Instrumental Variables

The instrument (Z) must be strongly correlated with the endogenous treatment variable (X). This is the most testable assumption.

• Statistical Test: A first-stage regression of X on Z should yield a statistically significant coefficient. Weak instruments lead to biased estimates.
• Example: In estimating the effect of education (X) on earnings (Y), using the proximity to a college (Z) as an instrument relies on the assumption that proximity affects the likelihood of attending college.

The instrument (Z) must affect the outcome (Y) only through its effect on the treatment (X). It cannot have a direct path to Y.

• This is a non-testable assumption that must be justified on theoretical or logical grounds.
• Violation Example: Using rainfall as an instrument for agricultural productivity to estimate its effect on conflict assumes rainfall only affects conflict via crop yields. If rainfall also affects terrain mobility for armies (a direct path), the assumption is violated.

The instrument (Z) must be independent of all unobserved confounders (U) that affect both the treatment (X) and the outcome (Y). Essentially, Z is as-good-as-randomly assigned.

• Formally: Z ⊥ U. This ensures the variation in X driven by Z is exogenous.
• Violation Example: Using family background as an instrument for education assumes it is unrelated to unobserved traits like ambition. Since family background likely correlates with many unobserved factors affecting earnings, this assumption is often questionable.

When estimating a Local Average Treatment Effect, the instrument must not cause any unit to take the opposite of their intended treatment. This assumption defines the complier subpopulation.

• For a binary instrument and treatment, it assumes no defiers (units who do the opposite of what the instrument encourages).
• Example: With a scholarship (Z) encouraging college enrollment (X), monotonicity assumes no student who would enroll without the scholarship would decide not to enroll if offered the scholarship.

A weak instrument (low relevance) causes severe finite-sample bias in IV estimators, making them unreliable.

• Diagnostic: The first-stage F-statistic. A common rule-of-thumb is F > 10 to reject the null of a weak instrument.
• Consequence: Weak instruments amplify any small violation of the exogeneity assumption, leading to estimates that can be more biased than ordinary least squares.

When you have more instruments than endogenous variables, you can test the validity of the instrument set. This tests whether the instruments are consistent with each other, providing indirect evidence for the exogeneity assumption.

• Common Test: Sargan-Hansen J-test. A statistically significant result suggests at least one instrument is invalid (violates exogeneity).
• Limitation: The test cannot identify which instrument is invalid, only that the set is inconsistent.

Causal inference is the overarching discipline of drawing conclusions about cause-and-effect relationships from observational data. Unlike purely predictive modeling, it seeks to answer "what if" questions about interventions.

• Core Goal: Estimate the Average Treatment Effect of a policy, feature, or treatment.
• Key Challenge: Overcoming confounding, where a third variable influences both the treatment assignment and the outcome.
• Methods: Includes Instrumental Variables, Propensity Score Matching, Regression Discontinuity, and difference-in-differences.
• Application: Essential for evaluating the true impact of product changes, marketing campaigns, or economic policies where A/B testing is impossible.

Propensity score matching is a quasi-experimental method used to estimate causal effects by creating a synthetic control group from observational data.

• Mechanism: Calculates the probability (propensity) that a unit would receive the treatment based on observed covariates. Treated units are then matched with untreated units that have a similar propensity score.
• Goal: To mimic randomization by balancing the distribution of observed confounders between treatment and control groups.
• Contrast with IV: While PSM addresses observed confounding, instrumental variables are designed to handle unobserved confounding by leveraging an external instrument.
• Use Case: Commonly used in healthcare and social sciences to evaluate treatments when random assignment is unethical.

The Average Treatment Effect is the primary target quantity in causal inference, representing the average causal effect of a treatment or intervention across a population.

• Formal Definition: ATE = E[Y(1) - Y(0)], where Y(1) is the potential outcome under treatment and Y(0) is the potential outcome under control.
• Estimation Challenge: We never observe both potential outcomes for the same individual (the fundamental problem of causal inference).
• Role of IV: Instrumental variables, when valid, can provide a Local Average Treatment Effect estimate for the subpopulation of compliers—those whose treatment status is influenced by the instrument.
• Business Context: In A/B testing, the ATE is the measured lift in a key metric (e.g., conversion rate) caused by the new variant.

A confounding variable (or confounder) is an extraneous factor that distorts the apparent relationship between a treatment and an outcome, creating spurious correlation.

• Definition: A variable that influences both the independent variable (treatment) and the dependent variable (outcome).
• Problem: Leads to omitted variable bias in standard regression, making it impossible to isolate the true causal effect.
• Example: In studying the effect of education (treatment) on income (outcome), innate ability is a confounder (it affects both years of schooling and earning potential).
• Instrumental Variables Solution: IV methods use an instrument that is correlated with the treatment but uncorrelated with the confounder, thereby breaking the confounding link to provide a consistent causal estimate.

Two-Stage Least Squares is the most common estimation technique for instrumental variables regression in linear models.

• Stage 1: Regress the endogenous treatment variable (X) on the instrumental variable(s) (Z) and any control variables. Obtain the predicted values of X (X̂).
• Stage 2: Regress the outcome variable (Y) on the predicted treatment (X̂) and the control variables. The coefficient on X̂ is the IV estimate of the causal effect.
• Intuition: The first stage purges the treatment variable of its correlation with the error term (which contains the confounders), leaving only the variation explained by the instrument.
• Software: Standard in econometric packages (e.g., ivreg in R, IV2SLS in Python's linearmodels).

The Local Average Treatment Effect is the causal effect estimated by an instrumental variables analysis, which applies specifically to the subpopulation of compliers.

• Complier Definition: Individuals whose treatment status is actually changed by the instrument. This contrasts with always-takers, never-takers, and defiers.
• Key Insight: The LATE may differ from the Average Treatment Effect on the entire population or on the treated (ATT).
• Interpretation: The IV estimate answers, "What is the effect of the treatment for those who were induced to take it by the instrument?"
• Example: In a study using a scholarship lottery as an instrument for college attendance, the LATE measures the effect of college on earnings for students who attended only because they won the scholarship.