Local Average Treatment Effect (LATE) is a crucial concept in causal inference. It estimates the causal effect of a treatment for a specific subpopulation induced to take up the treatment by an instrumental variable. LATE is particularly useful when there's non-compliance or self-selection into treatment.

LATE differs from the Average Treatment Effect (ATE) by focusing on compliers rather than the entire population. It relies on key assumptions like relevance, exclusion restriction, and monotonicity. Understanding LATE is essential for researchers estimating causal effects in complex real-world scenarios.

Definition of LATE

• Local Average Treatment Effect (LATE) estimates the causal effect of a treatment for a specific subpopulation of individuals who are induced to take up the treatment by a change in an instrumental variable
• LATE is a key concept in causal inference that allows researchers to estimate causal effects in the presence of non-compliance or self-selection into treatment
• LATE is particularly relevant when there is heterogeneity in treatment effects and the subpopulation of interest is those who would change their treatment status based on the instrumental variable

LATE vs ATE

• LATE differs from the Average Treatment Effect (ATE) because it focuses on a specific subpopulation (compliers) rather than the entire population
• ATE estimates the average causal effect of a treatment across all individuals in a population, regardless of their compliance with treatment assignment
• LATE is more relevant when there is heterogeneity in treatment effects and the subpopulation of interest is those who would change their treatment status based on the instrumental variable

Assumptions for LATE

• LATE relies on several key assumptions, including relevance, exclusion restriction, and monotonicity
• Relevance assumes that the instrumental variable is correlated with the treatment variable
• Exclusion restriction assumes that the instrumental variable only affects the outcome through its effect on the treatment variable
• Monotonicity assumes that there are no defiers in the population (individuals who always do the opposite of their treatment assignment)

Interpretation of LATE

• LATE estimates the average causal effect of a treatment for the subpopulation of compliers
• Compliers are individuals who would take up the treatment if assigned to the treatment group and would not take up the treatment if assigned to the control group
• LATE can be interpreted as the causal effect of the treatment for the subpopulation of individuals who are induced to take up the treatment by a change in the instrumental variable

Instrumental variables

• Instrumental variables (IVs) are a key tool in causal inference for estimating causal effects in the presence of unmeasured confounding or self-selection into treatment
• An instrumental variable is a variable that affects the treatment variable but does not directly affect the outcome variable, except through its effect on the treatment
• IVs allow researchers to isolate the causal effect of a treatment by exploiting exogenous variation in treatment assignment

Relevance condition

• The relevance condition requires that the instrumental variable is correlated with the treatment variable
• This means that the instrumental variable must have a non-zero effect on the probability of receiving the treatment
• The strength of the correlation between the instrumental variable and the treatment variable determines the strength of the instrument

Exclusion restriction

• The exclusion restriction assumes that the instrumental variable only affects the outcome through its effect on the treatment variable
• This means that there should be no direct effect of the instrumental variable on the outcome, and no unmeasured confounding between the instrumental variable and the outcome
• Violations of the exclusion restriction can lead to biased estimates of the causal effect

Monotonicity assumption

• The monotonicity assumption requires that there are no defiers in the population
• Defiers are individuals who always do the opposite of their treatment assignment (take up treatment when assigned to control, and do not take up treatment when assigned to treatment)
• Monotonicity is necessary for the interpretation of LATE as a causal effect for compliers

Strength of instruments

• The strength of an instrumental variable is determined by the magnitude of its correlation with the treatment variable
• Weak instruments (those with low correlation) can lead to biased estimates and large standard errors
• Strong instruments (those with high correlation) provide more precise estimates of the causal effect
• The F-statistic from the first stage regression of the treatment on the instrument is often used to assess instrument strength (F > 10 is a common rule of thumb)

Compliers vs defiers

• Compliers and defiers are two key subpopulations in the context of LATE estimation
• The behavior of these subpopulations has important implications for the interpretation and validity of LATE estimates

Definition of compliers

• Compliers are individuals who would take up the treatment if assigned to the treatment group and would not take up the treatment if assigned to the control group
• In other words, compliers are those whose treatment status is determined by their assignment to treatment or control
• LATE estimates the causal effect of the treatment for the subpopulation of compliers

Definition of defiers

• Defiers are individuals who always do the opposite of their treatment assignment
• They take up the treatment when assigned to the control group and do not take up the treatment when assigned to the treatment group
• The presence of defiers violates the monotonicity assumption required for LATE estimation

