# Documentation

## Forecast a Conditional Variance Model

This example shows how to forecast a conditional variance model using forecast.

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.

r  = 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:
----------------------------------------
Conditional Probability Distribution: Gaussian

Standard          t
Parameter       Value          Error       Statistic
-----------   -----------   ------------   -----------
Constant       0.010868    0.00129723        8.37786
GARCH{1}       0.804518     0.0160384        50.1619
ARCH{1}       0.154325     0.0138523        11.1408

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, forecast infers 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,'Y0',pR);
sig2 = EstMdl.Constant/(1-EstMdl.GARCH{1}-EstMdl.ARCH{1});

figure
plot(v,'r','LineWidth',2)
hold on
plot(ones(200,1)*sig2,'k--','LineWidth',1.5)
xlim([0,200])
title('Forecast Conditional Variance')
legend('Forecast','Theoretical','Location','SouthEast')
hold off

The MMSE forecasts converge to the theoretical unconditional variance after about 160 steps.