Fixed and Random Effects in Classical and Bayesian Regression*

This paper proposes a common and tractable framework for analyzing fixed and random effects models, in particular constant‐slope variable‐intercept designs. It is shown that, regardless of whether effects (i) are treated as parameters or as an error term, (ii) are estimated in different stages of a hierarchical model, or whether (iii) correlation between effects and regressors is allowed, when the same prior information on idiosyncratic parameters is introduced into all estimation methods, the resulting common slope estimator is also the same across methods. These results are illustrated using the Grünfeld investment data with different prior distributions. Random effects estimates are shown to be more efficient than fixed effects estimates. This efficiency gain, however, comes at the cost of neglecting information obtained in the computation of the prior unknown variance of idiosyncratic parameters.