Maximum Likelihood Estimation of Fractional Ornstein-Uhlenbeck Process with Discretely Sampled Data

This paper first derives two analytic formulae for the autocovariance of the discretely sampled fractional Ornstein-Uhlenbeck (fOU) process. Utilizing the analytic formulae, two main applications are demonstrated: (i) investigation of the accuracy of the likelihood approximation by the Whittle method; (ii) the optimal forecasts with fOU based on discretely sampled data. The finite sample performance of the Whittle method and the derived analytic formula motivate us to introduce a feasible exact maximum likelihood (ML) method to estimate the fOU process. The long-span asymptotic theory of the ML estimator is established, where the convergence rate is a smooth function of the Hurst parameter (i.e., H) and the limiting distribution is always Gaussian, facilitating statistical inference. The asymptotic theory is different from that of some existing estimators studied in the literature, which are discontinuous at H = 3/4 and involve non-standard limiting distributions. The simulation results indicate that the ML method provides more accurate parameter estimates than all the existing methods, and the proposed optimal forecast formula offers a more precise forecast than the existing formula. The fOU process is applied to fit daily realized volatility (RV) and daily trading volume series. When forecasting RVs, it is found that the forecasts generated using the optimal forecast formula together with the ML estimates outperform those generated from all possible combinations of alternative estimation methods and alternative forecast formula.