On Testing Equal Conditional Predictive Ability Under Measurement Error

TL;DR

This paper characterizes loss functions that are exactly robust to measurement error, allowing reliable forecast comparisons using proxies, and shows that better proxies improve the power of predictive ability tests.

Contribution

It introduces the concept of exact robustness to measurement error for loss functions and fully characterizes this class as the Bregman class, enabling more accurate forecast evaluation.

Findings

Exactly robust loss functions are unaffected by measurement error on average.

More precise proxies increase the power of predictive ability tests.

Simulations and empirical analysis demonstrate the practical benefits of using robust loss functions and better proxies.

Abstract

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. 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 non-robust loss functions. An empirical application to US GDP growth rates demonstrates that it is easier to discriminate between forecasts issued at different horizons if a better proxy for GDP growth is used.