I have linear regression model clustered on both firm and data. when I add two more variables to the regression model, one of the newly added variable is significant, but the R squared remains the same as the model without these two variables. Could this be the case? If so, why?
If you are adding new variables and not getting any change in $R^2$, the most likely explanation is that your new variables are linear combinations of the existing variables, so that your design matrix does not have full rank. I would recommend finding the rank of your design matrix, both before and after the addition of your variables. This can be done in
R using the
rank function, but you need to be careful with it, because rank calculations can be subject to tolerance issues in computation.
It is theoretically possible for new explanatory variables to add nothing to the $R^2$ of a regression, even if they are not linearly combinations of existing variables. However, this requires some exact vector angle conditions that are essentially impossible for random data. You can construct this by design, but it does not occur at random with continuous data (and even for discrete data it occurs with vanishingly small probability). For this reason, I strongly suspect that you are adding variables that are just linear combinations of your existing explanatory variables.