A Source of Long Memory in Volatility

The long memory characteristic of financial market volatility is well documented and has important implications for volatility forecasting and option pricing. When fitted to the same data, different volatility models calculate the unconditional variance differently and could have very different volatility persistent parameters. Hence, they produce very different volatility forecasts even when the projection is just beyond a few days. The popular GARCH and GJR models have short memory. This paper compares the out-of-sample forecasting performance of four long memory volatility models, viz. fractional integrated (FI), break, component and regime switching. Using S&P 500 returns, we find structural break model to produce the best in-sample fit and out-of-sample forecasts, if future volatility breaks are known. Without knowing the future breaks, GJR produced the best short horizon forecasts. For volatility forecasts of 10 days and beyond, FI dominates. The FI model projects the future unconditional variance from the exponentially weighted sum of an infinite number of past shocks. The persistence parameters then control how fast the forecasts converge to this unconditional variance. As the fractional differencing parameter gets closer to and exceeds 0.5, volatility is non-stationary. The success of the FI model in forecasting S&P 500 volatility suggests that the latter should be treated as nonstationary. Which volatility model is best for forecasting is an empirical issue. A best model for S&P 500 need not be the best for the other series, and may not always be the best, all the time, for forecasting S&P 500 volatility. Unusual events such as the 1987 crash, for example, call for unusual treatments to get better forecasting performance.