# Local average treatment effect

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The local average treatment effect (LATE), also known as the complier average causal effect (CACE), was first introduced into the econometrics literature by Guido W. Imbens and Joshua D. Angrist in 1994. It is the treatment effect for the subset of the sample that takes the treatment if and only if they were assigned to the treatment, otherwise known as the compliers. It is not to be confused with the average treatment effect (ATE), which is the average subject-level treatment effect; the LATE is only the ATE among the compliers. The LATE can be estimated by a ratio of the estimated intent-to-treat effect and the estimated proportion of compliers, or alternatively through an instrumental variable estimator. Since the LATE may not be equivalent to the ATE, we should be cautious of extrapolation.

## General definition

The ATE is the difference between the expected value of the treatment group and the expected value of the control group. In an experimental setting, random assignment allows us to assume that the treatment group and control group have the same expected potential outcomes when treated (or untreated). This can be expressed as:

$E[Y_{i}(1)|D_{i}=1]=E[Y_{i}(1)|D_{i}=0]=E[Y_{i}(1)]$ $E[Y_{i}(0)|D_{i}=1]=E[Y_{i}(0)|D_{i}=0]=E[Y_{i}(0)]$ In an ideal experiment, all subjects assigned to treatment are treated, while those that are assigned to control will remain untreated. In reality, however, the compliance rate is never perfect, which prevents researchers from identifying the ATE. In such cases, estimating the LATE becomes the more feasible option. The LATE is the average treatment effect among a specific subset of the subjects, who in this case would be the compliers.

## Potential outcome framework and notation

Let $Y_{i}(d)$ denotes the potential outcome of subject i, where d is the binary indicator of subject $i$ ’s treatment status. $Y_{i}(1)$ denotes the treated potential outcome for subject i, while $Y_{i}(0)$ denotes the untreated potential outcome. The causal effect of the treatment on subject $i$ is $Y_{i}(1)-Y_{i}(0)$ . However, we can never observe both $Y_{i}(1)$ and $Y_{i}(0)$ for the same subject. At any given time, we can only observe a subject in its treated $Y_{i}(1)$ or untreated $Y_{i}(0)$ state.

Through random assignment, the expected untreated potential outcome of the control group is the same as that of the treatment group, and the expected treated potential outcome of treatment group is the same as that of the control group. The random assignment assumption thus allows us to take the difference between the average outcome in the treatment group and the average outcome in the control group as the overall average treatment effect, such that:

$ATE=E[Y_{i}(1)-Y_{i}(0)]=E[Y_{i}(1)]-E[Y_{i}(0)]=E[Y_{i}(1)|D_{i}=1]-E[Y_{i}(0)|D_{i}=0]$ ### Noncompliance framework

Quite often will researchers encounter noncompliance problems in their experiments, whereby subjects fail to comply with their experimental assignments. Some subjects will not take the treatment when assigned to the treatment group, so their potential outcome of $Y_{i}(1)$ will not be revealed, while some subjects assigned to the control group will take the treatment, so they will not reveal their $Y_{i}(0)$ .

Given noncompliance, the population in an experiment can be divided into four subgroups: compliers, always-takers, never-takers and defiers. We then introduce $z$ as a binary indicator of experimental assignment, such that when $z_{i}=1$ , subject $i$ is assigned to treatment, and when $z_{i}=0$ , subject $i$ is assigned to control. Thus, $d_{i}(z)$ represents whether subject $i$ is actually treated or not when treatment assignment is $z_{i}$ .

Compliers are subjects who will take the treatment if and only if they were assigned to the treatment group, i.e. the subpopulation with $d_{i}(1)=1$ and $d_{i}(0)=0$ .

Noncompliers are composed of the three remaining subgroups:

• Always-takers are subjects who will always take the treatment even if they were assigned to the control group, i.e. the subpopulation with $d_{i}(z)=1$ • Never-takers are subjects who will never take the treatment even if they were assigned to the treatment group, i.e. the subpopulation with $d_{i}(z)=0$ • Defiers are subjects who will do the opposite of their treatment assignment status, i.e. the subpopulation with $d_{i}(1)=0$ and $d_{i}(0)=1$ Non-compliance can take two forms. In the case of one-sided non-compliance, a number of the subjects who were assigned to the treatment group remain untreated. Subjects are thus divided into compliers and never-takers, such that $d_{i}(0)=0$ for all $i$ , while $d_{i}(1)=$ 0 or 1. In the case of two-sided non-compliance, a number of the subjects assigned to the treatment group fail to receive the treatment, while a number of the subjects assigned to the control group receive the treatment. In this case, subjects are divided into the four subgroups, such that both $d_{i}(0)$ and $d_{i}(1)$ can be 0 or 1.

