Instrument Variable

category_specifier : "Causal Inference"

Reference Docs: Omitted Variable Bias | Endogeneity and Exogeneity

Motivation

💡I can’t run an experiment, but I want to know the causal effect.

You lack a clean A/B test or random assignment.
You suspect non-random selection into treatment.
IV is a method born out of necessity - when experiments aren’t feasible and OLS is unreliable, IV becomes a powerful tool for credible causal inference.

IV introduces a third variable (Z), the instrument, which:

\[ Cov(Z, X) ≠ 0 \]

\[ Cov(Z, e) = 0 \]

This allows you to use only the part of X that is “as good as random” (driven by Z), isolating a causal estimate of \(\beta_1\).

Use IV when you suspect the explanatory variable (X) is correlated with the error term (e) in your model:

\[ Y = \beta_0 + \beta_1 X + e \quad \text{where } \text{Cov}(X, e) \neq 0 \]

This breaks the OLS assumption and leads to biased estimates.

In a causal graph:

IV_summary

A common IV estimation technique:

Regress X on Z:

\(X = \pi_0 + \pi_1 Z + u\)

Use predicted \(\hat{X}\)

Regress Y on \(\hat{X}\)

\(Y = \beta_0 + \beta_1 \hat{X} + \epsilon\)

The estimated \(\beta_1\) is now unbiased, assuming instrument validity.

Demand estimation: Use weather shocks or tax variation to instrument for prices.
Crime and incarceration: Use prison overcrowding litigation as an instrument.
Class size and student performance: Use earthquake-driven classroom reallocations.
Birth weight and smoking: Use state-level cigarette prices as an instrument for maternal smoking.

Relevance: Look at F-stat in the first stage. Rule of thumb: F > 10 is strong.
Exogeneity: Testable only if you have more instruments than endogenous variables using the J-test (Sargan test).

Can include multiple instruments and control variables.
Must ensure the model is at least exactly identified (number of instruments ≥ number of endogenous regressors).
Weak instruments bias TSLS toward OLS - use with caution!