David F. Hendry 31is to explicitly remove temporal dependence, as (1.4) showed, where a martin-
gale difference process is created by the sequential conditioning. In principle, the
same concepts apply to cross sections. It must be stressed that “random sampling”
by itself does not justify factorizing a joint density or likelihood function. As an
extreme form of cross-section dependence, put 1,000 copies of the number “1” in
a hat, then draw a random sample of 100 therefrom: one learns nothing after the
first draw, although all are “randomly drawn.” Sequential factorization correctly
reveals that difficulty. Denote the randomly-drawn data sample by(r 1 ...rN); then
for any ordering whenτis the mean value of all the numbers in the hat:
Dr(
r 1 ...rN|τ)
=∏Ni= 1Dri(
ri|ri− 1 ...r 1 ;τ)
=Dr 1(
r 1 ;τ)
, (1.11)since all the other probabilities are precisely unity. Asr 1 =1, we correctly deduce
τ=1. Certainly, the otherN−1 draws add the information that all the numbers
are unity, but would do so even if not randomly drawn.
More generally, the order of an independent sample does not matter, so unlike
(1.11), for any ordering the joint density should factorize as:
Dr(
r 1 ...rN|τ)
=∏Ni= 1Dri(
ri|ri− 1 ...r 1 ;τ)
=∏Ni= 1Dri(
ri|τ). (1.12)
Consequently, potential dependence is testable by conditioning ons“neighbors”
after a suitable exogenous ordering to check if their influence is non-zero; i.e., to
see whether:
∏Ni= 1Dri(
ri|ri− 1 ...ri−s;τ)
=∏Ni= 1Dri(
ri|τ). (1.13)
Suitable tests for the absence of dependence would seem essential before too great
a weight is placed on results that base (1.12) on the claim of random sampling,
especially when the units are large entities like countries. More generally, when all
units are affected in part by macro-forces and their attendant non-stationarities,
dependence like (1.13) is likely. If an ordering based on an outside variable is avail-
able, then models ofDri
(
ri|ri− 1 ...ri−s;τ)
could be investigated directly, similar
to some cases of spatial dependence (see Anselin, 2006).
There is a large literature on panel data analysis recently discussed in Choi (2006)
and Baltagi (2006).
1.4.5 Conditioning variables
“I’m afraid he’ll catch cold with lying on the damp grass,” said Alice, who
was a very thoughtful little girl. (Lewis Carroll, 1899)Instrumental variables are a key part of any conditioning set, so require weak
exogeneity as well as correlation with the relevant endogenous variables (or
the auxiliary assumptions of orthogonality to any unknown vector of excluded
influences and independence from the “true” model’s errors).