Estimating Stochastic Differential Equations Efficiently by Minimum Chi-Squared

A. Ronald Gallant, Jonathan R. Long
Biometrika, Vol. 84, No. 1. (Mar., 1997), pp. 125-141.


We propose a minimum chi-square estimator for the parameters of an ergodic system of stochastic differential equations with partially observed state. We prove that the efficiency of the estimator approaches that of maximum likelihood as the number of moment functions entering the chi-square criterion increases and as the number of past observations entering each moment function increases. The minimized criterion is asymptotically chi-squared and can be used to test system adequacy. When a fitted system is rejected, inspecting studentized moments suggests how the fitted system might be modified to improve the fit. The method and diagnostic tests are applied to daily observations on the U.S. dollar to Deutschmark exchange rate from 1977 to 1992.

Keywords: Diffusions, Efficiency, Estimation, Exchange rate, Minimum chi-squared, Partially observed state, Simulation, Specification test, Stochastic differential equation.