Modeling and forecasting long range dependence in volatility

Abstract

This thesis conducts three exercises on volatility modeling of financial assets. We are essentially interested in the estimation and forecasting of daily volatility, a measure of the strength of price movements over daily intervals. Two of the exercises are in the realm of high frequency data: modeling and forecasting realized volatility which is constructed from intra-day returns. The other exercise is concerned with discrete stochastic volatility modeling using daily returns. The main focus of each exercise is to represent the high degree of volatility persistence, which is an important stylized fact of daily volatility. In the first exercise, daily realized volatility of the Yen/USD exchange rate is modeled through an autoregressive and moving-average fractionally integrated (ARFIMA) process. We differ from previous studies by averaging across a set of ARFIMA and ARMA models with different orders of autoregressive and moving-average polynomials. The vehicle used to execute this averaging exercise is Bayesian model averaging, through which part of the uncertainty introduced by model selection is integrated out. We examine the practical usefulness of our method by conducting a rolling-sample estimation, and the results indicate the weighted average forecast out-performs that of a single model at long-term horizons by providing smaller mean squared forecast errors. The second exercise is concerned with Bayesian estimation of a long memory stochastic volatility (SV) model. We use a high-order moving-average process to approximate the fractional integration specified for the latent log volatility. As such, the long memory SV model can be expressed in a state-space form, which facilitates the implementation of Markov chain Monte Carlo (MCMC) simulation when parameters and latent volatility are estimated. We update the set of memory parameter and volatility of volatility parameter in one block in the MCMC algorithm, by using the hessian matrix. A Monte Carlo study indicates in general, when the posterior mean is treated as a point estimator of parameters, our Bayesian method compares well with classical methods. Furthermore, the Bayesian estimator tends to outperform the popular frequency quasi maximum likelihood estimator, according to the root mean square error criterion, with small and medium sample size. An empirical analysis of the daily Yen/USD exchange rate spanning 26 years is conducted, and the degree of persistency in volatility is found to be consistent with that from the first exercise when high frequency data are used. In the third exercise, we look at the long memory property from a different angle. There has been a large literature using specifications other than fractional integration to mimic the long memory property in time series analysis, although there are few applications to realized volatility. In this exercise, regime switching models are fitted to daily realized volatility of the JPY/USD exchange rate from 1996 to 2009. Both in-sample fit and out-of-sample forecasting are used to compare across the three types of models, including ARFIMA, regime switching and sum of short memory processes. An extensive recursive estimation over one year suggests that regime switching is superior in capturing the dynamics of the time series examined, and generating more accurate out-of-sample forecasts.