Instrumental Variables & Treatment Effects Skill

When to Use IV vs PSM

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

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

Related Skills & Commands

• ols-regression: Compare OLS and IV estimates to illustrate endogeneity bias
• did-analysis: DID is an alternative when you have a natural experiment but no instrument
• matching: For expanded coverage of PSM, CEM, and doubly-robust methods
• ml-causal: DML can handle IV with high-dimensional controls
• /diagnose: Run IV-specific diagnostics (first-stage F, Wu-Hausman, overidentification)
• /robustness: Compare 2SLS, LIML, OLS, and alternative instrument sets