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

By Lemma 15(d),

supEkQ1,iTk⁴<M,
i,T

and by Assumption 6(iii) and (v),

E kQ2,ik⁴ < M.

Thus,

ɪ XhE ∣∣Q2,i∣∣4E ∣∣Q1,iτIl4i1/2 → 0, i

which implies

(n X^vec (Qι>iτ Q2,i ))

2

By the Chebychev’s inequality, then, we have

N X Qι,iτ Q2,i →p 0^,
i

→ 0.

as (N, T →∞) . Finally,
(N,T→∞),

in view of (62) we have the desired result that as

N X Qι,iTQ2,iT →p 0. ¥
i

Part (b)

From (47) , we write

sup sup E ∣∣Dt (wi — w)∣∣⁴

N,T 1≤i≤N

sup sup E ∣I1,i,NT + I2,i,NT + I3,i,NT ∣⁴
N,T 1≤i≤N

/

Mi

∖

sup_N,T sup_1≤i≤N E ∣I1,i,NT ∣⁴
+ sup_N,T sup_1≤i≤N E ∣I2,i,N T ∣
+sup_N,Tsup_1≤i≤NE∣I3,i,NT∣⁴

(63)

for some constant M1 . Thus, we can complete the proof by showing that each
of the three terms in the right-hand side of the inequality (63) is bounded.

For some constant M2 ,

sup sup E ∣I1,i,N T ∣⁴
N,T 1≤i≤N

M2

s      supi,τ E ∣D1T (xi,i — Exι,i)k4

+ ^suPi,T E ° ^d√²T √T (^x2,i ^- E^x2,i) J]
_k +suPi,τ E U D√3T√T (χ33_iti — EFzi x≡,θ°°⁴

(64)

44