In
econometrics
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. ...
and related 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 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. The LATE is similar to the ATE, but excludes non-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.
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.
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."
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. Baker, Kramer, Lindeman (2016) summarized the history of its development. Various papers called both Imbens and Angrist (1994) and Baker and Lindeman (1994) seminal.
General definition
The typical terminology of the
Rubin causal model
The Rubin causal model (RCM), also known as the Neyman–Rubin causal model, is an approach to the statistical analysis of cause and effect based on the framework of potential outcomes, named after Donald Rubin. The name "Rubin causal model" was ...
is used to measure the LATE, with units indexed
and a binary treatment indicator,
for unit
. The term
is used to denote the potential outcome of unit
under treatment
.
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.
Potential outcome framework
The treatment effect for subject
is
. It is impossible to simultaneously observe
and
for the same subject. At any given time, only a subject in its treated
or untreated
state can be observed.
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 the treatment group is the same as that of the control group. The random assignment assumption thus allows one 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:
Non-compliance framework
Researchers frequently encounter non-compliance problems in their experiments, whereby subjects fail to comply with their experimental assignments. In an experiment with non-compliance, the subjects can be divided into four subgroups: compliers, always-takers, never-takers and defiers. The term
represents whether subject
is complies with their treatment protocol when treatment assignment is
.
Compliers are subjects who will take the treatment if and only if they were assigned to the treatment group, i.e. the subpopulation with
and
.
Non-compliers 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
* Never-takers are subjects who will never take the treatment even if they were assigned to the treatment group, i.e. the subpopulation with
* Defiers are subjects who will do the opposite of their treatment assignment status, i.e. the subpopulation with
and
Non-compliance can take two forms: one-sided (always-takers and never-takers) and two-sided (defiers). 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
for all
, while
or
. 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
and
can be 0 or 1.
Given non-compliance, certain assumptions are required to estimate the LATE. Under one-sided non-compliance, non-interference and excludability is assumed. Under two-sided non-compliance, non-interference, excludability, and
monotonicity is assumed.
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,
, of subject
depends only on the subject's own treatment assignment status,
. The treatment assignment status of other subjects will not affect the treatment status of subject
. Formally, if
, then
, where
denotes the vector of treatment assignment status for all individuals.
** The second part of this assumption stipulates that subject
'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
's outcomes. Formally, if
and
, then
.
** 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,
, not treatment assignment,
. Formally
. So under this assumption, only
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 each subject
,
. This states that if a subject were moved from the control to treatment group,
would either remain unchanged or increase. The monotonicity assumption rules out defiers, since their potential outcomes are characterized by
.
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
, whereby