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I am wondering if anyone can point me to a paper/lecture notes on the rationale behind first running an OLS on a set of variables, and then in a second regression using the residuals of that regression as the dependent variable to regress on several new (but related) independent variables. To specify, this is not aiming for an IV/2SLS approach - there's no instrument here that's uncorrelated with the dependent variable in the "first stage." Instead, it aims to kind of set a standard across the sample set with the first result, and then attribute the differences from the market-wide (ie the residuals) on the 2nd set of variables.

First off, wouldn't this approach by definition limit the first regression to one independent variable for a regular OLS? Otherwise the dependent variable in the 2nd would be an nx2 matrix...

Overall, is there any purpose to such a process? I believe I have seen it done before, but my searches have come up mostly fruitless. Closest conversation I've found is this Stata thread: http://www.stata.com/statalist/archive/2008-03/msg00264.html

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    $\begingroup$ are you talking about something along these lines? "In analysing multivariable datasets it is common that in looking at the effect of some variable ($x_1$) on a dependent variable of interest ($y$), the effects of a third continuous variable ($x_2$) are to be controlled for, for instance because its effects may confound those of $x_1$. In such circumstances it has become common to perform a regression of $y$ on $x_2$ and use the residuals from this regression in testing for the effects of $x_1$" found it here $\endgroup$
    – Robson
    Jun 30 '15 at 22:41
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    $\begingroup$ Yes, this should be helpful for background info - thank you. $\endgroup$
    – Z_D
    Jul 1 '15 at 15:34
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    $\begingroup$ Important to add here that the quote is only paraphrasing what other people say / do - the linked article explains why there is NO justification for this approach. $\endgroup$ Oct 27 '17 at 13:35
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What you're describing is called 'two-stage residual inclusion' or 'control function' approach. In a linear model, it gives exactly the same estimator as "traditional" two-stage least squares, as for instance derived by Hansen, pp. 335-38. But in both cases, you need a valid IV. So your idea of doing this without an instrument in the first stage won't work.

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  • $\begingroup$ I've tried both methods often enough, obtaining exactly the same results from each, to know that they do work. A reason for using the residual method is to allow for graphs that show a given X's relationship to Y after controlling for other predictors. Sometimes the partial plot one obtains via the more standard, condensed method isn't quite what's desired. $\endgroup$
    – rolando2
    Jul 2 '21 at 19:39

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