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Instrumental Variables and Two-Stage Least Squares (IV/2SLS)
IV estimation addresses endogeneity — the core challenge in causal inference when
the error term is correlated with the regressor of interest. A valid instrument Z must
satisfy: (1) Relevance: Cov(Z, X) ≠ 0 (testable), and (2) Exclusion restriction:
Cov(Z, u) = 0 (untestable, requires economic reasoning).
Conceptual Framework
The Endogeneity Problem
OLS: y = Xβ + u, but Cov(X, u) ≠ 0 ⟹ E[β̂_OLS] ≠ β.
Sources of endogeneity:
Omitted variables (unmeasured ability in earnings equations)
Simultaneous causality (price and quantity in supply/demand)
Measurement error in X (attenuation bias toward zero)
2SLS Estimator
First stage: X̂ = Z π₁ + W π₂ + v (W = exogenous controls)
Second stage: y = X̂ β + W γ + u
The 2SLS estimator: β̂_IV = (X̂'X)⁻¹ X̂'y
Weak Instruments
The Stock-Yogo (2005) rule of thumb: first-stage F-statistic ≥ 10 for acceptable
size distortion. More precisely:
F < 10: IV estimate may be nearly as biased as OLS (towards OLS)
F ≥ 10: bias ≤ 10% of OLS bias (approximately)
LIML is more robust than 2SLS with weak instruments
"""
First-stage F-statistic and partial R² for each endogenous variable.
Uses the Sanderson-Windmeijer (2016) approach for multiple endogenous regressors.
"""
"WEAK instrument (F<10) — IV estimates unreliable"
return
def
weak_instrument_test
result, first_stage_f: float = None
dict
"""
Report Stock-Yogo critical values for weak instrument rejection.
Stock & Yogo (2005) critical values for 5% Wald test size:
b = 0.10 (10% maximal relative bias): F > 16.38 (1 instrument)
b = 0.15: F > 8.96
b = 0.20: F > 6.66
b = 0.25: F > 5.53
Rule of thumb (Staiger-Stock): F > 10
"""
# try to extract F from linearmodels result
if
is
None
try
"f.stat"
0
except
None
"10% max relative bias (F>16.38)"
16.38
if
else
None
"15% max relative bias (F>8.96)"
8.96
if
else
None
"20% max relative bias (F>6.66)"
6.66
if
else
None
"Rule of thumb F>10"
10
if
else
None
return
"first_stage_F"
"stock_yogo_tests"
"recommendation"
"Proceed with IV estimates"
if
or
0
10
else
"Consider LIML or Anderson-Rubin confidence sets (weak instrument)"
"""
Wu-Hausman test: H0 = OLS is consistent (X is exogenous).
Method (Durbin-Wu-Hausman control function approach):
1. Regress each endogenous variable on all instruments and exogenous controls.
2. Save residuals v̂.
3. Include v̂ in OLS of y on X and controls.
4. Test H0: coefficient on v̂ = 0 via F-test.
Under H0: OLS is consistent; reject ⟹ IV is preferred.
"""
1
for
in
f"resid_{col}"
1
"HC3"
# F-test on residual coefficients
f"resid_{col}"
for
in
try
f"({name} = 0)"
for
in
float
float
except
# manual F-test
len
len
1
# restricted model (without residuals)
1
1
return
"F_statistic"
"p_value"
"reject_H0"
0.05
"interpretation"
f"Endogeneity confirmed (p={p_val:.4f}) — use IV/2SLS"
if
0.05
else
f"Cannot reject exogeneity (p={p_val:.4f}) — OLS may be consistent"
"""
Sargan-Hansen J-test for overidentifying restrictions.
H0: all instruments are valid (uncorrelated with structural error).
Rejection suggests at least one instrument is endogenous or mis-specified.
Only valid in over-identified models (# instruments > # endogenous regressors).
J = n · R² from regressing 2SLS residuals on all instruments and exogenous vars.
J ~ χ²(m - k) under H0, where m = # instruments, k = # endogenous variables.
"""
1
1
if
return
"error"
"Model is exactly identified — Sargan test requires over-identification"
"""
Side-by-side OLS vs 2SLS estimates for each regressor.
OLS is biased if endogeneity exists; IV corrects this at the cost of larger SE.
The direction of divergence between OLS and IV reveals the nature of the bias.
"""
f" F = {info['F_statistic']:.2f} (p={info['F_p_value']:.4f})"
f" Partial R² = {info['partial_R2']:.4f}"
f" {info['interpretation']}"
""
"hausman"
if
"F_statistic"
in
"Wu-Hausman Test:"
f" F = {wh['F_statistic']:.4f} p = {wh['p_value']:.4f}"
f" {wh['interpretation']}"
""
"sargan"
if
"J_statistic"
in
"Sargan-Hansen J-Test:"
f" J = {sg['J_statistic']:.4f} df = {sg['df']} p = {sg['p_value']:.4f}"
f" {sg['interpretation']}"
"="
60
return
"\n"
Example A — Returns to Education (Card 1995)
Card (1995) used geographic proximity to a college as an instrument for education.
