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

is possible to show that

σ_v²ι⁰Ψxι > 0,

as (N, T →∞) . So, the asymptotic normality in (79) holds if

as (N, T → ∞). Let P_i,N_τ = ^1q,t. Then, E (P_i,N_τ) = 0 and Pi EPi²NT = 1.
Thus, by the central limit theorem of the double indexed process (e.g., see
Theorem 2 in Phillips and Moon, 1999), we can claim that (81) holds, if we can
show that

^X EPi,NT¹ ©^|Pi2NT I > ^ε} → ⁰ for all ^ε > ^{0,             (82)}

i

as (N, T →∞).

Now, in view of (80) , condition (82) holds if

supEkι⁰Qi2Tk⁴ ≤ supE kQi2T k⁴ < ∞.                (83)

Note for some constant M1 that

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