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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.

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

model = arima('ARLags',1,'Variance',garch(1,1),...
              'Distribution','t');
fit = estimate(model,r,'Variance0',{'Constant0',0.001});
[E0,V0] = infer(fit,r);
 
    ARIMA(1,0,0) Model:
    --------------------
    Conditional Probability Distribution: t

                                  Standard          t     
     Parameter       Value          Error       Statistic 
    -----------   -----------   ------------   -----------
     Constant     0.00103504   0.000170394        6.07439
        AR{1}       0.145478     0.0193135        7.53245
          DoF        7.39555      0.905084        8.17112
 
 
    GARCH(1,1) Conditional Variance Model:
    ----------------------------------------
    Conditional Probability Distribution: t

                                  Standard          t     
     Parameter       Value          Error       Statistic 
    -----------   -----------   ------------   -----------
     Constant     1.6756e-06   6.55736e-07        2.55529
     GARCH{1}       0.890901      0.012025        74.0876
      ARCH{1}       0.105624     0.0124898        8.45681
          DoF        7.39555      0.905084        8.17112

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(fit,1000,'Y0',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).

See Also

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