Instrumental Variables & Treatment Effects Skill

This skill covers IV/2SLS estimation and propensity score matching (PSM) for causal inference when treatment is endogenous. It helps identify valid instruments, run 2SLS, test instrument validity, and implement PSM.

When to Use IV vs PSM

Method	Use When
IV / 2SLS	Treatment is endogenous; a valid instrument exists
PSM	Selection on observables assumption is credible; rich covariate data
OLS + controls	Selection on observables, limited instruments

IV / 2SLS Framework

Conditions for a Valid Instrument Z for endogenous X

1. Relevance: Cov(Z, X) ≠ 0 — Z must be correlated with the endogenous regressor
2. Exclusion restriction: Cov(Z, ε) = 0 — Z affects Y only through X (cannot be tested directly)
3. Independence: Z is as-good-as-randomly assigned (exogenous)

Two-Stage Least Squares Procedure

Stage 1: Regress endogenous X on instruments Z and exogenous controls W

• X̂ = γ₀ + γ₁Z + γ₂W + v
• Check F-statistic > 10 (Stock-Yogo rule of thumb); ideally > 16.4 (5% bias threshold)

Stage 2: Regress Y on predicted X̂ and controls W

• Y = β₀ + β₁X̂ + β₂W + ε
• SE must be corrected for the two-stage estimation (done automatically by software)

Quick Code Templates

# Python (linearmodels)
from linearmodels.iv import IV2SLS

# Formula: dependent ~ exogenous [endogenous ~ instruments]
model = IV2SLS.from_formula(
    'y ~ 1 + w1 + w2 + [x_endog ~ z1 + z2]', data=df
)
result = model.fit(cov_type='robust')
print(result.summary)

# First-stage diagnostics
print(result.first_stage.diagnostics)
# Check: partial F-stat, Shea partial R²

# R (AER)
library(AER)
iv_model <- ivreg(y ~ x_endog + w1 + w2 | z1 + z2 + w1 + w2, data = df)
summary(iv_model, diagnostics = TRUE)
# Shows: weak instruments F-test, Wu-Hausman endogeneity test, Sargan overID test

* Stata
ivregress 2sls y w1 w2 (x_endog = z1 z2), robust first
estat firststage      // First-stage diagnostics
estat endogenous      // Wu-Hausman test
estat overid          // Sargan-Hansen overidentification test

Key Diagnostic Tests

Test	Null Hypothesis	Interpretation
First-stage F-stat	Instruments are weak	F > 10 → relevant instruments
Wu-Hausman	X is exogenous (OLS consistent)	p < 0.05 → endogeneity confirmed, use IV
Sargan-Hansen	All instruments valid (overID only)	p > 0.05 → instruments pass overID test
Anderson-Rubin	Robust to weak instruments	Use when F-stat is borderline

Propensity Score Matching (PSM)

Assumptions

1. Conditional independence (unconfoundedness): Treatment T ⊥ Y(0), Y(1) | X
2. Common support (overlap): 0 < P(T=1|X) < 1 for all X

PSM Procedure

# Python
from sklearn.linear_model import LogisticRegression
import numpy as np

# Step 1: Estimate propensity scores
lr = LogisticRegression(max_iter=1000)
lr.fit(df[covariates], df['treatment'])
df['pscore'] = lr.predict_proba(df[covariates])[:, 1]

# Step 2: Check common support
import matplotlib.pyplot as plt
df.groupby('treatment')['pscore'].plot.hist(alpha=0.5, bins=30)

# Step 3: Match (nearest neighbor, 1:1 without replacement)
treated = df[df['treatment'] == 1].copy()
control = df[df['treatment'] == 0].copy()

from sklearn.neighbors import NearestNeighbors
nn = NearestNeighbors(n_neighbors=1)
nn.fit(control[['pscore']])
distances, indices = nn.kneighbors(treated[['pscore']])

matched_control = control.iloc[indices.flatten()].copy()
matched_df = pd.concat([treated, matched_control])

# Step 4: Estimate ATT
att = matched_df.groupby('treatment')['y'].mean().diff().iloc[-1]
print(f"ATT: {att:.4f}")

# R (MatchIt)
library(MatchIt)
match_out <- matchit(treatment ~ x1 + x2 + x3, data = df,
                     method = "nearest", ratio = 1, replace = FALSE)
summary(match_out)

# Covariate balance
plot(match_out, type = "jitter")
plot(summary(match_out))

# Estimate ATT
matched_data <- match.data(match_out)
att_model <- lm(y ~ treatment, data = matched_data, weights = weights)
coeftest(att_model, vcov = vcovCL(att_model, ~subclass))

* Stata (psmatch2 from SSC)
psmatch2 treatment x1 x2 x3, outcome(y) neighbor(1) common
pstest x1 x2 x3

Reporting IV Results

1. Always show first-stage results with F-statistic
2. Report OLS alongside IV to illustrate endogeneity bias direction
3. State the exclusion restriction argument explicitly — this cannot be statistically tested
4. Interpret LATE not ATE: IV estimates are local to compliers (those induced by instrument)
5. Overidentification test: report Sargan p-value when instruments > endogenous regressors

For weak-instrument robust inference (Anderson-Rubin confidence sets, LIML), control function approach, shift-share (Bartik) instruments, judge/examiner designs, and sensitivity analysis for PSM, see references/iv-reference.md.

Common Pitfalls

• Using 2SLS with weak instruments without robust inference: When F < 10, use LIML or Anderson-Rubin confidence sets instead of 2SLS
• Not arguing for exclusion restriction: The exclusion restriction cannot be tested statistically — you must make a convincing argument
• Confusing LATE with ATE: IV estimates the local average treatment effect for compliers, not the population average
• Clustering SE at the wrong level in Bartik IV: With shift-share instruments, inference should account for the exposure shares structure
• Over-identifying without caution: Adding more instruments improves efficiency but only if all are valid — a significant Sargan test means at least one instrument is invalid
• Using PSM without checking common support: If treated and control propensity score distributions barely overlap, matching is unreliable