  Difference-in-differences (DiD) is a popular method to evaluate treatment
effects of real-world policy interventions. Several approaches have previously
developed under alternative identifying assumptions in settings where pre- and
post-treatment outcome measurements are available. However, these approaches
suffer from several limitations, either (i) they only apply to continuous
outcomes and the average treatment effect on the treated, or (ii) they depend
on the scale of outcome, or (iii) they assume the absence of unmeasured
confounding given pre-treatment covariate and outcome measurements, or (iv)
they lack semiparametric efficiency theory. In this paper, we develop a new
framework for causal identification and inference in DiD settings that
satisfies (i)-(iv), making it universally applicable, unlike existing DiD
methods. Key to our framework is an odds ratio equi-confounding (OREC)
assumption, which states that the generalized odds ratio relating treatment and
treatment-free potential outcome is stable across pre- and post-treatment
periods. Under the OREC assumption, we establish nonparametric identification
for any potential treatment effect on the treated in view, which in principle
would be identifiable under the stronger assumption of no unmeasured
confounding. Moreover, we develop a consistent, asymptotically linear, and
semiparametric efficient estimator of treatment effects on the treated by
leveraging recent learning theory. We illustrate our framework with extensive
simulation studies and two well-established real-world applications in labor
economics and traffic safety evaluation.
