Why OLS fails under endogeneity
Consider $y = X\beta + \varepsilon$. OLS requires $\mathbb{E}[X^\top \varepsilon] = 0$. Endogeneity — omitted variable bias, simultaneity, or measurement error — violates this, making $\hat{\beta}_{OLS}$ inconsistent.
The IV estimator
An instrument $Z$ must satisfy two conditions:
- Relevance: $\text{Cov}(Z, X) \neq 0$ — the instrument is correlated with the endogenous regressor.
- Exclusion restriction: $\text{Cov}(Z, \varepsilon) = 0$ — the instrument affects $y$ only through $X$.
The IV estimator (just-identified case):
\[\hat{\beta}_{IV} = (Z^\top X)^{-1} Z^\top y\]Two-Stage Least Squares (2SLS)
In practice, 2SLS is used for over-identified systems ($\dim Z > \dim X$):
Stage 1: Regress $X$ on $Z$: $\hat{X} = Z(Z^\top Z)^{-1} Z^\top X$
Stage 2: Regress $y$ on $\hat{X}$: $\hat{\beta}_{2SLS} = (\hat{X}^\top X)^{-1} \hat{X}^\top y$
This is numerically equivalent to the GMM estimator with moment conditions $\mathbb{E}[Z^\top \varepsilon] = 0$.
Weak instruments
When the first-stage $F$-statistic is below 10–16 (rule of thumb), weak instrument bias becomes severe. Use Anderson–Rubin confidence sets for inference that is robust to weak instruments.