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

where Γjl,i (t, s)=E xj,it ^- EFz_i xj,it xl,is ^- EFz_i xl,is , for j, l =2,3. Essen-
tially, the Γ_i is the unconditional mean of the conditional variance-covariance
matrix of (x⁰_2,it , x⁰_3,it)⁰. We also define the unconditional variance-covariance
matrix of (x⁰_1,it,x⁰_2,it,x⁰_3,it)⁰ by

Γi (t, s) = [Γjl_,i (t, s)]jl,

where Γji,i (t,s) = E (xj,it - Exj,it) (xι,is - Exι,is) , for j,l = 1, 2, 3. Observe
that Γ₂₂,_i (t,s) = Γ₂₂,_i (t,s), since x₂,_it and z_i are independent. With this
notation, we make the following assumption on the convergence of variances
and covariances:

Assumption 8 (convergence of covariances): As (N, T →∞),

∕i∙) ɪ P ɪ PP μ ^r22,i (^t>^s) ^r23,i (^{t, s)} ∖   .i ^γ22 ^Г23 A

⁽ⁱ⁾    ...    Γ23,i (t,s) Γ33,i (t,s) ) V Г2з Γ33 )■

(a) N PiT Pt Γi (t,t) → φ.

Note that the variance matrix [Γjl]j,l=2,3 is the cross section average of the long-
run variance-covariance matrix of (x^ i_t,x'3 i_tt)¹'■ For future use, we partition the
two limits in the assumption conformably to (x⁰_21,it,x⁰_22,it,x⁰_31,it,x⁰_32,it,x⁰_33,it)⁰ as
follows:

Finally, we make a formal definition of the random effects assumption, which
is a more rigorous version of (3).

Assumption 9 (random effects): Conditional on F_w, {ui}_i=1,...,N is i.i.d. with
mean zero, variance σ²_u and finite κ_u ≡ kui k_{F ,4} .

To investigate the power property of the Hausman test, we also need to
define an alternative hypothesis which states a particular direction of model
misspecification. Among many alternatives, we here consider a simpler one.
Specifically, we consider an alternative hypothesis under which the conditional
mean of ui is a linear function of DT wei . Abusing the conventional definition
of fixed effects (that indicates nonzero-correlations between wi =(x^'_it,z_i^')^' and
ui ), we refer to this alternative as the fixed effects assumption:

Assumption 10 (fixed effects): Conditional on F_w, the {u_i}_i=1,...,N is i.i.d.
with mean we_i^' DT λ and variance σ²_u, where λ is a (k +g) × 1 nonrandom nonzero
vector.

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