Estimation of a diffusion process for the United States short interest rate using a semigroup pseudo likelihood

In this paper, a new pseudo maximum likelihood estimator is developed for discretely observed diffusion processes and applied to the estimation of a diffusion process for the US short interest rate. The estimator exploits a prior result that parameters appearing in the mean and the variance of a distribution can be estimated consistently using pseudo maximum likelihood estimation if the pseudo density is a member of the quadratic exponential family and if the mean and the variance of the pseudo density are correctly specified. Correct specification of the first two conditional moments is achieved through the use of a semigroup representation of the expectation operator.;The resulting semigroup pseudo maximum likelihood estimator has a number of favorable properties. It is a strongly consistent estimator of the true parameters for fixed sampling intervals of any length. Nonlinear drift and diffusion coefficients are estimated easily. It makes extensive use of the transition information in the data, so that the drift and diffusion coefficients are fully identified. When the number of terms in the semigroup series expansion is controlled relative to the number of observations, the estimator is asymptotically normal. Evaluation of the objective function is straightforward, requiring no simulation or other numerical procedures.;After theoretical results for the new estimator are developed, the estimator is applied to a stationary scalar diffusion process for the US short interest rate using historical data. The drift coefficient is found to be nonlinear, with zero drift over much of the support of the process. The diffusion coefficient is found to be monotonically increasing and convex in the levels of the short rate.