why is sigma u in xtreg output different from standard deviation of predicted u?

[Katarzyna Kopycka]

Dear Statalist,
my question concerns how xtreg calculates sigma u (and sigma e). My idea was that sigma u is the standard deviation of u. But when I predict the u with the predict postestimation command, and then ask for standard deviation using summarize I get a number different from sigma u in the xtreg output. Why?
Furthermore, when I calculate the ICC in an ANOVA fashion (code provided below), i.e. SSB/SSW+SSB I get a similar but still different value than rho in the xtreg output. Why?
I provide the data (which is generated data using rnormal) and a list of commands for replication of what I did.
Kind regards, Katarzyna

* Example generated by -dataex-. For more info, type help dataex
clear
input float result
  54.4773
 48.74695
 77.07542
  64.7085
 95.48249
 54.09767
 80.53448
 55.39391
 54.34201
 65.56054
 42.70115
 65.54634
 44.12673
 53.39897
 82.55605
 63.11832
 52.41323
 61.91088
 78.43945
 57.30145
 21.37305
 48.21943
 85.63551
 84.51431
33.619602
 69.19332
 63.55545
 62.24532
  73.2518
 60.65511
 60.29387
 53.87441
 42.28486
 64.07081
 67.38633
 48.02635
 43.04099
 93.99119
 58.77192
 88.25754
 55.99929
 45.92612
 60.34822
 71.17495
70.307594
 53.73354
 63.25698
 52.06905
 56.19698
 70.90791
end

sort res
gen school=1 in 1/20
replace school=2 in 21/35
replace school=3 in 36/50
*introduce some variability
foreach u in 19 16 13 {
replace school=2 in `u'
}
foreach u in 15 20 27 31 38 {
replace school=3 in `u'
}
foreach u in 21 26 {
replace school=1 in `u'
}

bysort sc: egen mean=mean(res)
egen grmean=mean(res)

gen sqdiff=(res-mean)^2
egen ssw=total(sqdif)

gen sqdiffgrmean=(mean-grmean)^2
egen ssb=total(sqdiffgr)

list ssb ssw in 1
display 6938.091/(6938.091+4138.738)

xtset sch
xtreg res, re
xtreg res, fe
predict uest, u
predict eest, e
sum uest eest

xtreg res, fe

Fixed-effects (within) regression               Number of obs     =         50
Group variable: school                          Number of groups  =          3

R-squared:                                      Obs per group:
     Within  =      .                                         min =         14
     Between = 0.0001                                         avg =       16.7
     Overall =      .                                         max =         19

                                                F(0, 47)          =       0.00
corr(u_i, Xb) =      .                          Prob > F          =          .

------------------------------------------------------------------------------
      result | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       _cons |   61.48227   1.327089    46.33   0.000     58.81252    64.15203
-------------+----------------------------------------------------------------
     sigma_u |  13.890912
     sigma_e |  9.3839355
         rho |  .68664295   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(2, 47) = 39.39                      Prob > F = 0.0000

. predict uest, u

. predict eest, e

. sum uest eest

    Variable |        Obs        Mean    Std. dev.       Min        Max
-------------+---------------------------------------------------------
        uest |         50   -4.29e-08    11.89932   -14.3086   13.46764
        eest |         50    2.59e-08    9.190431  -25.80062   20.53258

[Carlo Lazzaro]

Katarzyna:
the Methods and formula section of the -xtreg- entry in Stata .pdf manual can give you some clues on that.

[Erik Ruzek]

One thing to note is that you are running descriptive statistics on uest and it is giving you 50 observations. In your regression model, you only have three schools. This means that you only have 3 values of u. So to get the observed distribution of uest, I would use egen's tag function and re-run the summary command:

bysort school: egen pick1sch = tag(school)
sum uest if pick1sch==1

[Katarzyna Kopycka]

Thanks Erik, that's right of course.