Confounding and regression adjustment in difference-in-differences studies

To define confounding bias in difference-in-difference studies and compare regression- and matching-based estimators designed to correct bias due to observed confounders.

We simulated data from linear models that incorporated different confounding relationships: time-invariant covariates with a time-varying effect on the outcome, time-varying covariates with a constant effect on the outcome, and time-varying covariates with a time-varying effect on the outcome. We considered a simple setting that is common in the applied literature: treatment is introduced at a single time point and there is no unobserved treatment effect heterogeneity.

We compared the bias and root mean squared error of treatment effect estimates from six model specifications, including simple linear regression models and matching techniques.

Simulation code is provided for replication.

Confounders in difference-in-differences are covariates that change differently over time in the treated and comparison group or have a time-varying effect on the outcome. When such a confounding variable is measured, appropriately adjusting for this confounder (ie, including the confounder in a regression model that is consistent with the causal model) can provide unbiased estimates with optimal SE. However, when a time-varying confounder is affected by treatment, recovering an unbiased causal effect using difference-in-differences is difficult.

Confounding in difference-in-differences is more complicated than in cross-sectional settings, from which techniques and intuition to address observed confounding cannot be imported wholesale. Instead, analysts should begin by postulating a causal model that relates covariates, both time-varying and those with time-varying effects on the outcome, to treatment. This causal model will then guide the specification of an appropriate analytical model (eg, using regression or matching) that can produce unbiased treatment effect estimates. We emphasize the importance of thoughtful incorporation of covariates to address confounding bias in difference-in-difference studies.