# Re: st: Basic question on Hausman test.

 From Mark Schaffer To statalist@hsphsun2.harvard.edu, Lucio Vinhas de Souza Subject Re: st: Basic question on Hausman test. Date Sat, 27 Sep 2003 13:05:20 +0100 (BST)

```Lucio,

The null is that the two estimation methods are both OK and that therefore
they should yield coefficients that are "similar".  The alternative
hypothesis is that the fixed effects estimation is OK and the random
effects estimation is not; if this is the case, then we would expect to
see differences between the two sets of coefficients.

This is because the random effects estimator makes an assumption (the
random effects are orthogonal to the regressors) that the fixed effects
estimator does not.  If this assumption is wrong, the random effects
estimator will be inconsistent, but the fixed effects estimator is
unaffected.  Hence, if the assumption is wrong, this will be reflected in
a difference between the two set of coefficients.  The bigger the
difference (the less similar are the two sets of coefficients), the bigger
the Hausman statistic.

A large and significant Hausman statistic means a large and significant
difference, and so you reject the null that the two methods are OK in
favour of the alternative hypothesis that one is OK (fixed effects) and
one isn't (random effects).

Your Hausman stat is very big, and you can see why - the differences
between some of the coefficients are big enough to be visible to the naked
eye, so to speak - and so you can reject random effects as inconsistent
and go with fixed effects instead.

BTW, xthausman after random effects will do the test for you in one step.

Cheers,
Mark

Quoting Lucio Vinhas de Souza <lvdesouza@yahoo.com>:

> Dear all,
>
> I have a very basic question concerning a Hausman
> test. I am comparing a fixed effects panel estimation
> with a random effects one (see below). How do I
> interpret the results of the Hausman test? Do they
> mean that the random effects estimates are
> inconsistent?
>
>
> Lucio Vinhas de Souza
> **************************************
> . xtreg ltrade  lgdp lpop eud emud trend, fe
>
> Fixed-effects (within) regression               Number
> of obs      =     57442
> Group variable (i) : ipair                      Number
> of groups   =      2611
>
> R-sq:  within  = 0.1548                         Obs
> per group: min =        22
>        between = 0.3077
>         avg =      22.0
>        overall = 0.2112
>         max =        22
>
> F(5,54826)         =   2008.23
> corr(u_i, Xb)  = 0.2545
> Prob > F           =    0.0000
>
> -------------------------------------------------------
>       ltrade |      Coef.   Std. Err.      t    P>|t|
>    [95% Conf. Interval]
> -------------+-----------------------------------------
>        lgdp |   .0754704   .0292365     2.58   0.010
>   .0181668    .1327741
>         lpop |   .5473182   .1313844     4.17   0.000
>    .2898038    .8048326
>          eud |  -.2723743   .0951406    -2.86   0.004
>   -.4588506    -.085898
>         emud |  -.9780319   .1085947    -9.01   0.000
>   -1.190878   -.7651856
>        trend |   .1153878   .0018864    61.17   0.000
>    .1116905    .1190851
>        _cons |  -10.33135   2.421705    -4.27   0.000
>   -15.07791   -5.584793
> -------------+-----------------------------------------
>      sigma_u |  2.9860951
>      sigma_e |  1.8353774
>          rho |   .7258032   (fraction of variance due
> to u_i)
> -------------------------------------------------------
> F test that all u_i=0:     F(2610, 54826) =    45.08
>       Prob > F = 0.0000
>
> . hausman, save
>
> . xtreg ltrade  lgdp lpop eud emud trend
>
> Random-effects GLS regression                   Number
> of obs      =     57442
> Group variable (i) : ipair                      Number
> of groups   =      2611
>
> R-sq:  within  = 0.1537                         Obs
> per group: min =        22
>        between = 0.3468
>         avg =      22.0
>        overall = 0.2963
>         max =        22
>
> Random effects u_i ~ Gaussian                   Wald
> chi2(6)       =  11354.00
> corr(u_i, X)       = 0 (assumed)                Prob >
> chi2        =    0.0000
>
> -------------------------------------------------------
>       ltrade |      Coef.   Std. Err.      z    P>|z|
>    [95% Conf. Interval]
> -------------+-----------------------------------------
>        lgdp |   .2138072    .026484     8.07   0.000
>   .1618996    .2657149
>         lpop |   1.477494   .0498542    29.64   0.000
>    1.379781    1.575206
>          eud |   .0097496   .0884326     0.11   0.912
>   -.1635752    .1830744
>         emud |  -1.025233   .1084758    -9.45   0.000
>   -1.237842   -.8126247
>        trend |   .1032162    .001403    73.57   0.000
>    .1004664     .105966
>        _cons |  -25.08318   1.038565   -24.15   0.000
>   -27.11873   -23.04763
> -------------+-----------------------------------------
>      sigma_u |  2.5927197
>      sigma_e |  1.8353942
>          rho |  .66616628   (fraction of variance due
> to u_i)
> -------------------------------------------------------
>
> . hausman
>
>  ---- Coefficients ----
>        (b)        (B)        (b-B)
> qrt(diag(V_b-V_B))
>  |     Prior       Current  Difference    S.E.
> -------------+-----------------------------------------
> lpop | .5473182     1.477494     -.9301754
> .1215583
> eud |  -.2723743     .0097496    -.2821239
> .0350914
> emud |  -.9780319    -1.025233    .0472016
> .0050788
> trend |   .1153878     .1032162   .0121716
> .001261
> -------------------------------------------------------
> b= less efficient estimates obtained previously from
> xtreg
> B= fully efficient estimates obtained from xtreg
>
> Test:  Ho:  difference in coefficients not systematic
> chi2(  5) = (b-B)'[(V_b-V_B)^(-1)](b-B)=   167.24
> Prob>chi2 =     0.0000
>
>
>
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Prof. Mark Schaffer
Director, CERT
Department of Economics
School of Management & Languages
Heriot-Watt University, Edinburgh EH14 4AS
tel +44-131-451-3494 / fax +44-131-451-3008
email: m.e.schaffer@hw.ac.uk
web: http://www.sml.hw.ac.uk/ecomes

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