######################################################################### # Code Examples for Basic Garch Simulation, ARCH Testing, GARCH Fitting ######################################################################### module(finmetrics) #Ford Daily Returns for 2000 days. class(ford.s) ford.s@title start(ford.s) end(ford.s) nrow(ford.s) #A look at the data plot(ford.s,reference.grid=F) #Now a look at the ACF and the ACF of the SQUARES. par(mfrow=c(1,2)) tmp = acf(ford.s,lag=12) tmp = acf(ford.s^2, lag=12) par(mfrow=c(1,1)) #We see that the SQUARES have more lags of significance. #To some extent this is a "cheat" since it is not clear #that the squared data will conform to the assumption that #one needs to set significance levels on the estimated correlations. #Still the picture is suggestive. #Even without the validity of the confidence intervals, we have some #pretty suggestive evidence that Ford Retruns are NOT INDEPENDENT. #Thus, another instance of the Black Scholes (AKA Samuelson) model #does not hold. ####################################### #Simulation of an ARCH(1) model sim.arch1=simulate.garch(model=list(a.value=0.01, arch=0.8), n=250, rseed=196) #Now lets look at what it holds... names(sim.arch1) #And look at plots of these components par(mfrow=c(2,1)) tsplot(sim.arch1\$et, main="Simulated ARCH(1) errors", ylab="e(t)") tsplot(sim.arch1\$sigma.t, main="Simulated ARCH(1) volatility", ylab="sigma(t)") par(mfrow=c(1,1)) #And a summary summaryStats(sim.arch1\$et) #Kurtosis is a little high, but not brutal. Also, we might ask, #what sample kurtosis even means here. By construction, these guys are not independent. #On the other hand, the simulation function returns both the error term and the conditional standard #deviations. In theory (i.e. if the model was exactly appropriate) these values #would be perfectly independent standard normals. Let's look. ratio=sim.arch1\$et/sim.arch1\$sigma.t normalTest(ratio, method="sw") normalTest(ratio, method="jb") #Golly, for once we find that jb does not reject the normality hypothesis. #Well, after all, this is simulated data. We just got back what we baked into the cake. ##################################### # Back to the real data and "testing for arch" # More precisely we test that 0=a_1=...=a_p where p=lag.n archTest(ford.s, lag.n=12) #We get a brutally small p-value, so it looks like there is an arch effect. #Let estimate a univarite GARCH model for the series. #What makes the model "Generalized" is that sigma.t is #modeled by lags in sigma.t as well as lags in epsilon.t #We'll try one of each. ford.mod11 = garch(ford.s~1, ~garch(1,1)) class(ford.mod11) #Lets Look ford.mod11 #and check the names... names(ford.mod11) #Tidier to use the summary "method" summary(ford.mod11) #This has some weird inferences... #Shapiro-Wilk is nonsignificant but Jarque-Berra is highly significant. #This "may" mean that the residuals are "normal in the middle" but #exhibit skewness or kurtosis. Also, the JB test is VERY sensitive #to an outlier the series. ############################################################## #Now lets look at some individual components of the series. ford.mod11\$asymp.sd coef(ford.mod11) vcov(ford.mod11) vcov(ford.mod11,method="qmle") residuals(ford.mod11,standardize=T)[1:5] sigma.t(ford.mod11)[1:5] # garch diagnostics summary(ford.mod11) autocorTest(residuals(ford.mod11,standardize=T),lag=12) autocorTest(residuals(ford.mod11,standardize=T)^2,lag=12) archTest(residuals(ford.mod11,standardize=T),lag=12) #And a bucket of potential plots.... plot(ford.mod11) ################################################################################# # Now lets look at the idea of combining the garch errors with a model for the mean ford.meanmodel = garch(ford.s~ar(1), ~garch(1,1)) summary(ford.meanmodel) #Now explore this model... replacing ar(1) with ar(2), ma(2), arma(1,1) #Do the changes in the coefficients make sense? Consider size, p-value, signs, etc. #How would you choose among these models? #How would you summarize what you learn from this exploration.