The HPPANEL procedure outputs one specification test for random effects: the Hausman (1978) specification test (m statistic) can be used to test hypotheses in terms of bias or inconsistency of an estimator. This test was also proposed by Wu (1973) and further extended in Hausman and Taylor (1982). Hausman’s m statistic is as follows.
Consider two estimators, and
, which under the null hypothesis are both consistent, but only
is asymptotically efficient. Under the alternative hypothesis, only
is consistent. The m statistic is
where and
are consistent estimates of the asymptotic covariance matrices of
and
. Then
is distributed as
with
degrees of freedom, where
is the dimension of
and
.
In the random-effects specification, the null hypothesis of no correlation between effects and regressors implies that the
OLS estimates of the slope parameters are consistent and inefficient but the GLS estimates of the slope parameters are consistent
and efficient. This facilitates a Hausman specification test. The reported degrees of freedom for the statistic are equal to the number of slope parameters. If the null hypothesis holds, the random-effects specification should
be used.