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



Assumption 2 (about x1,it):

(i) For some q>1, κx1 supi,t kx1,it - Ex1,itk4q < ∞.

(ii) Let Xh,ι,it be the hth element of xι,it. Then, Exh,ι,it ~ tmh,1 for all i and
h =1, ..., k1 , where mh,1 0.

Assumption 3 (about x2,it): For some q>1,

(i) {x2,it - Ex2,it}t is an α-mixing process for all i, and is independent of
Fzi for all i. Let αi be the mixing coefficient of x2,it. Then, supi αi (d) is
of size
3ɪ, i.e., supi αi (d) = O (d-p-3q--1), for some p > 0.

(ii) E(x21,it) = Θ21,it and E(x22,it) = Θ22,t, where supi,t kΘ21,itk, supt kΘ22,tk
< ∞
, and Θ2i,it 6= Θ2i,jt if i 6= j.

(iii) κx2 supi,t kx2,it —Ex2,itk4q < ∞.

Assumption 4 (about x3,it): For some q>1,

(i) x3,it — EFz x3,it t is conditional α-mixing for all i. Let αFz be the con-
ditional
α-mixing coefficient of x3,it on Fzi. Then, supi αzi (d) is of size
3q--i a.s., i.e., supi αzi (d) = O (d-p-3q--τ) a.s., for some p > 0. Also,
(q-    q       q           q-1 2

P=1 d suPi (αFzi (d) q ) ) < ∞.

(ii) E (x3,it) = Θ3,it, where supi,t kΘ3,itk < ∞.

(iii) E supi,t °x3,it — EFzix3,it°8Fqz ,4q < ∞.

(iv) Let xh,3k,it be the hth element of x3k,it, where k =1, 2, 3. Then, conditional
on
zi ,

(iv.1) (EFzi xh,31,it Exh,3ι,it) ~ t mh,31 a.s., where 1 < mh,31 ≤ ∞ for
h =1, ..., k31 (here, mh,3i = implies that EFzi xh,3i,it —Exh,3i,it =
0
a.s.);

(iv.2) (EFzixh,32,it Exh,32,it) ~ t-2 a.s. for h =l,...,k32;

(iv.3) (EFzi xh,33,it Exh,33,it) ~ t-mh,33 a.s., where 0 ≤ mh,33 2 for

h=1,...,k33.

Assumption 5 (about zi):

(i) {zi}i is i.i.d. over i with E(zi)=Θz , and kzi k4q < ∞ for some q>1.

(ii) The support of the density of zi is countable for all i.

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