Binary-instrument principal strata and LATE

In the setup of Binary instrumental-variable potential-outcome setup, the four possible joint values of $(A(0),A(1))$ define the following principal strata (subsets of the target population):

• Always-takers: $A(0)=A(1)=1$.
• Never-takers: $A(0)=A(1)=0$.
• Compliers: $A(0)=0$ and $A(1)=1$; write this event as $C=\{A(1)>A(0)\}$.
• Defiers: $A(0)=1$ and $A(1)=0$.

Under exclusion, the common potential outcomes $Y(a)$ are defined as in Binary instrumental-variable potential-outcome setup. When $Y(0),Y(1)$ are integrable, the average treatment effect is $\mathrm{ATE}:=E[Y(1)-Y(0)]$. If $P(C)>0$, the local average treatment effect (LATE) is ${\tau }_C:=E[Y(1)-Y(0)\mid C]\mathrm{.}$ Complier status depends on the joint pair of potential treatments and is not defined by the single treatment value observed for a unit. Monotonicity excludes defiers; it does not assert that the LATE equals the population ATE.