Dinghai Xu

Continuous Empirical Characteristic Function Estimation of Mixtures of Normal Parameters

This article develops an efficient method for estimating the discrete mixtures of normal family based on the continuous empirical characteristic function (CECF). An iterated estimation procedure based on the closed form objective distance function is proposed to improve the estimation efficiency. The results from the Monte Carlo simulation reveal that the CECF estimator produces good finite sample properties. In particular, it outperforms the discrete type of methods when the maximum likelihood estimation fails to converge. An empirical example is provided for illustrative purposes.

An Empirical Characteristic Function Approach to VaR under a Mixture of Normal Distribution with Time-Varying Volatility

Calculation of risk measures, such as Value-at-Risk and expected shortfall, requires knowledge of the underlying asset’s or portfolio’s returns distribution. To be realistic, this must be allowed to change over time. GARCH can be a good way to model the random evolution of an asset’s volatility, but standard GARCH assumes the innovation at each time step comes from a normal distribution. The resulting conditionally Gaussian returns therefore have normal, not fat, tails. One way to fatten the tails of the returns distribution is to use a fat-tailed forcing process, such as a Student-t with a low number of degrees of freedom. An alternative approach is to model the returns process as a mixture of normals, but if the distributions that are mixed have constant parameters, the time variation in volatility disappears. In this article, Xu and Wirjanto describe how timevarying fat-tailed densities can be formed by mixing GARCH processes together. Performance comparisons against other models in calculating the tail risks for exchange rates on four currencies show that the GARCH mixture model works very well.

GMM estimation of a realized stochastic volatility model: A Monte Carlo study

ABSTRACT This article investigates alternative generalized method of moments (GMM) estimation procedures of a stochastic volatility model with realized volatility measures. The extended model can accommodate a more general correlation structure. General closed form moment conditions are derived to examine the model properties and to evaluate the performance of various GMM estimation procedures under Monte Carlo environment, including standard GMM, principal component GMM, robust GMM and regularized GMM. An application to five company stocks and one stock index is also provided for an empirical demonstration.