It is well known that volatility asymmetry exists in financial markets. This paper reviews and investigates recently developed techniques for Bayesian estimation and model selection applied to a large group of modern asymmetric heteroskedastic models. These include the GJR-GARCH, threshold autoregression with GARCH errors, threshold GARCH and Double threshold heteroskedastic model with auxiliary threshold variables. Further we briefly review recent methods for Bayesian model selection, such as: reversible jump Markov chain Monte Carlo, Monte Carlo estimation via independent sampling from each model and importance sampling methods. Seven heteroskedastic models are then compared, for three long series of daily Asian market returns, in a model selection study illustrating the preferred model selection method. Major evidence of nonlinearity in mean and volatility is found, with the preferred model having a weighted threshold variable of local and international market news.