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Bayesian inference with an adaptive proposal density for GARCH models

We perform the Bayesian inference of a GARCH model by the Metropolis-Hastings algorithm with an adaptive proposal density. The adaptive proposal density is assumed to be the Student's t-distribution and the distribution parameters are evaluated by using the data sampled during the simulation. We apply the method for the QGARCH model which is one of asymmetric GARCH models and make empirical studies for for Nikkei 225, DAX and Hang indexes. We find that autocorrelation times from our method are very small, thus the method is very efficient for generating uncorrelated Monte Carlo data. The results from the QGARCH model show that all the three indexes show the leverage effect, i.e. the volatility is high after negative observations.