It is very unlikely that you would ever want to choose between these two extremes. You would likely only go with a BE model if 90%+ of the variance (as indexed by the intraclass correlation coefficient) is between units (just pulling a number out of thin air here).
Alternatively, you can use a RE model to get both the within and between coefficients in the same model.
For each predictor that varies within clusters, calculate the mean of that variable for each cluster:
bysort cluster: egen cmn_x = mean(x)
Then run the RE model:
xtreg y x cmn_x, i(cluster) re
The coefficient for cmn_x is the difference in size of the coefficient of the between effect relative to the within effect. The test of significance for cmn_x tells you whether the between effect is different from the within effect. You can easily recover the pure between effect:
lincom x + cmn_x
As an exmaple, we can use the admissions.dta from the Stata 15 manual, which has data on admissions from a set of departments within a university:
use http://www.stata-press.com/data/r15/admissions.dta, clear
describe
Contains data from http://www.stata-press.com/data/r15/admissions.dta
obs: 46 Graduate school admissions data
vars: 5 25 Feb 2016 09:28
size: 644 (_dta has notes)
------------------------------------------------------------------------------------------
storage display value
variable name type format label variable label
--------------------------------------------------------------------------------------
department long %8.0g dept department id
nadmitted byte %8.0g number of admissions
napplied float %9.0g number of applications
female byte %8.0g =1 if female, =0 if male
The between effect model predicting # of admissions from # applied:
xtreg nadmitted napplied, i(department) be
Between regression (regression on group means) Number of obs = 46
Group variable: department Number of groups = 23
R-sq: Obs per group:
within = 0.7392 min = 2
between = 0.4398 avg = 2.0
overall = 0.5302 max = 2
F(1,21) = 16.49
sd(u_i + avg(e_i.))= 7.859423 Prob > F = 0.0006
------------------------------------------------------------------------------
nadmitted | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
napplied | .180098 .0443518 4.06 0.001 .0878634 .2723326
_cons | 7.719926 2.586917 2.98 0.007 2.340136 13.09971
------------------------------------------------------------------------------
Now the within effect:
xtreg nadmitted napplied, i(department) fe
Fixed-effects (within) regression Number of obs = 46
Group variable: department Number of groups = 23
R-sq: Obs per group:
within = 0.7392 min = 2
between = 0.4398 avg = 2.0
overall = 0.5302 max = 2
F(1,22) = 62.37
corr(u_i, Xb) = -0.2430 Prob > F = 0.0000
------------------------------------------------------------------------------
nadmitted | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
napplied | .2428959 .0307572 7.90 0.000 .1791093 .3066825
_cons | 4.88583 1.594119 3.06 0.006 1.579828 8.191831
-------------+----------------------------------------------------------------
sigma_u | 8.0368968
sigma_e | 5.3163754
rho | .69561491 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(22, 22) = 4.30 Prob > F = 0.0006
Now put them together in the RE model (switching to mixed for REML b/c of small sample size), first calculating the cluster mean of napplied:
bysort department: egen cmn_app=mean(napplied)
mixed nadmitted napplied cmn_app || department:, reml
Mixed-effects REML regression Number of obs = 46
Group variable: department Number of groups = 23
Obs per group:
min = 2
avg = 2.0
max = 2
Wald chi2(2) = 78.85
Log restricted-likelihood = -160.93643 Prob > chi2 = 0.0000
------------------------------------------------------------------------------
nadmitted | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
napplied | .2428959 .0307572 7.90 0.000 .1826128 .3031789
cmn_app | -.0627979 .053973 -1.16 0.245 -.1685831 .0429873
_cons | 7.719926 2.586918 2.98 0.003 2.64966 12.79019
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Random-effects Parameters | Estimate Std. Err. [95% Conf. Interval]
-----------------------------+------------------------------------------------
department: Identity |
var(_cons) | 47.63861 19.5332 21.3276 106.4085
-----------------------------+------------------------------------------------
var(Residual) | 28.26384 8.521869 15.65252 51.03618
------------------------------------------------------------------------------
LR test vs. linear model: chibar2(01) = 10.87 Prob >= chibar2 = 0.0005
Recover the pure between effect:
lincom napplied+cmn_app
( 1) [nadmitted]napplied + [nadmitted]cmn_app = 0
------------------------------------------------------------------------------
nadmitted | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | .180098 .0443518 4.06 0.000 .0931701 .2670259
------------------------------------------------------------------------------
There you have it! One model to get both effects and a test of whether the between effect is different than the within effect.
