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

N ∑E D1τ ⅛∑(x1≠

Exι,it)

i
1 ,, „ ,,2 „

≤ N IID1TIl supE

T X (^x1,it ^- E^x1,it)
t

→ 0,

(59)

where the last convergence result holds by Lemma 15(a). Similarly, by Lemma
15(b),

EN X D2kT T X (^x2k,it ^- E^x2k,it)

∙¹ sup E —= X (x2k,it — Ex2k,it)
^N i,t T T t

for k = 1, 2. Finally, by Lemma 15(d), we have

^e N∑D³^kτTXC

^x3k,it

EFz_i ^x3k,it)

N s^uτ^p^-TD^3kτ

^E^-T X(^x3k.it
^t T t

→ 0,

(60)

EFz_i ^x3k,it)

(61)

for k = 1, 2, 3. The results (59), (60) and (61) imply that I₃,_νt →_p 0.
Proof of (51): Notice that by (48) and (50),

^N X I1,i,NT^ ¹3,NT

^N X I2,i,NT^ ⁱ3,nt

N X Iι,i,NT III3,NT∣∣² = Op ⁽¹⁾ Op ⁽¹⁾ ;

N X I2,i,NT ∣∣I3,NTIl² = ^op ⁽¹⁾^op ⁽¹⁾ .

Thus, as (Ν,T →∞) ,

^N X I1,i,NT^ ¹3,NT^,

^NF X I2,i,NT^ ¹3,NT →p 0.

We now consider the (k,l)^th term of N P_i I1,i,NTI'0iNT. By the Cauchy-
Schwarz inequality,

(N X

I1,i,NT ¹2 ,i,NT

k,l

≤ (N X[(I1,i,NT)k]²^ (N X[(I2,i,NT)ι]2