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

7. Treatment of Quality Change
tives—the Carli index—equally weights each price
change, but the ratio of arithmetic means—the
Dutot index—weights price changes according to
the prices of the product in the base period relative
to the sum of the base period prices. Item 1 has a
relatively low price, and thus weight, in the base
period 1 of 4, but this product has the highest price
increase, one of 6/5. Therefore, the Dutot index is
lower than the Carli index.
7.93 As noted above, it is also possible to refine
the imputation method by targeting the imputation:
including the weight for the unavailable
products in groupings likely to experience similar
price changes—say by product type, industry, and
geographical region. Any stratification system
used in the selection of establishments would facilitate
this. For example, in Table 7.1(b) assume
that the price change of the missing product 2 in
March is more likely to follow price changes of
product 1, and product 6 is more likely to experience
price changes similar to products 5 and 8. For
March compared with February, with weights all
equal to unity, the geometric mean of price ratios
(Jevons) is
(7.10)
N
2 3 3 2 1/ N
J( , ) = ∏ (
n
/
n)
n=
1
P p p p p
3 2 3 2 3 2
( p1 / p1 ) ( p5 / p5 p8 / p8
)
= ⎡ × ×
⎢⎣
2 3/2
( 6/5) ( 12/11 10/10)
= ⎡ × ×
⎣
= 1.1041.
2 3/ 2
⎤
⎦
1 5
Note the weights used: for product type A, the single
price represents 2 prices; for product type B,
the prices represent three or 3/2 = 1.5 each.
7.94 The ratio of average (mean) prices—the
Dutot index—is
(7.11)
N
N
2 3 3 2
D( , ) = ( ∑ n
/ )/( ∑ n
/ )
n= 1 n=
1
3 3 3
= ( 2p1 + 1.5p5 + 1.5 p8)
/5
2 2 2
÷ 2p1 + 1.5p5 + 1.5 p8
/5
P p p p N p N
( )
( 2 6 1.5 12 1.5 10)
÷ ( 2× 5+ 1.5× 11 + 1.5×
10)
= × + × + ×
= 1.0843
⎤
⎥⎦
.
1/5
7.95 The average (mean) of price ratios – the
Carli index - is:
(7.12)
N
P ( p ,p ) = ∑ ( p /p )/ N
2 3 3 2
C i i
i=
1
3 2 3 2 3 2
= 2 ( p1 / p1 ) + 3 ⎡ ( p5 / p5 + p8 / p87 )/2⎤
5 5 ⎣
⎦
2 3
= ( 6 / 5) + ⎡ ( 12 /11 + 10 /10)/ 2
5 5
⎣
⎤⎦
= 1.1073.
Alternatively, and more simply, imputed figures
could be entered in Table 7.1B for products 2 and
6 in March using just the price movements of A
and B for products 2 and 6 respectively, and indices
calculated accordingly. Using a Jevons index
for product 2, the imputed value in March would
be 6/5× 6= 7.2, and for product 6 it would be
[(12/11) x (10/10)] 1/2 x 12= 12.533. It is thus apparent
that not only does the choice of formula
matter, as discussed in Chapter 20, but so too may
the targeting of the imputation. In practice, the
sample of products in a targeted subgroup may be
too small. An appropriate stratum is required with
a sufficiently large sample size, but there may be a
trade-off between the efficiency gains from the
larger sample and the representativity of price
changes achieved by that sample. Stratification by
industry and region may be preferred to industry
alone, if regional differences in price changes are
expected, but the resulting sample size may be too
small. In general, the stratum used for the target
should be based on the analyst’s knowledge of the
industry and an understanding of similarities of
price changes between and within strata. It also
should be based on the reliability of the available
sample to be representative of price changes.
7.96 The underlying assumptions of these
methods require some analysis since—as discussed
by Triplett (1999 and 2002)—they are often misunderstood.
Consider i = 1.... m products where,
t
as before, pm
is the price of product m in period t,
t 1
and p +
n
is the price of a replacement product n in
period t + 1. Now n replaces m but is of a different
quality. As before, let A(z) be the quality adjustment
to p + n
t 1
, which equates its quality services or
utility to p t+
1
m
such that the quality adjusted
* t+ 1 t+
1
price pm
= A( z ) pn
. For the imputation
method to work, the average price changes of the i
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