Producer Price Index Manual: Theory and Practice ... - METAC

1. An Introduction to PPI Methodology
derlying unknown theoretical PPI—and certainly a
much closer approximation than either the
Laspeyres or the Paasche indices can yield on their
own.
1.103 This intuition is corroborated by the following
line of reasoning. Diewert (1976) noted that
a homogeneous quadratic is a flexible functional
form that can provide a second-order approximation
to other twice-differentiable functions around the
same point. He then described an index number
formula that is exactly equal to a theoretical one
based on the underlying aggregator function as superlative
when that functional form is also flexible—for
example, a homogeneous quadratic. The
derivation of these results, and further explanation,
are given in detail in Section B.3 of Chapter 17. In
contrast to the theoretical index itself, a superlative
index is an actual index number that can be calculated.
The practical significance of these results is
that they provide a theoretical justification for expecting
a superlative index to provide a fairly close
approximation to the unknown underlying theoretical
index in a wide range of circumstances.
E.3.1 Superlative indices as symmetric
indices
1.104 The Fisher index is not the only example of
a superlative index. In fact, there is a whole family
of superlative indices. It is shown in Section B.4 of
Chapter 17 that any quadratic mean of order r is a
superlative index for each value of r ≠ 0. A quadratic
mean of order r price index P r is defined as
follows:
(1.15) P
r
≡
n
r
i=
1
r
∑
n
∑
i=
1
t
0
⎛ p ⎞
i
si
⎜ 0 ⎟
⎝ pi
⎠
0
t
⎛ p ⎞
i
si
⎜ t ⎟
⎝ pi
⎠
r 2
r 2
where s i 0 and s i
t
are defined as in equation (1.2)
above.
1.105 The symmetry of the numerator and denominator
of equation (1.11) should be noted. A
distinctive feature of equation (1.11) is that it treats
the price changes and revenue shares in both periods
symmetrically whatever value is assigned to the
parameter r. Three special cases are of interest:
,
• When r = 2, equation (1.1) reduces to the
Fisher price index;
• When r = 1, it is equivalent to the Walsh price
index;
• In the limit as r → 0, it equals the Törnqvist index.
1.106 These indices were introduced earlier as
examples of indices that treat the information available
in both periods symmetrically. Each was originally
proposed long before the concept of a superlative
index was developed.
E.3.2 The choice of superlative index
1.107 Section B.6 of chapter 17 addresses the
question of which superlative formula to choose in
practice. Because each may be expected to approximate
to the same underlying theoretical output
index, it may be inferred that they ought also to approximate
to each other. That they are all symmetric
indices reinforces this conclusion. These conjectures
tend to be borne out in practice by numerical
calculations. It seems that the numerical values of
the different superlative indices tend to be very
close to each other, but only so long as the value of
the parameter r does not lie far outside the range 0
to 2. However, in principle, there is no limit on the
value of the parameter r, and in Section B.5.1 of
Chapter 17, it has shown that as the value of r becomes
progressively larger, the formula tends to assign
increasing weight to the extreme price relatives
and the resulting superlative indices may diverge
significantly from each other. Only when the absolute
value of r is very small, as in the case of the
three commonly used superlative indices—Fisher,
Walsh, and Törnqvist—is the choice of superlative
index unimportant.
1.108 Both the Fisher and the Walsh indices date
back nearly a century. The Fisher index owes its
popularity to the axiomatic, or test, approach, which
Fisher (1922) himself was instrumental in developing.
As shown above, it appears to dominate other
indices from an axiomatic viewpoint. That it is also
a superlative index whose use can be justified on
grounds of economic theory suggests that, from a
theoretical point of view, it may be impossible to
improve on the Fisher index for PPI purposes.
1.109 However, the Walsh index has the attraction
of being not merely a superlative index, but
also a conceptually simple pure price index based
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