Copyrights to papers in the Duke Economics Working Paper Archive remain with the authors or their assignees. Archive users may download papers and produce them for their own personal use, but downloading of papers for any other activity, including reposting to other electronic bulletin boards or archives, may not be done without the written consent of the authors.

Duke Economics Working Paper #96-32

Estimating Stochastic Differential Equations Efficiently by Minimum Chi-Square

A. Ronald Gallant
Jonathan R. Long


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. Key Words: Diffusions, efficiency, estimation, exchange rate, minimum chi-square, partially observed state, simulation, specification test, stochastic differential equation.

Published in Biometrika, Volume 84, 1997, pp. 125-141.