Volatility Puzzle: Long Memory or Antipersistency

Abstract

The log realized volatility (RV) is often modeled as an autoregressive fractionally integrated moving average model ARFIMA([Formula: see text]). 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([Formula: see text]) 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 [Formula: see text] (ARFIMA([Formula: see text]) with α close to 0 and d close to 0.5) from [Formula: see text] (ARFIMA([Formula: see text]) 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. This paper was accepted by Lukas Schmid, finance.