On Recoding Ordered Treatments as Binary Indicators

Researchers using instrumental variables to investigate the effects of ordered treatments (e.g., years of education, months of healthcare coverage) often recode treatment into a binary indicator for any exposure (e.g., any college, any healthcare coverage). The resulting estimand is difficult to interpret unless the instruments only shift compliers from no treatment to some positive quantity and not from some treatment to more -- i.e., there are extensive margin compliers only (EMCO). When EMCO holds, recoded endogenous variables capture a weighted average of treatment effects across complier groups that can be partially unbundled into each group's treated and untreated means. Invoking EMCO along with the standard Local Average Treatment Effect assumptions is equivalent to assuming choices are determined by a simple two-factor selection model in which agents first decide whether to participate in treatment at all and then decide how much. The instruments must only impact relative utility in the first step. Although EMCO constrains unobserved counterfactual choices, it places testable restrictions on the joint distribution of outcomes, treatments, and instruments.