Load the data and specify the model.

Load the Deutschmark/British pound foreign exchange rate data included with the toolbox, and convert to returns. For numerical stability, convert returns to percentage returns.

loadData_MarkPoundr = price2ret(Data); pR = 100*r; T = length(r);

Specify and fit a GARCH(1,1) model.

Mdl = garch(1,1); EstMdl = estimate(Mdl,pR);

GARCH(1,1) Conditional Variance Model (Gaussian Distribution): Value StandardError TStatistic PValue ________ _____________ __________ __________ Constant 0.010868 0.0012972 8.3779 5.3898e-17 GARCH{1} 0.80452 0.016038 50.162 0 ARCH{1} 0.15432 0.013852 11.141 7.9447e-29

Generate MMSE forecasts.

Use the fitted model to generate MMSE forecasts over a 200-period horizon. Use the observed return series as presample data. By default,forecastinfers the corresponding presample conditional variances. Compare the asymptote of the variance forecast to the theoretical unconditional variance of the GARCH(1,1) model.

v = forecast(EstMdl,200,pR); sig2 = EstMdl.Constant/(1-EstMdl.GARCH{1}-EstMdl.ARCH{1}); figure plot(v,'r','LineWidth',2) holdonplot(ones(200,1)*sig2,'k--','LineWidth',1.5) xlim([0,200]) title('Forecast Conditional Variance') legend('Forecast','Theoretical','Location','SouthEast') holdoff

患者的预测理论unc收敛onditional variance after about 160 steps.