## Specification Tests

The TSCSREG procedure outputs the results of one specification test
for fixed effects and one specification test for random effects.

For fixed effects, let be the *n* dimensional
vector of fixed effects parameters. The specification test reported
is the conventional *F*-statistic for the hypothesis
.
The *F*-statistic with *n*, *M*-*K* degrees of freedom is computed as

where is the estimated
covariance matrix of the fixed effects parameters.

Hausman's (1978) specification test or *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 *m* is distributed with *k* degrees of freedom, where
*k* 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 statistic has degrees
of freedom equal to the number of slope parameters.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.