About

If an auto regressive moving average model (ARMA model) is assumed for the error variance, the model is a generalized auto regressive conditional heteroskedasticity (GARCH) .

The process $\epsilon_t$ is GARCH(p,q) Model if $\mathbb{E}[\epsilon_t|\mathcal{F}_{t-1}]=0$ and $$\sigma_t^2=\omega+\sum_{i=1}^{p}\alpha_i\epsilon_{t-i}^2+\sum_{j=1}^{q}\beta_i\sigma_{t-j}^2$$ such that

  1. $Var(\epsilon_t|\mathcal{F}_{t-1})=\sigma_t^2$
  2. $\sigma_t^2$ and $Z_t=\frac{\epsilon_t}{\sigma_t}$ are i.i.d.
  3. $P(\epsilon_t^2|1,\epsilon_{t-1},\epsilon_{t-2},...,\epsilon_{t-1}^2,\epsilon_{t-2}^2,...)=\sigma_{t}^2$.
history | excerpt history