Manipulation testing based on Benford's Law for discrete scores

2026-07-15

AI Summary

This study addresses the challenge of detecting asymmetric or systematic manipulation of the running variable in regression discontinuity designs—issues often overlooked by conventional methods. Building on Benford’s Law, the authors develop a novel diagnostic framework that innovatively extends the law from the digit level to the probability level. They introduce a directional density decomposition and an adaptive nonparametric bandwidth selection mechanism, thereby overcoming the symmetry assumption inherent in the McCrary test and enabling precise identification of manipulation sources. By integrating Nigrini’s (2012) threshold adaptation technique, they construct two complementary robust statistical tests. Empirical evidence demonstrates that this framework effectively uncovers subtle manipulations missed by traditional approaches, substantially enhancing the diagnostic power and reliability of regression discontinuity designs.

Abstract

This paper addresses the problem of running variable manipulation in Regression Discontinuity Designs. Leveraging the observation that manipulation often alters the density balance around the cutoff, we detect these structural imbalances using Benford's Law -a natural statistical regularity widely applied in fraud detection. Our framework serves as a vital precautionary safeguard alongside traditional McCrary-type tests. It eliminates researcher-chosen parameters that can skew outcomes, while delivering a deeper diagnostic breakdown of the density's behavior. Crucially, whereas the classic McCrary test can overlook systemic imbalances due to its rigid symmetric setup, our method separates the data into directional components. This allows researchers to pinpoint the exact origin of a deviation and spot hidden manipulation that standard frameworks fail to capture. To achieve this, we introduce an innovative method for selecting a bandwidth consistent with BL, and construct two distinct, complementary tests using threshold values adapted from Nigrini (2012) that successfully transition the law's application from digits to probabilities. Empirical applications confirm the enhanced protective value of this diagnostic framework.

Problem

• Regression Discontinuity Designs
• running variable manipulation
• Benford's Law
• density discontinuity
• fraud detection

Innovation

• Benford's Law
• Regression Discontinuity Design
• manipulation test
• density discontinuity
• bandwidth selection