It seem as though nearly all regressions require some special technique such as 2SLS, clustering, or hetero-skedastic standard errors, none of which are available here. An easy way to obtain corrected standard errors for a manual implementation of 2SLS is to regress the 2nd stage residuals (calculated with the real, not predicted data) on the independent variables. Those standard errors are unbiased for the coefficients of the 2nd stage regression.
I have not tested -rollreg-, but here is a description. And there is -rolling-.