IV and LATE

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 $B$.

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.

For a setup of Binary instrumental-variable potential-outcome setup with integrable observed outcome $Y$, define the reduced-form outcome contrast and first-stage treatment contrast by ${\mathrm{\Delta }}_Y:=E[Y\mid Z=1]-E[Y\mid Z=0],{\mathrm{\Delta }}_A:=E[A\mid Z=1]-E[A\mid Z=0]\mathrm{.}$ Whenever ${\mathrm{\Delta }}_A\mathrm{\ne }0$, the Wald ratio is the observed-data functional ${\psi }_W:=\frac{{\mathrm{\Delta }}_Y}{{\mathrm{\Delta }}_A}\mathrm{.}$ This definition alone does not assert that the ratio equals a treatment effect.

In an IV setup satisfying consistency, exclusion, and joint instrument independence, let $\tau =Y(1)-Y(0)$. Then $E[A\mid Z=1]-E[A\mid Z=0]=E[A(1)-A(0)]$ and $E[Y\mid Z=1]-E[Y\mid Z=0]=E[(A(1)-A(0))\tau ]$. No monotonicity or relevance condition is needed for these equalities; positivity of both instrument atoms and integrability are part of the setup.

In the binary setup of Binary-instrument principal strata and LATE, assume monotonicity $A(1)\ge A(0)$ almost surely. Then the always-taker, never-taker and complier events are pairwise disjoint and cover $\mathrm{\Omega }$ up to a null set; the defier event has probability zero.

Under consistency, joint instrument independence and monotonicity in the binary IV setup, $E[A\mid Z=1]-E[A\mid Z=0]=P(C),$ where $C=\{A(1)>A(0)\}$ is the complier event. If the instrument is relevant, then $P(C)>0$.

Under consistency, exclusion, joint instrument independence and monotonicity in the binary IV setup, with $\tau =Y(1)-Y(0)$ and $C=\{A(1)>A(0)\}$, $E[Y\mid Z=1]-E[Y\mid Z=0]=E[{\mathbf{1}}_C\tau ]\mathrm{.}$ If $P(C)>0$, the right-hand side is $P(C)E[\tau \mid C]$.

Let $\mathrm{\Theta }$ be a nonempty set of full-data laws satisfying the binary IV setup, consistency, exclusion and joint instrument independence. Each $\theta \in \mathrm{\Theta }$ induces an observed-data law $P_{\theta }$ for $(Z,A,Y)$, and let ${\psi }_{\mathrm{A}\mathrm{T}\mathrm{E}}(\theta )=E_{\theta }[Y(1)-Y(0)]$. Suppose relevance holds and there is a constant ${\tau }_0\in \mathbb{R}$ such that $Y(1)-Y(0)={\tau }_0$ almost surely under every $\theta \in \mathrm{\Theta }$. Then ${\psi }_W(P_{\theta })=\frac{E_{\theta }[Y\mid Z=1]-E_{\theta }[Y\mid Z=0]}{E_{\theta }[A\mid Z=1]-E_{\theta }[A\mid Z=0]}={\tau }_0={\psi }_{\mathrm{A}\mathrm{T}\mathrm{E}}(\theta )\mathrm{.}$ Consequently, the ATE is identifiable in the induced observed-data model. Monotonicity is not assumed.

Let $\mathrm{\Theta }$ be a nonempty class of full-data laws satisfying the binary IV setup, consistency, exclusion, joint instrument independence, relevance and monotonicity. Each $\theta \in \mathrm{\Theta }$ induces the observed-data law $P_{\theta }$ of $(Z,A,Y)$. Define the estimand ${\psi }_C(\theta )=E_{\theta }[Y(1)-Y(0)\mid C]$, where $C=\{A(1)>A(0)\}$. Then the Wald ratio is defined and ${\psi }_W(P_{\theta })=\frac{E_{\theta }[Y\mid Z=1]-E_{\theta }[Y\mid Z=0]}{E_{\theta }[A\mid Z=1]-E_{\theta }[A\mid Z=0]}={\psi }_C(\theta )\mathrm{.}$ Thus the LATE is identifiable in the induced observed-data model. This does not assert that the LATE equals the population ATE.