Instrumental Variable

The instrument (Z) must be correlated with the treatment variable (X). This is the only testable assumption. A weak correlation leads to weak instrument bias, where small measurement errors in X are amplified, causing large standard errors and unreliable estimates.

• Statistical Test: The first-stage F-statistic in a Two-Stage Least Squares (2SLS) regression. A common rule of thumb is F > 10 to avoid weak instrument problems.
• Example: Using geographic distance to the nearest college as an instrument for years of education. Distance must predict educational attainment.

The instrument (Z) must affect the outcome (Y) only through its effect on the treatment (X). There must be no direct causal path from Z to Y, nor any path through unobserved confounders. This is the most critical and untestable assumption.

• Violation Example: Using rainfall as an instrument for agricultural productivity to estimate its effect on conflict. Rainfall could affect conflict through channels other than productivity (e.g., flooding disrupting travel).
• Design Imperative: Justification relies on strong theoretical reasoning and research design, not statistical tests.

The instrument (Z) must be independent of all unobserved confounders (U) affecting both X and Y. Formally, Z is as-good-as-randomly assigned. This ensures Z does not share common causes with Y.

• Graphical Condition: In a causal graph, there are no backdoor paths between Z and Y. All paths from Z to Y are blocked unless they go through X.
• Design-Based Instruments: Natural experiments (lotteries, policy changes in some areas) are often used to satisfy this assumption by mimicking random assignment.

When treatment effects are heterogeneous, a fourth assumption is required to interpret the IV estimate as a Local Average Treatment Effect (LATE). It states that the instrument moves all units in the same direction (or not at all); there are no defiers.

• Defiers: Units who would take the treatment if not encouraged by the instrument, but would not take it if encouraged. Monotonicity assumes this group does not exist.
• Interpretation: Under monotonicity, the IV estimator identifies the average treatment effect only for the compilers—the subset of the population whose treatment status is changed by the instrument.

While the core assumptions are largely untestable, econometric practice employs several diagnostic checks to assess instrument validity and estimator robustness.

• Overidentification Test (Sargan-Hansen J-test): Used when multiple instruments are available. Tests whether all instruments are exogenous (uncorrelated with the error term). A rejection suggests at least one instrument is invalid.
• Weak Instrument Diagnostics: Examine the first-stage F-statistic, partial R², and compare 2SLS to Limited Information Maximum Likelihood (LIML) estimates, which are less biased with weak instruments.
• Falsification Tests: Test if the instrument predicts placebo outcomes it should not affect, given the exclusion restriction.

Understanding typical failures of IV assumptions is crucial for robust research design.

• Invalid Instruments: The most common failure is a violation of the exclusion restriction, where Z has a direct effect on Y. This leads to biased estimates.
• Weak Instruments: Low correlation between Z and X causes estimates to be biased towards the OLS estimate and have unreliable confidence intervals.
• Heterogeneous Treatment Effects: Without monotonicity, the IV estimate is a complex weighted average of effects, not easily interpretable as an average causal effect for a clear subpopulation.
• Violation of Linearity/Additivity: Standard 2SLS assumes a linear, constant-effects model. Nonlinear models or effect heterogeneity require more complex IV methods.

A seminal study by Angrist (1990) used the Vietnam War draft lottery as an instrument for military service to estimate its effect on lifetime earnings. The random lottery number assignment was correlated with service (men with low numbers were more likely to be drafted) but, by design, affected earnings only through service, not through other confounding factors like ambition or education. This allowed estimation of the Local Average Treatment Effect (LATE) of military service on earnings for the subpopulation of 'compliers'—those who served because of the draft.

To estimate the causal return to education on wages, researchers have used geographic proximity to a college as an instrument for years of schooling. Living closer to a college reduces tuition costs, making college attendance more likely, but distance itself is plausibly uncorrelated with innate ability (the unmeasured confounder). This satisfies the exclusion restriction if distance affects wages only by influencing education levels, not through local labor markets. This method helped isolate the true economic return to schooling.

In health economics, a physician's preferred prescribing preference is used as an instrument for a specific drug treatment to estimate its effect on patient outcomes. For example, some doctors have a stronger preference for prescribing a new statin. This preference influences whether a patient receives the drug but is assumed to be independent of that specific patient's unobserved health factors. The key assumption is that the doctor's preference only affects patient health through the drug prescription channel.

