Manipulation Tests in Regression Discontinuity Design: The Need for Equivalence Testing

Researchers applying regression discontinuity design (RDD) often test for endogenous running variable (RV) manipulation around treatment cutoffs, but misinterpret statistically insignificant RV manipulation as evidence of negligible RV manipulation. I introduce novel procedures that can provide statistically significant evidence that RV manipulation around a cutoff is bounded beneath practically negligible levels. The procedures augment classic RV density tests with an equivalence testing framework, along with bootstrap methods for (cluster-)robust inference. I apply these procedures to replication data from 36 RDD publications, conducting 45 equivalence-based RV manipulation tests. Over 44% of RV density discontinuities at the cutoff cannot be significantly bounded beneath a 50% upward jump. Obtaining equivalence testing failure rates beneath 5% requires arguing that a 350% upward RV density jump at the cutoff is practically equal to zero. My results imply that meaningful RV manipulation around treatment cutoffs cannot be ruled out in many published RDD papers, and that standard tests frequently misclassify the practical significance of RV manipulation. I provide research guidelines and help researchers conduct more credible equivalence-based manipulation testing in future RDD research. The lddtest estimation routine is available in R, Stata, and Python.