Endogenous Judge Decision Quality, Monotonicity, and Treatment Effects

I consider a setting where judges differ in the quality of their decisions and sort unobservable types into two groups. I identify properties of evidence generation processes which lead to violations of monotonicity, as defined in instrumental variable designs exploiting variations in judge propensities. When monotonicity is violated due to variations in judge decision quality, the standard local average treatment effect (LATE) estimand produces the wrong sign under simple conditions that I identify. Moreover, variations in judge decision quality make it impossible to use existing outcomes tests to make inferences regarding judge decision criteria, and therefore discriminatory practices adopted by judges. I then consider the impact of policy responses that adjust the cost of one type of decision error relative to another. When judge decision quality is determined endogenously, an increase in the relative cost of false positives can increase judges' correct positive rates only if higher judge decision quality is associated with a reduction in both types of decision errors. This provides an additional testable implication of monotonicity violations based on judges' responses to policy changes. I also show that when the LATE estimand is valid, policies can generate unintended consequences only if the signs of treatment effects vary across subjects of different types. Thus, I identify conditions under which LATEs as well as average treatment effects by type can be ascertained even when monotonicity is violated, and propose a test of these conditions