0
$\begingroup$

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?
$\endgroup$

1 Answer 1

2
$\begingroup$

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.

$\endgroup$
7
  • $\begingroup$ 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? $\endgroup$
    – CBBAM
    Apr 27, 2021 at 8:01
  • 1
    $\begingroup$ 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. $\endgroup$
    – Count
    Apr 27, 2021 at 10:29
  • 1
    $\begingroup$ 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: $E(\sigma_{T+l}^2\vert {\cal F_{T}})$. $\endgroup$
    – Count
    Apr 27, 2021 at 10:38
  • $\begingroup$ @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 $\beta=(X^\top X)^{-1}X^\top y$? $\endgroup$ Apr 27, 2021 at 13:39
  • $\begingroup$ @RichardHardy, I corrected my answer, I mean it not possible to estimate a GARCH model via OLS, my thoughts were with ARCH models at this moment. Sorry for the confusion. $\endgroup$
    – Count
    Apr 27, 2021 at 15:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.