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Instrumental Variables

An instrumental variable (IV) shifts treatment receipt without otherwise changing the outcome. A strong association with treatment is only relevance; causal interpretation also needs independence, exclusion, monotonicity, a defined treatment, and an interference boundary.

Prerequisites​

Complete Regression Discontinuity and revisit Noncompliance, ITT, and Treatment on the Treated.

Learning goals​

  • decompose a Wald estimate into reduced form and first stage.
  • interpret the result as a local average treatment effect (LATE) for compliers under explicit assumptions.
  • challenge exclusion and monotonicity with design knowledge, falsification checks, and sensitivity analysis.

Assignment mechanism and estimand​

Suppose a lottery assigns an encouragement to enroll in Study Sprint, but enrollment remains voluntary. The instrument is lottery offer, treatment is enrollment, and outcome is the six-week score. The intention-to-treat effect of the offer is directly tied to randomized assignment. Dividing it by the offer's effect on enrollment produces the Wald estimand.

For a LATE interpretation, the offer must be independent of potential outcomes and potential receipt; affect scores only through defined enrollment (exclusion); not make anyone less likely to enroll (monotonicity); have a nonzero first stage; and avoid cross-student spillovers. The target is compliers induced by this encouragement, not all students or all enrollees.

Worked Wald estimate and exclusion failure​

This deterministic simulation starts with arm-level probabilities and means. The valid arithmetic yields a six-point complier effect. The counterexample adds a direct two-point encouragement effect, which the same Wald ratio incorrectly attributes to enrollment.

enrollment = {"control": 0.25, "offer": 0.75}
outcome = {"control": 55.0, "offer": 58.0}
first_stage = enrollment["offer"] - enrollment["control"]
reduced_form = outcome["offer"] - outcome["control"]
wald = reduced_form / first_stage
direct_encouragement_effect = 2.0
contaminated_wald = (reduced_form + direct_encouragement_effect) / first_stage
print(f"first stage={first_stage:.2f}")
print(f"offer reduced form={reduced_form:.2f}")
print(f"complier LATE={wald:.2f}")
print(f"exclusion-failure apparent LATE={contaminated_wald:.2f}")

The first stage is necessary but says nothing about whether the encouragement directly improves confidence, provides study tips, changes measurement, or spills over. In the failure fixture, a two-point direct path inflates the apparent enrollment effect from six to ten.

Diagnostics and falsification​

Show treatment receipt by instrument arm with uncertainty and weak-instrument diagnostics appropriate to the estimator. Verify allocation, attrition, and baseline balance when the instrument is randomized. Test outcomes and behaviors measured before encouragement, treatment versions the instrument should not alter, and channels that would violate exclusion.

Exclusion is usually defended with institutional knowledge, not tested from the main variables. Compare alternative instruments only if each has a credible causal story; agreement is not proof when they share violations. Examine whether the encouragement can create defiers and whether compliers can be characterized without pretending their identities are observed.

Weak instruments amplify noise and finite-sample bias; conventional normal intervals can mislead. A precisely estimated first stage still does not validate exclusion. If the instrument itself is a policy of interest, report its ITT effect even when the receipt effect is not identified.

Failure modes​

  • The encouragement directly changes effort or expectations.
  • Lottery information spills over to control students.
  • Treatment has multiple versions induced differently by the instrument.
  • Some students avoid enrollment because they receive the encouragement.
  • Attrition differs by instrument arm or receipt status.

Common mistakes​

  • Calling any variable correlated with treatment an instrument.
  • treating first-stage strength as proof of validity.
  • labeling LATE as the population ATE or effect on all treated students.
  • controlling for post-instrument variables that lie on causal paths.
  • hiding the reduced form when the ratio is unstable.

Practice​

For a proposed instrument, draw every plausible path to the outcome that does not pass through treatment. Name the complier population, the monotonicity claim, the ITT policy interpretation, and the action if exclusion is implausible.

Next steps​

Synthetic Controls constructs a counterfactual trajectory for an aggregate treated unit from a weighted donor pool.

Primary sources​

Checked 2026-10-15: Angrist, Imbens, and Rubin, “Identification of Causal Effects Using Instrumental Variables”, Imbens and Angrist, LATE identification, and Staiger and Stock, weak instruments.