# Coefficients of univariate regressions equal to those in the multivariate regression

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?

• I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? Apr 17 at 10:51