Here, DT wei = [(Dx,Txei)0, zei] can be viewed as a vector of detrended regressors.
Thus, Assumption 10 indicates non-zero correlations between the effect ui and
detrended regressors. The term WiDTλ can be replaced by λo + wiDDTλ, where
λo is any constant scalar. We use the term w'iDTλ instead of λo+wiDTλ simply
for convenience.
A sequence of local versions of the fixed effects hypothesis is given:
Assumption 11 (local alternatives to random effects): Conditional on Fw, the
sequence {ui}i=1 N is i.i.d. with mean wiiDTλ∕∖∕N, variance σ2l, and κ4u =
EFw (ui
EFw ui)4 < ∞, where λ 6= 0(k+g)×1 is a nonrandom vector in Rk+g .
Under this Assumption, E (DTwiui) = √√= E DTwWiW0DT^ λ → 0(k+g)×ι, as
(N, T →∞).
Although Assumptions 10 and 11 are convenient to analyze the power prop-
erties of the Hausman test, they are somewhat restrictive. Specifically, under
these alternative hypotheses, the Hausman test lacks power to detect correla-
tions between the effect ui and the time invariant regressors zi . To see this,
suppose we partition λ into (λ0x , λ0z)0 corresponding to wei = (xe0i , zei0)0. Assume
that λx = 0k×1; that is, the xit (t =1, ..., T) are not correlated with ui, con-
ditional on zi . For simplicity, assume that T is fixed. For this case, under the
fixed effects assumption, the between estimators of β and γ, βb and γbb , are
equivalent to least squares on the model
Vi = β'xi + (γ + λz )'zi + (u* + ¾),
where u* = ui - wiDTλ = ui- λ'zei. From this, we can easily see that bb and
γbb are asymptotically unbiased estimators of β and (γ + λz), respectively. That
is, γbb is not an asymptotically unbiased estimator of γ . As we have discussed
in Section 2.2, the asymptotic distribution of the Hausman statistic depends on
that of βb , not of γbb . Thus, the Hausman test does not have power to detect
the violations of the random effects assumption that do not bias βb (regardless
of the size of T ). Accordingly, under our fixed effects and the local alternative
assumptions (Assumptions 10 and 11), the Hausman test possesses no power to
detect nonzero correlations between zi and ui . This problem arises of course
because we assume that the conditional mean of the effect Ui is a linear function
of wi. When the conditional mean of the effect is a nonlinear function of wi,
the Hausman test can possess power to detect nonzero correlations between ui
and zi.17
The following lemmas provide some results that are useful to derive the
asymptotic distributions of the within, between, and GLS estimators of β and
γ.
17 Even if the conditional mean of ui is linear in wei , the Hausman test may have power
to detect non-zero λz , if λx and λz are not functionally independent. For example, consider
a model with scalar xit and zi . Suppose that xit and zi have a common factor fi ; that is,
xit = fi + eit and zi = fi + ηi. (This is the case discussed below Assumption 5.) Assume
E(ui | fi, ηi,⅛) = c¾∙ Assume that fi, ηi and eit are normal, mutually independent, and
i.i.d. over different i and t with zero means, and variances σf2 , σ2η ,and σe2 , respectively. Note
that under given assumptions, xit is not correlated with ui, while zi is. For this case, however,
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