International Journal of Forecasting 35 (2019) 0–12 Contents lists available at ScienceDirect International Journal of Forecasting journal homepage: www.elsevier.com/locate/ijforecast Unrestricted and controlled identification of loss functions: Possibility and impossibility results Robert P. Lieli a,∗ , Maxwell B. Stinchcombe b , Viola M. Grolmusz a a Department of Economics, Central European University, Budapest, Hungary b Department of Economics, University of Texas at Austin, United States article info Keywords: Loss functions Bregman loss functions GPL loss functions Osband’s principle Identification Point forecasts abstract The property that the conditional mean is the unrestricted optimal forecast characterizes the Bregman class of loss functions, while the property that the α-quantile is the unre- stricted optimal forecast characterizes the generalized α-piecewise linear (α-GPL) class. However, in settings where the forecaster’s choice of forecasts is limited to the support of the predictive distribution, different Bregman losses lead to different forecasts. This is not true for the α-GPL class: the failure of identification is more fundamental. Motivated by these examples, we state simple conditions that can be used to ascertain whether loss functions that are consistent for the same statistical functional become identifiable when off-support forecasts are disallowed. We also study the identifying power of unrestricted forecasts within the class of smooth, convex loss functions. For any such loss ℓ, the set of losses that are consistent for the same statistical functional as ℓ is a tiny subset of this class in a precise mathematical sense. Finally, we illustrate the identification problem that is posed by the non-uniqueness of consistent losses for the moment-based loss function estimation methods proposed in the literature. © 2019 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved. 1. Introduction 1.1. Overview Under the assumption that point forecasts are con- structed so as to minimize expected loss, the mean of the conditional distribution of the target variable is the unrestricted optimal forecast for any Bregman loss. (In statistical parlance, Bregman losses are consistent for the mean.) The converse of this statement is also true: if the conditional mean is the optimal forecast under a given loss function for a sufficiently rich set of distributions, then that loss function must belong to the Bregman class. These loss functions go back to Bregman (1967) and Savage (1971); more recently, Banerjee, Guo, and Wang (2005), Gneiting ∗ Corresponding author. E-mail addresses: lielir@ceu.edu (R.P. Lieli), maxwell.stinchcombe@austin.utexas.edu (M.B. Stinchcombe), Grolmusz_Viola@phd.ceu.edu (V.M. Grolmusz). (2011a) and Patton (2011, 2016) have used Bregman losses to make various points about the construction and evalua- tion of point forecasts. There are also other loss functions for which the op- timal point forecast is given by a well-known statistical functional other than the mean. In the case of asymmetric absolute loss, the optimal forecast is a fixed quantile of the predictive distribution, and the quantile is determined by the marginal losses for positive vs. negative forecast er- rors. Each asymmetric absolute loss function for which the α-quantile is the optimal forecast belongs to a larger class of generalized α-piecewise linear (α-GPL) loss functions, a class that is characterized by the property that each mem- ber of the class induces the same α-quantile as the optimal forecast (see Saerens, 2000, for the characterization result; and Gneiting, 2011b; Komunjer, 2005; Lieli & Stinchcombe, 2013, for identification issues). The Bregman class and the α-GPL classes pose a chal- lenge for the literature that is concerned, either directly or indirectly, with recovering loss functions from forecasts, https://doi.org/10.1016/j.ijforecast.2018.11.007 0169-2070/© 2019 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.