Reversals of signal-posterior monotonicity imply a bias of screening

TL;DR

This paper strengthens previous results on how screening biases can cause nonmonotonic relationships between signals and posteriors, extending the analysis to various distributions and signals.

Contribution

It enhances prior findings by demonstrating that reversals of signal-posterior monotonicity occur under broader conditions, including exponential and Pareto distributions.

Findings

Reversals of monotonicity can occur with exponential and Pareto distributions.

The nonmonotonicity results apply to a wide range of signals.

Screening biases can lead to non-intuitive distributional relationships.

Abstract

This note strengthens the main result of Lagziel and Lehrer (2019) (LL) "A bias in screening" using Chambers Healy (2011) (CH) "Reversals of signal-posterior monotonicity for any bounded prior". LL show that the conditional expectation of an unobserved variable of interest, given that a noisy signal of it exceeds a cutoff, may decrease in the cutoff. CH prove that the distribution of a variable conditional on a lower signal may first order stochastically dominate the distribution conditional on a higher signal. The nonmonotonicity result is also extended to the empirically relevant exponential and Pareto distributions, and to a wide range of signals.