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SKILL.md
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
Relevance: Cov(Z, X) ≠ 0 — Z must be correlated with the endogenous regressor
Exclusion restriction: Cov(Z, ε) = 0 — Z affects Y only through X (cannot be tested directly)
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
* 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
Conditional independence (unconfoundedness): Treatment T ⊥ Y(0), Y(1) | X
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
Always show first-stage results with F-statistic
Report OLS alongside IV to illustrate endogeneity bias direction
State the exclusion restriction argument explicitly — this cannot be statistically tested
Interpret LATE not ATE: IV estimates are local to compliers (those induced by instrument)
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
