Quasi‐maximum likelihood estimation of discretely observed diffusions

Abstract

This paper introduces quasi-maximum likelihood estimator for discretely observed diffusions when a closed-form transition density is unavailable. Higher order Wagner-Platen strong approximation is used to derive the first two conditional moments and a normal density function is used in estimation. Simulation study shows that the proposed estimator has high numerical precision and good numerical robustness. This method is applicable to a large class of diffusions.Keywords Quasi-maximum likelihood estimator, diffusions, Wagner-Platen approximation. * The author thanks a co-editor and three referees for outstanding comments and suggestions. Thanks also go to Eckhard Platen, Yongjun Tang, and seminar participants at the SNDE 17th annual symposium for helpful comments. All remaining errors are mine.

Quasi-maximum likelihood estimation of multivariate diffusions

“…Results in Tables 3B and 3C further show that QMLE is numerically robust to large sampling intervals and to data with high volatility, which correspond to larger deviations from normality. This is consistent with the findings in Huang (2010).…”

“…We also note that the Euler method, which is based on order 0.5 strong approximation, is a special case of QMLE proposed in this paper. In practice, sampling interval is never zero and higher order QMLE usually outperforms Euler method when sampling interval is relatively large and the diffusion process has non-normal transition density (see Tables 3A, 3B, 3C, and other simulation results in Huang, 2010).…”

A quasi-maximum likelihood method for estimating the parameters of multivariate diffusions

“…There is therefore no need to rely on approximations such as those proposed by Kessler (1997) and Huang (2011) to generate these moments. Although this point seems obvious, it is our contention that it has not been sufficiently emphasised.…”

“…As these moments are determined by the initial point of each transition and do not change as the process evolves, discrete maximum likelihood is generally not a consistent estimation method. Elerian (1998), Shoji and Ozaki (1998), Kessler (1997) and Huang (2011) all develop ways of improving the Gaussian approximation.…”