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)
    }