Instrumental Variables (IV)

Definition

Purpose

1. Estimate causal relationships in observational data when experiments are impossible.
2. Address omitted variable bias and reverse causality.
3. Isolate the "clean" variation in the endogenous variable.

When to Use

Theoretical Background

The Problem

In the model $Y=\beta X+u$, if $Cov(X,u)\ne 0$, OLS is biased.

Example: Effect of Education on Wages.

• Endogeneity: Ability affects both Education and Wages. $Cov(Edu,u)\ne 0$.
• OLS Bias: The coefficient on Education captures Ability, not just Education's effect.

The Solution: Instrument ($Z$)

An instrument $Z$ must satisfy:

1. Relevance: $Z$ is correlated with $X$. ($Cov(Z,X)\ne 0$).
2. Exclusion Restriction: $Z$ affects $Y$ only through $X$. ($Cov(Z,u)=0$).

Example Instrument: Distance to nearest college.

• Relevant: Distance affects Education (closer = more likely to attend).
• Exclusion: Distance does not directly affect Wages (only through Education).

Two-Stage Least Squares (2SLS)

1. Stage 1: Regress $X$ on $Z$ (and controls). Obtain $\hat{X}$.
2. Stage 2: Regress $Y$ on $\hat{X}$. Coefficient is the IV estimate.

Assumptions

• Relevance: Instrument predicts the endogenous variable strongly. (Test: First-stage F-statistic > 10).
• Exclusion Restriction: Instrument only affects $Y$ through $X$. (Cannot be statistically tested; relies on theory).
• Independence: Instrument is not correlated with the error term.

Limitations

Python Implementation

from linearmodels.iv import IV2SLS
import pandas as pd

# Data: Y = Wage, X = Education (Endog), Z = Distance (Instrument), W = Experience (Control)
# Formula: dependent ~ exogenous + [endogenous ~ instruments]
model = IV2SLS.from_formula('Wage ~ 1 + Experience + [Education ~ Distance]', data=df)
results = model.fit()
print(results.summary)

# First-Stage Diagnostics
print(results.first_stage.diagnostics)
# Check F-statistic > 10

R Implementation

library(AER)

# Formula: Y ~ Exog + Endog | Exog + Instruments
model <- ivreg(Wage ~ Experience + Education | Experience + Distance, data = df)

summary(model, diagnostics = TRUE)

# Diagnostics:
# - Weak Instruments: F-stat should be > 10.
# - Wu-Hausman: Tests if OLS is consistent. If significant, IV is needed.

Interpretation Guide

Related Concepts

• Multiple Linear Regression - The biased OLS baseline.
• Propensity Score Matching (PSM) - Alternative for selection bias.
• Difference-in-Differences (DiD)
• Regression Discontinuity Design (RDD)