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When I am running a 2 stage least squares, where do I put the control variables? Should I put the control variables in the first stage? The second stage? Both?

Can someone explain why?

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2 Answers 2

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You put the controls in both the first and second stages.

For intuition, I guess the best explanation I can give would be that you want to get a more precise beta in the first stage, and relevant controls help with precision. In the second stage, those same relevant controls that mediate the selection-into-treatment that motivates most IV would probably also mediate the outcome of that treatment.

To put it differently, if you forget 2SLS and just think about it as the IV estimator $(Z'X)^{-1}Z'Y$, and you want an estimate of the coefficients of your control variables, you're going to have to put them into Z as well as X.

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In 2SLS, the variables are classified as endogenous and exogenous (and of course dependent). In the first stage, you regress each endogenous variable against the exogeneous variables and use the fitted value as instrument for the endogenous variable in the second stage along with exogenous variables.

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