Instrumental Variables (IV) / Two-Stage Least Squares (2SLS)

Created:December 27, 2025

Last Updated:December 27, 2025

Instrumental Variables (IV) estimation using Two-Stage Least Squares (2SLS) is a powerful technique for estimating causal effects when your treatment variable is endogenous (correlated with unobserved factors). By using an instrument that affects treatment but not the outcome directly, IV methods can recover unbiased causal estimates.

What You'll Get:

Two-Stage Least Squares Estimates: Unbiased causal effect estimates
First Stage Results: Verify instrument strength with F-statistics
Diagnostic Tests: Weak instrument tests, endogeneity tests
Comparison with OLS: See the bias from ignoring endogeneity
Visualizations: First stage and reduced form plots
APA-Ready Report: Publication-quality results

💡 Key Assumptions: For IV to work, your instrument must satisfy: (1) Relevance - strongly predicts treatment; (2) Exclusion - affects outcome only through treatment; (3) Exogeneity - uncorrelated with unobserved confounders.

Ready to estimate causal effects with IV? to see how IV/2SLS works, or upload your own data with an outcome, treatment, and instrument variable.

Calculator

1. Load Your Data

2. Select Columns & Options

Outcome Variable (Y):

Treatment/Endogenous (X):

Instrument (Z):

Confidence Level:

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What is Instrumental Variables (IV) Estimation?

Definition

Instrumental Variables (IV) is a method for estimating causal effects when the treatment variable is endogenous (correlated with unobserved factors). An instrument is a variable that affects the treatment but has no direct effect on the outcome. Two-Stage Least Squares (2SLS) is the most common IV estimator.

The 2SLS Procedure

First Stage: Predict Treatment with Instrument

X_i = \pi_0 + \pi_1 Z_i + \nu_i

Where X = treatment, Z = instrument

Second Stage: Estimate Effect Using Predicted Treatment

Y_i = \beta_0 + \beta_1 \hat{X}_i + \varepsilon_i

$\beta_1$ is the IV estimate of the causal effect

IV Assumptions (Critical!)

Relevance:Instrument must strongly predict treatment (First-stage F > 10)
Exclusion Restriction: Instrument affects outcome only through treatment
Exogeneity: Instrument uncorrelated with unobserved confounders
Monotonicity: Instrument affects everyone in same direction (for LATE)

When to Use IV

Returns to Schooling: Use quarter of birth, distance to college

Supply and Demand: Use weather shocks, policy changes

Program Evaluation: Use lottery-based assignment, eligibility cutoffs

Health Economics: Use physician preference, distance to facility

Code Examples

library(tidyverse)
library(AER)  # For ivreg and IV diagnostics
library(lmtest)  # For robust standard errors

# Example: Effect of education on earnings using quarter of birth as instrument
# (Angrist & Krueger 1991 setup)

# Fixed dataset for verification (30 observations)
data <- tibble(
  id = 1:30,
  qob1 = c(1, 0, 0, 1, 0, 0, 1, 0, 1, 0,
           0, 1, 0, 0, 1, 0, 1, 0, 0, 1,
           0, 1, 0, 0, 1, 0, 0, 1, 0, 1),
  education = c(12.5, 14.2, 13.8, 11.9, 15.1, 13.5, 12.2, 14.8, 12.8, 13.9,
                14.5, 11.8, 15.2, 13.1, 12.4, 14.3, 12.1, 15.5, 13.7, 12.6,
                14.1, 11.5, 13.3, 15.8, 12.9, 14.6, 13.2, 11.7, 14.9, 12.3),
  earnings = c(42500, 58200, 51800, 38900, 62100, 49500, 41200, 57800, 43800, 52900,
               55200, 37800, 63200, 47100, 42400, 53300, 40100, 64500, 50700, 43600,
               54100, 36500, 48300, 65800, 44900, 56600, 48200, 37100, 57900, 42100)
)

# Method 1: Two-Stage Least Squares using ivreg
iv_model <- ivreg(earnings ~ education | qob1, data = data)
summary(iv_model, diagnostics = TRUE)

# Method 2: Manual 2SLS
# First stage: Regress endogenous variable on instrument
first_stage <- lm(education ~ qob1, data = data)
summary(first_stage)

