Parameter Estimation with Out-of-Sample Objective

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 in forecasting problems and whenever an estimated model is to be applied to a draw from the general population. A natural candidate for solving max_T EQ(Y; T), is the innate estimator,  \hat\theta  = arg max_\theta  Q(X ; \theta ). While the innate estimator has certain advantages, we show, under suitable regularity conditions, that the asymptotically eficient estimator takes the form \tilde\theta  = arg max_\theta  \tilde{Q}(X ; \theta), where \tilde{Q} 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 truth, causing an improper mapping to be used under misspecification. The theoretical results are illustrated with two applications, one involving a Gaussian likelihood and an asymmetric loss function; and another is the problem of making multi-period ahead forecasts.