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I am trying to fit a regression model in which the dependent variable is a continuous positive variable (costs, no zero costs). In literature I often read that considering the lack of negative values, a ordinary least square regression is not appropriate. Generalized linear models with for example a Gamma distribution is often recommended over OLS. This leads to the following questions

  1. To get unbiased regression estimates the assumptions of the OLS do not specify anything regarding the specification of the error distribution. Hence, even though the errors are clearly not normally distributed, an ordinary least square regression would still provide you with unbiased estimates. In case the sample is sufficiently large and the errors are homoscedastic (or use robust standard errors when they are not), why would one prefer a glm?
  1. I fitted a GLM with a gaussian family as well as with a Gamma distribution, with both the identity link function as well as the log link function. However, the regression estimates for the different families (with the same link function) are substantially different (even when considering robust standard errors). Given that both should give unbiased estimates why would this happen?
  2. Some papers say that fitting non negative data with an identity link function is inappropriate as it might lead to negative predictions. However, when given the regression coefficients negative predictions are impossible would this still be argument? I am asking as I am interested in the additive effect of my variables. Moreover, the AIC of the Gamma regression with the identity link is the lowest. The error plots for the gamma regression with the identity link also look superior compared to the others.
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This is a very broad question, and similar questions have been posted before. So this is mostly and enlarged comments giving links to earlier posts which can help you. But first, shortly, glm's are a modeling framework for many different regression models, and the choice mostly depends on your goals and what kind of parameters you find interpretable in your context. AS to the question in title, estimated differ because this are different models!

So:

(many other relevant posts, search our site)

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  • $\begingroup$ Thanks for your reply. The first link indicates that you would move to a GLM framework when your SE are too big? If you are fine with your SE and your not trying to detect significance, then OLS should be fine and consistent? Q2 is not answered with any of these links (Why the regression estimates can be so different when using the same link function but a different family (the third link only says something about the p values)). From the last link it seems like an identity link can be fine (costs per person, seems to be an intensive variable), but what about negative coefficients (Q3)? $\endgroup$
    – Leslie
    Feb 23 at 11:45

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