Given non-compliance, we require certain assumptions to estimate the LATE. Under one-sided non-compliance, we assume non-interference and excludability. Under two-sided non-compliance, we assume non-interference, excludability, and monotonicity.

### Assumptions under one-sided non-compliance

• The non-interference assumption, otherwise known as the Stable Unit Treatment Value Assumption (SUTVA), is composed of two parts.
• The first part of this assumption stipulates that the actual treatment status, $d_{i}$ , of subject $i$ depends only on the subject's own treatment assignment status, $z_{i}$ . The treatment assignment status of other subjects will not affect the treatment status of subject $i$ . Formally, if $z_{i}=z_{i}'$ , then $D_{i}(\mathbf {z} )=D_{i}(\mathbf {z} ')$ , where $\mathbf {z}$ denotes the vector of treatment assignment status for all individuals.
• The second part of this assumption stipulates that subject $i$ 's potential outcomes are affected by its own treatment assignment, and the treatment it receives as a consequence of that assignment. The treatment assignment and treatment status of other subjects will not affect subject $i$ 's outcomes. Formally, if $z_{i}=z_{i}'$ and $d_{i}=d_{i}'$ , then $Y_{i}(z,d)=Y_{i}(z',d)$ .
• The plausibility of the non-interference assumption must be assessed on a case-by-case basis.
• The excludability assumption requires that potential outcomes respond to treatment itself, $d_{i}$ , not treatment assignment, $z_{i}$ . Formally $Y_{i}(z,d)=Y_{i}(d)$ . So under this assumption, only $d$ matters. The plausibility of the excludability assumption must also be assessed on a case-by-case basis.

### Assumptions under two-sided non-compliance

• All of the above, and
• The monotonicity assumption, i.e. for all subject $i$ , $d_{i}(1)\geq d_{i}(0)$ . This states that whenever a subject moves from the control to treatment group, $d_{i}$ either remains unchanged or increases. The monotonicity assumption rules out defiers, since their potential outcomes are characterized by $d_{i}(1) . It should be noted that monotonicity cannot be tested, so like the non-interference and excludability assumptions, its validity must be determined on a case-by-case basis.

## Identification

The $LATE={\frac {ITT}{ITT_{D}}}$ , whereby

$ITT=E[Y_{i}(z=1)]-E[Y_{i}(z=0)]$ $ITT_{D}=E[d_{i}(z=1)]-E[d_{i}(z=0)]$ The $ITT$ measures the average effect of experimental assignment on outcomes without accounting for the proportion of the group that was actually treated (i.e. average of those assigned to treatment minus the average of those assigned to control). In experiments with full compliance, the $ITT=ATE$ .

The $ITT_{D}$ measures the proportion of subjects who are treated when they are assigned to the treatment group, minus the proportion who would have been treated even if they had been assigned to the control group, i.e. $ITT_{D}$ = the share of compliers.

### Proof

Under one-sided noncompliance , all subjects assigned to control group will not take the treatment, therefore: $E[d_{i}(z=0)]=0$ ,

so that $ITT_{D}=E[d_{i}(z=1)]==P[d_{i}(1)=1]$ If all subjects were assigned to treatment, the expected potential outcomes would be a weighted average of the treated potential outcomes among compliers, and the untreated potential outcomes among never-takers, such that

{\begin{aligned}E[Y_{i}(z=1)]=E[Y_{i}(d(1),z=1)]}=E[Y_{i}(z=1,d=1)|d_{i}(1)=1]*P[d_{i}(1)=1]&\\+E[Y_{i}(z=1,d=0)|d_{i}(1)=0]*(1-P[d_{i}(1)=1])\end{aligned}} If all subjects were assigned to control, however, the expected potential outcomes would be a weighted average of the untreated potential outcomes among compliers and never-takers, such that

{\begin{aligned}E[Y_{i}(z=0)]=E[Y_{i}(d=0,z=0)]}=E[Y_{i}(z=0,d=0)|d_{i}(1)=1]*P[d_{i}(1)=1]&\\+E[Y_{i}(z=0,d=0)|d_{i}(1)=0]*(1-P[d_{i}(1)=1])\end{aligned}} Through substitution, we can express the ITT as a weighted average of the ITT among the two subpopulations (compliers and never-takers), such that

