## 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.

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