How should parameter estimation be tailored to the objective?

Abstract

We study parameter estimation from the sample X, when the objective is to maximize the expected value of a criterion function, Q, for a distinct sample, Y. This is the situation that arises when a model is estimated for the purpose of describing other data than those used for estimation, such as in forecasting problems. A natural candidate for solving maxT∈σ(X)EQ(Y,T) is the innate estimator, θˆ=argmaxθQ(X,θ). While the innate estimator has certain advantages, we show that the asymptotically efficient estimator takes the form θ̃=argmaxθQ̃(X,θ), where Q̃ is defined from a likelihood function in conjunction with Q. The likelihood-based estimator is, however, fragile, as misspecification is harmful in two ways. First, the likelihood-based estimator may be inefficient under misspecification. Second, and more importantly, the likelihood approach requires a parameter transformation that depends on the true model, causing an improper mapping to be used under misspecification.