When simulating realizations from GARCH, EGARCH, or GJR processes, you need presample conditional variances and presample innovations to initialize the variance equation.
For the GARCH(P,Q) and GJR(P,Q) models, P presample variances and Q presample innovations are needed. For an EGARCH(P,Q) model, max(P,Q) presample variances and Q presample innovations are needed.
You can either specify your own presample data, or let
simulate automatically generate presample data.
If you let
simulate generate presample data, then:
Presample variances are set to the theoretical unconditional variance of the model being simulated.
Presample innovations are random draws from the innovation distribution with the theoretical unconditional variance.
If you are specifying your own presample data, note that
simulate assumes presample innovations with mean zero. If your observed series is an innovation series plus an offset, subtract the offset from the observed series before using it as a presample innovation series.
When you specify adequate presample variances and innovations, the first conditional variance in the simulation recursion is the same for all sample paths. The first simulated innovations are random, however, because they are random draws from the innovation distribution (with common variance, specified by the presample variances and innovations).