  The difference-in-differences (DID) design is widely used in observational
studies to estimate the causal effect of a treatment when repeated observations
over time are available. Yet, almost all existing methods assume linearity in
the potential outcome (parallel trends assumption) and target the additive
effect. In social science research, however, many outcomes of interest are
measured on an ordinal scale. This makes the linearity assumption inappropriate
because the difference between two ordinal potential outcomes is not well
defined. In this paper, I propose a method to draw causal inferences for
ordinal outcomes under the DID design. Unlike existing methods, the proposed
method utilizes the latent variable framework to handle the non-numeric nature
of the outcome, enabling identification and estimation of causal effects based
on the assumption on the quantile of the latent continuous variable. The paper
also proposes an equivalence-based test to assess the plausibility of the key
identification assumption when additional pre-treatment periods are available.
The proposed method is applied to a study estimating the causal effect of mass
shootings on the public's support for gun control. I find little evidence for a
uniform shift toward pro-gun control policies as found in the previous study,
but find that the effect is concentrated on left-leaning respondents who
experienced the shooting for the first time in more than a decade.
