r-squared goes down when I remove largest var with the largest p-value

Question

re: linear multivariate regression --

Does anyone know if your r-squared value can go down when you remove the variable with the largest p-value from a multivariate regression? Cause that's what's happening to me and I am baffled. In fact, it goes down when I remove anything from my model.

I'm a student; in class we've only seen examples where r-squared goes up after removing the var with the lowest p-value, so I'm confused.

Answer 1

$R^2$ is a monotonic strictly increasing function of the number of variables in the model (unless the added variable is a linear transformation of the others). It is not hard to prove it, but intuitively - any new variable, event if it is not significant in any plausible level, adds another dimension to the column space of the data matrix $X'X$, as such the distance between the predicted and the observed values, $|| \hat{Y} - Y || ^2 $, necessary getting smaller.