A semi-parametric GARCH (1, 1) estimator under serially dependent innovations

The main objective of this study is to derive semi parametric GARCH (1, 1) estimator under serially dependent innovations. The specific objectives are to show that the derived estimator is not only consistent but also asymptotically normal. Normally, the GARCH (1, 1) estimator is derived through quasi-maximum likelihood estimation technique and then consistency and asymptotic normality are proved using the weak law of large numbers and Linde-berg central limit theorem respectively. In this study, we apply the quasi-maximum likelihood estimation technique to derive the GARCH (1, 1) estimator under the assumption that the innovations are serially dependent. Allowing serial dependence of the innovations has however brought problems in terms of methodology. Firstly, we cannot split the joint probability distribution into a product of marginal distributions as is normally done. Rather, the study splits the joint distribution into a product of conditional densities to get around this problem. Secondly, we cannot use the weak laws of large numbers or/and the Linde-berg central limit theorem. We therefore employ the martingale techniques to achieve the specific objectives. Having derived the semi parametric GARCH (1, 1) estimator, we have therefore shown that the derived estimator not only converges almost surely to the true population parameter but also converges in distribution to the normal distribution with the highest possible convergence rate similar to that of parametric estimators