Forecasting volatility using GARCH

Question

How do I forecast volatility using GARCH in STATA after estimating the conditional volatility?

Answer 1

ARCH postestimation help file explains it all. You will most likely need

     predict hat_volatility, variance

or

     predict hat_volatility_factor, het

depending on what exactly you mean by ``volatility''. The former is the full prediction, the latter is the multiplier that goes in front of the $\hat\sigma^2$.

Answer 2

I can't speak to whether stata has a built-in function for forecasting specific GARCH models. I imagine they do, but you can dig through the help manuals to find it.

More generally, assume a Garch(1,1) model has been fit to $X$, a variable with a constant mean of zero. This model is of the form:$$\sigma_{t}^{2}=\beta_{0}+\beta_{1}\sigma_{t-1}^{2}+\beta_{2}\varepsilon_{t-1}^{2}$$ The next step is to scale the variable $X$ by the conditional standard deviation $\sigma_{t}$ to obtain $\varepsilon_{t}$, the scaled residuals. You would then simulate $\varepsilon_{t}$ under the relevant assumption (so for instance, if you estimate the Garch model with t errors, then you would simulate from that distribution). The future value of volatility would then be: $$\sigma_{t+1}^{2}=\beta_{0}+\beta_{1}\sigma_{t}^{2}+\beta_{2}\varepsilon_{t}^{2}$$ The values in $t+1$ are known, but thereafter you would use the simulated values. If you just want to get a single value for future volatility, rather than a large panel of simulated values, you can simply set $\varepsilon$ to be a vector of zeros.