Total Loss Functions for Measuring the Accuracy of Nonnegative Cross-Sectional Predictions

TL;DR

This paper characterizes the form of total loss functions for evaluating the accuracy of nonnegative cross-sectional predictions, showing that only additive, multiplicative, and certain L-type functions are admissible under broad assumptions.

Contribution

It provides a theoretical characterization of total loss functions, demonstrating that only additive, multiplicative, and specific L-type functions satisfy key axioms and assumptions.

Findings

Additive total loss functions are always admissible.

An isomorphism exists between additive and multiplicative loss functions.

Additive loss functions satisfy von Neumann-Morgenstern utility axioms.

Abstract

The total loss function associated with a set of cross-sectional predictions, that is, estimates or forecasts, summarizes the set's overall accuracy. Its arguments are the individual cross-sectional units' loss functions. Under general assumptions, including impartiality, about the forms of the individual loss functions, and the specific assumptions that the total loss function is anonymous and monotonic, only the additive, multiplicative and L-type (with restrictions) total loss functions are found to be admissible. The first two total loss functions correspond to different interpretations of economic utility. An isomorphism exists between these two total loss functions. Thus, the additive total loss function can always be used. This isomorphism can also be used to explore the properties of various combinations of total and individual loss functions. Moreover, the additive loss function obeys the von Neumann-Morgenstern expected utility axioms.