# Two questions on GARCH

Suppose we have a GARCH (1,1) equation as follows:

$$\sigma_t^2 = \alpha_0 + \alpha\epsilon_{t-1}^2 + \beta\sigma_{t-1}^2$$

1. How are the lagged variance terms calculated?
2. Suppose I am also implementing an ARMA-GARCH process sequentially, so I am using the residuals from ARMA to optimize GARCH. (I believe) Then it is assumed that the $$\epsilon_i$$ in both ARMA and GARCH follow the same distribution. If this is the case, since $$\epsilon_i$$ and $$\sigma_i$$ are known, why can't this parameter identification be down through OLS?

In general the GARCH(1,1) model is given by: \begin{align} r_t&=\mu_t+\epsilon_t \\ \epsilon_t&=\sigma_tz_t \quad z_t \overset{iid}{\sim} D(0,1) \\ \sigma_t^2&=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2 \end{align} Where $$\mu_t=E(r_t \vert {\cal F_{t-1}})$$ is the conditional mean of $$r_t$$. The conditional variance of $$r_t$$ is given by $$\sigma_t^2$$, so the lagged variance is included through the second term in the variance equation $$\beta_1\sigma_{t-1}^2$$. When you specify an ARMA-GARCH model, you model $$\mu_t$$ via an ARMA model. The residuals for fitting the GARCH model are then given by $$r_t-\hat{\mu}_t$$, where $$\hat{\mu}_t$$ are the fitted values from the ARMA model. It is not possible to estimate the GARCH parameters, i.e. $$\alpha_0$$, $$\alpha_1$$ and $$\beta_1$$, via OLS due to the iterative character of the variance equation. Therefore the parameters are usually estimated via ML.
• Thanks a lot! I have two more questions if you don't mind. First, I am still unclear how the conditional variance associated with $\beta_1$ is computed in practice, could you elaborate on that? Secondly, after the ARMA GARCH model has been fitted, how do we predict values? As in, how do we choose a value for the $\epsilon_i$ if they are random? Commented Apr 27, 2021 at 8:01
• Sure, assume that you estimated a GARCH(1,1) model. Than you need an inital value for $\sigma_0^2$. There are different possiblities but one commonly used is: $\sigma_0^2=\frac{1}{T}\sum_{t=1}^Tr_t^2$ (because this is a consistent estimator). Now you can calculate $\sigma_1^2=\hat{\alpha}_0+\hat{\alpha}_1\epsilon_0^2+\hat{\beta}_1\sigma_0^2$ and so on. In most cases you have a very large $T$ and the effect of $\sigma_0^2$ is negligible. Commented Apr 27, 2021 at 10:29
• Related to your second question: You observe the returns $r_t$ and you fitted the values for the conditional mean quation $\hat{\mu}_t$. You can use these values to calculate the $\hat{\epsilon}_t$ as $\hat{\epsilon}_t=r_t-\hat{\mu}_t$. When it comes to prediction you have to differ between predictions for $r_t$ or the conditional variance $\sigma_t^2$. In most cases you are interested in predicions for $\sigma_t^2$. The MSE optimal $l$-step ahead prediction is given by: $E(\sigma_{T+l}^2\vert {\cal F_{T}})$. Commented Apr 27, 2021 at 10:38
• @Jonas_Dim, it is worth noting that the estimator of $\sigma_0^2$ you provide above is consistent under the assumption that $\mu_t\equiv 0$ but not necessarily otherwise. Could you also indicate briefly how GARCH can be estimated by OLS? E.g. provide a formula for the estimator like you would for a linear model where $\beta=(X^\top X)^{-1}X^\top y$? Commented Apr 27, 2021 at 13:39