Economics Commons^™

Different Strokes For Different Folks: Long Memory And Roughness, Shuping Shi, Jun Yu

SMU Economics and Statistics Working Paper Series

The log realized volatility of financial assets is often modeled as an autoregressive fractionally integrated moving average model (ARFIMA) process, denoted by ARFIMA(p, d, q), with p = 1 and q = 0. Two conflicting results have been found in the literature regarding the dynamics. One stream shows that the data series has a long memory (i.e., the fractional parameter d > 0) with strong mean reversion (i.e., the autoregressive coefficient |α1| ≈ 0). The other stream suggests that the volatil-ity is rough (i.e., d < 0) with highly persistent dynamic (i.e., α1 → 1). To consolidate the findings, this paper first examines the finite sample properties of alternative estimation methods employed in the literature for the ARFIMA(1, d, 0) model and then applies the outperforming techniques to a wide range of financial assets. The candidate methods include two parametric maximum likeli-hood (ML) methods (the maximum time-domain modified profile likelihood (MPL) and maximum frequency-domain likelihood) and two semiparametric methods (the local Whittle method and log periodogram estimation method). The two parametric methods work well across all parameter set-tings, with the MPL method outperforming. In contrast, the two semiparametric methods have a very large upward bias for d and an equally large downward bias for α1 when α1 is close to unity. The poor performance of the semiparametric methods in the presence of a highly persistent dynamic might lead to a false conclusion of long memory. In the empirical applications, we find that the log realized volatilities of exchange rate futures over the past decade have a long memory, where the point estimate of d is between 0.4 and 0.5 and the estimate of α1 is near zero. For other finan-cial assets considered (including stock indices and industry indices), we find that they have rough volatility, with the point estimate of d being negative and the point estimates of α1 close to unity.