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

9. PPI Calculation in Practice
(iii) The Lowe index reduces to the Laspeyres index
when b = 0 and to the Paasche index when
b = t.
9.90 In practice, the situation is more complicated
for actual PPIs because the duration of the
reference period b is typically much longer than periods
0 and t. The weights w j usually refer to the
revenues over a year, or longer, while the price reference
period is usually a month in some later year.
For example, a monthly index may be compiled
from January 2003 onward with December 2002 as
the price reference month, but the latest available
weights during the year 2003 may refer to the year
2000, or even some earlier year.
9.91 Conceptually, a typical PPI may be viewed
as a Lowe index that measures the change from
month-to-month in the total revenue of an annual
basket of goods and services that may date back
several years before the price reference period. Because
it uses the fixed basket of an earlier period, it
is sometimes loosely described as a “Laspeyrestype”
index, but this description is unwarranted. A
true Laspeyres index would require the basket to be
that purchased in the price reference month,
whereas in most PPIs the basket not only refers to a
different period from the price reference month but
to a period of a year or more. When the weights are
annual and the prices are monthly, it is not possible,
even retrospectively, to calculate a monthly
Laspeyres price index.
9.92 A Lowe index that uses quantities derived
from an earlier period than the price reference period
is likely to exceed the Laspeyres (see Section
D.1 of Chapter 15), and by a progressively larger
amount, the further back in time the weight reference
period is. The Lowe index is likely to have an
even greater upward bias than the Laspeyres index
as compared with some target superlative index and
the underlying economic index. Inevitably, the
quantities in any basket index become increasingly
out of date and irrelevant the further back in time
the period to which they relate. To minimize the resulting
bias the weights should be updated more
frequently, preferably annually.
9.93 A statistical office may not wish to estimate
an economic index and may prefer to choose
some basket index as its target index. In that case, if
the theoretically attractive Walsh index were to be
selected as the target index, a Lowe index would
have the same bias, as just described, given that the
Walsh index is also a superlative index.
C.5 Factoring the Young index
9.94 It is possible to calculate the change in a
higher-level Young index between two consecutive
periods, such as t – 1 and t, as a weighted average
of the individual price indices between t – 1 and t,
provided that the weights are updated to take into
account the price changes between the price reference
period 0 and the previous period, t – 1. This
makes it possible to factor equation (9.10) into the
product of two component indices in the following
way:
∑
0: 0: −1 ( −1) −1:
(9.14) I t = I t ⋅ w b t ⋅
t t
i
I
i
,
( −1) 0: −1 0: −1
where w bt = w b ⋅I t w b ⋅I t .
∑
i i i i i
I 0:t-1 is the Young index for period t – 1. The weight
w i
b(t-1)
is the original weight for elementary aggregate
i price updated by multiplying it by the elementary
price index for i between 0 and t – 1, the
adjusted weights being rescaled to sum to unity.
The price updated weights are hybrid weights because
they implicitly revalue the quantities of b at
the prices of t – 1 instead of at the average prices of
b. Such hybrid weights do not measure the actual
revenue shares of any period.
9.95 The index for period t can thus be calculated
by multiplying the already calculated index
for t – 1 by a separate Young index between t – 1
and t with hybrid price-updated weights. In effect,
the higher-level index is calculated as a chained index
in which the index is moved forward period by
period. This method gives more flexibility to introduce
replacement products and makes it easier to
monitor the movements of the recorded prices for
errors, since month-to-month movements are
smaller and less variable than the total changes
since the price reference period.
9.96 Price updating may also occur between the
weight reference period to the price reference period,
as explained in the next section.
C.6 Price-updating from weight reference
period to price reference period
9.97 When the weight reference period b and
the price reference period 0 are different, as is normally
the case, the statistical office has to decide
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