# Documentation

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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});```
``` ARIMA(1,0,0) Model: -------------------- Conditional Probability Distribution: t Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant 0.00102703 0.00017 6.04137 AR{1} 0.145703 0.0192289 7.57732 DoF 7.37236 0.898406 8.20604 GARCH(1,1) Conditional Variance Model: ---------------------------------------- Conditional Probability Distribution: t Standard t Parameter Value Error Statistic ----------- ----------- ------------ ----------- Constant 1.6663e-06 6.51093e-07 2.55923 GARCH{1} 0.891952 0.011931 74.7591 ARCH{1} 0.103971 0.0123007 8.45245 DoF 7.37236 0.898406 8.20604 ```
`[E0,V0] = infer(fit,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(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).