Given data that arrives at a daily frequency, I aggregated it to a weekly frequency, and estimated an OLS regression on it. Given that there are roughly 5 trading days per week, I can construct 5 different OLS models using 5 different starting points. For example - one model uses returns from Monday-Monday, the next Tuesday-Tuesday, and so on.
Assuming I believe there are no seasonal effects (e.g. models trained using Monday-Monday returns should be no different than Tuesday-Tuesday), is there a correct way to combine the predictions/coefficients of these 5 (or in the general case, N) models? I am inclined to think quick and simple averaging of coefficients would work. In that case, is there a proper way to combine the standard errors and residual standard errors across models? I ask because I am interested in constructing confidence/predictive intervals for forecasts. I hesitate to estimate the model using the full dataset, because this will cause overlaps in my endogenous variable, and I am not well equipped/don't know how to deal with that.
Of course this question could be asked more generally for any (non-linear) kind of model, but it seems like OLS/linear models would have the most hope for a theoretically sound procedure/heuristic.