Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes

Stochastic differential equations often provide a convenient way to describe the dynamics of economic and financial data, and a great deal of effort has been expended searching for efficient ways to estimate models based on them. Maximum likelihood is typically the estimator of choice; however, since the transition density is generally unknown, one is forced to approximate it. The simulation-based approach suggested by Pedersen (1995) has great theoretical appeal, but previously available implementations have been computationally costly. We examine a variety of numerical techniques designed to improve the performance of this approach. Synthetic data generated by a Cox-Ingersoll-Ross model with parameters calibrated to match monthly observations of the U.S. short-term interest rate are used as a test case. Since the likelihood function of this process is known, the quality of the approximations can be easily evaluated. On datasets with 1,000 observations, we are able to approximate the maximum likelihood estimator with negligible error in well under 1 min. This represents something on the order of a 10,000-fold reduction in computational effort as compared to implementations without these enhancements. With other parameter settings designed to stress the methodology, performance remains strong. These ideas are easily generalized to multivariate settings and (with some additional work) to latent variable models. To illustrate, we estimate a simple stochastic volatility model of the U.S. short-term interest rate.