# Check first stage F-statistic (should be > 10 for strong instrument)
first_stage_f <- summary(first_stage)$fstatistic[1]
cat("First Stage F-statistic:", first_stage_f, "
")

# Get fitted values from first stage
education_hat <- fitted(first_stage)

# Second stage: Regress outcome on fitted values
second_stage <- lm(earnings ~ education_hat, data = data)
summary(second_stage)

# Compare with OLS (biased due to endogeneity)
ols_model <- lm(earnings ~ education, data = data)

# Compare coefficients
cat("Coefficient Comparison:")
cat("OLS estimate:", coef(ols_model)["education"])
cat("IV estimate:", coef(iv_model)["education"])

# Visualize first stage
ggplot(data, aes(x = qob1, y = education)) +
  geom_jitter(alpha = 0.3, width = 0.1) +
  geom_smooth(method = "lm", se = TRUE, color = "red") +
  labs(title = "First Stage: Instrument vs Endogenous Variable",
       x = "Born in Quarter 1",
       y = "Years of Education") +
  theme_minimal()

# Test for endogeneity (Hausman test)
# If significant, IV is needed
cf_test <- coef(iv_model)["education"] - coef(ols_model)["education"]
cat("Hausman test (coefficient difference):", cf_test)

Python

import numpy as np
import pandas as pd
from scipy import stats
import statsmodels.api as sm
from statsmodels.sandbox.regression.gmm import IV2SLS
import matplotlib.pyplot as plt
import seaborn as sns

# Example: Effect of education on earnings using quarter of birth as instrument
# (Angrist & Krueger 1991 setup)

# Fixed dataset for verification (30 observations - same as R)
data = pd.DataFrame({
    'id': range(1, 31),
    'qob1': [1, 0, 0, 1, 0, 0, 1, 0, 1, 0,
             0, 1, 0, 0, 1, 0, 1, 0, 0, 1,
             0, 1, 0, 0, 1, 0, 0, 1, 0, 1],
    'education': [12.5, 14.2, 13.8, 11.9, 15.1, 13.5, 12.2, 14.8, 12.8, 13.9,
                  14.5, 11.8, 15.2, 13.1, 12.4, 14.3, 12.1, 15.5, 13.7, 12.6,
                  14.1, 11.5, 13.3, 15.8, 12.9, 14.6, 13.2, 11.7, 14.9, 12.3],
    'earnings': [42500, 58200, 51800, 38900, 62100, 49500, 41200, 57800, 43800, 52900,
                 55200, 37800, 63200, 47100, 42400, 53300, 40100, 64500, 50700, 43600,
                 54100, 36500, 48300, 65800, 44900, 56600, 48200, 37100, 57900, 42100]
})

# Method 1: Two-Stage Least Squares using IV2SLS
exog = sm.add_constant(data['qob1'])
endog = data['education']
outcome = data['earnings']

iv_model = IV2SLS(outcome, exog, endog, exog).fit()
print("2SLS Results:")
print(iv_model.summary())

# Method 2: Manual 2SLS
# First stage: Regress endogenous variable on instrument
X_first = sm.add_constant(data['qob1'])
first_stage = sm.OLS(data['education'], X_first).fit()
print("First Stage Results:")
print(first_stage.summary())

# Check first stage F-statistic (should be > 10 for strong instrument)
first_stage_f = first_stage.fvalue
print(f"First Stage F-statistic: {first_stage_f:.2f}")

if first_stage_f < 10:
    print("Warning: Weak instrument (F < 10)")
else:
    print("Instrument appears strong (F >= 10)")

# Get fitted values from first stage
education_hat = first_stage.fittedvalues

# Second stage: Regress outcome on fitted values
X_second = sm.add_constant(education_hat)
second_stage = sm.OLS(data['earnings'], X_second).fit()
print("Second Stage Results:")
print(second_stage.summary())

# Compare with OLS (biased due to endogeneity)
X_ols = sm.add_constant(data['education'])
ols_model = sm.OLS(data['earnings'], X_ols).fit()
print("OLS Results (Biased):")
print(ols_model.summary())

# Compare coefficients
print(f"Coefficient Comparison:")
print(f"OLS estimate: {ols_model.params['education']:.2f}")
print(f"IV estimate: {second_stage.params['education_hat']:.2f}")
print(f"Difference: {second_stage.params['education_hat'] - ols_model.params['education']:.2f}")

