Instrumental Variable

The instrument Z must have a statistically significant, non-zero effect on the endogenous treatment X. This is empirically testable via the first-stage F-statistic.

• Rule of thumb: F-statistic > 10 to avoid weak instrument bias
• Weak instruments amplify finite-sample bias and produce unreliable standard errors
• A weak first stage means the IV estimator is effectively unidentified, even in large samples
• This condition is also called cov(Z, X) ≠ 0

The instrument Z must affect the outcome Y only through its effect on the treatment X. There can be no direct path from Z to Y, nor any indirect path that bypasses X.

• This is the untestable core assumption of IV analysis
• Violations occur if Z affects Y through omitted channels (e.g., policy changes that shift multiple behaviors)
• Requires strong domain knowledge and institutional context to defend
• Mathematically: cov(Z, ε) = 0 where ε is the structural error term

The instrument Z must be as good as randomly assigned with respect to all confounders U. Z must be independent of both observed and unobserved factors that jointly influence X and Y.

• In randomized experiments, random assignment to treatment is the perfect instrument
• In observational studies, natural experiments (e.g., draft lotteries, weather shocks) approximate this
• Independence implies cov(Z, U) = 0 for all confounders U
• This is distinct from the exclusion restriction—independence is about confounders, not the outcome

The instrument must shift all units in the same direction—or at least not shift any unit in the opposite direction. This identifies the Local Average Treatment Effect (LATE) for compliers.

• Compliers: Units whose treatment status changes as Z changes in the expected direction
• Always-takers: Units always treated regardless of Z
• Never-takers: Units never treated regardless of Z
• Defiers: Units that do the opposite of what Z encourages—assumed to not exist
• Without monotonicity, the IV estimand is a weighted average with potentially negative weights

The workhorse estimator for IV models. Stage 1 regresses the endogenous treatment X on the instrument Z and all exogenous controls to obtain predicted values X̂. Stage 2 regresses the outcome Y on X̂ and the same controls.

• Stage 1: X = π₀ + π₁Z + γW + ν
• Stage 2: Y = β₀ + β₁X̂ + δW + ε
• The coefficient β₁ is the consistent IV estimate of the causal effect
• Standard errors must be corrected—naive OLS in Stage 2 underestimates variance
• Modern implementations use GMM for efficiency with heteroskedastic errors

In supply chain causal inference, valid instruments are rare but powerful. Example: using supplier geographic weather shocks as an instrument for shipment delays.

• Z: Rainfall anomaly at supplier location (exogenous, random)
• X: Actual shipment delay (endogenous—affected by unobserved supplier quality)
• Y: Factory downtime cost (outcome)
• Relevance: Heavy rain causes delays
• Exclusion: Rain at supplier only affects factory costs through delays, not directly
• Independence: Weather is orthogonal to supplier management quality
• This isolates the true causal cost of delays, not the confounded correlation