, 1)
with estimated parameter and standard error 0.4451982 -------------------------------------------------------------Estimated -------------------------------------------------------------Value Coefficients: Std.Error t value Pr(>|t|) C 0.059501 0.010533 5.649 8.476e-009 A 0.005365 0.001468 3.654 1.305e-004 ARCH(1) 0.047170 0.005091 9.265 0.000e+000 GARCH(1) 0.948307 0.005243 180.874 0.000e+000 -------------------------------------------------------------AIC(5) = 13974.61 BIC(5) = 14007.54 Normality Test: -------------------------------------------------------------Jarque-Bera P-value 11092 0 Ljung-Box test for standardized residuals: -------------------------------------------------------------Statistic P-value Chi^2-d.f. 32.22 0.001281 12 Ljung-Box test for squared standardized residuals: -------------------------------------------------------------Statistic P-value Chi^2-d.f. 19.63 0.07445 12 Lagrange multiplier test: -------------------------------------------------------------Lag 1 Lag 2 L...