Instrumental Variables — IV Estimation

Instrumental variables (IV) methods address endogeneity — when a covariate is correlated with the error term. IV uses an external variable (the instrument) to isolate exogenous variation in the treatment.

The IV Approach

Required Conditions

Relevance

The instrument must be strongly correlated with the endogenous variable.

Exogeneity (Exclusion Restriction)

The instrument must be uncorrelated with the error term — it affects only through .

Two-Stage Least Squares (2SLS)

The most common IV estimation method.

Stage 1

Regress on (and any exogenous covariates):

Stage 2

Regress on :

IV Estimator Formula

Weak Instruments

Testing for Weak Instruments

| F-statistic | Interpretation |

|------------|---------------|

| F > 10 | Rule of thumb: instrument is strong |

| F < 10 | Potentially weak; use weak-IV robust methods |

Overidentification

When you have more instruments than endogenous variables, you can test whether the instruments are valid.

| Decision | Interpretation |

|---------|---------------|

| Reject | At least one instrument is invalid |

| Fail to reject | Instruments are jointly valid |

Hausman Test for Endogeneity

If is rejected, endogeneity is present and IV is preferred.

Python Implementation

import numpy as np

import pandas as pd

import statsmodels.api as sm

from linearmodels.iv import IV2SLS

import matplotlib.pyplot as plt



np.random.seed(42)



# Simulate endogeneity

n = 1000

Z = np.random.randn(n)  # Instrument

U = np.random.randn(n)  # Unobserved confounder

X = 0.8 * Z + 0.5 * U + np.random.randn(n) * 0.5  # Endogenous

Y = 2.0 * X + 1.5 * U + np.random.randn(n)  # Outcome



# Naive OLS (biased)

ols = sm.OLS(Y, sm.add_constant(X)).fit()

print(f"OLS estimate: {ols.params[1]:.3f} (true: 2.0)")



# IV/2SLS

iv_model = IV2SLS.from_formula('Y ~ 1 + [X ~ Z]', 

    data=pd.DataFrame({'Y': Y, 'X': X, 'Z': Z})).fit()

print(f"IV estimate: {iv_model.params['X']:.3f} (true: 2.0)")



# First-stage F-stat

first_stage = sm.OLS(X, sm.add_constant(Z)).fit()

f_stat = first_stage.fvalue

print(f"\nFirst-stage F-statistic: {f_stat:.1f}")

print(f"Weak instrument: {'Yes' if f_stat < 10 else 'No'}")



# Hausman test

diff = ols.params[1] - iv_model.params['X']

var_diff = ols.bse[1]**2 - iv_model.std_errors['X']**2

hausman_stat = diff**2 / var_diff

print(f"\nHausman test: {hausman_stat:.3f} (p ~ {1 - stats.chi2.cdf(hausman_stat, 1):.3f})")

Worked Example

Key Takeaways

Related Topics

• See Causal Inference for the potential outcomes framework

• See Regression Discontinuity for another quasi-experimental method

• See Difference-in-Differences for policy evaluation