Variable selection: Adding a variable to regression model only increases r by 2% - should I leave that variable out?

sorry not the smartest question, tried to google this and search forum, but questions remain.

So x1 correlates with y by r=0.8. If I now add x2 and x3 to the model, the whole model correlates with r=0.82 to y. So I only gain 0.02 of r by adding x2 and x3. So are x2 and x3 not important for the model?! What I would like to know which share of the r is explained by which variable?

Details: x2 and x3 both come with significant p values in the model. So this means it's highly probable that in the overall population x2 and x3 also would have an observable effect on y. But at the same time they only explain 0.02 of the r if I add them. How can both be true at the same time? x2 and x3 alone correlate with r=0.50 to y. So they seem to be important? x1 and x3 correlate by r=0.45 with each other. The other variables barely correlate with each other.

edit: My goal is to understand which variables out of 10 actually explain my dependent variable. After some more googling I understand that you absolutely can't use stepwise (forward or backwards) selection for this. Is that correct? If yes which method for variable selection is the next easiest to use? (preferably in excel)

• It is only a convention, but it is a very, very strong convention, never to report correlations as percentages. Your correlation $r$ changed from 0.80 to 0.82, a change of 0.02. On the other hand, if by $r$ you really mean $r^2$ or $R^2$, that is a different story, and percents and proportions are equally conventional but calling $R^2$ by the name $r$ is a major error, akin to confusing lengths and areas. Dec 24, 2022 at 11:59
• sorry for potential confusion. By r I mean correlation coefficient, which I hope is the correct usage. Dec 24, 2022 at 12:08
• That is fine, but people never, ever report correlation as a percentage in my reading. Asking why not would be a good question, and at least one excellent reason is to avoid any confusion whatsoever with $R^2$ which often is discussed and reported in percent terms. in terms of both $R^2$ and change in $R^2$. I strongly recommend editing your question now, because many people read questions but are less likely to read comments. Also, if you fix your question, this exchange of comments can be deleted. Dec 24, 2022 at 12:16