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

4             ∖4     1 T-1 s s         Q_-1

4!8 f sup ∣∣Xi_t∣∣_4qj sup T∑∑∑α,(s) q

^{,              i,T}    s=1 p=0 k=0

4∞                    ₁

sup ∣Xit ∣_4q   X(s +1)2 sup ai (s) ^q < M,

^i,t                  s=1              ⁱ

where the last bound holds because supi ai (s) is of size —3 ^-ɪ (see Assumption
3(i)). By the similar fashion, we can also show that

and we have all the required result. ¥

Part (c)

Let Yi_t = xh_,3_,i_t — EF_z xh_,3_,i_t. Using the arguments similar to those used in
the proof of Part (b), we can show that under Assumption 4,

for some constant M. By Assumption 4(i), P∞= s2 supi aFZ (s) ^q < ∞ a.s..
Finally, since the terms supi _t kYitkF 4_q and PS=1 s2 supi αF_z. (s) ^q are F∞^o-
measurable, we have the desired result by choosing

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