Continuously updated extremum estimators

An important class of structural econometric models (nonlinear rational expectations, option pricing, auction models, ...) characterize observable variables as highly nonlinear transforma- tions of some latent variables. These transformations are one-to-one, but they depend on the unknown distribution of the latent variables through the equilibrium of the game and/or the learning process. Therefore numerical complexity of the equilibrium definition generates sub- stantial obstacles for the direct implementation of maximum likelihood inference. This is a particular case of argmax estimators based on a untractable sample based criterion Q[exp.T][thêta,lambda(thêta)] contaminated by the occurences of [thêta] in a ‘nuisance function’ [lambda(thêta)] to which corresponds a simple criterion Q[exp.T][thêta,lambda(thêta.exp0)] with [thêta.exp.0] the true, unknown value of the parameter. The natural idea is to replace the unknown value [lambda(thêta.exp0)] by some ’good proxy’, say [lambda(thêta.exp1)], to maximize Q[exp.T][thêta,lambda(thêta.exp1)] with respect to [thêta] and to get a new, updated estimate which, in turn, can be used for approxi- mating [lambda(thêta.exp0)],... Such steps can be considered for a fixed sample size (iterative M-estimators), or each time new data arrive (recursive estimators). In this paper we present and analyze several iterative and recursive (Robbins-Monro type) estimation procedures of this kind which, at the end, are applied to the class of structural econometric models which motivated this work.