On the recoverability of forecasters' preferences

We study the problem of identifying a forecaster's loss function from observations on forecasts, realizations, and the forecaster's information set. Essentially different loss functions can lead to the same forecasts in all situations, though within the class of all continuous loss functions, this is strongly nongeneric. With the small set of exceptional cases ruled out, generic nonparametric preference recovery is theoretically possible, but identification depends critically on the amount of variation in the conditional distributions of the process being forecast. There exist processes with sufficient variability to guarantee identification, and much of this variation is also necessary for a process to have universal identifying power. We also briefly address the case in which the econometrician does not fully observe the conditional distributions used by the forecaster, and in this context we provide a practically useful set identification result for loss functions used in forecasting binary variables.