A bracketing relationship between difference-in-differences and lagged-dependent regression estimates

A bracketing relationship between difference-in-differences and lagged-dependent-variable adjustment

Difference-in-differences is a widely-used evaluation strategy that draws causal inference from observational panel data.Its causal identificationrelies on the assumption of parallel trends, which is scale dependent and may be questionable in some applications.A common alternative is a regressionmodel that adjusts for the lagged dependent variable, which rests on the assumption of ignorability conditional on past outcomes.We show that the difference-in-differences and lagged-dependent-variable regression estimates have a bracketing relationship.Namely, for a true positive effect, if ignorability is correct, then mistakenly assuming parallel trends will overestimate the effect ; in contrast, if the parallel trends assumption is correct, then mistakenly assuming ignorability will underestimate the effect.We show that the same bracketing relationship holds in general nonparametric (model-free) settings.We also extend the result to semiparametric estimation based on inverse probability weighting.