A famous study used rainfall variation in India as an instrument for economic growth to estimate the impact of growth on government spending. Higher rainfall leads to better agricultural yields and economic growth, but rainfall is exogenous to political decisions. This allowed researchers to isolate the causal effect of economic conditions on fiscal policy, separating it from reverse causality where spending might also stimulate growth. The instrument's strength relies on agriculture's historical share of GDP.

The occurrence of twin births has been used as an instrument for family size (number of children) to estimate its causal effect on a mother's labor force participation. A twin birth represents a quasi-random shock to family size that is largely uncorrelated with parental preferences for work. This helps address the confounding where parents who choose to have more children may also have different labor market attitudes. The analysis estimates the labor supply effect for mothers who had more children due to a twin birth.

In studies of incarceration effects, the random assignment of defendants to judges with varying sentencing leniency serves as an instrument for receiving a prison sentence. A defendant assigned to a stricter judge is more likely to be incarcerated, but the judge assignment is random with respect to defendant characteristics. This setup allows estimation of the causal effect of incarceration on future recidivism, addressing the severe confounding where the most likely-to-reoffend defendants receive the harshest sentences.

For an instrumental variable (Z) to provide a valid causal estimate for the effect of a treatment (X) on an outcome (Y), three assumptions must hold:

• Relevance: Z must be correlated with the treatment X. This is empirically testable.
• Exclusion Restriction: Z must affect the outcome Y only through its effect on X. It cannot have a direct path to Y.
• Exogeneity/Exchangeability: Z must be independent of any unmeasured confounders (U) affecting both X and Y. This is the critical, untestable assumption.

Violation of any assumption, particularly exogeneity, invalidates the IV estimate, making the choice of instrument paramount.

Two-Stage Least Squares is the standard econometric method for implementing instrumental variable estimation.

• Stage 1: Regress the endogenous treatment variable (X) on the instrument (Z) and any included covariates. This generates the predicted values of X (X̂), which represent the variation in X explained only by the exogenous instrument.
• Stage 2: Regress the outcome (Y) on the predicted treatment (X̂) from Stage 1 and the covariates. The coefficient on X̂ is the IV estimate of the causal effect.

This process isolates the 'clean' variation in X, purging it of correlation with unobserved confounders.

The Local Average Treatment Effect is the causal effect estimated by an instrumental variable. It is a critical interpretation.

• An IV does not estimate the Average Treatment Effect (ATE) for the entire population.
• It estimates the effect only for the 'compliers'—the subpopulation whose treatment status is changed by the instrument.
• For example, in a study using a lottery for military draft as an instrument for military service, the LATE is the effect of service only on those who served because they lost the lottery, not on those who would have served anyway or never served.
• This makes the LATE highly context-dependent on the chosen instrument.

A weak instrument is one that has only a very small correlation with the treatment variable (X). This poses severe problems:

• Bias: 2SLS estimates become biased towards the biased ordinary least squares (OLS) estimate, even in large samples.
• Inference Failure: Standard errors become extremely large, and confidence intervals become unreliable and too narrow, leading to incorrect conclusions.
• Rule of Thumb: The first-stage F-statistic is used to test for weak instruments. An F-statistic below 10 is a common warning sign of a weak instrument problem, requiring alternative estimators like Limited Information Maximum Likelihood (LIML).

When you have more instruments than endogenous variables, you can perform an overidentification test (e.g., Sargan or Hansen's J test).

• Purpose: To test the validity of the overidentifying restrictions—the assumption that all instruments are exogenous.
• Logic: If all instruments are valid, the estimates using different subsets of instruments should be statistically similar. A significant test statistic suggests that at least one of the instruments is invalid (violates the exogeneity assumption).
• Caution: A passing test does not prove all instruments are valid; it merely fails to reject their joint validity. It is a necessary but not sufficient condition.

The frontdoor criterion is an alternative graphical identification strategy to the instrumental variable approach when faced with unmeasured confounding.

• Mechanism: It requires a mediator variable (M) that:
  1. Intercepts all directed paths from the treatment (X) to the outcome (Y).
  2. Is not influenced by the unmeasured confounder (U).
  3. Has no direct path from X to Y that bypasses M.
• Estimation: The causal effect is identified by combining the effect of X on M and the effect of M on Y (adjusting for X).
• Contrast with IV: While IV uses an external variable (Z) affecting X, the frontdoor criterion uses an internal mechanism (M) through which X operates. It is useful when a valid instrument cannot be found.