OLS and QMLE Estimation for GARCH Models
Summary
The document considers whether ordinary least squares can estimate the parameters of a GARCH model, which are commonly fitted using quasi-maximum likelihood. It summarizes a reference describing explicit OLS estimation for ARCH models through an autoregressive representation, and notes that OLS can also be defined for GARCH models even though the estimator is not explicit when variance lags are present.
The cited discussion favors QMLE because OLS is less efficient and its asymptotic normality requires strong moment conditions, including finite eighth moments in the ARCH setup described. A second answer highlights a further complication: conditional variance is not directly observed, and estimating it from returns introduces measurement error that can undermine OLS regressor exogeneity. Thus, OLS is possible under restrictive assumptions, but the document presents it as a less attractive general approach than likelihood-based estimation. It does not provide a full derivation, implementation procedure, or empirical comparison for a particular dataset.
Key ideas
- OLS estimators can be constructed for ARCH and GARCH models under suitable assumptions.
- ARCH models admit an explicit autoregressive representation that can support OLS estimation.
- For GARCH models, the OLS estimator is generally not explicit when variance lags are included.
- The cited reference favors QMLE on efficiency grounds and because OLS needs restrictive moment conditions.
- Variance estimates from observed returns can introduce measurement error and threaten OLS exogeneity.
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Full text
# GARCH parameter estimation by linear regression?
# 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?
## Answer by Pleb (score 3, accepted)
https://quant.stackexchange.com/a/65703
#### 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.
## Answer by Kashyap (score 0)
https://quant.stackexchange.com/a/85722
We do not directly observe the variance. In quant finance context, what we observe is the price or the returns. We can estimate the variance but this will not be without measurement errors. These measurement errors become part of the error term (residuals) in the OLS framework and cause violation of exogeneity of the regressors condition, that is, $Cov(X, \epsilon) \neq 0$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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