A Regime Switching Long Memory Model to Forecast Realized Volatility

It is well known that accurately measuring and forecasting financial volatility plays a central role in many pricing and risk management problems. With high frequency intra-day data becoming widely available more accurate estimates of volatility can be obtained. Realized volatility, i.e. the sum of intra-day squared returns, reduces the noise in the volatility estimate considerably compared to other volatility measures such as squared or absolute daily returns. Thus volatility becomes, in some sense, ‘observable’ and can be modelled directly rather than being treated as a latent variable as in classical GARCH or stochastic volatility models. \nRecent studies have documented some properties of realized volatility. One of the most relevant is that its dynamics exhibits long memory or high persistence. Linear fractionally integrated models are generally used to capture this long range dependence. The literature has also documented non linear effects in volatility, such as leverage effects and regime changes. Moreover it has been found that break, nonlinearities and long memory are confounding factors and distinguishing between them can be rather troublesome. Some recent literature in this field suggest the existence of both non linear and long memory components in many economics and financial time series, thus it appears of obvious interest to joint modelling both features into a single time series model. In this way it is also possible to see whether the benefits of combining long memory and nonlinearities will improve accuracy in forecasting volatility. For these reasons, we propose a simple model that simultaneously captures long memory and nonlinearities in which level and persistence shifts through a Markov switching approach. We use this model, named as Markov switching-long memory (MS-LM), to describe the dynamic characteristics of the logarithmic realized volatility of the S&P500 stock index. To estimate the model we use Bayesian methods. In particular Markov Chain Monte Carlo (MCMC) techniques are employed to estimate all the unknown quantities of the model, i.e. probabilities and sizes of shifts together with the other parameters. The algorithm proposed here extends the standard procedures implemented for Markov switching models to take in account the additional long memory parameter. More precisely we adopt a multi-move Gibbs sampler to simulate the state process and a Metropolis-Hastings scheme for the long memory parameters. Once applied the MS-LM model to S&P500 realized volatility, we compare its performance in term of fit and forecasting with those of MS and linear fractionally integrated models. Forecasting accuracy are based on the weighted likelihood ratio test together with other standard forecasting evaluation tests. All models have been estimated using the MCMC algorithm previously introduced. Also Bayesian predictive densities have been obtained within such algorithm. The in-sample results show that both long memory and nonlinearities are significant and improve the description of the data. The out-sample results, obtained using several forecast horizons, show that introducing these nonlinearities produces superior long range forecasts over those obtained from nested models. \n