Hence, provided that E[(∆sξ.t)0(∆sξ.t)] and E[(ξ.t — ξ)0(ξ.t — ξ)] have rank K,
consistent estimators of β can be obtained by applying OLS on (12) or on (13),
which give, respectively,
(16) βb∆s =
(17) βbBP =
T
X (δsX∙t) 0 (δsX∙t)
t=s+1
T
Σ(x∙t- x) 0(x∙t- x)
t=1
T
X (δsX∙t) 0 (δsy-t)
t=s+1
-1 T
Σ(x ∙t
t=1
- x) 0(y∙t - y)
s=1,...,T—1,
The latter is the ‘between period’ (BP) estimator. The consistency of these esti-
mators simply relies on the fact that averages of a large number of repeated mea-
surements of an error-ridden variable give, under weak conditions, an error-free
measure of the true average at the limit, provided that this average shows variation
along the remaining dimension, i.e., across periods. Basic to these conclusions is
the assumption that the measurement error has no period specific component. If
such a component is present, it will not vanish when taking plims of period means,
i.e., plim(υ.t) will no longer be zero, (14) and (15) will no longer hold, and so βδs
and βb BP will be inconsistent.
Table 24.1 reports between period estimates of β based on levels (column 2)
and on differences (column 5) - the latter removing the effect of technical changes
represented by a log-linear trend - as well as seven-period difference estimates
(column 7) for the four sectors and the two inputs.5 Since K = 1 in this application,
the estimators read
βBP
b
βBPDC
∑t(x∙t - x)(y∙t - y)
P t ( X ∙t — x)2
∑t (∆ x ∙t — ∆ x)(∆ y∙t — ∆ y)
P t (∆ X .t — ∆ X)2
β∆7 =
y∙ 8 — y∙ 1
χ∙ 8 — χ∙ 1
Rows 1 and 3 can be interpreted as estimates of 1 /μ and rows 2 and 4 as estimates
of μ. This way of running original and reverse regressions in an EIV context can be
related to Frisch’s confluence analysis [Frisch (1934, sections 5, 10, 11, and 14)], in
which he proposed taking regressions in different directions, e.g., in the directions
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