Bayesian inference with an adaptive proposal density for GARCH models

📝 Abstract

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.

💡 Analysis

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.

📄 Content

arXiv:0908.2982v1 [q-fin.CP] 20 Aug 2009 Bayesian inference with an adaptive proposal density for GARCH models Tetsuya Takaishi Hiroshima University of Economics, Hiroshima 731-0192, JAPAN E-mail: takaishi@hiroshima-u.ac.jp Abstract. 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.

1. Introduction In empirical finance volatility of asset returns is an important value to measure risk. In order to forecast future volatility it is desirable to use appropriate models which have the properties of volatility of asset returns. Many empirical studies suggest that the distribution of asset returns is leptokurtic. Furthermore the volatility is not constant, but changes over time. There are periods when volatility is very high or very low. This property of the volatility is called volatility clustering. The Autoregressive Conditional Heteroscedasticity (ARCH) model[1] and its generalization, the Generalized ARCH (GARCH) model[2] are designed to capture the property of the volatility clustering. Moreover the distributions of returns from those models show fat-tailed distributions and they are suggested to be Student’s t-(Tsallis) distributions[3, 4]. There are many extensions of the GARCH model to include additional properties of the volatility. An example of the properties of the volatility is that the volatility response is high after negative news (returns), which is known as the leverage effect, first observed by Black[5]. In order to cope with this fact, some models[6, 7, 8, 9, 10, 11] which introduce asymmetry into the volatility response function are proposed. In this study among them we focus on the Quadratic GARCH(QGARCH) model[10, 11] which adds an additional term proportional to a return to the volatility response function. To utilize the GARCH models we need to infer GARCH parameters from financial time series data. In general the Maximum Likelihood (ML) estimation is favored to the inference of GARCH models. Although implementation of the ML method is straightforward, there exist practical difficulties in estimating GARCH parameters by the ML technique. The model parameters must be positive to ensure a positive volatility and the stationarity condition for volatility is also required. The ML method with such requirements is performed via a constrained optimization technique which can be cumbersome. Forethermore the output of the ML method is often sensitive to starting values. Another estimation technique is the Bayesian inference which does not have the difficulties seen in the ML method. Usually the Bayesian inference is performed by Markov Chain Monte Carlo (MCMC) methods which have been common in the recent computer development. Various MCMC methods for the Bayesian inference of the GARCH models have been proposed[12, 13, 14, 15, 16, 17, 18, 19]. In a survey on the MCMC methods of the GARCH models[17] it is shown that Acceptance-Rejection/ Metropolis-Hastings (AR/MH) algorithm with a multivariate Student’s t-distribution works better than the other algorithms. The multivariate Student’s t-distribution is used as a proposal density of the MH algorithm and the parameters to specify the Student’s t-distribution are determined by the Maximum Likelihood (ML) technique. Recently an alternative method to estimate those parameters without relying on the ML technique was proposed[20, 21, 22]. The method is called ”adaptive construction scheme”, where the parameters of the multivariate Student’s t-distribution are determined by using the pre-sampled data by an MCMC method. And the parameters are updated adaptively during the MCMC simulation. The adaptive construction scheme was tested for GARCH and QGARCH models[20, 21, 22] and it is shown that the adaptive construction scheme can significantly reduce the correlation between sampled data. In this paper first we describe the adaptive construction scheme for the GARCH models. Then we make empirical studies with the QGARCH model for three major stock indexes, Nikkei 225, DAX and Hang Seng.
2. GARCH Model Let xt be an asset return observed at time t. We transform xt to yt as yt = xt −¯x, (1) where ¯x is the average over N observations, i.e. ¯x = 1 N N X i=1 xi. In the GARCH model yt is assumed to be decomposed as yt = σtǫt, (2) where ǫt is an identically distributed random variable with zero mean and

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