Efficient Method of Moments

We describe a computationally intensive methodology for the estimation and analysis of partially observable nonlinear systems. An example from epidemiology is the SEIR model, which is a system of differential equations with random coefficients that describe a population in terms of four state variables: those susceptible to disease, those exposed to it, those infected by it, and those recovered from it. Only those infected by the disease are known to public health officials. An example from finance is the continuous-time stochastic volatility model, which is a system of stochastic differential equations that describes a security's price and instantaneous variance. Only the security's price can be observed directly. System parameters are estimated by a variant of simulated method of moments known as efficient method of moments (EMM). The idea is to match moments implied by the system to moments implied by the transition density for observables. System analysis is accomplished by reprojection. Reprojection is carried out by projecting a long simulation from the estimated system onto an appropriate representation of a relationship of interest. A general purpose representation is a Hermite expansion of the conditional density of state variables given observables. A reprojection density thus obtained embodies all constraints implied by the nonlinear system and is analytically convenient. As an instance, nonlinear filtering can be accomplished by computing the conditional mean of the reprojection density and evaluating it at observed values from the data. Ideas are illustrated by using the methodology to assess the dynamics of a stochastic volatility model for daily Microsoft closing prices.