Volatility Puzzle: Long Memory or Antipersistency

The log realized volatility (RV) is often modeled as an autoregressive fractionally integrated moving average model ARFIMA( 1 , d , 0 ). Two conflicting empirical results have been found in the literature. One stream shows that log RV has a long memory (i.e., the fractional parameter d > 0). The other stream suggests that the autoregressive coefficient α is near unity with antipersistent errors (i.e., d < 0). This paper explains how these conflicting empirical findings can coexist in the context of ARFIMA( 1 , d , 0 ) model by examining the finite sample properties of popular estimation methods, including semiparametric methods and parametric maximum likelihood methods. The finite sample results suggest that it is challenging to distinguish Model 1 (ARFIMA( 1 , d , 0 ) with α close to 0 and d close to 0.5) from Model 2 (ARFIMA( 1 , d , 0 ) with α close to unity and d close to –0.5). An intuitive explanation is given. For the 10 financial assets considered, despite that no definitive conclusions can be drawn regarding the data-generating process, we find that the frequency domain maximum likelihood (or Whittle) method can generate the most accurate out-of-sample forecasts.