I would like some help with a GARCH(1,1) volatility modeling.
I am working with the assumption the volatility is the weighted sum of three factors: Long run variance + $n-1$ squared return + $n-1$ variance
If that is accurate, my doubt is, what is the difference between the 1st and 3rd parts of the equation? I was reading it as $n-1$ variance being the historical variance of the moving window I am using. However, that seems to me the same as long run variance.
Can anyone clarify that for me?
A GARCH(1,1) model is \begin{aligned} y_t &= \mu_t + u_t, \\ \mu_t &= \dots \text{(e.g. a constant or an ARMA equation without the term $u_t$)}, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \beta_1 \sigma_{t-1}^2, \\ \varepsilon_t &\sim i.i.d(0,1). \\ \end{aligned} The three components in the conditional variance equation you refer to are $\omega$, $u_{t-1}^2$, and $\sigma_{t-1}^2$. Your question seems to be, how is $\omega$ different from $\sigma_{t-1}^2$?
First, note that $\omega$ is not the long-run variance; the latter actually is $\sigma_{LR}^2:=\frac{\omega}{1-(\alpha_1+\beta_1)}$. $\omega$ is an offset term, the lowest value the variance can achieve in any time period, and is related to the long-run variance as $\omega=\sigma_{LR}^2(1-(\alpha_1+\beta_1))$.
Second, $\sigma_{t-1}^2$ is not the historical variance of the moving window; it is instantaneous variance at time $t-1$.