Markov Regime Switching Stochastic Volatility

This is a project on modeling time-varying volatility of S&P 500 weely return for the years 1990 to 2012 using Bayesian methods. First, MCMC on the log-stochastic volatility (SV) model is implemented with simulation results analyzed. Second, I generalize the SV model to encompass regime-switching properties with the markov switching log-stochastic volatility (MSSV) model, under which, high-volatility regime is able to overlap with economic recession periods. Finally, simulation study is performed. 1 The log-stochastic volatility model (SV) The SV model with an AR(1) specification for the evolution of the log-variance, yt+1 = u + √ Vt t+1 log(Vt+1) = αv + βvlog(Vt) + σv t+1 where y denotes S&P 500 weekly return rate, has been widely used to model US stock price return (See, for example, Hamilton and Susmel 1994). S&P 500 weekly return data 1990-2012 is plotted as following in table 1, and the sample mean and standard error are 0.14% and 2.4%. Figure 1. S&P 500 weekly rate of return Using Gibbs MCMC together with random walk Metropolis, I estimate parameters in the SV model as follows in table 1: