# GARCH parameter estimation by linear regression?

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$. Can I also use linear regression or ordinary least square method to estimate the parameter tuple? • We do neither directly observe$\sigma_t^2$nor$\epsilon_t^2$. If you had those, you could come up with an auxiliary equation like$y=Xb+\varepsilon\$ - but that would not really be of help, IMO. Jun 25 at 7:01

### 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.

• I won't blame you if I buy it and don't like it but what's your review of the book. If you don't want to state a public review, my email is my name with a 2 at the end and then then that big g company. Thanks. Jun 25 at 11:35
• +1. Thank you so much. I will accept after a while if no more answer appears. Also, could you please taka a look at quant.stackexchange.com/q/65704/6686 ?
– Hans
Jun 25 at 11:58
• I will take a look at it later :-). @markleeds IMHO the book is on the theoretical side with some decent examples given throughout the book. I usually pick it up when I need to (re-)understand some theoretical concepts regarding estimation and/or inference of (G)ARCH models. I don't own a copy of the book myself, but rent it online through the national library of my home country. I would advise you to do the same and skim it through, before buying it :-).
– Pleb
Jun 25 at 14:01
• @Pleb; That's helpful. I'll put it on my "maybe get in the future" books list and look into doing what you do.. Everything is so dang expensive and Amazon changed the way they do "Look Inside". it's very limited now. Jun 25 at 14:21
• Accepted. Thank you.
– Hans
Jun 27 at 21:34