A Maximum Likelihood Approach for Stochastic Volatility Models
Moshe Fridman and Lawrence Harris, forthcoming, Journal of Business and Economic Statistics, 1998.
Abstract
The paper presents a fast new numeric algorithm for evaluating the N-fold integral that appears in the likelihood function of stochastic volatility models. The method allows these models to be estimated using standard numeric maximum likelihood techniques. The method is much faster than currently used methods and it is easily implemented for a wide variety of models. It is easily adapted for filtering, smoothing and forecasting.
Using the new method, the paper estimates and compares several alternative models for daily stock index returns. In an attempt to compare the relative utility of stochastic methods and GARCH methods of characterizing volatility, a model in which both effects are nested is estimated. The results suggest that the data are well characterized by the stochastic volatility model: The GARCH effect adds little value when stochastic volatility is modeled.
The paper also examines alternative models for the conditional process. These included a normal model with both mean and variance subordinated to a common stochastic process, and a conditional Student-t model with stochastic variance.
These models were not as useful in characterizing the data as were variations in the distribution of the stochastic volatility process. The stochastic model that best fits daily stock index returns data characterizes log volatility as a AR(1) process with innovations given by a signed power transform of the normal distribution. The estimated power coefficient of 2.6 implies very fat tails for this distribution. Interestingly, when the stochastic innovations are allowed to have such large tails, the estimated AR persistence drops significantly.
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Last revised 11/19/97.