一篇关于 HAUSMAN TEST 为负的原因

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It is hard to discuss the Hausman test without being specific about how the
test is performed. Let B be the parameter estimates from a fully efficient
estimator (random-effects regression in this case) and b be the estimates from
a less efficient estimator (fixed-effects regression), but one that is
consistent in the face of one or more violated assumptions, in this case that
the effects are correlated with one or more of the regressors. If the
assumption is violated then we expect that the estimates from the two
estimators will not be the same, b~=B.

The Hausman test is essentially a Wald test that (b-B)==0 for all coefficients
where the covariance matrix for b-B is taken as the difference of the
covariance matrices (VCEs) for b and B. What is amazing about the test is
that we can just subtract these two covariance matrices to get an estimate of
the covariance matrix of (b-B) without even considering that the VCEs of the
two estimators might be correlated -- they are after all estimated on the same
data. We can just subtract, but only because the the VCE of the fully
efficient estimator is uncorrelated with the VCEs of all other estimators, see
Hausman and Taylor (1981), "panel data and unobservable individual effects",
econometrica, 49, 1337-1398). The VCE of the efficient estimator will also be
smaller than the less efficient estimator. Taken together, these results
imply that the subtraction of the two VCE (V_b-V_B) will be positive definite
(PD) and that we need not consider the covariance between the two VCEs.

These results, however, hold only asymtotically. For any given finite sample
we have no reason to believe that (V_b-V_B) will be PD. So, it is amazing
that we can just subtract these two matrices, but the price we pay is that we
can only do so safely if we have an infinite amount of data. The Hausman
test, unlike most tests, relies on asymptotic arguments not only for its
distribution, but for its ability to be computed! Let's discuss what we do
what we do when (V_b-V_B) in not PD in the context of Eric's results.

Aside: If anyone is interested in a Hausman-like test that drops the
assumption that either estimator is fully efficient, actually estimates the
covariance between the VCEs, and can always be computed, see Weesie (2000)
"Seemingly unrelated est. and cluster-adjusted sandwich estimator", STB
Reprints Vol 9, pp 231-248. The test unfortunately requires the scores from
the estimator, and -xtreg, fe- does not directly produce these.

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