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Loss functions are widely used to compare several competing forecasts.However, forecast comparisons are often based on mismeasured proxy variablesfor the true target. We introduce the concept of exact robustness tomeasurement error for loss functions and fully characterize this class of lossfunctions as the Bregman class. For such exactly robust loss functions,forecast loss differences are on average unaffected by the use of proxyvariables and, thus, inference on conditional predictive ability can be carriedout as usual. Moreover, we show that more precise proxies give predictiveability tests higher power in discriminating between competing forecasts.Simulations illustrate the different behavior of exactly robust and non-robustloss functions. An empirical application to US GDP growth rates demonstratesthat it is easier to discriminate between forecasts issued at differenthorizons if a better proxy for GDP growth is used.

On Testing Equal Conditional Predictive Ability Under Measurement Error