Instrumental Variables — IV Estimation

Statistics

Isolating Exogenous Variation to Solve Endogeneity

Instrumental variables exploit external variation that affects the treatment but not the outcome directly. Two-stage least squares uses this exogenous variation to produce consistent causal estimates even when OLS fails.

Economics — Estimate returns to education using quarter of birth as an instrument
Healthcare — Assess treatment effects when?? assignment is non-random
Political Science — Evaluate policy impacts with institutional instruments

A valid instrument creates a natural experiment that breaks the correlation between treatment and error.

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.

DfEndogeneity

A variable $X$ is endogenous if it is correlated with the error term: $\text{Cov}(X, \varepsilon) \neq 0$ . This causes OLS to be biased and inconsistent.

The IV Approach

Intuition

An instrument $Z$ affects the outcome $Y$ only through the endogenous variable $X$ . By using only the variation in $X$ that comes from $Z$ (which is exogenous), we can estimate the causal effect of $X$ on $Y$ .

Required Conditions

Relevance

Relevance Condition

\text{Cov}(Z, X) \neq 0

Here,

$Z$ =Instrumental variable
$X$ =Endogenous regressor

The instrument must be strongly correlated with the endogenous variable.

Exogeneity (Exclusion Restriction)

Exogeneity Condition

\text{Cov}(Z, \varepsilon) = 0

Here,

$\varepsilon$ =Error term in the structural equation

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

Untestable Assumption

The exclusion restriction cannot be tested directly. It must be justified theoretically or by the research design. This is the main challenge of IV estimation.

Two-Stage Least Squares (2SLS)

The most common IV estimation method.

Stage 1

Regress $X$ on $Z$ (and any exogenous covariates):

Stage 1

X = \pi_0 + \pi_1 Z + \nu

Here,

$\pi_1$ =First-stage coefficient
$\hat{X}$ =Fitted values from first stage

Stage 2

Regress $Y$ on $\hat{X}$ :

Stage 2

Y = \beta_0 + \beta_1 \hat{X} + \varepsilon

Here,

$\beta_1$ =IV estimate of the causal effect

IV Estimator Formula

IV Estimator

\hat{\beta}_{IV} = \frac{\text{Cov}(Z, Y)}{\text{Cov}(Z, X)}

Here,

$\text{Cov}(Z, Y)$ =Covariance between instrument and outcome
$\text{Cov}(Z, X)$ =Covariance between instrument and treatment

Weak Instruments

Weak Instrument Problem

If $\text{Cov}(Z, X)$ is small (weak first stage), the IV estimator has:

Large variance (imprecise estimates)
Bias toward OLS in finite samples
Invalid confidence intervals

Testing for Weak Instruments

First-Stage F-Statistic

F = \frac{\pi_1^2 / \text{Var}(\hat{\pi}_1)}{1}

Here,

$F$ =First-stage F-statistic

| 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.

Sargan/Hansen J-Test

J = n \times R^2_{aux}

Here,

$R^2_{aux}$ =R-squared from auxiliary regression of 2SLS residuals on all instruments

| Decision | Interpretation |

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

| Reject $H_0$ | At least one instrument is invalid |

| Fail to reject $H_0$ | Instruments are jointly valid |

Hausman Test for Endogeneity

Hausman Endogeneity Test

H = (\hat{\beta}_{OLS} - \hat{\beta}_{IV})'(\text{Var}(\hat{\beta}_{OLS}) - \text{Var}(\hat{\beta}_{IV}))^{-1}(\hat{\beta}_{OLS} - \hat{\beta}_{IV})

Here,

$H$ =Test statistic ($\chi^2$ under $H_0$: no endogeneity)

If $H_0$ 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

Example: Returns to Education

Estimating the causal effect of education on earnings:

Endogenous variable: Years of education
Instrument: Quarter of birth (in states with compulsory schooling laws)
Confounder: Ability (unobserved)

| Method | Estimate | SE |

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

| OLS | 0.12 | 0.01 |

| IV (2SLS) | 0.08 | 0.03 |

First-stage F: 45.2 (strong instrument)

The IV estimate is smaller than OLS, suggesting OLS has upward ability bias. The 95% CI [0.02, 0.14] excludes zero — education has a significant causal effect.

Key Takeaways

Summary: Instrumental Variables

IV addresses endogeneity when $\text{Cov}(X, \varepsilon) \neq 0$
An instrument must satisfy relevance ( $\text{Cov}(Z,X) \neq 0$ ) and exogeneity ( $\text{Cov}(Z,\varepsilon) = 0$ )
2SLS is the standard estimation method
Check the first-stage F-statistic (>10 indicates strong instrument)
The exclusion restriction is untestable — must be justified theoretically
Use the Sargan/Hansen test when overidentified (more instruments than endogenous variables)
IV estimates are less precise than OLS but consistent when endogeneity is present

Instrumental Variables — IV Estimation

Instrumental Variables — IV Estimation

Isolating Exogenous Variation to Solve Endogeneity

DfEndogeneity

The IV Approach

Required Conditions

Relevance

Relevance Condition

Exogeneity (Exclusion Restriction)

Exogeneity Condition

Two-Stage Least Squares (2SLS)

Stage 1

Stage 1

Stage 2

Stage 2

IV Estimator Formula

IV Estimator

Weak Instruments

Testing for Weak Instruments

First-Stage F-Statistic

Overidentification

Sargan/Hansen J-Test

Hausman Test for Endogeneity

Hausman Endogeneity Test

Python Implementation

Worked Example

Example: Returns to Education

Key Takeaways

Summary: Instrumental Variables

Related Topics

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