  Difference-in-differences is a widely-used evaluation strategy that draws
causal inference from observational panel data. Its causal identification
relies on the assumption of parallel trends, which is scale dependent and may
be questionable in some applications. A common alternative is a regression
model that adjusts for the lagged dependent variable, which rests on the
assumption of ignorability conditional on past outcomes. In the context of
linear models, \citet{APbook} 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. We provide
three examples to illustrate the theoretical results with replication files in
\citet{ding2019bracketingData}.
