What happens to R squared when you take out a variable from a regression? Im assuming the model & estimations would be less accurate, causing the residuals to be larger, therefore, it makes R^2 larger. Just want to make sure and see if anyone has any insight for me. Thanks!
 A: Removal of a variable from regression cannot increase R squared because adding a new variable cannot decrease residual sum of squares (R squared = 1 - residual sum of squares/total sum of squares). But it doesn't mean that you should add in your model as mny variable as possible. In order to determine the effecteveness of added variable use adjusted R squared or information criteria (Akaike's or Schwarz's).
A: Taking out a variable will remove some of the "wiggle room" for the model to fit the data, so yes, the fitted points probably won't be as close to the data, so the $R^2$ will probably be lower.
However, keep in mind that this does not necessarily mean that the model or estimates are less accurate because you took out a variable. That variable could have been completely meaningless, but the $R^2$ could still be higher because adding an extra variable gives your model another opportunity to overfit.
A: The answer to your question is context dependent. Just about anything can happen
1) The trivial case is the quality of the fit reducing. This happens, when you remove a variable which explains part of the variance not explained by any other variable (you have written "makes $R^{2}$ larger which is not correct.. Recall $R^{2}$ = $ 1 - \frac {S_{Res}}{S_{tot}}$ so when $S_{Res}$ increases, $R^{2}$ will decrease). 
2) The quality of the fit remains the same: Consider the case where you are fitting a regression using multiple variables and you decide to remove a variable X which is perfectly correlated with another variable Y (which you do not remove). In this case, you would expect the quality of the fit to remain the same.
3) Suppose we have 100 observations and 300 variables we are using to predict. You do not have enough predictive power in this case BUT if you know before hand (say) 295 are useless and remove them your $R^{2}$ may increase
