I'm looking for advice/opinion on whether the co-efficient(s) resulting from a geographically weighted regression analysis can subsequently be entered into an OLS regression as the dependent variable (i.e. to test for factors influencing the observed spatial variation). Are any major assumptions violated in doing so and, if so, are there any acceptable work-around?
Regressing on OLS estimates runs against the (classical) assumptions of the model, which hold that the estimates are unknown constants that are estimated with data (which are random). So, if I understand the regression you are suggesting running, you would get some estimates but they would not have a formal interpretation. (Bayesians have the opposite view, that our estimates are random variables given the data.) Depending on the type of patterns you wish to test for, you may wish to look at spatial autocorrelation models (http://stats.idre.ucla.edu/other/mult-pkg/faq/general/faq-how-can-i-detectaddress-spatial-autocorrelation-in-my-data/) or hierarchical models (https://www.r-bloggers.com/hierarchical-linear-models-and-lmer/).