📘 MORISTAT: Advanced Econometrics

What Is Endogeneity?

Endogeneity occurs when an explanatory variable is correlated with the error term:

When this happens, OLS estimates are:

• Biased: E(b) ≠ β (even in finite samples)
• Inconsistent: plim(b) ≠ β (does not converge to true value)

Why Does Endogeneity Arise?

Consequences of Endogeneity:

The IV Solution

An Instrumental Variable (IV) is a variable Z that satisfies two key conditions:

What Makes a Good Instrument?

Examples of Instruments in Economics

The IV Estimator (Simple Case)

For the model: Yᵢ = β₀ + β₁Xᵢ + εᵢ

With instrument Zᵢ (correlated with X, uncorrelated with ε):

Why Does This Work?

If Cov(Z, ε) = 0 (exogeneity), then:

The General 2SLS Setup

Consider the structural model:

Where:

• Y₁ = dependent variable
• Y₂ = endogenous variable (correlated with ε)
• X₁, X₂ = exogenous variables (uncorrelated with ε)
• Z₁, Z₂ = instruments (correlated with Y₂, uncorrelated with ε)

Stage 1: First-Stage Regression

Regress the endogenous variable Y₂ on all exogenous variables and instruments:

Obtain the predicted values:

Stage 2: Second-Stage Regression

Regress Y₁ on the predicted values from Stage 1:

Matrix Form of 2SLS

Where:

• P_Z = Z(Z'Z)⁻¹Z' (projection matrix onto Z)
• Z = instruments (including all exogenous variables)
• X = all regressors (endogenous + exogenous)

Hausman-Wu Test

Purpose: Test whether OLS and 2SLS estimates are significantly different.

Null Hypothesis: H₀: Cov(X, ε) = 0 (OLS is consistent)

Alternative: H₁: Cov(X, ε) ≠ 0 (OLS is inconsistent)

Procedure:

1. Run first-stage regression: X = Zπ + v
2. Obtain residuals: v̂ = X - Zπ̂
3. Run augmented regression: Y = Xβ + v̂γ + ε
4. Test H₀: γ = 0 using a t-test or F-test

Interpretation:

Durbin-Wu-Hausman Test (Alternative)

1. Estimate both OLS and 2SLS
2. Compute: H = (β̂_2SLS - β̂_OLS)'[Var(β̂_2SLS) - Var(β̂_OLS)]⁻¹(β̂_2SLS - β̂_OLS)
3. Under H₀, H ~ χ²(k)

When to Use OLS vs 2SLS

Relevance Test (First-Stage F-Statistic)

Purpose: Test whether instruments are correlated with the endogenous variable.

Rule of Thumb: First-stage F-statistic > 10 → Instruments are relevant.

Overidentification Test (Sargan-Hansen J-Test)

Purpose: Test whether the instruments are exogenous (valid).

Requirement: At least as many instruments as endogenous variables (overidentified model).

Procedure:

1. Estimate 2SLS and obtain residuals: ε̂_2SLS
2. Regress ε̂_2SLS on all instruments and exogenous variables
3. Compute J = n × R² from this regression
4. Under H₀, J ~ χ²(m) where m = (# instruments - # endogenous variables)

Interpretation:

Weak Instrument Diagnostics

• Angrist-Pischke F-Statistic: Test for weak instruments in the presence of multiple endogenous variables.
• Cragg-Donald F-Statistic: Test for weak instruments (similar to first-stage F).
• Stock-Yogo Critical Values: Provide critical values for weak instrument tests.

Reporting 2SLS Results

Coefficient Table with Both OLS and 2SLS

First-Stage Results

Diagnostic Tests Summary

Checklist for IV/2SLS Analysis

• Identify endogenous variables and their sources
• Find plausible instruments (economic theory + intuition)
• Test relevance (first-stage F-statistic > 10)
• Test exogeneity (Sargan-Hansen J-test)
• Test for endogeneity (Hausman-Wu test)
• If endogeneity detected and instruments valid → Use 2SLS
• If instruments weak → Use LIML or weak-IV robust methods
• Report both OLS and 2SLS results
• Interpret results cautiously

Common Mistakes to Avoid

What IV Estimates Tell You

IV estimates identify the Local Average Treatment Effect (LATE) — the average treatment effect for those whose treatment status is affected by the instrument.

Where a complier is an individual whose treatment status is determined by the instrument.

Why LATE Matters

• IV estimates do not estimate the average treatment effect for the whole population
• They estimate the effect for compliers only
• This may be different from the effect for always-takers or never-takers

When LATE = ATE

Limited Information Maximum Likelihood (LIML)

Why use LIML? When instruments are weak, 2SLS can be severely biased. LIML is less biased but has larger variance.

When to use LIML:

• Weak instruments (first-stage F < 10)
• Many instruments (overfit concerns)
• Small samples

GMM Estimation

Generalized Method of Moments is more flexible than 2SLS:

• Allows for heteroskedasticity
• More efficient with many instruments
• Robust to weak instruments

Efficient GMM:

Where W is the optimal weighting matrix.

Dynamic Panel IV

For models with lagged dependent variables:

Arellano-Bond GMM: Uses lagged values as instruments.

Judging IV Validity

Finding Instruments

Good sources of instruments:

• Random assignment (natural experiments)
• Policy reforms (exogenous variation)
• Geographic variation (border effects)
• Historical factors (path dependence)
• Technical changes (exogenous shocks)

Tips for finding instruments:

1. Understand the institutional context
2. Look for exogenous variation in treatment
3. Use economic theory to justify instruments
4. Document plausibility of exogeneity

Sensitivity Analysis

Always test the robustness of your IV results:

1. Overidentification test (Sargan-Hansen)
2. Weak instrument test (First-stage F)
3. Endogeneity test (Hausman-Wu)
4. Specification tests (Different instruments)
5. Placebo tests (Pretend instrument is invalid)

Reporting IV Results

Must include:

1. First-stage regression results
2. First-stage F-statistic
3. Second-stage coefficient estimates
4. Hausman-Wu test result
5. Sargan-Hansen J-test result
6. Comparison with OLS

Should include:

1. Balance tests (covariates across instrument)
2. Reduced-form results
3. Sensitivity analysis
4. Interpretive discussion of LATE

IV/2SLS Formula Summary

Diagnostic Tests Summary

Key Takeaways

1. Endogeneity is pervasive — Omitted variables, measurement error, and simultaneity are common in economics
2. IV methods address endogeneity — Use instruments that are relevant and exogenous
3. 2SLS is the workhorse — Two-stage procedure to get consistent estimates
4. Always test instruments — Relevance (F > 10) and exogeneity (Sargan-Hansen)
5. IV estimates are LATE — Interpret as the effect for compliers
6. Diagnostics are essential — Hausman test, first-stage F, overidentification test
7. Weak instruments are dangerous — Use alternative methods if F < 10
8. Report OLS and IV — Compare to assess bias

Checklist Before Proceeding

• Understand the sources of endogeneity
• Can identify good instruments
• Know how to perform 2SLS
• Can interpret IV results
• Know the diagnostic tests
• Understand LATE interpretation
• Ready for Panel Data (Chapter 3)