Large-N and Large-T Properties of Panel Data Estimators and the Hausman Test

Here, D_T we_i = [(D_x,Txe_i)⁰, ze_i] can be viewed as a vector of detrended regressors.
Thus, Assumption 10 indicates non-zero correlations between the effect u_i and
detrended regressors. The term WiDTλ can be replaced by λ_o + wiDDTλ, where
λ_o is any constant scalar. We use the term w'_iDTλ instead of λ_o+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 F_w, the
sequence {u_i}_i=₁ N is i.i.d. with mean wi_iDTλ∕∖∕N, variance σ2_l, and κ4_u =

EF_w ui)⁴ < ∞, where λ 6= 0(_k+g)_×1 is a nonrandom vector in R^k+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 (λ⁰_x , λ⁰_z)⁰ corresponding to wei = (xe⁰_i , ze_i⁰)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, β_b and γb_b , are
equivalent to least squares on the model

Vi = β'x_i + (γ + λz )'z_i + (u* + ¾),

where u* = u_i - wiDTλ = u_i- λ'_ze_i. From this, we can easily see that bb and
γb_b are asymptotically unbiased estimators of β and (γ + λz), respectively. That
is, γb_b 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 β_b , not of γb_b . Thus, the Hausman test does not have power to detect
the violations of the random effects assumption that do not bias β_b (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 w_i. When the conditional mean of the effect is a nonlinear function of w_i,
the Hausman test can possess power to detect nonzero correlations between u_i
and z_i.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 u_i is linear in we_i , 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 xi_t and zi . Suppose that xi_t and zi have a common factor fi ; that is,
xit = fi + eit and zi = fi + η_i. (This is the case discussed below Assumption 5.) Assume
E(ui | fi, ηi,⅛) = c¾∙ Assume that fi, η_i and eit are normal, mutually independent, and
i.i.d. over different i and t with zero means, and variances σ_f² , σ²_η ,and σ_e² , respectively. Note
that under given assumptions, xi_t is not correlated with ui, while zi is. For this case, however,

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