Rationale: people who grew up near a college face lower costs of attending, so they
get more schooling — but proximity does not directly affect wages.
# example_a_card_education.py"""
Card (1995) style IV: proximity to college as instrument for education.
Simulated data replicating the key features of the NLS Young Men survey.
"""import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from iv_2sls import run_iv2sls, compare_ols_iv, weak_instrument_test
rng = np.random.default_rng(42)
n = 3010# approximate NLS sample size# ── latent ability (unobserved) ──
ability = rng.normal(0, 1, n)
# ── instrument: distance to nearest 4-year college (binary: close = 1) ──
near_college = rng.binomial(1, 0.40, n).astype(float)
# ── background controls ──
exp = rng.uniform(1, 25, n)
exp2 = exp**2
black = rng.binomial(1, 0.25, n).astype(float)
south = rng.binomial(1, 0.35, n).astype(float)
urban = rng.binomial(1, 0.60, n).astype(float)
# ── education: near_college raises schooling by ~0.8 years ──
educ_noise = rng.normal(0, 1.5, n)
educ = (
12 + 0.8 * near_college
+ 2.0 * ability # ability raises schooling
- 0.3 * black
+ educ_noise
).clip(8, 20)
# ── log wage: true return to education = 0.10 ──
wage_noise = rng.normal(0, 0.25, n)
lnwage = (
4.0 + 0.10 * educ
+ 0.04 * exp - 0.0007 * exp2
+ 0.4 * ability # endogeneity: ability affects both educ and wage
- 0.15 * black
+ 0.10 * urban
+ wage_noise
)
df = pd.DataFrame({
"lnwage": lnwage,
"educ": educ,
"exp": exp,
"exp2": exp2,
"black": black,
"south": south,
"urban": urban,
"near_college": near_college,
})
y = df["lnwage"]
endog = df[["educ"]]
instruments = df[["near_college"]]
exog = sm.add_constant(df[["exp", "exp2", "black", "south", "urban"]])
import statsmodels.api as sm
exog_df = pd.DataFrame(
sm.add_constant(df[["exp", "exp2", "black", "south", "urban"]]),
index=df.index, columns=["const","exp","exp2","black","south","urban"],
)
# ── run IV ──
iv_out = run_iv2sls(y, endog, instruments, exog_df, cov_type="robust")
print(iv_out["summary"])
# ── OLS vs IV comparison ──
compare = compare_ols_iv(y, endog, instruments, exog_df)
print("\nOLS vs IV Comparison:")
print(compare[["ols_coef","ols_se","iv_coef","iv_se","bias_ols"]].round(4))
# ── weak instrument check ──
fs_f = iv_out["diagnostics"]["first_stage"]["educ"]["F_statistic"]
wi = weak_instrument_test(iv_out["result"], first_stage_f=fs_f)
print(f"\nFirst-stage F = {fs_f:.1f}")
print("Stock-Yogo tests:")
for k, v in wi["stock_yogo_tests"].items():
print(f" {k}: {'PASS'if v else'FAIL'}")
print("Recommendation:", wi["recommendation"])
# ── visualise OLS vs IV ──
fig, ax = plt.subplots(figsize=(8, 5))
ols_coef = float(compare.loc["educ", "ols_coef"])
iv_coef = float(compare.loc["educ", "iv_coef"])
ols_se = float(compare.loc["educ", "ols_se"])
iv_se = float(compare.loc["educ", "iv_se"])
true_val = 0.10
estimators = ["True value", "OLS", "IV (2SLS)"]
coefs = [true_val, ols_coef, iv_coef]
errors = [0, ols_se, iv_se]
colors = ["#636363", "#d7191c", "#2c7bb6"]
ax.barh(estimators, coefs, xerr=[1.96*e for e in errors],
color=colors, height=0.4, capsize=5, alpha=0.8)
ax.axvline(true_val, color="#636363", linestyle="--", linewidth=1.5)
ax.set_xlabel("Return to schooling (log wage per year of education)")
ax.set_title("OLS vs IV Estimates of Return to Education\n(Card 1995 style)")
plt.tight_layout()
fig.savefig("iv_card.png", dpi=150, bbox_inches="tight")
print("\nSaved iv_card.png")
Example B — Rainfall as IV for Conflict
Instruments rainfall (a supply shock to agricultural income) for conflict intensity,
following Miguel, Satyanath & Sergenti (2004). Lower rainfall → income shock → more conflict.
# example_b_rainfall_conflict.py"""
IV: rainfall as instrument for income shocks predicting conflict onset.
Panel data: country-year observations.
Exclusion restriction: rainfall affects conflict only through income.
