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

NT(β (βg — β)

Ntt (βw - β)

+   1 σ²v    (P2,2 ^- P2,1)

√T ^σU (q2,2 ^- q2,ι + ^σ2)

⁺o^p √Jf) ;

_n 0  ^σV    ^(p2,2 ^{- p}2,1)

, -U (q2,2 - q2,ι + σ^)²

σ⁴    (p2 2 - p²_{2 1} )

plimN,τ→∞NT²[Var(βw) - Var(βg)] = -^v ,     , ₂   ,1 _2λ2 .      (26)

^{w            g}      σ²_u (q2,2 - q₂²_,1 + σ_e²)²

Note that (22) - (26) are essentially the same as (13) - (17) except that they do
not include the time trend T^m.

CASE 2.2: Now, we consider the case in which the Θ_2,i are constant over
different i; that is, p2_,2 - p₂²_,1 =0. If the model contains no intercept term
(ζ =0) and it is estimated with this restriction, all of the results (22) - (26) are
still valid with p2,2 replacing (p2,2 - p²_2,1). However, if the model contains an
intercept term, the assumption of cross-sectional homoskedasticity affects the
convergence rates of the between estimator and the Hausman statistic, while
it does not to the within and GLS estimators. To see this, we assume that
Θ2,i =0, for all i, without loss of generality. Then, we can show

-—^ ^   ^         —²       —²        /n ^

NT3(β(βw - βg)   =  --v₇-------2r—. ʌNr (βb -β) + op(i)

^{w g}             -²_u (q22,2 - q₂²_2,1 + -_e²)   T ^b

=⇒ N (0, ^-v ₇------^-2- . J ;          (28)

-²_u (q22,2 - q₂²_2,1 + -_e²)²

-4 -2

plimN,τ→∞NT³[Var(^βw) - Var(^βg)] = -27-------₂---:—2τ2,     (29)

^{w            g}     -²_u(q22,2 - q₂²_2,1 +-_e²)²

as (N, T → ∞). Observe that the between estimator is ypN-consistent as in
CASE 1.2.

CASE 3: So far, we have considered the cases in which the time-varying
regressor xit and the time invariant regressor zi are uncorrelated. We now

13

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