where θτ = ʌ/ɪr^/(TσU + σ2v). The variance-covariance matrix of this estimator
is given:
V ar(bδg) = σv2 [Pi,t weitwei0t +Tθ2TPiweiwei0]-1. (4)
For notational convenience, we assume that σ2u and σ2v are known, while in
practice they must be estimated.4
An important advantage of the GLS estimator over the within estimator is
that it allows researchers to estimate γ . In addition, the GLS estimator of β
is more efficient than the within estimator of β, because [Var(βw) - Var(βg)]
is positive definite so long as θT > 0.5 Despite these desirable properties, it is
important to notice that the consistency of the GLS estimator crucially depends
on the RE assumption (3). Accordingly, the legitimacy of the RE assumption
should be tested to justify the use of GLS. In the literature, a Hausman test
(1978) has been widely used for this purpose. The statistic used for this test is
a distance measure between the within and GLS estimators of β:
HMNT ≡ (βbw - βbg)0[V ar(βbw) - V ar(βbg)]-1(βbw - βbg). (5)
For the cases in which T is fixed and N →∞, the RE assumption warrants that
the Hausman statistic HMNT is asymptotically χ2 -distributed with degrees of
freedom equal to k. This result is a direct outcome of the fact that for fixed T,
the GLS estimator βg is asymptotically more efficient than the within estimator
βbw, and that the difference between the two estimators is asymptotically normal;
specifically, as N →∞,
NTβ(βw — βg) =⇒ N(0,plimN→∞NT[Vαr(bw) - Var(bg)]), (6)
where u=⇒, means “converges in distribution.”
An important condition that guarantees (6) is that θT > 0. If θT = 0, then
the within and GLS estimators become identical and the Hausman statistic is
not defined. Observe now that θT → 0asT →∞. This observation naturally
raises several issues related with the asymptotic properties of the Hausman test
as T →∞. In order to clarify the nature of the problem, consider model (1),
but without the time-invariant regressors and the overall intercept term (ζ).
Assume that xit contains a single time-varying regressor which is independently
and identically distributed over different i and t. For this simple model, we
can easily show plimN,T →∞N T V ar(β w) = plimN,T→∞NTVar(β g), using the
fact that θT → 0asT →∞. This asymptotic equality immediately implies
that the within and GLS estimators ofβ are asymptotically equivalent; that
is, plimN,τ →∞ N NT (βw — bg ) = 0k×ι. This preliminary finding raises several
questions. First, does this equivalence result hold for the general cases with
4 The conventional estimates are given:
σ2v = Pi,t(yu - ⅛)2∕[n(t -1)];bU = Pi(ei - Wibb)2∕N -92v/τ.
5 This efficiency gain of course results from the fact that the GLS estimator utilizes between
variations in x⅛.