Comment on Garland B. Durham and A. Ronald Gallant’s “Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes”

This paper proposes an interesting approach for estimating the parameters of nonlinear diffusion models with discretely sampled data. The parameter estimates are obtained by maximizing an approximate likelihood function that is obtained by a Monte Carlo importance sampling method. As the authors point out, the elements of their approach are not substantially new. In particular, the idea of approximating the transition density of the process by integrating out the “missing values” between each successive observation is due to Pedersen (1995); the use importance sampling to estimate the likelihood in which the missing values are drawn from a “tied-down” distribution is due to Elerian, Chib, and Shephard (2001); and finally, the idea of transforming the process to one with a constant diffusion coefficient in order to improve the accuracy of the Euler approximation scheme is due to Doss (1977). As we see it, the paper makes two main contributions. First, it provides a detailed comparison of various extant methods for estimating the likelihood function of univariate diffusion models. Elerian (1999) has done related work along the same lines. Taken together these two papers have enhanced our understanding of what methods are effective for approximating the likelihood function. We are pleased that the value of such comparative work has been recognized by this journal. Second, the paper provides a new proposal density for the importance sampling step