Density Discontinuity Regression

TL;DR

This paper introduces a Bayesian method to analyze how density discontinuities at policy thresholds vary with observable factors, capturing heterogeneity in strategic behavior across contexts.

Contribution

It develops an interpretable Bayesian framework that jointly estimates density ratios and shapes as functions of covariates, improving upon existing methods.

Findings

Discovered variation in strategic behavior at thresholds across different firm characteristics.

Identified stronger discontinuities for proposals with negative recommendations and in larger firms.

Provided a new tool for quantifying heterogeneous responses to threshold-based policies.

Abstract

Many policies hinge on a continuous variable exceeding a threshold, prompting strategic behavior by agents to stay on the favorable side. This creates density discontinuities at cutoffs, evident in contexts like taxable income, corporate regulations, and academic grading. Existing methods detect these discontinuities, but systematic approaches to examine how they vary with observable characteristics are lacking. We propose a novel, interpretable Bayesian framework that jointly estimates both the log-density ratio at the cutoff and the local shape of the density, as functions of covariates, within a data-driven window. This formulation yields regression-style estimates of covariate effects on the discontinuity. An adaptive window selection balances bias and variance. Our approach improves upon common methods that target only the log-density ratio around the threshold while ignoring the local density shape. We constrain the density jump to be non-negative, reflecting that agents would not aim to be on the losing side of the threshold. Applied to corporate shareholder voting data, our method identifies substantial variation in strategic behavior, notably stronger discontinuities for proposals facing negative recommendations from Institutional Shareholder Services, larger firms, and firms with lower analyst coverage. Overall, our method provides an interpretable framework to quantify heterogeneous agent responses to threshold-based policies.