Semiparametric Estimation in Continuous-Time: Asymptotics for Integrated Volatility Functionals With Small and Large Bandwidths

Abstract

This paper studies the estimation of integrated volatility functionals, which is essentially a semiparametric two-step estimation problem in the nonstationary continuous-time setting. Different from the classic i.i.d. or stationary setting, a faster-than-$n^{1/4}$ convergence rate for the first-step nonparametric estimator can not be achieved here. By removing various biases in the second step, we establish a new stochastic equicontinuity condition and show that the proposed estimator is root-$n$ consistent and asymptotically mixed Gaussian, for a wider range of bandwidths than existing ones. Moreover, we employ matrix calculus to obtain a new analytical bias correction and variance estimation method that has computational advantage over existing analytical and Jackknife/simulation-based methods. Comprehensive simulation studies demonstrate that our method has good finite sample performance for a variety of volatility functionals, including quadraticity, determinant, continuous beta, and eigenvalues.