I have bee wondering why in a multivariate OLS-Regression it is not possible for R² to decrease when increasing the number of explanatory variables. The Point is that for example in the model Y=ß0+ß1x1+ß2x2+u, x1 and x2 are also correlated. Thus when using partialling out I only take the part of x1 that is not correlated or let's say orthogonal to x2. I also only take the part of x2 that is orthogonal to x1 before regressing. But what if the explanatory value is very small of x1 keeping x2 constant and of x2 keeping x1 constant as compared to only including one of the two explanatory variables into the model. To better illustrate my thought let's assume a circle C that represents the variation in Y and a circle A that represents the variation in x1 and a circle B that represents the Variation in x2. All of the three circles overlap. In a univariate Regression with only x1 as explanatory variable R² represents the fraction of the area Where A and C overlap to the area of C. In the Regression with x1 and x2 A and B usually overlap as well. Partialling out would mean to eliminate the area where A, B overlap and thus where A, B and C overlap. Thus, if x1 and x2 are strongly correlated the remaining area where A overlaps with C only and B overlaps with C only could be smaller that the area where A and C overlap in the univariate case. Thus it should be possible that R² decreases when increasing the number of explanatory variables.. But it never does..
My question is why and if there is some technique to divide the area where A,B and C overlap to the explanatory variables, how do they do it (e.g. 50:50?)?