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

for some constant M₂ < ∞. By Assumptions 2 and 6(i), for t = [Tr] and finite
constant M₂ , we have

sup sup E kD1Tx1,itk⁴i,T 1≤t≤T

≤ sup sup M3 (∣∣D1TIl⁴E ∣∣(xι,it
i,T 1≤t≤T

→ sup sup ∣τ1 (r) Θ1_,i ∣⁴ < ∞.

i r∈[0,1]

Ex1,it)k⁴+ kD1TEx1,itk⁴

Next, similarly, by Assumptions 3(ii) and (iii),

sup sup E ∣x2,it ∣⁴ ≤ sup sup M3 E∣x2,it

i,T 1≤t≤T i,T 1≤t≤T

< ∞,

Ex2,itk⁴+kEx2,itk⁴

and by Assumptions 4(ii) and (iii) and 6(iii) and (iv),

sup sup Ekx3,itk⁴ ≤

i,T 1≤t≤T

sup
i,T

sup M3 E ^°x3,it - EF_zi x3,it^°⁴ + ∣Ex3,it∣⁴

1≤t≤T ⁱ

Therefore,

sup sup
i,T 1≤t≤T

which yields (83). ¥

Part (c)

By Lemma 16(a). ¥

Part (d)

By Lemma 16(d). ¥

∞.

E (^suΡi,t °^x3,it ^- EFzi ^x3,it°⁴)
+ supi,t E ° ° EFz_i x3,it - Ex3,it°⁴+ sup_i,t ∣Ex3,it ∣⁴

kGx,T (xit
t

Xi)k2

²<M,

Proof of Lemma 3

By Lemma 16(c). ¥

Proof of Lemma 4

Write

., Pi Dtwi≈i=N Pi D

Twei

(ui

Ef.Ui) + (N Pi DT⅛W,'DT) λ. (85)

Notice that by Assumption 10,

∣Ef ɪ P

Fw ° _N

_i DT wei (ui

EFwui)

² σ² 1

) = Nutr^E\N Pi Dt^υjiW^j0 DTJ ■