# Visualize first stage
plt.figure(figsize=(10, 6))
sns.scatterplot(data=data, x='qob1', y='education', alpha=0.5)
sns.regplot(data=data, x='qob1', y='education', scatter=False, color='red')
plt.xlabel('Born in Quarter 1')
plt.ylabel('Years of Education')
plt.title('First Stage: Instrument vs Endogenous Variable')
plt.tight_layout()
plt.show()

# Visualize reduced form (instrument -> outcome)
plt.figure(figsize=(10, 6))
sns.scatterplot(data=data, x='qob1', y='earnings', alpha=0.5)
sns.regplot(data=data, x='qob1', y='earnings', scatter=False, color='green')
plt.xlabel('Born in Quarter 1')
plt.ylabel('Earnings')
plt.title('Reduced Form: Instrument vs Outcome')
plt.tight_layout()
plt.show()

Interpreting Results

IV Estimate: Causal effect for those affected by instrument (LATE)
First-Stage F:Should be > 10; if < 10, weak instrument bias
IV vs OLS: Large difference suggests endogeneity was important
Standard Errors: IV estimates have larger SE than OLS

Common Pitfalls

Weak Instruments:First-stage F < 10 leads to biased, inconsistent estimates
Exclusion Violation: Instrument directly affects outcome (untestable!)
Non-compliance: Not everyone follows instrument (affects interpretation)
LATE vs ATE: IV estimates local effect, not average treatment effect

Verification

library(tidyverse) library(AER) # For ivreg and IV diagnostics library(lmtest) # For robust standard errors # Example: Effect of education on earnings using quarter of birth as instrument # (Angrist & Krueger 1991 setup) # Fixed dataset for verification (30 observations) data <- tibble( id = 1:30, qob1 = c(1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1), education = c(12.5, 14.2, 13.8, 11.9, 15.1, 13.5, 12.2, 14.8, 12.8, 13.9, 14.5, 11.8, 15.2, 13.1, 12.4, 14.3, 12.1, 15.5, 13.7, 12.6, 14.1, 11.5, 13.3, 15.8, 12.9, 14.6, 13.2, 11.7, 14.9, 12.3), earnings = c(42500, 58200, 51800, 38900, 62100, 49500, 41200, 57800, 43800, 52900, 55200, 37800, 63200, 47100, 42400, 53300, 40100, 64500, 50700, 43600, 54100, 36500, 48300, 65800, 44900, 56600, 48200, 37100, 57900, 42100) ) # Method 1: Two-Stage Least Squares using ivreg iv_model <- ivreg(earnings ~ education | qob1, data = data) summary(iv_model, diagnostics = TRUE) # Method 2: Manual 2SLS # First stage: Regress endogenous variable on instrument first_stage <- lm(education ~ qob1, data = data) summary(first_stage) # Check first stage F-statistic (should be > 10 for strong instrument) first_stage_f <- summary(first_stage)$fstatistic[1] cat("First Stage F-statistic:", first_stage_f, " ") # Get fitted values from first stage education_hat <- fitted(first_stage) # Second stage: Regress outcome on fitted values second_stage <- lm(earnings ~ education_hat, data = data) summary(second_stage) # Compare with OLS (biased due to endogeneity) ols_model <- lm(earnings ~ education, data = data) # Compare coefficients cat("Coefficient Comparison:") cat("OLS estimate:", coef(ols_model)["education"]) cat("IV estimate:", coef(iv_model)["education"]) # Visualize first stage ggplot(data, aes(x = qob1, y = education)) + geom_jitter(alpha = 0.3, width = 0.1) + geom_smooth(method = "lm", se = TRUE, color = "red") + labs(title = "First Stage: Instrument vs Endogenous Variable", x = "Born in Quarter 1", y = "Years of Education") + theme_minimal() # Test for endogeneity (Hausman test) # If significant, IV is needed cf_test <- coef(iv_model)["education"] - coef(ols_model)["education"] cat("Hausman test (coefficient difference):", cf_test)

