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

(iii) Uniformly in i and r ∈ [0, 1],

D31T EFz_i x31,it ^- Ex31,it ^→ 0k31×1 a.s.;

D32τ (EFzix32,it — Ex32,it) → √1rIk32g32,i (zi) a.s.;

D33T EFz_i x33,it ^- Ex33,it → τ33 (r) g33,i (zi) a.s.,
where

g32,i = (g1,32,i, ..., gk32,32,i)⁰ ; g33,i = (g1,33,i, ..., gk33,33,i)⁰ ,
and g32_,i (zi) and g33_,i (zi) are zero-mean functions of zi with

0 <Esup kg3k,i (zi)k^4q < ∞, for some q>1,

i

and g3k,i 6= g3k,j for i 6= j, and τ33 (r) = diag (r^-m1,33, ...r^-mk33^,33).

(iv) There exist τ (r) and G_i (z_i) such that

∖∖D3τ (Er,.x3,it — Ex₃,it) k ≤ e(r)Gei(zi),

where R τ (r)^4q dr < ∞ and E supi (Gi (zi)^4q < ∞ for some q > 1.

(v) Uniformly in (i, j) and r ∈ [0, 1];

D31T (Ex31,it ^— Ex31,jt) ^→ 0k31 ×1,

D32T (E^x32,it — E^x32,jt) → √r^lk32 ^μgg₃∙_2i — ^μ_fl32j) ,

D33T (E^x33,it — E^x33,jt) → ^τ33 ^(r) (μg33i — Mg33j) ^,

^{with su}Pi kμg32i^{k, su}Pi ∖∖μg33i^k < ∞.

Some remarks would be useful to understand Assumption 6. First, to have
an intuition about what the assumPtion imPlies, we consider, as an illustrative
example, the simple model in CASE 3 in Section 2.2, in which x3,i_t = ∖∖zz∕l^m +
ei_t, where ei_t is indePendent of zi and i.i.d. across i. For this case,

D3τ (EFzi X3,it — Exз,it¢ = D3τ∏i (zi — Ezi) /^tm;

D3T (Ex3,it — Ex3,jt) = D3T (ΠiEzi — ΠjEzj) /t^m.

Thus,

g3k,i (zi)  = Πi (zi — Ezi);

^μg3k,i = ^niEz^i.

Second, Assumption 6(iii) makes the restriction that E sup_i ∖g3k_,i (zi)∖^4q is
strictly positive, for k =2, 3. This restriction is made to warrant that g3k,i (zi) 6=
0 a.s. If g_3k,i (z_i) = 0 a.s.,¹⁵then

D3kTEFzi (x3k,it — Ex3k,it) ~ τ3k (r) g32,i (zi) = 0 a.s.,

¹⁵An example is the case in which x3_,it = eit Πizi /t^m ,where eit is independent of zi with
mean zero.

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