Binary instrumental-variable potential-outcome setup

Let $(\mathrm{\Omega },\mathcal{F},P)$ be a probability space for a target population of units. A real random variable is a measurable map to $\mathbb{R}$ with its Borel sigma-algebra (Random elements and real random variables); expectations are integrals with respect to $P$ (Expectation of a nonnegative or integrable random variable). A unit's potential outcomes depend only on that unit's treatment receipt and instrument assignment; this is the no-interference and single-treatment-version scope used here.

An IV potential-outcome setup consists of binary random variables $Z,A:\mathrm{\Omega }\to \{0,1\}$, an outcome $Y:\mathrm{\Omega }\to \mathbb{R}$, binary potential treatments $A(z):\mathrm{\Omega }\to \{0,1\}$ for $z\in \{0,1\}$, and real potential outcomes $Y(a,z):\mathrm{\Omega }\to \mathbb{R}$ for $a,z\in \{0,1\}$. Each potential outcome is integrable. The instrument has positive assignment probability at both levels, $0<P(Z=1)<1$.

All of $Z,A,Y,A(z),Y(a,z)$ are measurable random variables, with the discrete sigma-algebra on $\{0,1\}$ and the Borel sigma-algebra on $\mathbb{R}$.

Consistency means $A=A(Z)$ and $Y=Y(A,Z)$ almost surely. The exclusion restriction means $Y(a,1)=Y(a,0)$ almost surely for each $a$; under this condition write the common value as $Y(a)$. Joint instrument independence means that $Z$ is independent of the random element $(A(0),A(1),(Y(a,z){)}_{a,z\in \{0,1\}})$, using the independence definition in Independent random elements. Under exclusion this is equivalent to independence of $Z$ and $(A(0),A(1),Y(0),Y(1))$. Thus joint instrument independence is defined even when exclusion fails. Relevance means ${\mathrm{\Delta }}_A:=E[A\mid Z=1]-E[A\mid Z=0]\mathrm{\ne }0.$ Monotonicity means $A(1)\ge A(0)$ almost surely. These are distinct model conditions: a setup need not satisfy all of them.

For an integrable real random variable $X$ and a measurable event $B$ with $P(B)>0$, define its event-conditioned mean by $E[X\mid B]:=\frac{E[{\mathbf{1}}_{B}X]}{P(B)}\mathrm{.}$ In particular $E[X\mid Z=z]$ means $E[X\mid \{Z=z\}]$. This is the finite-atom conditional mean used throughout this pair. The indicator is ${\mathbf{1}}_B(\omega )=1$ on $B$ and $0$ off $B$.