What is the best way to combine regression betas from separate data sets?
For example, a data set is split in two based on some fundamental characteristic, and the same two factor regression is run on each set. The weight placed each data set is x% and (1-x%) based on the fundamental characteristic used to split the data set. How do I combine the results of the two regressions to get an aggregate representation of the exposures to the two factors? Taking a simple weighted sum of the two results doesn't seem correct.
The objective is to characterize the variation captured by the factor exposures. Combining the data sets and running a single regression is not an option due to data limitations.