where Ez(∙) to the conditional expectation with respect to a sigma field Z. The
inequality (34) is a conditional version of Theorem 1 and equation (1.3) in Rio
(1993), or Theorem 1.1 and Corollary 1.1. of Bosq (1996). The inequality (35) is
a conditional version of the mixing inequality in Corollary A.2 of Hall and Heyde
(1980, p. 278). Observe that both of the inequalities indicate covZ(xt,xt+d) → 0
as d →∞, so long as the sequence {xt} is conditionally α-mixing. We may
obtain these inequalities by modifying the method used in Rio (1993) or Hall
and Heyde (1980) with the conditional arguments. However, to do so requires
αZ (d) to be measurable.13
It is difficult to derive the sufficient and necessary conditions that warrant
Z -measurability of the conditional mixing coefficient, αZ (d)orαZ(G, H). Thus,
we here only consider a sufficient condition. Stated formally:
Theorem 1 Suppose that the sigma field Z is generated by a countable partition
Π = {Π1,..., Πi,...} of Ω with P (Πi) > 0 for all i. Then, αz (G, H) in (31) is
measurable with respect to the sigma field Z.
When Z is the sigma field generated by a time-invariant regressor z , the
restriction on Z imposed by Theorem 1 is satisfied if z is a discrete random
variable, i.e., the supports of z are countable. This condition would not be
too restrictive in practice. In many empirical studies, time invariant regressors
generally consist of dummy variables (such as gender, race, or region), or dis-
crete variables (such as years of schooling). Such variables easily satisfy the
requirement of Theorem 1.
4 Main Results
This section derives for the general model (1) the asymptotic distributions of
the within, between, GLS estimators and the Hausman statistic. In Section
2, we have considered independently several simple models in which regressors
are of particular characteristics. The general model we consider in this section
contains all of the different types of regressors analyzed in Section 2. More
detailed assumptions are introduced below.
From now on, the following notation is repeatedly used. The expression
‘Ap” means “converges in probability,” while u⇒, means “converges in distri-
bution” as in Section 2.2. For any matrix A, the norm ∣∣A∣∣ signifies y⅞r(AA0).
When B is a random matrix with E kB kp < ∞, then kB kp denotes (E kB kp)1/p .
We use EF (∙) to denote the conditional expectation operator with respect to a
sigma field F. We also define ∣∣B∣∣Fp = (Ef ||B||p)1/p . The notation Xn ~
aN indicates that there exists n and finite constants d1 and d2 such that
13Admittedly, we here do not attempt to determine whether or not measurability of αZ (d)
is a necessary condition for the conditional mixing inequalities (34) and (35). It might be
possible to derive the inequalities with some alternative methods that do not require the
measurability assumption. Thus, we would like to emphasize that measurability of αZ (d) is
a sufficient, but not necessarily a necessary condition.
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