Two-stage least squares (2SLS) — the recipe.

hard

Answer

$Stage 1: regress T on instrument Z (and controls X) \to predicted T̂.$
$Stage 2: regress Y on T̂ (and X) \to coefficient on T̂ is IV estimate.$
${\beta}_{\mathrm{IV}}\; = \;\operatorname{Cov}(Z, \;Y) / \operatorname{Cov}(Z, \;T)$
$Use robust or cluster SEs; standard 'ivreg2' / 'AER' / linearmodels packages compute correct SEs automatically.$
$Never use naive 2-step OLS SEs from stage 2 — they're wrong.$

Check yourself — multiple choice

Random
Stage 1: T ~ Z + X → T̂; stage 2: Y ~ T̂ + X → β on T̂ is IV estimate; use ivreg2/linearmodels for correct SEs (naive stage-2 SEs wrong)
Same as OLS
Not real

2SLS: regress T on Z (stage 1), then Y on T̂ (stage 2); use proper IV SEs.

#causal-inference#regression

215 interview questions in this topic.