Combining Long Memory and Level Shifts in Modeling and Forecasting the Volatility of Asset Returns

We consider modeling and forecasting a variety of asset return volatility series by adding a random level shift component to the usual long-memory ARFIMA model. We propose a parametric state space model with an accompanying estimation and forecasting framework that combines long memory and level shifts by decomposing the underlying process into a simple mixture model and ARFIMA dynamics. The Kalman filter is used to construct the likelihood function after augmenting the probability of states by a mixture of normally distributed processes. The forecasts are constructed by exploiting the information in the Kalman recursions. The adequacy of the estimation methodology is shown through a simulation study. We apply our model to volatility series categorized in two groups: high frequency based series (tick-by-tick SPY trades and realized volatility on the S&P 500 and 30-year Treasury Bond futures) and longer spans of log-absolute daily returns (S&P 500 returns, Dollar-Aus and Dollar-Yen exchange rates). The full sample estimates show that level shifts are present in all series. A genuine long-memory component is present in measures of volatility constructed using high-frequency data. On the other hand, volatility series proxied by log daily absolute returns are characterized by a remaining short-memory component that is nearly uncorrelated once the level shifts are accounted for. We conduct extensive out-of-sample forecast evaluations and compare the results with four popular competing models. Interestingly, our ARFIMA model with random level shifts is the only model that consistently belongs to the 10% Model Con dence Set of Hansen et al. (2011) for both pairwise and joint comparisons. It does so for all series, forecasting periods, forecast horizons, forecast evaluation criteria and volatility measures. The gains in forecast accuracy can be very pronounced, especially at longer horizons.