I recently saw a paper where they run a regression of the covariate of interest $X_{1}$ on the outcome $Y\in\{0,1\}$ without further controls $$ Y = \alpha_{1} + \beta_{1} X_{1} + \epsilon_{1} \tag{1}\label{1} $$ where $Y$ and $X_{1}$ are such that one may wonder whether $E[\epsilon_{1} |X_{1}]=0 $ holds. Then they argue that they decompose $E[Y|X_{1}]$ into an observed and an unobserved part by regressing $$ X_{1} = \alpha_{2} + \beta Z + \epsilon_{2} \tag{2}\label{2} $$ where $Z$ is a matrix of predictors of $X_{1}$ and $\beta$ the respective parameters (not necessarily instruments, indeed I would suppose most $Z_{1}, ..., Z_{n}$ would not satisfy the mean independence restriction $E[\epsilon_{1}|Z] = 0$) and calculate two sets of predicted values, namely $\widehat{X_{o}}$ based on the linear prediction of $ \eqref{2} $, as well as $\widehat{X_{u}}$ based on the residuals of the linear prediction in $ \eqref{2} $. Then they estimate the following two models $$ Y = \alpha_{3} + \beta_{2} \widehat{X_{o}} + \epsilon_{3} \tag{3.1}\label{3.1} $$ $$ Y = \alpha_{4} + \beta_{3} \widehat{X_{u}} + \epsilon_{4} \tag{3.2}\label{3.2} $$ to argue that $\beta_{2}$ gives the observable effect of $X_{1}$ on $Y$ and $\beta_{3}$ gives the unobservable effect of $X_{1}$ on $Y$ and go on to make claims which one is more important. I am puzzled by this procedure, and not sure what to make of it. It looks like control function approach, but there is no guarantee or even mention that $E[\epsilon_{1}|Z] = 0$.

Would be interested to hear what others think about this approach of making claims about the portion of observed to unobserved effects? Sources on the method or other papers doing similar "decompositions" are very welcome as well.

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    $\begingroup$ When you write that $Y\in\{0,1\}$, am I understanding you correctly that $Y$ is binary? If so, I would first of all be interested in why the authors did not use logistic regression or any similar established GLM. Do they explain? $\endgroup$ – Stephan Kolassa Mar 31 at 15:23
  • $\begingroup$ Yes, $Y$ is binary, no, they don't do it and don't explain why, and yes, I agree with you that they should have :) $\endgroup$ – Papayapap Mar 31 at 15:24
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    $\begingroup$ Hm. I am also... puzzled. Can you link to the paper? $\endgroup$ – Stephan Kolassa Mar 31 at 15:31
  • $\begingroup$ As I think this is dubious, I prefer not shaming the authors publicly, I sent you the link via Linkedin. Hope that's fine for you. $\endgroup$ – Papayapap Mar 31 at 15:42
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    $\begingroup$ What exactly is the question here? We can't comment on whether we think mean independence holds without knowing what the variables are or where the data comes from. We can tell you anything about which variables are observable or not for the same reasons. One can certainly decompose a regression in a lot of different ways. Do you want us to comment on that? Maybe clarify your question, tell us more about the data, or at least tell us to make some assumptions. $\endgroup$ – jmbejara Apr 1 at 20:19

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