  We use the GARCH model with a fat-tailed error distribution described by a
rational function and apply it for the stock price data on the Tokyo Stock
Exchange. To determine the model parameters we perform the Bayesian inference
to the model. The Bayesian inference is implemented by the Metropolis-Hastings
algorithm with an adaptive multi-dimensional Student's t-proposal density. In
order to compare the model with the GARCH model with the standard normal errors
we calculate information criterions: AIC and DIC, and find that both criterions
favor the GARCH model with a rational error distribution. We also calculate the
accuracy of the volatility by using the realized volatility and find that a
good accuracy is obtained for the GARCH model with a rational error
distribution. Thus we conclude that the GARCH model with a rational error
distribution is superior to the GARCH model with the normal errors and it can
be used as an alternative GARCH model to those with other fat-tailed
distributions.