Implications for LATE

• The interpretation of LATE as a causal effect for compliers relies on the absence of defiers in the population
• If defiers are present, LATE estimates a weighted average of the causal effects for compliers and defiers, which is difficult to interpret
• The presence of defiers can also lead to biased estimates of the causal effect
• Researchers often argue for the plausibility of the monotonicity assumption in their specific context to justify the use of LATE

Estimation of LATE

• Several methods can be used to estimate LATE, including the Wald estimator and two-stage least squares (2SLS)
• These methods rely on the instrumental variable to isolate the causal effect of the treatment for compliers

Wald estimator

• The Wald estimator is a simple method for estimating LATE
• It is calculated as the ratio of the difference in outcomes between the treatment and control groups (reduced form) to the difference in treatment uptake between the treatment and control groups (first stage)
• The Wald estimator provides a consistent estimate of LATE under the assumptions of relevance, exclusion restriction, and monotonicity

Two-stage least squares

• Two-stage least squares (2SLS) is a more general method for estimating LATE that can accommodate multiple instruments and covariates
• In the first stage, the treatment variable is regressed on the instrumental variable(s) and any covariates
• In the second stage, the outcome variable is regressed on the predicted values of the treatment from the first stage and any covariates
• The coefficient on the predicted treatment in the second stage provides an estimate of LATE

Confidence intervals for LATE

• Confidence intervals for LATE can be constructed using standard errors from the 2SLS regression
• These confidence intervals reflect the uncertainty in the LATE estimate due to sampling variability
• It is important to note that confidence intervals for LATE only capture sampling uncertainty and do not account for uncertainty in the validity of the IV assumptions
• Researchers should assess the plausibility of the IV assumptions and consider sensitivity analyses to evaluate the robustness of their results

Applications of LATE

• LATE has been widely applied in various fields, including economics, public health, and social sciences
• Researchers use LATE to estimate causal effects in settings where there is non-compliance or self-selection into treatment

Examples in economics

• Angrist and Krueger (1991) used quarter of birth as an instrumental variable to estimate the causal effect of education on earnings
• Card (1995) used geographic proximity to college as an instrumental variable to estimate the causal effect of education on earnings
• Duflo and Saez (2003) used randomized information sessions as an instrumental variable to estimate the causal effect of retirement savings plan participation on savings

Examples in public health

• Greenland (2000) used physician prescribing preferences as an instrumental variable to estimate the causal effect of a drug treatment on patient outcomes
• Brookhart et al. (2006) used physician prescribing preferences as an instrumental variable to estimate the causal effect of COX-2 inhibitors on gastrointestinal complications
• Hernán and Robins (2006) used randomization as an instrumental variable to estimate the causal effect of hormone replacement therapy on cardiovascular outcomes

Limitations of LATE

• LATE estimates the causal effect for a specific subpopulation (compliers) and may not generalize to the entire population
• The validity of LATE estimates depends on the plausibility of the IV assumptions (relevance, exclusion restriction, monotonicity), which may be difficult to justify in some contexts
• LATE estimates can be sensitive to violations of the IV assumptions, and researchers should conduct sensitivity analyses to assess the robustness of their results
• LATE estimates may have limited policy relevance if the subpopulation of compliers is not representative of the target population for an intervention

Comparison to other estimands

• LATE is one of several estimands that can be used to estimate causal effects in the presence of non-compliance or self-selection into treatment
• Other estimands include the Average Treatment Effect (ATE), Average Treatment Effect on the Treated (ATT), and Average Treatment Effect on the Untreated (ATU)

LATE vs ATT

• ATT estimates the average causal effect of a treatment for those who actually received the treatment
• LATE estimates the average causal effect of a treatment for those who would receive the treatment if assigned to the treatment group (compliers)
• ATT and LATE may differ if there is selection bias in who receives the treatment

LATE vs ATC

• ATC (Average Treatment effect on the Compliers) is another name for LATE
• Both ATC and LATE refer to the average causal effect of a treatment for the subpopulation of compliers
• The terms are often used interchangeably in the literature

When to use LATE

• LATE is appropriate when there is non-compliance or self-selection into treatment and the researcher has a valid instrumental variable
• LATE is particularly useful when there is heterogeneity in treatment effects and the subpopulation of interest is those who would change their treatment status based on the instrumental variable
• Researchers should carefully consider the plausibility of the IV assumptions and the policy relevance of the complier subpopulation when deciding whether to use LATE
• If the IV assumptions are not plausible or the complier subpopulation is not of interest, other estimands (such as ATE or ATT) may be more appropriate