panel data - within-group estimate - individual fixed effects retrieved I am analyzing panel data.
First, I have to decide whether to use a random or fixed effect estimator.
The Hausman test suggests to use the fixed effect estimator (also named within group estimator). Thus, this is what I am using.
However, this method eliminates the individual fixed effects, that is, the Ui's, which is what I am more interested about. Hence, I proceed with a second step.
Second, I run a between-group estimator where I regress the predicted individual effects, that is the Ui's predicted from the first step, against a list of time invariant fixed effects I am interested in
My questions are three:


*

*I was taught this two-step method in a graduate summer course, and no book I have read mentions it, do you know what is its name?

*Is it meaningful to cluster the standard errors on individuals in the first step? (i.e. within-group estimator) 

*Is it meaningful to use a variable both as a covariate and as cluster for the standard error in the same model?

 A: In addition to Andy W's answer, the procedure that was suggested to you is similar to the Fixed Effects Vector Decomposition (FEVD) proposed by Plümber and Troeger (2007). It's not quite the same but very alike to their three-step method which goes as follows:


*

*estimate the unit fixed effects

*decompose the fixed effects into the time-invariant factors and an error term

*estimate 1. again by pooled OLS including the time-invariant variables and the error from 2.


This procedure was heavily criticized by Greene (2011) and Breusch et al. (2011) so I would be careful with such types of estimation strategies. The point about the lower/higher level effects mentioned by Andy W is one of the set of critique points in these two papers.
If it helps you, I have written another post in a related question on how to keep time-invariant variables in fixed effects regressions. I hope you will find this useful.
A: For 2, assuming that "individuals" are the cluster, no you shouldn't cluster the standard errors on the first step, and the same logic then extends to your question 3. For 1, this is sometimes called the between effects estimator in economics. See a Stata FAQ on it, and Snijders and Bosker's Multilevel modeling book has a pretty brief section explaining it as well.
That being said, I personally see no reason for it in favor of random effects modeling. Like Andrew Gelman says, "If you get to the point of asking, just do it." All the Hausman test tells you is if the between estimators are equal to the within estimators, which is not a terribly interesting question in and of itself. Most study designs should dictate the use of fixed effects or random effects, and here it appears you are really interested in the random effects.
