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*Duke Economics Working Paper #96-32*

# Estimating Stochastic Differential Equations Efficiently by
Minimum Chi-Square

##
A. Ronald Gallant

and

Jonathan R. Long

### Abstract

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.