On testing equal conditional predictive ability under measurement error

Loss functions are widely used to compare several competing forecasts. However, forecast comparisons are often based on mismeasured proxy variables for the true target. We introduce the concept of exact robustness to measurement error for loss functions and fully characterize this class of loss functions as the Bregman class. Hence, only conditional mean forecasts can be evaluated exactly robustly. For such exactly robust loss functions, forecast loss differences are on average unaffected by the use of proxy variables and, thus, inference on conditional predictive ability can be carried out as usual. Moreover, we show that more precise proxies give predictive ability tests higher power in discriminating between competing forecasts. Simulations illustrate the different behavior of exactly robust and nonrobust loss functions. An empirical application to U.S. GDP growth rates demonstrates the nonrobustness of quantile forecasts. It also shows that it is easier to discriminate between mean forecasts issued at different horizons if a better proxy for GDP growth is used.