Hi.
I am puzzled by the results I get from running Breusch Pagan and
Hausman tests on my (unbalanced) panel data sample.
The results of the tests are quite sensitive to the correction applied
to the standard errors, and somehow weird (at least, to me).
The regressors I use are a rate of growth (difa142), and two
interaction terms, that is in2t=var*difa142 and in2t2=(var^2)*difa142
I write some comments and questions between the results. I believe
that these results exhibit a good deal of the problems discussed
separately by listservers. However, being them all together, I am not
sure how to interpret the results and how to procede.
I estimate xtreg difa298 difa142 in2t in2t2, re cluster(co)
Random-effects GLS regression Number of obs = 772
Group variable (i): co Number of groups = 73
R-sq: within = 0.3139 Obs per group: min = 7
between = 0.6914 avg = 10.6
overall = 0.3632 max = 11
Random effects u_i ~ Gaussian Wald chi2(3) = 97.53
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
(Std. Err. adjusted for 73 clusters in co)
------------------------------------------------------------------------------
| Robust
difa298 | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
difa142 | 1.010826 .1579594 6.40 0.000 .7012313 1.320421
in2t | -4.775301 2.948644 -1.62 0.105 -10.55454 1.003935
in2t2 | 24.42007 14.02081 1.74 0.082 -3.060215 51.90035
_cons | -.0658962 .1802725 -0.37 0.715 -.4192238 .2874315
-------------+----------------------------------------------------------------
sigma_u | 0
sigma_e | 6.3054774
rho | 0 (fraction of variance due to u_i)
------------------------------------------------------------------------------
I run BP xttest0
Breusch and Pagan Lagrangian multiplier test for random effects:
difa298[co,t] = Xb + u[co] + e[co,t]
Estimated results:
| Var sd = sqrt(Var)
---------+-----------------------------
difa298 | 60.12951 7.754322
e | 39.75904 6.305477
u | 0 0
Test: Var(u) = 0
chi2(1) = 5.04
Prob > chi2 = 0.0248
The BP test suggests the existence of an individual effect even if
sigma_u is 0. Why?
I believe that a RE with rho=0 is an OLS!
(Btw, I checked for serial correlation with -xtserial- and serial
autocorrelation is rejected)
The Hausman test (i.e. hausman fixed random) does not produce proper
results because the difference of the variances is not positive
definite.
Test: Ho: difference in coefficients not systematic
chi2(3) = (b-B)'[(V_b-V_B)^(-1)](b-B)
= -0.02 chi2<0 ==> model fitted on these
data fails to meet the asymptotic
assumptions of the Hausman test;
see suest for a generalized test
I follow Wooldridge's suggestion and calculate the Hausman test by hand.
I add to the regressors the averages over time of the time-varying
regressors and test their significance in a random effects model.
In the following case, I use the standard errors corrected by means of
the -cluster- option.
xtreg difa298 difa142 in2t in2t2 adifa142 ain2t ain2t2, re cluster(co)
Random-effects GLS regression Number of obs = 772
Group variable (i): co Number of groups = 73
R-sq: within = 0.3140 Obs per group: min = 7
between = 0.7037 avg = 10.6
overall = 0.3649 max = 11
Random effects u_i ~ Gaussian Wald chi2(6) = 468.98
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
(Std. Err. adjusted for 73 clusters in co)
------------------------------------------------------------------------------
| Robust
difa298 | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
difa142 | .9954723 .1916506 5.19 0.000 .6198441 1.3711
in2t | -4.693986 3.283152 -1.43 0.153 -11.12884 1.740873
in2t2 | 23.39439 14.53121 1.61 0.107 -5.086252 51.87504
adifa142 | .0635942 .2096473 0.30 0.762 -.3473069 .4744954
ain2t | -2.534238 5.876584 -0.43 0.666 -14.05213 8.983655
ain2t2 | 29.47941 31.16784 0.95 0.344 -31.60843 90.56725
_cons | -.0055123 .181651 -0.03 0.976 -.3615417 .3505171
-------------+----------------------------------------------------------------
sigma_u | 0
sigma_e | 6.3054774
rho | 0 (fraction of variance due to u_i)
------------------------------------------------------------------------------
Individually, the averages adifa142, ain2t, ain2t2 are massively insignificant.
However, the wald test below reject the hypothesis of them being jointly 0.
testparm adifa142 ain2t ain2t2
( 1) adifa142 = 0
( 2) ain2t = 0
( 3) ain2t2 = 0
chi2( 3) = 7.52
Prob > chi2 = 0.0570
This would be in favour of a fixed effect model (at almost 5% sig.lev.).
If I run the same r.e. model correcting with robust instead of cluster, I get
xtreg difa298 difa142 in2t in2t2 adifa142 ain2t ain2t2, re robust
Random-effects GLS regression Number of obs = 772
Group variable (i): co Number of groups = 73
R-sq: within = 0.3140 Obs per group: min = 7
between = 0.7037 avg = 10.6
overall = 0.3649 max = 11
Random effects u_i ~ Gaussian Wald chi2(6) = 214.91
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
------------------------------------------------------------------------------
| Robust
difa298 | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
difa142 | .9954723 .1559118 6.38 0.000 .6898907 1.301054
in2t | -4.693986 3.173922 -1.48 0.139 -10.91476 1.526787
in2t2 | 23.39439 13.06566 1.79 0.073 -2.213824 49.00261
adifa142 | .0635942 .2219267 0.29 0.774 -.371374 .4985625
ain2t | -2.534238 7.728836 -0.33 0.743 -17.68248 12.614
ain2t2 | 29.47941 62.91061 0.47 0.639 -93.82311 152.7819
_cons | -.0055123 .2277465 -0.02 0.981 -.4518874 .4408627
-------------+----------------------------------------------------------------
sigma_u | 0
sigma_e | 6.3054774
rho | 0 (fraction of variance due to u_i)
------------------------------------------------------------------------------
Each of the three averages is individually insignificant and, this
time, they are also jointly insignificant.
( 1) adifa142 = 0
( 2) ain2t = 0
( 3) ain2t2 = 0
chi2( 3) = 0.39
Prob > chi2 = 0.9418
The result of this test are at odds with those reported above with the
correction -cluster-.
I reckon that in2t and in2t2 (and therefore their averages too) are
very closely correlated. However I do not see why this should affect
only the results of the wald tests after the -cluster- corrected
standard errors regressions.
Has it anything ro do with the fact that these variables are time
invariant and the cluster option sucks up all the degrees of freedom
at the group level?
As I will show below, the within estimation would suggest a very low
correlation between u_i and xb. Such findings seem to support a RE
model. In addition, the RE and FE models produce very similar
coefficients, as it can be seen below.
xtreg difa298 difa142 in2t in2t2 adifa142 ain2t ain2t2, fe cluster(co)
Fixed-effects (within) regression Number of obs = 728
Group variable (i): co Number of groups = 69
R-sq: within = 0.3705 Obs per group: min = 7
between = 0.7088 avg = 10.6
overall = 0.4242 max = 11
F(3,68) = 32.44
corr(u_i, Xb) = 0.0700 Prob > F = 0.0000
(Std. Err. adjusted for 69 clusters in co)
------------------------------------------------------------------------------
| Robust
difa300 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
difa142 | .9328421 .1833901 5.09 0.000 .5668929 1.298791
in2t | -4.626382 3.230345 -1.43 0.157 -11.07244 1.819671
in2t2 | 23.52781 14.22296 1.65 0.103 -4.853671 51.90928
_cons | -.1133588 .0220161 -5.15 0.000 -.1572912 -.0694264
-------------+----------------------------------------------------------------
sigma_u | 1.5468984
sigma_e | 5.3383765
rho | .07746212 (fraction of variance due to u_i)
------------------------------------------------------------------------------
In this model the estimated sigma_u is 1.54, not 0 as with the RE.
Should I use the FE measure of sigma_u and build the RE model by
quasi-demeaning the series by hand (and hence run the Hausman test by
adding the averages overt time as I have done before) ?
I would be extremely grateful if anyone could help me out interpreting
these results.
I am stuck since I can't choose which results to consider.
This is even worse when the BP test fails to reject the hypothesis of
no-individual effect, yet the hausman test is strongly in favour of a
within model.
many many thanks,
andrea
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