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

Professor Jun Yu received a Ph.D. in economics at the University of Western Ontario in 1998. He taught at the Business School of the University of Auckland between 1998 and 2003 and Singapore Management University (SMU) between 2004 and 2023. He is currently UMDF chair Professor of Finance and Economics at the University of Macau and Dean of the Faculty of Business Administration at the University of Macau. Before that, he was Lee Kong Chian Professor of Economics and Finance at SMU, director of Sim Kee Boon Institute for Financial Economics at SMU, and the lead principal investigator of the Centre for Research on the Economics of Ageing at SMU.

Professor Yu has published about 100 papers. Many of these publications are in leading journals in finance and economics. His articles for detecting the presence of asset price bubbles and estimating their origination and termination dates have initiated a new area of research on the econometric analysis of bubbles in financial assets and real estate. Many researchers have been attracted to work in this area using the methods developed in these articles. Many central banks have used these techniques for early warning signals. Several computer software packages have been written to implement these methods. In 2020, his co-authored textbook, titled “Financial Econometric Modeling”, was published by Oxford University Press. In honor of Professor Yu, an edited book titled “Financial Econometrics: Theory and Application” is forthcoming at the Cambridge University Press.

Professor Yu is an inaugural fellow of the Society of Financial Econometrics and a fellow of the Journal of Econometrics. He serves as Associate Editor of the Journal of Econometrics and Econometric Theory. 

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 smooth function of the Hurst parameter (i.e., $H$), and the asymptotic distribution is consistently Gaussian. These features greatly facilitates statistical inference. The asymptotic theory for ML estimators differs from that of alternative estimators in the literature, which is discontinuous in convergence rates at $H=3/4$ and involve non-standard limiting distributions. Simulation results indicate that the ML method offers more accurate parameter estimates, compared to the alternative estimators, and the derived asymptotic theory can closely approximates 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 the Hurst parameter always lies substantially below 1/2, supporting the roughness hypothesis documented in the literature. It is also found that the fOU process with ML parameter estimates and the conditional mean predictor delivers more accurate out-of-sample forecasts than the fOU process, when it is estimated by alternative estimation methods and alternative prediction formular, many popular models, including HAR, log-HAR, ARFIMA, Cauchy processes and Brownian semistationary processes with the power law kernel.