Efficient estimation of integrated volatility and related processes

We derive nonparametric efficiency bounds for regular estimators of integrated smooth transformations of instantaneous variances, in particular, integrated power variance. We find that realized variance attains the efficiency bound for integrated variance under both regular and irregular sampling schemes. For estimating higher powers such as integrated quarticity, the block-based procedures of Mykland and Zhang (2009) can get arbitrarily close to the nonparametric bounds, when observation times are equidistant. Moreover, the estimator in Jacod and Rosenbaum (2013), whose efficiency was documented for the submodel assuming constant volatility, is efficient also for nonconstant volatility paths. When the observation times are possibly random but predictable, we provide an estimator, similar to that of Kristensen (2010), which can get arbitrarily close to the nonparametric bound. Finally, parametric information about the functional form of volatility leads to a lower efficiency bound, unless the volatility process is piecewise constant.