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I am currently running fixed effect regressions with multiple dummy variables. These dummy variables are created by a grid of '1' '0':

e <- c("1","0")
r <- expand.grid(e, e, e, e, e)

By creation, the correlation of each dummy with the other dummies is 0.

I regress (multivariate) a variable on these 5 dummy variables while taking one dimension as a fixed effect:

feols(variable ~ dummy1 + dummy2 + dummy3 + dummy4 + dummy5 | dim1, data = x)

In addition, I perform 5 univariate regressions:

feols(variable ~ dummy1 | dim1, data = x)
feols(variable ~ dummy2 | dim1, data = x)
feols(variable ~ dummy3 | dim1, data = x)
feols(variable ~ dummy4 | dim1, data = x)
feols(variable ~ dummy5 | dim1, data = x)

The coefficients for each dummy are the same, both in its' univariate regression and in the multivariate regression.

Is there a proof that shows that this is always the case when the correlations amongst your independent variables are 0?

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  • $\begingroup$ I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? $\endgroup$
    – Richard Hardy
    Commented Apr 17, 2023 at 10:51

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Is there a proof that shows that this is always the case when the correlations amongst your independent variables are 0?

This is a known result; see e.g. Kennedy "A Guide to Econometrics" (6th ed., 2008) section 3.1. There must be a proof for it; it is given as exercise 3.15 in Davidson & MacKinnon "Econometric Theory and Methods (2004). It might be a corollary of the Frisch–Waugh–Lovell theorem.

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    $\begingroup$ It is quite surprising for me that searching for "orthogonal regressors" in almost 30 econometric and statistics textbooks did not result in much else beyond what I mentioned above... I suppose some of them do contain relevant material hidden under some different keywords, perhaps "uncorrelated regressors" or such. $\endgroup$
    – Richard Hardy
    Commented May 2, 2022 at 13:02

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