Two-Stage Least Squares

Watch both stages, and see why manual second-stage standard errors are wrong.

Day 64: 2SLS in Matrix Form

STAGE 1: x on z

STAGE 2: y on fitted x̂

First-stage coefficient π0.5Endogeneity corr(u, v)0.80Sample size500

first stage π̂

0.477

2SLS

0.387

truth 0.500

OLS

1.126

correct SE

0.1027

residuals y − b̂·x

manual 2nd-stage SE

0.1337

residuals y − b̂·x̂ (wrong)

Stage 1 projects the endogenous regressor on the instrument (left); stage 2 regresses the outcome on the fitted values (right). The slope is 2SLS. Standard errors must use residuals computed with the actual x, not the fitted x̂: compare the two.

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