Consistent estimation for fractional stochastic volatility model under high-frequency asymptotics

We develop a statistical theory for a continuous time approximately log-normal fractional stochastic volatility model to examine whether the volatility is rough, that is, whether the Hurst parameter is less than one half. We construct a quasi-likelihood estimator and apply it to realized volatility time series. Our quasi-likelihood is based on the error distribution of the realized volatility and a Whittle-type approximation to the auto-covariance of the log-volatility process. We prove the consistency of our estimator under high-frequency asymptotics, and examine by simulations its finite sample performance. Our empirical study suggests that the volatility of the time series examined is indeed rough.