Bayesian estimation of the parameters of the ARCH and GARCH models using lindley's approximation

Autoregressive conditionally heteroscedastic (ARCH) and Generalised ARCH (GARCH) models are used to analyze empirical financial data and capture various stylized facts in financial econometrics. The procedure that is most commonly used for estimating the unknown parameters of ARCH and GARCH model is the maximum likelihood estimation (MLE) method. In this study, it is assumed that the parameters of the ARCH and GARCH models are random variables having known prior probability density functions, and therefore they will be estimated using Bayesian methods. The Bayesian estimators are not in a closed form, and thus Lindley's approximation will be used to estimate them. The Bayesian estimators are derived under squared error loss (SEL) and linear exponential (LINEX) loss functions. Examples are given in order to illustrate the findings. Furthermore, Monte Carlo simulations are performed in order to compare the ML estimates to the Bayesian ones. Finally, conclusions on the findings are given.