Bayesian Fixed Effects Regression Panel Data

So I am new to bayesian and I was wondering what would be the equivalent of a Bayesian Fixed Eff Regression for Panel data? This is the following model I am referring to: Equation 1.

Would simply setting priors on the ai per subject allow one to identify the causal effect B of the treatment Xit?

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usually, choice of estimation strategy (e.g. fully Bayesian with MCMC, approximate Bayesian with ADVI, frequentist maximum likelihood) is only very weakly tied to the choice of the model. I.e. for most mathematical descriptions of a data generating process, one can perform both Bayesian and frequentist inference. And in many (but far from all) cases the results will not be substantially different - although there are models that will produce meaningless answers in a frequentist setting but can be useful with a Bayesian approach. So if you believe a fixed-effects regression captures your beliefs about the data well, then you can estimate it in Bayesian setting (e.g. via brsm or rstanarm::stan_lm) directly.

Does that answer your question?

Thanks! Yes this does answer it. Looks like then it is just as simple as a standard lm() but with a Bayesian approach via those 2 functions.

You will want to be careful with the priors. A key feature of the econometric fixed effects model is that the respondent-specific intercept is allowed to covary freely with all time-invariant variables. This is what makes the fixed effects model so powerful, because those intercepts stand in for all time-invariant variables (measured or unmeasured) with time-constant effects. Per Paul Allison (Fixed Effects Regression Models, 2009, page 21-22), a random effects model treats the random intercepts as uncorrelated with time-invariant variables, which is not what you want.

The only sure approach I can think of is to place wide, independent, uniform priors on the respondent-level intercepts, at least to align with the equation you cite and the traditional approach of including a dummy variable for each respondent. But there is probably some greater nuance here that might lead to more informed priors being appropriate.

Oh I see, thanks. Seems like its not so simple then to do a FE model with Bayesian. Wide Uniform priors would essentially be very similar to the frequentist approach

I apologize, I was too pessimistic in my response. The key motivating idea for a fixed effects model is that the intercept can freely correlate with the time-invariant variables. Mainly I wanted to warn against a typical Bayesian approach of random intercepts with a hierarchical prior, which assumes no correlation by default. I assume independent, informative priors would not be a problem.

The sometimes problematic assumption in hierarchical models of absence of correlation between random effects and fixed effects (which may cause biased fixed effects estimated if violated) can sometimes be managed with a so-called within-between approach. See:

This should be straightforward to implement in brms or Stan.

I don’t think prior choice has any bearing on the concern of biased fixed effect estimates in hierarchical models when aforementioned violates is violated. But maybe I’m misunderstanding @simonbrauer 's point.

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