Keywords: Asymptotic Properties of the OLS Estimator, CLT (Central Limit Theorem) for Martingale Increments, Constrained and unconstrained OLS, Ordinary Least Squares (OLS), Quasi-Least Squares (QLS).
Description: The simplest estimation method for ARCH models is that of the Ordinary Least Squares (OLS). This estimation procedure presents 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 quasilikelihood that will be presented in the next chapters; (ii) the method, in order to provide asymptotically normal estimators, requires moments of order 8 for the observed process. The extension of the OLS method, the Quasi-Least Squares (QLS) method, suppresses the first drawback and attenuates the second drawback of the OLS estimator: the QLS method provides estimators asymptotically as accurate as the quasi-maximum likelihood under the assumption that moments of order 4 exist. Note that the least-squares methods are interesting in practice because they provide initial estimators for the optimization procedure that is used in the quasi-maximum likelihood method.