Comparison between the Bayesian and Frequentist Estimators: Univariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH) Model

Abstract

The estimates of the Maximum Likelihood estimation method are the estimates of the global maximum likelihood function, by definition. However, the present study showed empirically that the likelihood function of the GARCH model is multimodal. Due to the presence of multimodality in the likelihood function leads to a difference in estimates at local and global maxima, and hence, Maximum Likelihood estimation methods can have unstable performance in such situations. Therefore, it will face the problem in inference and prediction, due to the difference in estimates at local and global maxima(s). Two estimation methods are chosen from the Frequentist and the Bayesian approach, respectively, to measure the significance of the difference in estimated parameters due to the presence of multimodality in the likelihood function. Besides, to calculate the level of difference, a standard method of Monte Carlo simulation method is used. The surface plot is constructed by changing the value of the Monte Carlo simulation method to evaluate their performance along the whole surface. these surfaces are then compared within each approach. Subsequently, the preferable algorithms are compared across the Bayesian and Frequentist approaches. For comparison, the present study has calculated bias and variance around the true data generating process. Empirically it is found that in case of Frequentist approach Differential Evolution (DE) algorithm is preferable estimation method for GARCH type models, as compared to Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. Because there is multimodality in the likelihood function of the GARCH model, and BFGS uses a single starting value to search maximum point in the likelihood function, and often this single starting value traps into local maxima. Therefore, the estimated parameter at the local and global maxima vary, and hence, inferences and predictions. Conversely, DE uses multiple starting values with multiple chains, due to which it automatically avoid local maxima and converges to global maxima. In the case of the Bayesian approach, Robust Adaptive Metropolis (RAM) is a preferable estimation for GARCH type models as compared to Metropolis Hasting (MH). Because RAM is based on the strategy of adaptive mechanism, i.e., the Markov Chain of the RAM move to the next point, after taking information from the previous point, and finally converge to some particular value of the estimate. While MH use chain of independent nature, i.e., it does not take information while moving from one point to another point in the Markov Chain. After confirming the best estimator from frequentist and the Bayesian approach, this study compared these approaches with each other. Empirically, it is found that the Bayesian approach (RAM) is the preferable estimation method than the Frequentist approach (DE) because the level of bias and variance around the true parameter for RAM is lower than DE. Pakistan Stock Exchange (PSX) is used as a real-world application. Empirically it is found that the Bayesian approach is preferable estimation method than the frequentist approach. Reasons are followed; first, in the frequentist approach estimated parameters are the point estimates, while in the case of the Bayesian approach, the complete distribution of the estimated parameter is obtained at the low cost of simulation. Second, the distribution of the point estimate is hypothetically assumed to be normal, while in case of Bayesian approach it is not valid, i.e., the distribution of the estimates could be skewed in either direction. Therefore, the frequentist approach either over or underestimate the true value of the parameter. Finally, the standard error of the estimates which are obtained through the DE algorithm is more precise as compared to the estimates of BFGS. Therefore, the forecasting based on DE is more accurate about risk and return.