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.

Derive the 2SLS estimator equations and prove the algebraic equality of the Wald Estimator $\hat{\beta}_{\text{IV}} = \frac{\text{Cov}(Y, Z)}{\text{Cov}(D, Z)}$?

Explain why IV identifies LATE strictly for Compliers and does not identify effects for Always-takers or Never-takers under monotonicity?

Why does first-stage $F$-statistic $< 10$ signal weak instrument pathology causing severe finite-sample bias and unbounded asymptotic variance?

Explain how notification annoyance directly violates the Exclusion Restriction when using app push prompts as instruments for feature adoption?

Contrast Intention-to-Treat (ITT) measuring macro-business ROI vs Treatment-on-Treated (TOT/LATE) measuring pure algorithmic efficacy?