The Bregman class of loss functions is characterized by the property that the con-ditional mean is the unrestricted optimal forecast for any Bregman loss. Similarly,generalized piecewise linear (GPL) loss functions all give rise to a given quantile asthe unrestricted optimal forecast. Using the identification theory in Lieli and Stinch-combe (2013), we argue that Bregman losses are still potentially distinguishable ifrestrictions are placed on the set of allowable forecasts; e.g., off-support forecasts areexcluded. In contrast, GPL loss functions remain observationally equivalent even insuch forecasting environments—here the failure of identification is more fundamental.Motivated by these examples, we conclude by asking partly open questions about thenonparametric identifiability of loss functions.