Degrees of freedom

[Jo Smith]

Dear all,
Under random effect model, the degrees of freedom is higher than fixed effect model.
Could you please confirm this as I am not sure- I found contradict views in econometric books.

[Clyde Schechter]

Show us the actual commands you ran and the actual output. There are lots of different degrees of freedom floating around in these models, and we can't guess which ones you're thinking about.

[Jo Smith]

Thanks Clyde, not sure if the tables will be clear for you- if not do not bother yourself with them.



xtreg lnauditfees b_ind rd b_meet b_size gend ac_ind ac_exp ac_meet ac_size non_fin_ins fin_ins family gov for_arb for_non_arb logsize lev cox loss risk roa big4 naf industry,

Random-effects GLS regression Number of obs = 690
Group variable: code Number of groups = 115

R-sq: within = 0.2364 Obs per group: min = 6
between = 0.7252 avg = 6.0
overall = 0.6963 max = 6

Wald chi2(24) = 448.61
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000

------------------------------------------------------------------------------
lnauditfees | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
b_ind | .302186 .107875 2.80 0.005 .0907549 .5136172
rd | -.0199833 .040691 -0.49 0.623 -.0997362 .0597697
b_meet | -.001416 .0046842 -0.30 0.762 -.0105969 .0077648
b_size | -.0067043 .0114312 -0.59 0.558 -.0291091 .0157005
gend | -.8902825 .2597646 -3.43 0.001 -1.399412 -.3811531
ac_ind | .3814208 .08931 4.27 0.000 .2063763 .5564653
ac_exp | .2410257 .0868015 2.78 0.005 .0708979 .4111535
ac_meet | -.0069895 .0097133 -0.72 0.472 -.0260272 .0120482
ac_size | .033932 .0609961 0.56 0.578 -.0856181 .153482
non_fin_ins | .1351616 .0972454 1.39 0.165 -.0554359 .3257591
fin_ins | .4740966 .1885661 2.51 0.012 .1045138 .8436795
family | .359431 .1309714 2.74 0.006 .1027318 .6161303
gov | 4.884256 .8961586 5.45 0.000 3.127817 6.640694
for_arb | .3369156 .1425939 2.36 0.018 .0574367 .6163946
for_non_arb | .6085307 .2355979 2.58 0.010 .1467672 1.070294
logsize | .1637895 .0228462 7.17 0.000 .1190117 .2085673
lev | .083703 .1087267 0.77 0.441 -.1293973 .2968034
cox | .0662836 .0101124 6.55 0.000 .0464637 .0861035
loss | -.0296293 .0190517 -1.56 0.120 -.0669699 .0077113
risk | -.0510675 .0664706 -0.77 0.442 -.1813474 .0792124
roa | -.2085926 .0945564 -2.21 0.027 -.3939199 -.0232654
big4 | .2519239 .0358092 7.04 0.000 .1817391 .3221086
naf | -.0864392 .1006981 -0.86 0.391 -.2838038 .1109253
industry | .0901792 .0781678 1.15 0.249 -.0630269 .2433853
_cons | 5.552787 .4218723 13.16 0.000 4.725932 6.379641
-------------+----------------------------------------------------------------
sigma_u | .36706632
sigma_e | .16508737
rho | .83175761 (fraction of variance due to u_i)

------------------------------------------------------------------------------------------------------------------------------------------------------
------------------------------------------------------------------------------------------------------------------------------------------------------

xtreg lnauditfees b_ind rd b_meet b_size gend ac_ind ac_exp ac_meet ac_size non_fin_ins fin_ins family gov for_arb for_non_arb logsize lev cox loss risk roa big4 naf industry,fe
note: gov omitted because of collinearity
note: industry omitted because of collinearity

Fixed-effects (within) regression Number of obs = 690
Group variable: code Number of groups = 115

R-sq: within = 0.2492 Obs per group: min = 6
between = 0.4858 avg = 6.0
overall = 0.4714 max = 6

