Forecast GJR Models

Open Live Script

This example shows how to generate MMSE forecasts from a GJR model usingforecast.

Step 1. Specify a GJR model.

Specify a GJR(1,1) model without a mean offset and $κ = 0.1$ , $γ_{1} = 0.7$ , $α_{1} = 0.2$ and $ξ_{1} = 0.1$ .

Mdl = gjr('Constant',0.1,'GARCH',0.7,...'ARCH',0.2,'Leverage',0.1);

Step 2. Generate MMSE forecasts.

Generate forecasts for a 100-period horizon with and without specifying a presample innovation and conditional variance. Plot the forecasts along with the theoretical unconditional variance of the model.

v1 = forecast(Mdl,100); v2 = forecast(Mdl,100,'Y0',1.4,'V0',2.1); denom = 1-Mdl.GARCH{1}-Mdl.ARCH{1}-0.5*Mdl.Leverage{1}; sig2 = Mdl.Constant/denom; figure plot(v1,'Color',[.9,.9,.9],'LineWidth',8) holdonplot(v2,'LineWidth',2) plot(ones(100,1)*sig2,'k--','LineWidth',1.5) xlim([0,100]) title('Forecast GJR Conditional Variance') legend('No Presamples','Presamples','Theoretical',...'Location','SouthEast') holdoff

Figure contains an axes object. The axes object with title Forecast GJR Conditional Variance contains 3 objects of type line. These objects represent No Presamples, Presamples, Theoretical.

v2(1)%显示预测条件的变化ance

ans = 1.9620

The forecasts generated without using presample data are equal to the theoretical unconditional variance. In the absence of presample data,forecastuses the unconditional variance for any required presample innovations and conditional variances.

In this example, for the given presample innovation and conditional variance, the starting forecast is

${σ_{}^{ˆ}}_{t + 1}^{2} = κ + γ_{1} σ_{t}^{2} + α_{1} ε_{t}^{2} = 0.1 + 0.7 (2.1) + 0.2 (1 . 4^{2}) = 1.962 .$

The leverage term is not included in the forecast since the presample innovation was positive (thus, the negative-innovation indicator is zero).

Forecast GJR Models

Step 1. Specify a GJR model.

Step 2. Generate MMSE forecasts.

See Also

Objects

Functions

Related Examples

More About