Maximum likelihood estimation of fractional Ornstein-Uhlenbeck process with discretely sampled data

摘要

This paper develops the long-span asymptotic theory for the maximum likelihood (ML) estimator of parameters in the discretely sampled fractional Ornstein-Uhlenbeck (fOU) process. It is shown that the convergence rate is a continuous function of the Hurst parameter (i.e., H), and the asymptotic distribution is always Gaussian, facilitating statistical inference. The asymptotic theory is different from that of existing estimators in the literature. Simulation results indicate that the ML method offers more accurate parameter estimates compared to the existing estimators and the derived asymptotic distribution can closely approximate the finite-sample distribution. In the empirical analysis, we fit the fOU process to daily logarithmic RV and daily logarithmic trading volume series for ten exchange-traded funds. The ML estimate of H always lies substantially below 1/2, supporting the roughness hypothesis documented in the literature. Moreover, the fOU process with the ML estimate and the conditional mean predictor outperformed two sets of competitors — the fOU process with existing estimates and predictive formula, and established models, including HAR, log-HAR, ARFIMA, Cauchy processes and Brownian semistationary processes with the power law kernel — in out-of-sample forecasting.