Fixed and random effects: what do statistics and econometrics say?

Dan Schrage writes with a question about how to model group-level variation:

I [Dan] am trying to better understand the recommendation in your new book to always use random effects (pg. 246) in modeling. (I’m following your definition #5 here of fixed and random effects, as is standard in econometrics.) In econometrics, as I’m sure you know, the classical advice (dating from at least Mundlak (1978)) is this: If unobserved heterogeneity is correlated with regressors in your model, use fixed effects; otherwise, use random effects since they’re more efficient. The idea is that random effects lumps this unobserved heterogeneity into the composite error term, and so if it the unobserved heterogeneity is correlated with the regressors, then the regressors are correlated with the error term, and this is bad news: estimators will be biased and inconsistent. So I’m trying to understand why your advice is essentially not to worry about this issue.

Is this just an argument about the bias/variance tradeoff, or is there something deeper here? Perhaps all of this just speaks to one of the common gaps between statistics and econometrics–econometricians tend to care a lot about asymptotic properties like consistency, and statisticians seem to care much less (correct me if I’m off here–I’m still trying to get a good sense of the divide). Econometricians are almost never willing to use an inconsistent estimator, no matter what the gain in efficiency.

My reply:

See this paper, which addresses this issue a bit. The short answer is that if correlation is a problem, you can include the group-level mean of your correlated x as another predictor.

Regarding the consistency issue: it’s a tricky question, because once you consider the option of including _any_ additional x-variable, you have to include it in some way, otherwise you have consistency problems. When I say any additional x-variable, this includes all possible interactions of existing x-variables in the model. (This includes, for example, interactions of all x-variables with your group indicators.) But in practice, econometricians don’t include all interactions (and neither to we). So consistency has its limitations as a principle.

There might be more to this issue, but this is what I know about it so far.

1 thought on “Fixed and random effects: what do statistics and econometrics say?

  1. Andrew,

    On the consistency issue, doesn't it also come down to which parameter you are really trying to estimate. Not including an interaction implies an assumption that there isn't an interaction, and you should be able to consistently estimate the effect as if there isn't an interaction, right.

    Setting interactions aside for a moment, let's consider the simpler case of adding a single predictor. If the predictor is not correlated with any of the predictors in your current model, it won't change the parameters you had been trying to estimate in your current model. However, if the additional predictor is a confounder, this implies that including the predictor in the model changes the actual parameter you are trying to estimate. So, I guess I am wondering if the consistency issue mentioned above comes down to what are your parameters of interest, since adding or leaving out terms can literally change the parameters of interest you are trying to estimate.

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