Simulated Score Methods and Indirect Inference for Continuous-time Models

We describe a simulated method of moments estimator that is implemented by choosing the vector valued moment function to be the expectation under the structural model of the score function of an auxiliary model, where the parameters of the auxiliary model are eliminated by replacing them with their quasi-maximum likelihood estimates. This leaves a moment vector depending only the parameters of the structural model. Structural parameter estimates are those parameter values that put the moment vector as closely to zero as possible in a suitable GMM metric. This methodology can also be interpreted as a practical computational strategy for implementing indirect inference. We argue that considerations from statistical science dictate that the auxiliary model should approximate the true data generating process as closely as possible and show that using the SNP model is one means to that end. When the view of close approximation is accepted in implementation, the methodology described is usually referred to as Efficient Method of Moments (EMM) in the literature because (i) the estimator is asymptotically as efficient as maximum likelihood under correct specification, and (ii) detection of model error is assured under incorrect specification. There are alternative views toward the desirability of close approximation to the data, which we discuss.