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



By the Cauchy-Schwarz inequality,

kQiT k2


T∑
t


(Dit (xι,it Exι,it)
x2,it - Ex2,it
x3,it - Ex3,it


kD1T (x1,it


22


- Ex1,it)k4 + kx2,it
+ kx3,it - Ex3,it k


- Ex2,it k4


Notice that by Assumptions 2(i), 3(iii),


supE kx1,it - Ex1,it k4, supE kx2,it -Ex2,itk4 < ∞,
i,t                                          i,t


(75)


and by Assumption 4(iii) and Assumption 6(iii),


sup E kx3,it - Ex3,itk4
i,t


≤ M1 sup EFzi°x3,it - EFzi x3,it°4 +M1E°EFzix3,it


- Ex3,it °


< ∞.


(76)


Thus, we have


sup E kQiT k2 < ∞.
i,T


From this,

supE ∣∣QII 1 {∣∣Q∣∣ > M} ≤ supi,τ E∣lQiTk0,
i,T                              M

as M →∞. By Corollary 14 and Assumption 8(ii), then,

NN Σ T Σ IIIι,iτ II10,iτ
it

1                   1              ( 0  0    0 ʌ

= N Σ QiTP^1T N Σ EQiT = I 0 φ22 φ23 ) .

i                , i                 0 Φ023 Φ33

Next, by Assumption 6(i) and (ii), Lemma 11, and Lemma 12, we have

N Σ T ΣIII2,iTIII2,iT
it

1    1    DDT (Exι,it - Exι,i)DDit (Exι,it - Exι,i)

N∑T∑ EX2,it — EX2,i        EX2,it — EX2,i

i t Exi,it — EX3,i       JEx3,it — EX3,i      )

[ Ro1 (τ 1 R τ il iimN 1 Pi θ1,iθ1,i∏τ 1 R τ 1/dr  0 0 ʌ

.(77)


000

000

50



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