R2 Addition in OLS Regression: Unrelated Variables?

[monsmatglad]

Hey. I am working with OLS regression. First I run 2 regression operations with each having just one independent variable. Then I run another regression using both the independent variables from the first two regressions. If the explanatory "power" (R^2) in the third regression was to be the sum of the R^2 from the two first regressions, would this require the independent variables to be completely unrelated?

 

[FactChecker]

The short answer is no.
You need to distinguish between the independent variables versus the sample data. Consider these two possibilities:

• Two independent uncorrelated variables are unlikely to be represented in the data as though they are exactly uncorrelated.
• Two correlated variables may be in a designed experiment where they appear in the data as though they are uncorrelated.

 

[monsmatglad]

yes, but what if the sample shows that the independent variables are in fact uncorrelated, regardless of how likely it is that this represents the population, would then R1^2 + R2^2 = R3^2?

Mons

 

[FactChecker]

I think it is a mistake to draw conclusions about two variables from their individual relationships to a third variable.
Suppose you have three uncorrelated variables, ε₁, ε₂, and ε₃.
Consider X₁ = ε₁ + ε₃, X₂ = ε₂ + ε₃, and Y = ε₁ + ε₂

X₁ and X₂ are correlated through ε₃, which is uncorrelated with Y.
The individual correlations of Y with X₁ and X₂ are through the uncorrelated variables ε₁ and ε₂, respectively.

 

[FactChecker]

yes, but what if the sample shows that the independent variables are in fact uncorrelated, regardless of how likely it is that this represents the population, would then R1^2 + R2^2 = R3^2?

This is not the same question as the original post. I think this may be true.