Instrumental Variables

Summary

IV methods solve the omitted variables problem by using a variable (the instrument) that is correlated with the treatment but uncorrelated with other determinants of the outcome. The key implementation is Two-Stage Least Squares (2SLS).

The IV Setup

When the causal model $Y_i=\alpha +\rho s_i+{\eta }_i$ suffers from $Cov(s_i,{\eta }_i)\ne 0$, an instrument $z_i$ satisfies:

1. Relevance (first stage): $Cov(s_i,z_i)\ne 0$ — the instrument affects treatment
2. Exclusion restriction: $Cov({\eta }_i,z_i)=0$ — the instrument only affects outcomes through treatment

The IV estimand:

$$\rho =\frac{Cov(Y_i,z_i)}{Cov(s_i,z_i)}=\frac{\text{reduced form}}{\text{first stage}}$$

Two-Stage Least Squares

Stage 1: Regress $s_i$ on $z_i$ and covariates $X_i$ → get fitted values ${\hat{s}}_i$

Stage 2: Regress $Y_i$ on ${\hat{s}}_i$ and $X_i$ → coefficient on ${\hat{s}}_i$ is the 2SLS estimate of $\rho$

Use Canned Software

Don’t literally run 2SLS in two steps — the standard errors will be wrong. Use built-in IV commands (e.g., ivregress in Stata).

The Wald Estimator

With a binary instrument:

$$\rho =\frac{E[Y_i|z_i=1]-E[Y_i|z_i=0]}{E[s_i|z_i=1]-E[s_i|z_i=0]}$$

The reduced-form difference in means, rescaled by the first-stage difference.

Key Examples

Returns to schooling (Angrist & Krueger, 1991)

• Instrument: Quarter of birth (affects schooling through compulsory attendance laws)
• First stage: Q1 births → ~0.15 fewer years of schooling
• Result: 2SLS estimates of ~0.08-0.11 (slightly above OLS ~0.07)

Vietnam-era military service (Angrist, 1990)

• Instrument: Draft lottery number (randomly assigned)
• First stage: Draft-eligible men 16pp more likely to serve
• Result: Military service reduced 1981 earnings by ~$2,700 (15% of mean)

Effect of family size on labor supply (Angrist & Evans, 1998)

• Instruments: Twins at second birth; same-sex sibling composition
• Different instruments give different estimates → suggests heterogeneous effects

Local Average Treatment Effects (LATE)

With heterogeneous effects, IV estimates the causal effect on compliers — those whose treatment status is changed by the instrument. This is a local rather than population-average effect.

See Also

• Local Average Treatment Effects
• Omitted Variables Bias
• Regression Discontinuity Designs — fuzzy RD is IV
• Differences-in-Differences — the main competing strategy for panel settings where IV instruments are unavailable
• Instrumental Variables and Principal Stratification — extends IV via principal stratification for non-compliance and censoring
• Mostly Harmless Econometrics - Overview
• Hierarchical Models — Bayesian partial pooling as an alternative approach to treatment effect heterogeneity
• Data Collection Models — ignorability through instrumental design vs. conditioning on observables
• Activity Bias in Advertising — real-world case where CIA fails and IV is the appropriate remedy
• Bayesian Propensity Score Weighting — Bayesian selection-on-observables alternative; compare with IV when exclusion restriction is questionable
• Parameter Estimation in Market Response — 2SLS used for price endogeneity in marketing mix models
• GMM Estimation and Instruments for Price Endogeneity — IV/GMM for price endogeneity in demand estimation