"""import numpy as np
import pandas as pd
import statsmodels.api as sm
from iv_2sls import run_iv2sls, _first_stage_stats, hausman_test_iv
rng = np.random.default_rng(77)
n_countries = 40
n_years = 20
N = n_countries * n_years
country_id = np.repeat(np.arange(n_countries), n_years)
year_id = np.tile(np.arange(n_years), n_countries)
# ── rainfall: exogenous weather shock ──
rainfall = rng.normal(0, 1, N) # standardised log deviation from mean# ── income growth: driven partly by rainfall (agricultural channel) ──
country_fe = rng.normal(0, 0.5, n_countries)[country_id] # fixed effect
income_growth = (
0.0 + 0.4 * rainfall
- 0.3 * country_fe # structural heterogeneity
+ rng.normal(0, 0.8, N)
)
# ── conflict: income shocks worsen conflict; latent unobservable correlated with both ──
u_struct = rng.normal(0, 0.5, N)
conflict = (
0.5 - 1.2 * income_growth # true causal effect
+ 1.5 * country_fe # omitted variable: country FE also affects conflict
+ u_struct
)
conflict_bin = (conflict > conflict.mean()).astype(float)
# ── population and ethnic fractionalization (controls) ──
log_pop = rng.normal(15, 2, N)
eth_frac = rng.beta(2, 2, n_countries)[country_id]
df = pd.DataFrame({
"conflict": conflict,
"conflict_bin": conflict_bin,
"income_growth": income_growth,
"rainfall": rainfall,
"log_pop": log_pop,
"eth_frac": eth_frac,
"country": country_id,
"year": year_id,
})
# add country fixed effects as dummies (partial out via demeaning)# Demeaning (within transformation) is simpler for illustration
df["conflict_dm"] = df["conflict"] - df.groupby("country")["conflict"].transform("mean")
df["income_growth_dm"] = df["income_growth"] - df.groupby("country")["income_growth"].transform("mean")
df["rainfall_dm"] = df["rainfall"] - df.groupby("country")["rainfall"].transform("mean")
df["log_pop_dm"] = df["log_pop"] - df.groupby("country")["log_pop"].transform("mean")
y = df["conflict_dm"]
endog_df = df[["income_growth_dm"]]
instr_df = df[["rainfall_dm"]]
exog_df = pd.DataFrame({
"const": 1.0,
"log_pop": df["log_pop_dm"],
}, index=df.index)
# ── IV estimation ──
iv_out = run_iv2sls(y, endog_df, instr_df, exog_df, cov_type="robust")
fs = iv_out["diagnostics"]["first_stage"]["income_growth_dm"]
wh = iv_out["diagnostics"]["hausman"]
print("=== Rainfall IV for Conflict ===")
print(f"First-stage F = {fs['F_statistic']:.2f} ({fs['interpretation']})")
print(f"Wu-Hausman: {wh['interpretation']}")
print()
try:
params = iv_out["result"].params
se = iv_out["result"].std_errors
pvals = iv_out["result"].pvalues
except AttributeError:
params = iv_out["result"].params
se = iv_out["result"].bse
pvals = iv_out["result"].pvalues
print("IV coefficient on income growth:", f"{params['income_growth_dm']:.3f}")
print("SE:", f"{se['income_growth_dm']:.3f}")
print("p-value:", f"{pvals['income_growth_dm']:.4f}")
print("\nInterpretation: A 1 std-dev decline in income growth increases conflict")
print(f"by approximately {abs(params['income_growth_dm']):.2f} standard deviations.")
IV Validity Checklist
Criterion
How to Verify
What Failure Means
Relevance
First-stage F ≥ 10 (Stock-Yogo)
Weak instrument bias toward OLS
Exclusion restriction
Economic theory + domain expertise
IV is invalid — no statistical fix
Monotonicity (fuzzy IV)
Verify no defiers in the population
LATE interpretation breaks down
Homogeneity (if targeting ATE)
Assess treatment heterogeneity
IV recovers LATE, not ATE
No defiance
Context knowledge
Negative first-stage for some units
LIML vs 2SLS
LIML (Limited Information Maximum Likelihood) is preferred over 2SLS when instruments are
weak:
LIML is median-unbiased even with weak instruments (2SLS is not).
LIML has heavier tails than 2SLS — confidence intervals are wider but more honest.
For strong instruments (F ≥ 10), 2SLS and LIML give nearly identical estimates.
# switch to LIML by passing estimator='liml':
iv_liml = run_iv2sls(y, endog_df, instr_df, exog_df,
cov_type="robust", estimator="liml")
References
Card, D. (1995). Using geographic variation in college proximity to estimate the return
to schooling. In Aspects of Labour Market Behaviour (pp. 201–222). U of Toronto Press.
Angrist, J. D., & Pischke, J.-S. (2009). Mostly Harmless Econometrics. Princeton UP.
Stock, J. H., & Yogo, M. (2005). Testing for weak instruments in linear IV regression.
In Identification and Inference for Econometric Models (pp. 80–108). Cambridge UP.
Miguel, E., Satyanath, S., & Sergenti, E. (2004). Economic shocks and civil conflict.
Journal of Political Economy, 112(4), 725–753.
Baum, C. F., Schaffer, M. E., & Stillman, S. (2007). Enhanced routines for instrumental
variables/generalized method of moments estimation and testing. Stata Journal, 7(4).
