# STATA: Interpreting sigma_u, sigma_e in Random Effects Modells, Difference betwenn sigma_u in Random- and Fixed-Effects-Modells

I am struggling with a pretty pragmatic problem. Like the heading is indicating I am doing longitudinal analysis in STATA. Therefore I need to model a fixed-effects-modell and a random-effects-modell an compare them.

You can find the results listet below.The depend Variable is the job prestige of italian and turkish migrants in germany and the independent variables are the speaking and the writing skills (correlation >.60) in another modell I also add the age at immigration as a timeinvariant variable.

Now I have to compare these two modells, which is okay, but there is point which is overhelming me: the sigma values in the random-effects-modell and the comparisn of sigma_u between random- and fixed-effects.

I know that rho in context of the random-effects-modell indicates the estimated proportion of the between-Variance at the total variance. It is calculated like this: sigma_u/sigma_u+ sigma_e

So sigma_u in the random-effects-modell has to be the between variance. But what is sigma_u in the fixed-effects-modell? And what is sigma_e in the random effects modell? Or how do I interpret them? Another smaller problem is, that the Fu test, the corr(u_i,Xb) and the Hausman test indicate that I have to use the fixed-effects-modell. But actually its not significant. Could a conclusion be that there are timeinvariant variables missing in the modell, so that the fixed-effects-modell is not significant but reliable while the random-effects-modell is significant but distorted?

Best wishes, Marcel

## 1 Answer

The online Stata documentation provides some answers to your questions: http://www.stata.com/manuals13/xtxtreg.pdf : p.8 introduces the notations, p.13 an example for the fixed-effect (FE) model, and p.16 for the random-effect (RE) model.

If the model is: $$y_{it} = X_{it}\beta + \eta_i + \varepsilon_{it}$$ Whether in the FE or RE model, $\sigma_u$ is the standard deviation of the time-invariant individual-specific term $\eta_i$, and $\sigma_e$ the standard deviation of the error term $\varepsilon_{it}$. In both case, your results suggest that a substantial part of the variance of the outcome (or, more precisely, what is left after you control for the observables) is due to unobserved heterogeneity across migrants.

About your second question, while I'm 100% sure to have understood you, I would say that, indeed, it seems preferable in your case to grant more trust to the FE model, especially if the tests you mention indicate to do so. Note that the FE model requires, in general, milder assumptions than the RE model, in which you have to specify the distribution for the $\eta$ as well as the way it co-varies with the observables. No such assumption in the FE model but this might come at the cost of precision.

• Nice answer, but I think the penultimate sentence needs some attention. – Dimitriy V. Masterov Sep 23 '16 at 18:26
• Thanks for the kind answer Roland! But I have one more question according to σu in FE and RE model. I found something in literature: - sigma_u in FE: mean variation of person-specific means. It's constant. In this case the mean difference of the unit specific mean from the total mean of the job prestige is 12.114 units. - sigma_u in RE: variation of the residual unit effects (=the not through independent variables explained difference between the unit specific means). So I can not really compare them between the models, but sigma_u in the RE should get smaller when variables are added? – Marcel_G Sep 25 '16 at 20:55
• In FE models, the covariates that are not varying over time are simply not identified. In RE models, it must be true that sigma_u is non-increasing when covariates are added, pretty much like the sum of squares of residuals. – Roland Sep 26 '16 at 7:06
• @Roland I think the second FE should be RE. – Dimitriy V. Masterov Sep 26 '16 at 15:10