Higher R2 by adding insignificant variables, how?

Hi!

I got a quick question about my statistical analysis for my thesis. To provide you with some background information: I am running a multifactor risk model to asses returns for certain stocks and including market returns, technology firms returns, and commodity prices (oil, natural gas and coal) as regressors.

However, I run into some trouble and let me explain why. The commodities natural gas and coal have been found to have no (significant) correlation with any of the other variables (nor dependent or independent). But running the multiple linear regression with these two variables (natural gas and coal) raises the R2 very slightly (from 0.527 to 0.528), while the coefficients are insignificant.

Can anyone maybe help explain to me how R2 can rise even though both extra included variables are insignificant and do not have correlation?

I am analysing with SPSS and Excel by the way.

Kind regards,

Joep

• There are a number of related threads, and yours is most likely a duplicate. You can find more detailed answers if you do a little search on Cross Validated. – Richard Hardy Jun 24 '17 at 16:38

$R^2$ will never decrease as the number of variables increase. This is because for any finite sample a small insignificant portion of the remaining variation (assuming $R^2$ is not already 1) can be explained by the added variable(s) even though the estimated coefficient is not statistically significant from 0. This is why we worry about overfitting a model when we base goodness of fit on $R^2$ alone.