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

Assumption 2 (about x_1,it):

(i) For some q>1, κ_x1 ≡ sup_i,t kx1,it - Ex1,itk_4q < ∞.

(ii) Let Xh,ι,i_t be the h^th element of xι,i_t. Then, Exh,ι,i_t ~ t^mh,1 for all i and
h =1, ..., k1 , where mh_,1 > 0.

Assumption 3 (about x_2,it): For some q>1,

(i) {x2,it - Ex2,it}_t is an α-mixing process for all i, and is independent of
F_zi for all i. Let α_i be the mixing coefficient of x_2,it. Then, sup_i α_i (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 sup_i,t kΘ21,itk, sup_t kΘ22,tk
< ∞, and Θ2i,it 6= Θ2i,jt if i 6= j.

(iii) κx₂ ≡ sup_i,t kx2,it —Ex2,itk_4q < ∞.

Assumption 4 (about x3_,i_t): For some q>1,

(i) x3_,it — EF_z x3_,it _t is conditional α-mixing for all i. Let αF_z be the con-
ditional α-mixing coefficient of x3_,it on F_zi. Then, sup_i α_zi (d) is of size
—3q--i a.s., i.e., supi α_zi (d) = O (d-^p-^3q--^τ) a.s., for some p > 0. Also,
(q-    q       q           q-1 ∖ ∖ 2

P∞=1 ^{d su}Pi (^αFz_i ^{(d) q} ) ) < ∞.

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

(iii) E sup_i,t ^°x3,it — EF_zix3,it^°8_F^q_{z ,4q} < ∞.

(iv) Let xh_,3k_,it be the h^th 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 EF_zi 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) (EFz_i xh,33,it — Exh,33,it) ~ t-m^h,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)=Θ_z , and kzi k_4q < ∞ for some q>1.

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

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