We develop a GMM approach for estimation of log-normal stochastic volatility models
driven by a fractional Brownian motion with unrestricted Hurst exponent. We show that
a parameter estimator based on the integrated variance is consistent and, under stronger
conditions, asymptotically normally distributed. We inspect the behavior of our procedure
when integrated variance is replaced with a noisy measure of volatility calculated from discrete
high-frequency data. The realized estimator contains sampling error, which skews the fractal
coefficient toward “illusive roughness.” We construct an analytical approach to control the
impact of measurement error without introducing nuisance parameters. In a simulation study,
we demonstrate convincing small sample properties of our approach based both on integrated
and realized variance over the entire memory spectrum. We show the bias correction attenuates
any systematic deviance in the parameter estimates. Our procedure is applied to empirical
high-frequency data from numerous leading equity indexes. With our robust approach the
Hurst index is estimated around 0.05, confirming roughness in stochastic volatility.