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I have two equations of the form:

$y = X\beta + Z\gamma + r\alpha +\epsilon_1$ (1)

$r = Z\delta +\epsilon_2$ (2)

The basic intuition here is $y$ is total cost data, $X$ is a matrix of variables relating to technologies, and $Z$ is a matrix of geographical attributes that can impact total cost. $r$ is an indicator that is determined in part by $Z$.

Right now, I'm using 2SLS to run this model. The intuition behind the approach seems fine to me. $r$ is only determined by geographical attributes in the regression so it should be okay to run $(2)$, determine $\hat r$, and plug that into $(1)$. However, I don't think I've encountered a regression like this before and am unsure if there would be a problem in using $Z$ as both an instrument in the first stage and an explanatory variable in the second stage of a 2SLS regression.

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    $\begingroup$ Welcome to CV, mirrror! $\endgroup$
    – Alexis
    Commented Apr 26, 2023 at 19:03

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Once you model $r$ using $(2)$ and substitute that into $(1)$ to obtain $(1')$, you will get perfect multicollinearity in $(1')$. You will not be able to estimate each of $\gamma$ and $\alpha$, only the linear combination $\gamma+\hat\delta\alpha$. $$ y=X\beta+Z\gamma+Z\hat\delta\alpha+u \tag{1'} \\ =X\beta+Z(\gamma+\hat\delta\alpha)+u $$ (I assume you meant $r=Z\color{red}{\delta}+\epsilon_2$ in $(2)$, as otherwise there is a restriction for a pair of coefficients to be equal across $(1)$ and $(2)$.)

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    $\begingroup$ +1 This would also be an explicit violation of the assumption that the instrument only affects the outcome through $r$, e.g., the "no pleiotropy" assumption of instrumental variables estimation in a Mendelian randomization context. See, e.g., Chapter 4: Instrumental Variables in Hernán and Robins' Causal Inference: What If?. $\endgroup$
    – Alexis
    Commented Apr 26, 2023 at 19:02
  • $\begingroup$ Thanks to you both. If I can ask a follow-up, Richard you mention that I would only be able to estimate the linear combination of $\gamma + \hat \delta$ - what does this mean? I partially asked my original question because I would expect running this regression would result in a perfect multicollinearity, and an error in R. But running the first stage, and using $\hat r$ in the second stage, I get coefficients for $\hat r$ and $Z$. Should I be getting an error, or I get coefficients that are biased and/or not consistent? $\endgroup$
    – mirrror
    Commented Apr 26, 2023 at 20:12
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    $\begingroup$ @mirrror, the estimate of $\gamma+\hat\delta\alpha$ should be fine (consistent and unbiased under the usual set of assumptions), but $\gamma$ and $\alpha$ individually are not identified individually, so they cannot be estimated consistently. Whether you get an error might be a matter of software. What you should not get is a perfectly usual output, especially the standard errors for $\gamma$ and $\alpha$. Regarding the linear combination, just look at the second equation in my answer. $\endgroup$ Commented Apr 27, 2023 at 5:19
  • $\begingroup$ @mirror, What do you think about my answer? If it is helpful and clear, you may accept it by clicking on the tick mark to the left. Otherwise, you may ask for further clarification. A helpful answer can also be upvoted by clicking on the upward-pointing arrow. This is how Cross Validated works. $\endgroup$ Commented Jun 4, 2023 at 12:54

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