Difference-in-Differences for Ordinal Outcomes: Application to the Effect of Mass Shootings on Attitudes toward Gun Control
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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.
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