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



By definition,


N1 X EFzi kRiTk
i

sup EFzi kRiT k ≤ sup EFzi kQiTk1{kQiTk >M}.       (39)

Also,



N X(PiT - EFzi PiT )
i


(⅛ X EFzi ° PiT - EFZi PiT ° 22

≤√⅛ μsiuτp EFzi kPiT k2)2


(40)

Choose M = N a, where 0 < a < 2. Then, (39) , (40) 0 α.s.. Consequently,
we have (38).
¥

Lemma 15 Suppose that Assumptions 1-8 hold. Let Fz = σ (z1, ..., zN , ...) .
For a generic constant M that is independent of N and T, and for some Fz -
measurable function Mz , the followings hold:

(a) suPi,τ TT Pt (xι,it - Exι,it) °4 M;

(b) suPi,T T Pt (χ2,it - Eχ2,it) ∣∣4 M;

(c) suPi,T 1T Pt (χ3,it - EFziχ3,i0 °F 4 < Mz, a.s.;

(d) suPi,T ° T Pt (χ3,it - EFzi χз,it¢ ∣∣4 M.

Proof

We here use q to denote the real number used in Assumptions 1-6, which is
strictly greater than 1.

Part (a)

Note that

sup
i,T


T X (x1,it - Ex1,it)

t                         °4


sup kx1,it - Ex1,itk4

i,T

sup kx1,it - Ex1,itk4q = κx2 ,
i,T                      q

where the first inequality holds by Minkowski’s inequality, the second inequality
holds by Liapunov’s inequality, and the last inequality holds by Assumption 2.
Choose M = κ
x2 . ¥

Part (b)

34



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