mixing model. For this model, we can establish the joint limits of the sample
averages of panel data whose time series are non-ergodic, as we show in Section
4 and Appendix.
Suppose that (Ω, F, P) is a basic probability space, and G, H, Z are sub-
sigma fields of F with Z ⊂ G and Z ⊂ H. Then, a conditional α-mixing coeffi-
cient between two sub-sigma fields G and H on Z is defined as
αz (G, H)= sup ∣PZ (G ∩ H ) - Pz (G) Pz (H )∣, (31)
G∈G, H∈H
where PZ (∙) denotes a conditional probability defined on the sigma field Z.10
This conditional α-mixing coefficient is a straightforward extension of the usual
α-mixing coefficient, except that it uses the conditional probability Pz (∙) in-
stead of the usual unconditional probability P (∙).
The general definition of the conditional α-mixing coefficient can apply to
our panel data model as follows. Suppressing the subscript i for convenience,
assume that {xt }t and z are scalar random variables, respectively defined in the
probability space (Ω, F, P), where supt E ∣xt∣2q < ∞, for some q > 1.11 Define
F-∞ = σ (..., xt-1 ,xt);Ft+d = σ (xt+m , xt+m+1 , ...);Z = σ(z),
where Z is assumed to be a non-trivial sigma field, i.e., in Z there exists a
subset A of Ω with 0 < P (A) < 1. Define
αz (d) = sup sup ∣PZ (G ∩ H) - Pz (G) Pz (H)∣. (32)
t G∈F-t∞,H∈Ft∞+d
With this definition, we will say that the sequence {xt} is conditionally α-mixing
if and only if
αz (d) → 0 a.s. , (33)
as d →∞, where the almost sure convergence of αz (d) holds with respect to
an outer probability measure P * of the probability space (Ω, F, P ).12
A technical problem in using the conditional α-mixing coefficient αz (d) (as
well as αz (G,H)) is that it is not necessarily measurable with respect to the
conditioning sigma field Z . This problem raises some technical difficulties in
deriving useful inequalities. For example, following the usual techniques related
to (unconditional) α-mixing coefficients, one may expect that the following
conditional versions of α-mixing inequalities hold:
1
|EZ (xtxt+d)-(EZxt) (EZxt+d)∣ ≤ 2-q~⅛z (d)^—~´(≡up(Ez ∣xt∣2q ´
q-1 t
(34)
|Ez (xtxt+d)-(Ezxt)(Ezxt+d)| ≤ 8
1
EZ ∣xt∣2qj) q
(35)
10 We also could define similar conditional mixing coefficients of β- mixing and φ- mixing.
11The xt need not be strictly stationary.
12For the details of the outer probability measure P *, readers may refer to Chapter 1.2 of
van der Vaart and Wellner (1996).
17