GARCH parameter estimation by linear regression?

Question

In estimating a GARCH(1,1) model, $$\sigma_{t+1}^2 = \omega+\alpha \epsilon_t^2+\beta\sigma_t^2$$ Usually the parameter tuple $(\omega,\alpha,\beta)$ is estimated by the quasi-maximal likelihood$. Can I also use linear regression or ordinary least square method to estimate the parameter tuple?

Answer 1

This is not an answer, but reference help:

In the book "GARCH models: structure, statistical inference and financial applications" (Chapter 6 & 7) by Christian Francq & Jean-Michel Zakoian, they derive an OLS estimator (unconstrained and constrained) for the ARCH(q) model by rewriting it into an explicit AR(q)-representation. However, in the start of the chapter they further emphasize that QMLE would be a better estimation scheme (p. 127):

This estimation procedure has the advantage of being numerically simple, but has two drawbacks: (i) the OLS estimator is not efficient and is outperformed by methods based on the likelihood or on the quasi-likelihood that will be presented in the next chapters; (ii) in order to provide asymptotically normal estimators, the method requires moments of order 8 for the observed process.

A couple of pages into the chapter, they argue that you can define an OLS estimator for the GARCH(p,q) model. Yet, it is not explicit, since you cannot derive an AR(q)-representation from the GARCH(p,q) model when $p>0$ (Remark 6.1):

An OLS estimator can also be defined for a GARCH(p,q) model, but the estimator is not explicit, because $\varepsilon_t^2$ does not satisfy an AR model when $p\neq0$.

Lastly, in exercise 7.5 (p. 181) they specify the assumptions for the OLS estimators (unrestricted and restricted) to be strongly consistent.

I stumbled upon these chapters some time ago, and thought it might be of some help. All in all, it seems that OLS estimators can be constructed for ARCH and GARCH models under restrictive setups. Nevertheless, the authors still emphasize the use of QMLE as opposed to OLS.