I have a simple theoretical question as I am a beginner in time series analysis.
The idea of modeling an AR-ARCH or GARCH is to model the mean and the variance. After modeling, I want to forecast the mean.
I know that I will need to forecast the variance too.
But when I write the formula, for example, for the one-step-ahead forecast of the conditional mean, the one-step-ahead forecast of the conditional variance, $\hat\sigma_{t+1}^2$, doesn´t show up.
That's where my doubts come from. How can I see the benefits in modeling and forecasting the variance when my goal is modeling and forecasting the mean?
And second: For what purpose would I use my estimated variance and make your prediction?
Would it be just to build a confidence interval for the expected mean? In this case, obviously, the confidence interval would not be a straight line because the conditional variance is non-constant.
Best Answer
If the conditional variance is non-constant but rather of GARCH type, (implicitly) assuming the cond. variance to be constant will yield inefficient estimates of the coefficients in the cond. mean model. That is, if the true data generating process is better approximated by an AR-GARCH model than by an AR model, using an AR model without a GARCH model will yield inefficient estimates of the AR coefficients. When you use the cond. mean model to forecast the cond. mean some periods ahead, the inefficiently estimated model will produce poorer forecasts than an efficiently estimated model would.
Essentially, you got the idea right. The confidence interval produced by a model with GARCH-type of cond. variance will be different than one produced by a model with constant cond. variance. But note that in any case the confidence interval will be expanding with time, so it will not be bordered by two straight horizontal lines even in the case of constant cond. variance.