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Unrestricted and controlled identification of loss functions: Possibility and impossibility results

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  • Lieli, Robert P.
  • Stinchcombe, Maxwell B.
  • Grolmusz, Viola M.

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 unrestricted 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.

Suggested Citation

  • Lieli, Robert P. & Stinchcombe, Maxwell B. & Grolmusz, Viola M., 2019. "Unrestricted and controlled identification of loss functions: Possibility and impossibility results," International Journal of Forecasting, Elsevier, vol. 35(3), pages 878-890.
    • Handle: RePEc:eee:intfor:v:35:y:2019:i:3:p:878-890
    DOI: 10.1016/j.ijforecast.2018.11.007
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    References listed on IDEAS

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