brycewang-stanford/iv-regression-guide

Instrumental Variables Regression Guide

A skill for applying instrumental variables (IV) estimation to address endogeneity in regression models. Covers the logic of IV, two-stage least squares (2SLS), instrument validity tests, weak instrument diagnostics, and reporting standards.

The Endogeneity Problem

Why OLS Fails

Ordinary Least Squares assumes:  E[u | X] = 0
(Regressors are uncorrelated with the error term)

This assumption is violated when:
  - Omitted variable bias: A confound affects both X and Y
  - Simultaneity: X affects Y and Y affects X
  - Measurement error: X is measured with noise

Consequence: OLS estimates are biased and inconsistent.
No amount of data will fix this.

The IV Solution

An instrumental variable Z satisfies two conditions:

1. Relevance:  Z is correlated with the endogenous regressor X
               Cov(Z, X) != 0

2. Exclusion:  Z affects Y ONLY through X (not directly)
               Cov(Z, u) = 0

   Z --> X --> Y
   Z -/-> Y  (no direct path)

Two-Stage Least Squares (2SLS)

How 2SLS Works

Stage 1: Regress the endogenous variable on the instrument(s)
         X = gamma_0 + gamma_1 * Z + controls + v
         Save the fitted values: X_hat

Stage 2: Regress the outcome on the fitted values
         Y = beta_0 + beta_1 * X_hat + controls + e

The coefficient beta_1 is the IV estimate of the causal effect.

Implementation in Python

from linearmodels.iv import IV2SLS
import pandas as pd


def run_2sls(data: pd.DataFrame, dependent: str,
             endogenous: str, instruments: list[str],
             controls: list[str] = None) -> dict:
    """
    Run a 2SLS instrumental variables regression.

    Args:
        data: DataFrame with all variables
        dependent: Name of the dependent variable (Y)
        endogenous: Name of the endogenous regressor (X)
        instruments: List of instrument variable names (Z)
        controls: List of exogenous control variable names
    """
    controls = controls or []
    exog_str = " + ".join(["1"] + controls) if controls else "1"
    endog_str = endogenous
    instr_str = " + ".join(instruments)

    formula = f"{dependent} ~ {exog_str} + [{endog_str} ~ {instr_str}]"

    model = IV2SLS.from_formula(formula, data)
    result = model.fit(cov_type="robust")

    return {
        "coefficients": dict(result.params),
        "std_errors": dict(result.std_errors),
        "p_values": dict(result.pvalues),
        "f_statistic_first_stage": result.first_stage.diagnostics,
        "summary": str(result.summary)
    }

Implementation in R

library(ivreg)

# 2SLS estimation
iv_model <- ivreg(
  log(wage) ~ education + experience | parent_education + experience,
  data = df
)

summary(iv_model, diagnostics = TRUE)

Instrument Validity Tests

First-Stage F-Statistic (Relevance)

def check_weak_instruments(first_stage_f: float) -> dict:
    """
    Evaluate instrument strength using first-stage F-statistic.

    Args:
        first_stage_f: F-statistic from the first-stage regression
    """
    return {
        "f_statistic": first_stage_f,
        "rule_of_thumb": (
            "Strong instruments" if first_stage_f > 10
            else "Potentially weak instruments"
        ),
        "interpretation": (
            "Stock & Yogo (2005) suggest F > 10 as a minimum for "
            "one endogenous variable. For more precise thresholds, "
            "consult the Stock-Yogo critical values table based on "
            "the number of instruments and desired maximal bias."
        ),
        "if_weak": [
            "Use LIML (Limited Information Maximum Likelihood) instead of 2SLS",
            "Report Anderson-Rubin confidence intervals (robust to weak IV)",
            "Consider finding stronger instruments",
            "Use the Lee et al. (2022) tF procedure for valid inference"
        ]
    }

Overidentification Test (Exclusion Restriction)

When you have more instruments than endogenous variables, the Hansen J test (or Sargan test) checks whether the extra instruments are valid:

H0: All instruments are valid (uncorrelated with the error)
H1: At least one instrument is invalid

If p < 0.05: Reject -> at least one instrument may violate exclusion
If p > 0.05: Fail to reject -> instruments appear valid
             (but this test has low power)

Classic IV Examples

Famous Instruments in Economics

Research Question          | Endogenous Var | Instrument
---------------------------|---------------|------------------
Returns to education       | Years of school| Quarter of birth (Angrist & Krueger)
Effect of institutions     | Institutions   | Settler mortality (Acemoglu et al.)
Colonial origins of trade  | Trade openness | Geography (Frankel & Romer)
Effect of military service | Veteran status | Draft lottery number (Angrist)
Price elasticity of demand | Price          | Supply shifters (cost, weather)

Reporting IV Results

Required Elements

1. Justify instrument choice with economic/theoretical reasoning
2. Report first-stage regression results:
   - Coefficient of Z on X with standard error
   - First-stage F-statistic
3. Report second-stage (2SLS) results:
   - IV coefficient with robust standard errors
   - Compare with OLS estimate (discuss direction of bias)
4. Report diagnostic tests:
   - Weak instrument test (F-statistic or Kleibergen-Paap)
   - Overidentification test if applicable (Hansen J)
   - Endogeneity test (Hausman or Durbin-Wu-Hausman)
5. Discuss threats to instrument validity
   - Can the exclusion restriction be challenged?
   - Are there plausible alternative channels?

Always present both OLS and IV estimates side by side. The comparison helps readers understand the direction and magnitude of endogeneity bias and assess whether the IV correction is meaningful.