Bayesian Estimation of GARCH(1,1) Model Using Tierney-Kadane’s Approximation

Abstract

The Generalized Autoregressive Conditionally Heteroscedastic (GARCH) process models the dependency of conditional second moments of financial time series. The maximum likelihood estimation (MLE) procedure is most commonly used for estimating the unknown parameters of a GARCH model. In this study, the parameters of the GARCH models with normal innovations are discussed for estimations using the Bayesian approach in which the parameters of the GARCH model are assumed as random variables having known prior probability density functions. The prior probability density functions of the parameters satisfy the conditions on GARCH parameters such as positivity and stationarity. The Bayesian estimators are not in a closed form. Thus Tierney-Kadane’s approximation that is a numerical integration method to calculate the ratio of two integrals is used to estimate. The Bayesian estimators are derived under squared error loss function. Finally, simulations are performed in order to compare the ML estimates to the Bayesian ones and furthermore, an example is given in order to illustrate the findings.