A Volatility Model for Financial Time Series in the Generalized Pearson Setting

Abstract

Our objective is to review the existing literature on modeling financial asset return distributions and propose additional models and techniques that provide a better fit for a given financial asset return series such as global stock indices, industry segments and foreign exchange. One possible way is to adjust the model within the whole family of parameters, or only within the family of parameters, which make economic sense. The ideal model specification is the one that can handle structural change and time dependence in conditional mean, variance, skewness, and kurtosis. This may have more economic appeal than assuming fundamental nonnormality. We postulate that the simultaneous estimation of time-varying first-four moments using a flexible family of probability distributions such as the Pearson type distributions might provide a better explanation of risks, and hence, robust design of portfolio allocation systems beyond the traditional first-two moments framework. In a study of fractual structures in exchange rates, Richards (2000) finds in a simulation experiment that the best performing model among non-linear time series models is a GARCH, in which a generalized error distribution was modified to allow for wider tails.In this research, we will combine an autoregressive conditional heteroskedasticity model with an asymmetric information structure due to Daniel B. Nelson (1991) with a flexible family of distributions, developed by Karl Pearson (1895). Within this framework, a nonlinear parametric model with time-varying higher moments is proposed. We, hereby, attempt to extend the time-varying conditional variance nature of traditional ARCH/GARCH-type models to include either time-varying skewness or kurtosis for it might improve our understanding of risks and risk premia seen in financial markets. To this end, we will explore the possible ill behavior of standardized residuals in many types of the time-varying conditional variance models and show that these residuals lead us to the modeling of time-varying skewness and kurtosis. We will fit the Pearson distribution directly to sample data by calculating the second, third and fourth central moments of the observed values and using the definitions of skewness and kurtosis. However, observed values of the third and fourth moments could be sensitive to outliers. This would question the validity of the equation, which takes advantage of the time-varying properties of kurtosis. Moreover, the sensitivity of higher moments’ estimates to a small number of extreme returns also means that ex-post returns may have quite different properties from ex-ante returns. We are also planning to incorporate the concepts of L-moments, introduced by Hosking (1990). L-moments are defined to be the expected values of linear expectations of the order statistics. They are less sensitive to outliers than ordinary moments, and often provide a better identification of the parent distribution that generates a particular data sample. One might use L-moments to fit a particular distribution to data by equating the first few sample and population L-moments, analogously to the method of moments. The resulting estimators of parameters and quantiles are sometimes more accurate than the maximum-likelihood estimates, in case of generalized extreme value distributions, Hosking et al. (1985), and some instances of the generalized Pareto distribution, Hosking and Wallis (1987). If the mean of the distribution exists, then all the higher-order L-moments do as well (see “Theorem 1” in Hosking (1996)). Thus, L-moments can describe fat-tailed distributions whose variance or higher-order regular moments may be infinite. Moreover, for standard errors of sample L-moments to be finite, the largest moment that needs to be finite is the second moment; i.e., variance. We believe the use of L-moments for a leptokurtic distribution such as the Pearson type-VII might be of some value due to the success of L-moments with other heavy-tailed distributions.