Local average treatment effect

In econometrics and related empirical fields, the local average treatment effect (LATE), also known as the complier average causal effect (CACE), is the effect of a treatment for subjects who comply with the experimental treatment assigned to their sample group. It is not to be confused with the average treatment effect (ATE), which includes compliers and non-compliers together. Compliance refers to the human-subject response to a proposed experimental treatment condition. Similar to the ATE, the LATE is calculated but does not include non-compliant parties. If the goal is to evaluate the effect of a treatment in ideal, compliant subjects, the LATE value will give a more precise estimate. However, it may lack external validity by ignoring the effect of non-compliance that is likely to occur in the real-world deployment of a treatment method. 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.

The LATE was first introduced in the econometrics literature by Guido W. Imbens and Joshua D. Angrist in 1994, who shared one half of the 2021 Nobel Memorial Prize in Economic Sciences.^[1]^[2] As summarized by the Nobel Committee, the LATE framework "significantly altered how researchers approach empirical questions using data generated from either natural experiments or randomized experiments with incomplete compliance to the assigned treatment. At the core, the LATE interpretation clarifies what can and cannot be learned from such experiments."^[2]

The phenomenon of non-compliant subjects (patients) is also known in medical research.^[3] In the biostatistics literature, Baker and Lindeman (1994) independently developed the LATE method for a binary outcome with the paired availability design and the key monotonicity assumption.^[4] Baker, Kramer, Lindeman (2016) summarized the history of its development.^[5] Various papers called both Imbens and Angrist (1994) and Baker and Lindeman (1994) seminal.^[6]^[7]^[8]^[9]

An early version of LATE involved one-sided noncompliance (and hence no monotonicity assumption). In 1983 Baker wrote a technical report describing LATE for one-sided noncompliance that was published in 2016 in a supplement.^[5] In 1984, Bloom published a paper on LATE with one-sided compliance.^[10] For a history of multiple discoveries involving LATE see Baker and Lindeman (2024).^[11]

General definition[edit]

The typical terminology of the Rubin causal model is used to measure the LATE, with units indexed $i=1,\ldots ,N$ and a binary treatment indicator, $z_{i}$ for unit $i$ . The term $Y_{i}(z_{i})$ is used to denote the potential outcome of unit $i$ under treatment $z_{i}$ .

In an ideal experiment, all subjects assigned to the treatment will comply with the treatment, while those that are assigned to control will remain untreated. In reality, however, the compliance rate is often imperfect, 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.

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 $d_{i}(0)=1$

Others: LATE in instrumental variable framework[edit]

LATE can be thought of through an IV framework.^[15] 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.

In addition to the potential outcomes framework mentioned above, LATE can also be estimated through the Structural Equation Modeling (SEM) framework, originally developed for econometric applications.

SEM is derived through the following equations:

$D_{i}=\alpha _{0}+\alpha _{1}Z_{i}+\xi _{1i}$

$Y_{i}=\beta _{0}+\beta _{1}Z_{i}+\xi _{2i}$

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 $\tau _{LATE}={\frac {\beta _{1}}{\alpha _{1}}}={\frac {Cov(Y,Z)/Var(Z)}{Cov(D,Z)/Var(Z)}}={\frac {Cov(Y,Z)}{Cov(D,Z)}}$

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, the potential outcomes framework is in more prevalent use today.

ICSW estimator

Angrist, Joshua D.; Fernández-Val, Iván (2013). Advances in Economics and Econometrics. Cambridge University Press. pp. 401–434. :10.1017/cbo9781139060035.012. ISBN 9781139060035.

Local average treatment effect

General definition[edit]

Others: LATE in instrumental variable framework[edit]

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