F(22,553) = 8.34
corr(u_i, Xb) = 0.1415 Prob > F = 0.0000

------------------------------------------------------------------------------
lnauditfees | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
b_ind | .2441196 .1171403 2.08 0.038 .0140253 .4742139
rd | -.0060516 .0457752 -0.13 0.895 -.0959662 .0838629
b_meet | -.0012673 .004839 -0.26 0.794 -.0107724 .0082377
b_size | -.007041 .0138119 -0.51 0.610 -.0341712 .0200893
gend | -.8503811 .299931 -2.84 0.005 -1.439524 -.2612378
ac_ind | .4484453 .1033747 4.34 0.000 .2453903 .6515003
ac_exp | .2518815 .1060375 2.38 0.018 .0435959 .4601671
ac_meet | -.004019 .0100093 -0.40 0.688 -.0236798 .0156418
ac_size | .0223809 .0705478 0.32 0.751 -.1161935 .1609554
non_fin_ins | .1385919 .1121531 1.24 0.217 -.0817062 .35889
fin_ins | .3893729 .3096154 1.26 0.209 -.2187932 .997539
family | .5291918 .170071 3.11 0.002 .1951276 .863256
gov | 0 (omitted)
for_arb | .2077524 .1728956 1.20 0.230 -.13186 .5473649
for_non_arb | -.1806559 .3569658 -0.51 0.613 -.8818306 .5205189
logsize | .1223611 .0298772 4.10 0.000 .0636745 .1810477
lev | .062224 .1169378 0.53 0.595 -.1674727 .2919206
cox | .0586556 .0165783 3.54 0.000 .0260915 .0912198
loss | -.0351273 .0193473 -1.82 0.070 -.0731305 .0028759
risk | -.0713014 .0766145 -0.93 0.352 -.2217924 .0791897
roa | -.1898521 .0959901 -1.98 0.048 -.378402 -.0013023
big4 | .2434706 .0389201 6.26 0.000 .1670212 .31992
naf | -.0259267 .1834614 -0.14 0.888 -.3862931 .3344397
industry | 0 (omitted)
_cons | 6.379406 .5603164 11.39 0.000 5.278797 7.480014
-------------+----------------------------------------------------------------
sigma_u | .49697455
sigma_e | .16508737
rho | .90061959 (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(114, 553) = 30.83 Prob > F = 0.0000

[Sergio Correia]

I'm not an expert in mixed/random models, but isn't it actually the opposite?
You lose DoF because you have to treat fixed effects as parameters to estimate.

Also, xtreg_re doesn't return e(df_r) so I'm not sure where do you get the degrees-of-freedom from?

[Clyde Schechter]

No mystery here. Stata tells you quite explicitly what's going on. You have 24 predictors specified in your commands for each model. In the -re- model, all 24 are used. And that is the number of model df reported by Stata. But in the output of the -fe- model, Stata tells you

note: gov omitted because of collinearity
note: industry omitted because of collinearity

The collinearity referred to is collinearity with the fixed effects. That is, at least within the estimation sample, gov and industry do not vary within panels. So they cannot be included in the model, and Stata drops them.

Anyway, with those two variables omitted from the estimation of the -fe- model, you are left with 24-2 = 22 predictors in use, and, just as you should expect, Stata reports 22 model degrees of freedom.

By the way, the inability of fixed-effects models to estimate the effects of variables that are constant within panels is one of its limitations, and until they get a lot of experience with this, people tend to stumble over this and be surprised when it bites them. The random-effects model is subject to no such constraint. But, as usual, there is no free lunch. The random-effects estimator is only consistent under additional assumptions not required for fixed-effects estimation, and when those assumptions are violated, the estimates of those effects that don't vary within panel are especially questionable.

[Jo Smith]

highly appreciated Clyde.
from your comment above you gave me a hint about a point I am struggling with.
A variable (foreign ownership ratio) when I collected my data manually I realised it slightly changes within the firms and it changes between firms. Do you think this is the most likely reason behind that this variable show insignificant correlation with outcome variables under FE model - while it shows significant relationship when I use RE model.

Many thanks for you.

[Clyde Schechter]

Yes, I think so. The fixed-effects regression estimates the within-firm associations between predictors and outcomes. The random effects regression assumes that the association is the same between firms as within firms and gives an estimate of that common coefficient. There is also a between estimator that estimates the association purely between firms. The random effects estimator is a weighted combination of the between and within.estimators.

If foreign ownership (I have to say I have no idea which of your variable names corresponds to this: there are two that begin with for_ but neither one sounds like ownership to me) varies very little within firms, then its within-firm relationship to any outcome will be close to zero. But that by no means precludes the possibility that between firms it is strongly associated with your outcome. However, if that is the case, your random effects model is probably invalid because, as I said, it presumes that the within- and between-firm associations to outcome are the same.

Another important point: in judging this you should not be relying on statistical significance results. Just because one thing is statistically significant and another is not does not imply that the two are different from each other in the sense of statistical significance. So in deciding whether your fixed and random effects estimates of the effect of foreign ownership are similar, you should be looking at the coefficients from the models and asking whether those look close or not. (Actually it's a little more complicated. You need to compare apples to apples, and since your fixed effects model dropped two variables, you really need to re-estimate the random effects model omitting the two variables that were dropped and then compare the foreign ownership coefficient of that model to the fixed-effects model's coefficient of foreign ownership.)

More generally, as noted earlier, consistency of the random effects estimator relies on additional assumptions not needed for consistency of the fixed effects estimator. So you probably ought to do a Hausman test to check that. If the Hausman test says the re model is inconsistent, you'd best not rely on it. If you specifically need an estimate of the effect of foreign ownership, and if that varies too little within firms to be of use, then you should use the between estimator for that purpose.

[Jo Smith]

Thanks very much Clyde.
Your explanation is very clear.
The Hausman test as you advised indicates that random is more appropriate. In this case, is validity of random still questionable? Or I can go ahead with my justification of the variable(foreign ownership) as mentioned earlier.

Thank you very much

[Clyde Schechter]

If the Hausman test suggests that random is OK, then go with it.

[Jo Smith]

Thanks very much Dr Clyde.