  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.
