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I am interested in estimating an endogenous treatment effects model of the following form: \begin{eqnarray} Y_i = \alpha + \beta_x X_i + \beta_{z1} Z_{1i} + e_i \\ X_i = a + \beta_{z2} Z_{2i} + v_i \end{eqnarray} where $Y$ is a continuous variable, $X$ is a binary variable that is endogenous in the first equation and $Z_1$ and $Z_2$ are exogenous variables. Furthermore, I have reason to believe that $Z_3$, a third exogenous variable, is a determinant of both $X$ and $Y$.

My question is the following. Should $Z_3$ appear in the second equation that explains $X$ only, or should it appear in the first equation that explains $Y$ as well?

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In structural models, you would add $Z_3$ to both equations. It will then have a direct effect $\beta_{z31}$, an indirect effect $(\beta_{x}\beta_{z32})$, and a total effect (sum of both) on $Y_i$. Omitting $Z_3$ from any oth the two structural equations will bias the parameters of the direct effects in the model.

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  • $\begingroup$ You're welcome. Note I edited the index of $\beta_{z3}$ to $\beta_{z31}$ and $\beta_{z32}$ because these are two different coefficients. $\endgroup$
    – tomka
    Commented Nov 30, 2013 at 13:05

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