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,itk4
i,T 1tT

sup sup M3 (∣∣D1TIl4E ∣∣(xι,it
i,T
1tT

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 Ex2,it

i,T 1tT                 i,T 1tT

< ,


Ex2,itk4+kEx2,itk4


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


sup sup Ekx3,itk4  

i,T 1tT


sup
i,T


sup M3 E °x3,it - EFzi x3,it°4 + Ex3,it4

1tT                       i


Therefore,


sup sup
i,T 1tT

which yields (83). ¥

Part (c)

By Lemma 16(a). ¥

Part (d)

By Lemma 16(d). ¥


M2


.


E (suΡi,t °x3,it - EFzi x3,it°4)
+ supi,t E ° ° EFzi x3,it - Ex3,it°4
+ supi,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 Dtwii=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

) = NutrE\N Pi DtυjiWj0 DTJ


54




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