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I am going through an old metrics tutorial using instrumental variables regression. In one question we are supposed to estimate the Average Treatment Effect (ATE) using a control function approach. We predict the residuals $\hat{R}$ from the first stage of the instrument $Z$ and other controls $X$ on the (multi-valued) treatment $T$

$T = \alpha_0 + \delta Z + \beta X + \varepsilon_1$

and then include the residuals in the structural equation

$Y = \alpha_0 + \gamma T + \eta \hat{R} + \beta X + \varepsilon_2$

The solution indicates that this gives the normal 2SLS estimate, and to get the Average Treatment Effect (ATE) we need to also include the interaction of the treatment with the predicted residual.

$Y = \alpha_0 + \phi T \times \hat{R} + \gamma T + \eta \hat{R} + \beta X + \varepsilon_3$

Does anybody know why including the interaction gives the ATE?

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