{\begin{alignedat}{2}ITT=E[Y_{i}(z=1)]-E[Y_{i}(z=0)]=E[Y_{i}(z=1,d=1)-Y_{i}(z=0,d=0)|d_{i}(1)=1]*P[d_{i}(1)=1]+&\\E[Y_{i}(z=1,d=1)-Y_{i}(z=0,d=0)|d_{i}(1)=0]*P[d_{i}(1)=0]\end{alignedat}} Given the exclusion and monotonicity assumption, the second half of this equation should be zero.

As such,

${\frac {ITT}{ITT_{D}}}={\frac {E[Y_{i}(z=1,d=1)-Y_{i}(z=0,d=0)|d_{i}(1)=1]*P[d_{i}(1)=1]}{P[d_{i}(1)=1]}}=E[Y_{i}(d=1)-Y_{i}(d=0)|d_{i}(1)=1]=LATE$ ## Application: hypothetical schedule of potential outcome under two-sided noncompliance

The table below lays out the hypothetical schedule of potential outcomes under two-sided noncompliance.

The ATE is calculated by the average of $Y_{i}(d=1)-Y_{i}(d=0)$ Hypothetical Schedule of Potential Outcome under Two-sided Noncompliance
Obeservation $Y_{i}(1)$ $Y_{i}(0)$ $Y_{i}(1)-Y_{i}(0)$ $d_{i}(z=0)$ $d_{i}(z=1)$ Type
1 4 7 3 0 1 Complier
2 3 5 2 0 0 Never-taker
3 1 5 4 0 1 Complier
4 5 8 3 1 1 Always-taker
5 4 10 6 0 1 Complier
6 2 8 6 0 0 Never-taker
7 6 10 4 0 1 Complier
8 5 9 4 0 1 Complier
9 2 5 3 1 1 Always-taker

$ATE={\frac {3+2+4+4+6+6+4+4+3}{9}}={\frac {35}{9}}=3.9$ LATE is calculated by ATE among compliers, so

$LATE={\frac {3+4+6+4+4}{5}}=4.2$ ITT is calculated by the average of $Y_{i}(z=1)-Y_{i}(z=0)$ ,

so $ITT={\frac {3+0+4+0+6+0+4+4+0}{9}}={\frac {21}{9}}=2.3$ $ITT_{D}$ is the share of compliers

$ITT_{D}={\frac {5}{9}}$ ${\frac {ITT}{ITT_{D}}}={\frac {21/9}{5/9}}={\frac {21}{5}}=4.2=LATE$ ## Others: LATE in instrumental variable framework

We can also think of  LATE through an IV framework. Treatment assignment $z_{i}$ is the instrument that drives the causal effect on outcome $Y_{i}$ through the variable of interest $d_{i}$ , such that $z_{i}$ only influences $Y_{i}$ through the endogenous variable $d_{i}$ , and through no other path. This would produce the treatment effect for compliers.

Except for the potential outcome framework mentioned about, LATE can also be estimated thorough Structural Equation Framework, which is built on specific equation with parameters and variables. SEM was first developed in the field of econometrics and has grown an enormous literature in the field.

SEM is derived through the following equations:

$D_{1}=\alpha _{0}+\alpha _{1}Z_{i}+\xi _{1i}$ $Y_{i}=\beta _{0}+\beta _{1}Z_{i}+\xi _{2i}$ The first equation captures the first stage effect of $z_{i}$ on $d_{i}$ , adjusting for variance, where

$\alpha _{1}=Cov(D,Z)/var(Z)$ The second equation $\beta _{1}$ captures the reduced form effect of $z_{i}$ on $Y_{i}$ ,

$\beta _{1}=Cov(Y,Z)/var(Z)$ The covariate-adjusted IV estimator is  the ratio $\tau _{LATE}={\frac {\alpha _{1}}{\beta _{1}}}={\frac {Cov(Y,Z)/Var(Z)}{Cov(D,Z)/Var(Z)}}={\frac {Cov(Y,Z)}{Cov(D,Z)}}$ Similar to the nonzero compliance assumption, the coefficient $\alpha _{1}$ in first stage regression needs to be significant to make $z$ a valid instrument.

However, because of SEM’s strict assumption of constant effect on every individual, potential outcome model is of more prevalent use today.