Instrumental Variables & 2SLS Estimation

Instrumental Variables (IV, Angrist & Imbens Nobel Prize framework) & Two-Stage Least Squares (2SLS) uniquely identifies causal treatment effects in the presence of unobserved confounding, omitted variable bias, or bidirectional reverse causality; An instrument $Z$ must satisfy 3 strict core conditions: 1) Relevance Condition ($\text{Cov}(Z, D) \neq 0$, verified by first-stage $F$-statistic $> 10$ to rule out weak instruments); 2) Independence / Exogeneity ($\text{Cov}(Z, \epsilon) = 0$, typically randomized assignments or natural experiments); 3) Exclusion Restriction (the instrument $Z$ affects outcome $Y$ strictly through treatment channel $D$, with zero direct pathways); The 2SLS estimator executes: Stage 1 regresses treatment $D$ onto $Z$ to purge endogeneity yielding $\hat{D}$; Stage 2 regresses outcome $Y$ onto exogenous $\hat{D}$, estimating the Local Average Treatment Effect (LATE) on Compliers.

Imperfect compliance in online experimentation (e.g. coupon assignment vs actual coupon redemption) and structural price elasticity modeling.

OLS, PSM, and DiD fail under unobserved confounding; IV isolates exogenous variation to identify unbiased causal treatment effects on compliant sub-populations.

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