Estimating GARCH(1,1) in the presence of missing data

Abstract

Maximum likelihood estimation of the famous $GARCH(1,1)$ model is generally straightforward, given the full observation series. However, when some observations are missing, the marginal likelihood of the observed data is intractable in most cases of interest, also intractable is the likelihood from temporally aggregated data. For both these problems, we propose to approximate the intractable likelihoods through sequential Monte Carlo (SMC). The SMC approximation is done in a smooth manner so that the resulting approximate likelihoods can be numerically optimized to obtain parameter estimates. In the case with data aggregation, the use of SMC is made possible by a novel state space representation of the aggregated GARCH series. Through extensive simulation experiments, the proposed method is found to be computationally feasible and produce more accurate estimators of the model parameters compared with other recently published methods, especially in the case with aggregated data. In addition, the Hessian matrix of the minus logarithm of the approximate likelihood can be inverted to produce fairly accurate standard error estimates. The proposed methodology is applied to the analysis of time series data on several exchange-traded funds on the Australian Stock Exchange with missing prices, due to interruptions such as scheduled trading holidays.