Stochastic Volatility with ARMA Extension Applied to Portfolio Choice

Empirical studies report that the Geometric Brownian motion does not describe accurately the returns on many securities. Some of the departures are skewness, excess kurtosis and time-varying volatility. The idea of modeling the time-varying volatility component of securities goes back to Merton's work in the 1970s. In the following decades, many authors worked on the applications of the stochastic volatility in the dynamic modeling of many time series. The stochastic volatility literature of the 1990s and of the last years pursued the estimation of a simple stochastic volatility model focusing on the efficiency of the Markov chain Monte Carlo (MCMC) method rather than on the adaptation of the model to accommodate the real features of the underlying process. In the last years, some attempts have been made to develop new models with more realistic features, but the issue of autoregressive moving average (ARMA) behavior of the returns in the stochastic volatility models has not yet been explored. The ARMA and stochastic volatility behavior of stock returns are well-documented in empirical studies and have separately received a great deal of attention in the financial econometrics literature. This paper develops a new MCMC procedure for inference in a model that combines stochastic volatility with the ARMA process, which will be named ARMA-SVOL. The Markov chain sampling schemes for a linear regression model with an ARMA (p,q) are modified and combined with the MCMC procedures for stochastic volatility sampling. This procedure is specified as a hierarchy of conditional probability distributions which provides an environment for the estimation of the ARMA-SVOL model and departures from standard distributional assumptions. The results of the MCMC estimation procedure for the ARMA-SVOL model are explored in two dimensions. First, a model of simulated time series data is estimated and sensitivity analysis for parameter and volatility inference is performed. Second, optimal portfolios are formed and out-of-sample portfolio returns are generated using a model with ARMA process in returns and stochastic volatility. The economic benefits of return predictability are studied by comparing the impact of market (ARMA process) versus volatility (ARMA-SVOL process) timing on the performance of optimal portfolio rules. The new MCMC procedure that is proposed here offers flexibility in the dynamic modeling of the securities returns dynamics. This method allows the error term not only to be ARMA(p,q) process, but also to have a time-varying variance. The application of this method to simulated data demonstrates how to conduct inference with samples of parameters obtained by the Markov chain sampling. Moreover, the application of this method to portfolio optimization rules is an important tool to compare the economic benefits of market versus vo