Measurement Error Sensitivity of Loss Functions for Distribution Forecasts

Abstract

I examine the sensitivity of loss functions—equivalently called scoring rules—for distribution forecasts in two dimensions: sensitivity to linear rescaling of the data and the influence of measurement error on the forecast evaluation outcome. First, I show that all commonly used scoring rules for distribution forecasts are robust to rescaling the data. Second, it is revealed that the forecast ranking based on the continuous ranked probability score is less sensitive to measurement error than the ranking based on the log score. The theoretical results are complemented by a simulation study aligned with frequently revised quarterly US GDP growth data and an empirical application forecasting realized variances of S&P 100 constituents. In line with its proven gross-error insensitivity, the ranking of the continuous ranked probability score is the most consistent between evaluations based on the true outcome and the observations with measurement error.

Keywords: Forecast evaluation, measurement error, distribution forecasts, proper scoring rules

JEL Classification: C50, C52, C53