Athanasia -

WLS is not for combing different strata that should use different models, but for accounting for the increase in variance of y with larger x that occurs naturally. OLS is just a special case of WLS, which is often used as a default, often incorrectly, as noted on page 5 in https://www.researchgate.net/publication/263036348_Properties_of_Weighted_Least_Squares_Regression_for_Cutoff_Sampling_in_Establishment_Surveys

I have seen WLS used less formally, assigning weight to indicate data quality, but again this is not for combining strata.

It is very important to use a model that is appropriate for the stratum. See https://www.researchgate.net/publication/261474154_On_Model-Failure_When_Estimating_from_Cutoff_Samples

The balance you have to consider is whether or not the increased sample size might be more than offset by bias and variance from combining groups of data that are too different. You could start your considerations by considering subject matter theory, and then look at a scatterplot where the data are on the same plot, but color-coded or otherwise delineated in your graph to see if combining the data make sense. Consider b estimates and their standard errors.

If you look at multiple regression, you could plot y vs predicted y to get some visual input.

Cheers - Jim 

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Athanasia -

I think perhaps that the first example, with graphics, in the paper at the following link, may fairly directly address your issue, as it shows how combining what might be thought of as different strata, would relate to heteroscedasticity:

 https://www.researchgate.net/publication/263032446_Weighting_in_Regression_for_Use_in_Survey_Methodology

It is, however, contrived to illustrate a point, without the usual error structure.

For a definition of heteroscedasticity, see this:

 https://www.researchgate.net/publication/262972023_HETEROSCEDASTICITY_AND_HOMOSCEDASTICITY

So, it can be seen that if you combine strata, where different models apply, a 'compromise' model of sorts might be used, with pronounced heteroscedasticity, but if the two models that should have been used are very different, results may not be very satisfactory.

Does this address your issue?

Jim 

PS -  Note that this example is for regression through the origin (continuous data), one regressor.  Is that what you are doing? 

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Many thanks for your answer. I will study these papers and  come back to you if I have any further questions.

I'm not certain if weighted least squares is needed, unless your sampling scheme is different in the two strat, in which case you would use inverse probability of sampling weights. . I would begin by fitting two separate identical regression models one for each stratum,, checking that the model coefficient estimates look similar (ie the 95% CIS overlap). If they do, then combine the two datasets and analyze as one, otherwise you can add separate inercepts for the two strata and interaction terms as needed.

Athanasia -

Heteroscedasticity is very natural, but by mixing data that should have been modeled separately, the graphics I supplied in example 1 in that 1997 paper using simple linear regression through (well, to) the origin showed artificially induced heteroscedasticity. If there were an intercept term, then the influence would be to decrease heteroscedasticity up to the point where the two linear regressions would have intersected, and increase after that point.

Again, I suggest you look at a scatterplot. You could color code or use shapes to differentiate points from each stratum. Consider Figure 4 in https://www.researchgate.net/publication/262066356_Quasi-Cutoff_Sampling_and_Simple_Small_Area_Estimation_with_Nonresponse. There it appears that it may be OK to combine data. But also consider an application such as https://www.researchgate.net/publication/262143393_A_Note_on_Estimating_Impact_of_Data_Collection_Mode_Change_for_a_Given_Key_Response_Item

If there is a non-zero intercept,  that complicates things, as noted above. If there is multiple regression, you could plot y vs predicted y to possibly get some ideas.  If there is nonlinearity, that further complicates matters: Usually I'd recommend against a transformation as just making results less understandable, but you might consider it for graphing here.

Jim 

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From the question description "...I need to combine data from two different strata in the same regression model. I have read that I can use Weighted Least Squares...." Such a combination of data can make estimated residuals larger for larger predicted y because the model for the combined data might be considered a compromise, and as the predicted y for the combined data becomes larger, the predicted y one might have from each of the two models for the actual strata could become further distant from each other. (Another description is above.)

This falls under "nonessential heteroscedasticity" in https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression