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st: Problems with BP and Hausman tests when s.e. are cluster corrected


From   "andrea fracasso" <andrea.fracasso@gmail.com>
To   statalist@hsphsun2.harvard.edu
Subject   st: Problems with BP and Hausman tests when s.e. are cluster corrected
Date   Thu, 29 Jun 2006 08:13:15 +0200

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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