A Closed-Form Estimator for the Garch(1,1)-Model

We propose a closed-form estimator for the linear GARCH(1,1) model. The estimator has the advantage over the often used quasi-maximum-likelihood estimator (QMLE) that it can be easily implemented, and does not require the use of any numerical optimisation procedures or the choice of initial values of the conditional variance process. We derive the asymptotic properties of the estimator, showing T^{(K-1)/K}-consistency for some K(1,2) when the 4th moment exists and the square root of T-asymptotic normality when the 8th moment exists. We demonstrate that a finite number of Newton-Raphson iterations using our estimator as starting point will yield asymptotically the same distribution as the QMLE when the 4th moment exists. A simulation study confirms our theoretical results