Identifying causal effects with subjective ordinal outcomes

Abstract: Survey questions often ask respondents to select from ordered scales where the meanings of the categories are subjective, leaving each individual free to apply their own definitions in answering. This paper studies the use of these responses as an outcome variable in causal inference, accounting for variation in interpretation of the categories across individuals. I find that when a continuous treatment variable is statistically independent of both i) potential outcomes; and ii) heterogeneity in reporting styles, a nonparametric regression of response category number on that treatment variable recovers a quantity proportional to an average causal effect among individuals who are on the margin between successive response categories. The magnitude of a given regression coefficient is not meaningful on its own, but the ratio of local regression derivatives with respect to two such treatment variables identifies the relative magnitudes of convex averages of their effects. These results can be seen as limiting cases of analogous results for binary treatment variables, though comparisons of magnitude involving discrete treatments are not as readily interpretable outside of the limit. I obtain a partial identification result for comparisons involving discrete treatments under further assumptions. An empirical application illustrates the results by revisiting the effects of income comparisons on subjective well-being, without assuming cardinality or interpersonal comparability of responses.