When Orthogonal Regressors Have Matching Univariate and Multivariate Slopes
Summary
The document asks why coefficients from separate regressions can match those from a regression containing several binary explanatory variables alongside fixed effects. The author reports that the dummy variables are uncorrelated by construction and observes matching coefficients in the univariate and multivariate specifications. The question is whether zero correlation is enough to guarantee this result.
The response identifies the result as a standard econometric property and points to a textbook discussion and an exercise as references. It suggests that the Frisch–Waugh–Lovell theorem may provide a route to a proof, but it does not present the proof or spell out the assumptions. In particular, the example includes fixed effects, so the relevant orthogonality must be understood after accounting for those effects; raw pairwise correlations alone may not establish the needed condition. The text provides a useful pointer, but leaves the formal argument and its precise conditions to the cited sources.
Key ideas
- Orthogonal explanatory variables can have the same estimated slopes in separate and joint regressions under suitable conditions.
- The document connects the result to standard econometrics references and the Frisch–Waugh–Lovell theorem.
- With fixed effects, the relevant orthogonality must account for the fixed effects.
- The response offers references rather than a proof or a complete statement of assumptions.
Tags
Full text
# Coefficients of univariate regressions equal to those in the multivariate regression
# 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?
## Answer by Richard Hardy (score 1)
https://quant.stackexchange.com/a/70707
> 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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