Instrumental variables and local effects

Let D(1) and D(0) denote treatment receipt under the two instrument assignments. Let Y(z,d) denote the outcome under instrument z and receipt d. Consistency and no interference connect actual observations to these quantities. Require a nondegenerate instrument assignment independent jointly of the potential receipt and outcome variables. Random encouragement can justify that independence when implemented correctly, but it does not alone justify the exclusion restriction: Y(z,d) must equal Y(d), so encouragement has no outcome effect except through receipt.

Monotonicity requires that encouragement never reduces receipt for any unit: D(1) is at least D(0). Units then fall into always-takers, never-takers, and compliers, with defiers excluded. Relevance requires a strictly positive first-stage difference, equivalent here to a positive complier fraction. These assumptions have distinct roles. An instrument can predict receipt strongly while violating exclusion, and weak prediction can make estimation unstable even if the causal assumptions are otherwise credible.

Under monotonicity, the difference in mean receipt between instrument arms is the probability of being a complier. Always-takers receive treatment in both arms and never-takers in neither. Under exclusion, their outcomes also do not change with the instrument, so they contribute zero to the instrument’s average outcome effect. Compliers contribute their treatment effect, multiplied by their population fraction. Dividing the outcome intention-to-treat contrast by the receipt contrast therefore cancels that fraction.

The ratio identifies the average effect within the complier principal stratum, whose membership is defined by two potential receipt values and is not generally observed for each unit. It need not equal the population ATE when compliance types have different effects. Changing the instrument can change which units are compliers and thus change the local estimand. This explains why different valid instruments can identify different treatment effects without contradiction; local population and treatment margin are part of the causal question.

Without monotonicity, the first stage is a difference between complier and defier fractions, and the numerator weights their effects with opposite signs. The ratio may then fail to be a meaningful average of treatment effects. Without exclusion, a direct instrument effect enters the numerator. Without independence, the two instrument arms can differ in potential outcomes before any encouragement acts. None of these failures is diagnosed by a small standard error or a large first-stage coefficient alone.

Even with valid identification, a near-zero first stage amplifies sampling error in the ratio. Ordinary normal approximations for a ratio can perform poorly under weak instruments, so identification of a nonzero population denominator and strong-instrument inference are different issues. If the intervention of interest is encouragement itself, the intention-to-treat contrast may already be the appropriate policy target and need not be divided. The correct analysis begins with the desired intervention and population, then states which assumptions support that particular interpretation.