Good Controls Gone Bad: Difference-in-Differences with Covariates

TL;DR

This paper identifies a key assumption in Difference-in-Differences methods involving covariates, demonstrates bias when violated, and proposes a new estimator, DID-INT, that remains unbiased under such violations.

Contribution

It introduces the two-way CCC assumption, analyzes bias in existing estimators, and proposes the DID-INT estimator for unbiased ATT estimation with covariates.

Findings

Standard TWFE and CS-DID estimators are biased when the two-way CCC assumption is violated.

The proposed DID-INT estimator provides unbiased ATT estimates under two-way CCC violations.

DID-INT can handle heterogeneous treatment effects and staggered treatment rollout.

Abstract

This paper introduces the two-way common causal covariates (CCC) assumption, which is necessary to get an unbiased estimate of the ATT when using time-varying covariates in existing Difference-in-Differences methods. The two-way CCC assumption implies that the effect of the covariates remain the same between groups and across time periods. This assumption has been implied in previous literature, but has not been explicitly addressed. Through theoretical proofs and a Monte Carlo simulation study, we show that the standard TWFE and the CS-DID estimators are biased when the two-way CCC assumption is violated. We propose a new estimator called the Intersection Difference-in-differences (DID-INT) which can provide an unbiased estimate of the ATT under two-way CCC violations. DID-INT can also identify the ATT under heterogeneous treatment effects and with staggered treatment rollout. The estimator relies on parallel trends of the residuals of the outcome variable, after appropriately adjusting for covariates. This covariate residualization can recover parallel trends that are hidden with conventional estimators.