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

for some finite constant M. But Assumption 4(i) and (iii) imply that the right-
hand side of the last inequality in (45) is finite. This completes the proof. ¥

7 Appendix B: Proofs of Main Results

Proof of Theorem 1

For the desired result, it is enough to prove that

{ω ∈ Ω∣αz (G, H) (ω) ≤ x} ∈ Z,

for all x ∈ R. Since the partition Π = {∏ι,..., ∏i, ...}of Ω generates the sigma
field Z , we have

sup   |(Pz (G ∩ H))(ω) - (PzG)(ω)(PzH)(ω)∣

G∈G, H∈H

where 1Π_i (ω) denotes the indicator function that equals one if ω ∈ Πi, and
otherwise equals zero. Let I = {1, 2, ..., i, ...} be the set of positive integers, and
let

Then, we have (46) because

{ω ∈ Ω∣αz (G,H) (ω) ≤ x} = ⅛∏ ∈ Z. ¥

Before we start proving the lemmas and theorems in Section 4, we introduce
some additional lemmas that are used repeatedly below. Recall that wi_t =
x⁰_1,it,x⁰_2,it , x⁰_3,it ,z_i⁰ . We also repeatedly use the diagonal matrix DT defined
in Section 4.

Lemma 16 Suppose that Assumptions 1-8 hold. Define Ξ = Ξ1 + Ξ2, where

<<   <   34   35   36   37   38   39   40   >   >>