I am running regressions with the same dependent and explanatory variables on about 500 sub-samples of the same population of observations. Now, I want to know of certain characteristics of the subsamples impact the regression results. Hence, I run a second-stage regression using (one) coefficient of the first stage regressions as dependent and regressing that on the sample characteristics. Is this valid? Given that my observations are themselves estimates, I am not sure if usual statistical inference is feasible. Also - does this qualify as meta-regression?