Forecast Conditional Mean and Variance Model

This example shows how to forecast responses and conditional variances from a composite conditional mean and variance model.

Step 1. Load the data and fit a model.

Load the NASDAQ data included with the toolbox. Fit a conditional mean and variance model to the data.

nasdaq = DataTable.NASDAQ;
r = price2ret(nasdaq);
N = length(r);

Mdl = arima('ARLags',1,'Variance',garch(1,1),...
'Distribution','t');
EstMdl = estimate(Mdl,r,'Variance0',{'Constant0',0.001});

ARIMA(1,0,0) Model (t Distribution):

Value      StandardError    TStatistic      PValue
_________    _____________    __________    __________

Constant    0.0009524     0.00017051        5.5855      2.3308e-08
AR{1}         0.13987       0.019051         7.342      2.1037e-13
DoF            8.3525         1.0273        8.1308       4.266e-16

GARCH(1,1) Conditional Variance Model (t Distribution):

Value       StandardError    TStatistic      PValue
__________    _____________    __________    __________

Constant    1.6076e-06     6.1538e-07        2.6123       0.0089925
GARCH{1}       0.89701       0.011191        80.153               0
ARCH{1}       0.095254       0.010975         8.679      3.9935e-18
DoF             8.3525         1.0273        8.1308       4.266e-16
[E0,V0] = infer(EstMdl,r);

Step 2. Forecast returns and conditional variances.

Use forecast to compute MMSE forecasts of the returns and conditional variances for a 1000-period future horizon. Use the observed returns and inferred residuals and conditional variances as presample data.

[Y,YMSE,V] = forecast(EstMdl,1000,r,'E0',E0,'V0',V0);
upper = Y + 1.96*sqrt(YMSE);
lower = Y - 1.96*sqrt(YMSE);

figure
subplot(2,1,1)
plot(r,'Color',[.75,.75,.75])
hold on
plot(N+1:N+1000,Y,'r','LineWidth',2)
plot(N+1:N+1000,[upper,lower],'k--','LineWidth',1.5)
xlim([0,N+1000])
title('Forecasted Returns')
hold off
subplot(2,1,2)
plot(V0,'Color',[.75,.75,.75])
hold on
plot(N+1:N+1000,V,'r','LineWidth',2);
xlim([0,N+1000])
title('Forecasted Conditional Variances')
hold off The conditional variance forecasts converge to the asymptotic variance of the GARCH conditional variance model. The forecasted returns converge to the estimated model constant (the unconditional mean of the AR conditional mean model).