import numpy as np import pandas as pd from scipy import stats import statsmodels.api as sm from statsmodels.sandbox.regression.gmm import IV2SLS import matplotlib.pyplot as plt import seaborn as sns # Example: Effect of education on earnings using quarter of birth as instrument # (Angrist & Krueger 1991 setup) # Fixed dataset for verification (30 observations - same as R) data = pd.DataFrame({ 'id': range(1, 31), 'qob1': [1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1], 'education': [12.5, 14.2, 13.8, 11.9, 15.1, 13.5, 12.2, 14.8, 12.8, 13.9, 14.5, 11.8, 15.2, 13.1, 12.4, 14.3, 12.1, 15.5, 13.7, 12.6, 14.1, 11.5, 13.3, 15.8, 12.9, 14.6, 13.2, 11.7, 14.9, 12.3], 'earnings': [42500, 58200, 51800, 38900, 62100, 49500, 41200, 57800, 43800, 52900, 55200, 37800, 63200, 47100, 42400, 53300, 40100, 64500, 50700, 43600, 54100, 36500, 48300, 65800, 44900, 56600, 48200, 37100, 57900, 42100] }) # Method 1: Two-Stage Least Squares using IV2SLS exog = sm.add_constant(data['qob1']) endog = data['education'] outcome = data['earnings'] iv_model = IV2SLS(outcome, exog, endog, exog).fit() print("2SLS Results:") print(iv_model.summary()) # Method 2: Manual 2SLS # First stage: Regress endogenous variable on instrument X_first = sm.add_constant(data['qob1']) first_stage = sm.OLS(data['education'], X_first).fit() print("First Stage Results:") print(first_stage.summary()) # Check first stage F-statistic (should be > 10 for strong instrument) first_stage_f = first_stage.fvalue print(f"First Stage F-statistic: {first_stage_f:.2f}") if first_stage_f < 10: print("Warning: Weak instrument (F < 10)") else: print("Instrument appears strong (F >= 10)") # Get fitted values from first stage education_hat = first_stage.fittedvalues # Second stage: Regress outcome on fitted values X_second = sm.add_constant(education_hat) second_stage = sm.OLS(data['earnings'], X_second).fit() print("Second Stage Results:") print(second_stage.summary()) # Compare with OLS (biased due to endogeneity) X_ols = sm.add_constant(data['education']) ols_model = sm.OLS(data['earnings'], X_ols).fit() print("OLS Results (Biased):") print(ols_model.summary()) # Compare coefficients print(f"Coefficient Comparison:") print(f"OLS estimate: {ols_model.params['education']:.2f}") print(f"IV estimate: {second_stage.params['education_hat']:.2f}") print(f"Difference: {second_stage.params['education_hat'] - ols_model.params['education']:.2f}") # Visualize first stage plt.figure(figsize=(10, 6)) sns.scatterplot(data=data, x='qob1', y='education', alpha=0.5) sns.regplot(data=data, x='qob1', y='education', scatter=False, color='red') plt.xlabel('Born in Quarter 1') plt.ylabel('Years of Education') plt.title('First Stage: Instrument vs Endogenous Variable') plt.tight_layout() plt.show() # Visualize reduced form (instrument -> outcome) plt.figure(figsize=(10, 6)) sns.scatterplot(data=data, x='qob1', y='earnings', alpha=0.5) sns.regplot(data=data, x='qob1', y='earnings', scatter=False, color='green') plt.xlabel('Born in Quarter 1') plt.ylabel('Earnings') plt.title('Reduced Form: Instrument vs Outcome') plt.tight_layout() plt.show()

Instrumental Variables (IV) / Two-Stage Least Squares (2SLS)

What You'll Get:

Calculator

1. Load Your Data

2. Select Columns & Options

Related Calculators

Difference-in-Differences Calculator

Linear Regression Calculator

Two-Sample T-Test Calculator

Learn More

What is Instrumental Variables (IV) Estimation?

Definition

The 2SLS Procedure

IV Assumptions (Critical!)

When to Use IV

Code Examples

Interpreting Results

Common Pitfalls

Verification

View Verification Details

Instrumental Variables (IV) / Two-Stage Least Squares (2SLS)

What You'll Get:

Calculator

1. Load Your Data

2. Select Columns & Options

Related Calculators

Difference-in-Differences Calculator

Linear Regression Calculator

Two-Sample T-Test Calculator

Learn More

What is Instrumental Variables (IV) Estimation?

Definition

The 2SLS Procedure

IV Assumptions (Critical!)

When to Use IV

Code Examples

Interpreting Results

Common Pitfalls

Verification

View Verification Details