(JBES Invited Paper with Discussion and Reply)
Numerical Techniques for Maximum Likelihood Estimation of
Continuous-Time Diffusion Processes
Garland B. Durham and A. Ronald Gallant
Journal of the Business and Economic Statistics,
Vol. 20, No. 3. (Jul., 2002), pp. 297-316.
Abstract
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 CIR 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 data sets with 1000 observations, we are able to
approximate the maximum likelihood estimator with negligible error in
well under one minute. 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.
Keywords:
Simulated maximum likelihood, Stochastic differential equations.