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



Write


N X 12,i,N I2 ,i,N
i

II1,1

II1,21

0

0

II1,32

II1,33

II1,z

II1,21

II21,21

0

0

II21,32

II21,33

II21,z

0

0

0

0

0

0

0

0

0

0

0

0

0

0

II10 ,32

II20 1,32

0

0

II32,32

II32,33

II32,z

II1,33

II20 1,33

0

0

II30 2,33

II33,33

II33,z

II10,z

II201,z

0

0

II302,z

II30 3,z

IIz,z

/


(53)


By Assumption 7(i), as N →∞,

II1,1

II10,21


II1,21

II21,21


ΓΘ11
Γ0Θ121


ΓΘ121

ΓΘ21,Θ21


(54)


By the weak law of large numbers (WLLN) and Assumption 7, as N →∞,we
have

(1132,32  1132,33 II32,z

II30 2,33 II33,33 II33,z

II30 2,z    II303,z    IIz,z


Γ

g32 ,g32

+ ɪ μ32,μ32

Γ

g32,g33

+ ɪ μ32,μ32

Γ
g32,z

Γ0

g32 ,g33

+ Γ0

ʃ μ32,μ32

Γ

g33,g33

μ33,μ33

Γg33,z     .      (55)

Γ0

g32,z

Γ0

g33,z

Γz,z


From Assumption 7 and WLLN with the assumption that Eg32,i (zi) = Eg33,i (zi)
= 0, it follows that

II1,32

II21,32


II1,33

II21,33


p


rθ132
rθ2132


rθ1,μ33
rθ2133


(56)


as N →∞. In addition, by WLLN with Assumption 5,

μ ⅛. ) p μ 0 ),                  (57)

as N →∞. The results (53)-(57) indicate that

N X I2,i,NI2,i,N p ξ2,                      (58)

i

as N →∞. Finally, by Lemma 12, (52) and (58) imply (49).

Proof of (50): Notice that

N ∑D1T 1∑(xι,it - Exι,it)
it

41



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