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

7. Treatment of Quality Change
are prices in periods t + 2 adjusted for the sum of
the changes in each quality characteristic weighted
by their coefficients derived from a linear hedonic
regression. Note that the summation is over the
same i in both periods since replacements are included
when a product is missing and equation
(7.30b) adjusts their prices for quality differences.
7.181 A Paasche upper bound is estimated as:
(7.31a)
where
(7.31b)
R p z
R( p( z) , S( v) )
t+ 2 t+
2
( ( ) , S( v) )
t t+
2
N
t+ 2 t+
2
∑ x ˆ
i
pi t
⎡ N
1
t 2
ˆ
i
+
⎛
=
pi
N ∑ si
⎜ t+
2
t+ 2 t i=
1 pi
xi
p ⎢ ⎝
∑
⎣
i
i=
1
⎞⎤
≥ = ⎢ ⎟
⎠⎥⎦
N
t+ 2 t+ 2 t+ 2 t+ 2 t+
2
i i i i i
i=
1
s = x p ∑ x p and
N
t t t t 2 t
pˆ i
≡ pi + ∑ βk( z +
ik
−zik)
i=
1
which are prices in periods t adjusted for the sum
of the changes in each quality characteristic
weighted by its respective coefficients derived
from a linear hedonic regression.
7.182 In Chapter 17, it is shown that Laspeyres,
P L , and Paasche, P P , price indices form bounds on
their respective true economic theoretic indexes.
Using reasoning similar to that in Chapter 17 applied
to equations (7.31a) and (7.32a) it can be
shown that under homothetic preferences:
(7.32)
0 1
PL
Pp ( , p, ) P
≤ α ≤ .
7.183 The approach is similar to that used for
adjustments to noncomparable replacement items
in equations (7.27). First, the SEHI approach uses
all of the data in each period, not just the matched
sample and selected replacements. Second, it uses
coefficients from hedonic regressions on changes
in the characteristics to adjust observed prices for
quality changes. Third, it incorporates a weighting
system using data on the value of output of each
model and their characteristics, rather than treating
each model as equally important. Finally, it has a
−1
,
direct correspondence to formulation defined from
economic theory.
7.184 Semi-logarithmic hedonic regressions
would supply a set of β coefficients suitable for
use with theses base and current period geometric
bounds:
(7.33a)
t+
2
2
si
N t+ t 2
N t 2
p ( ( ) , , ) ˆ
i R p z
+
+
q T pi
≥
≥
t t t
i= 1 pˆ i
R( p( z) , q, T)
i=
1 pi
⎛ ⎞ ⎛ ⎞
⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
∏ ∏ ,
where
(7.33b)
N
t t t t 2 t
pˆ i
≡ pi exp[ ∑βk( z +
ik
−zik)]
and
i=
1
N
t 2 t 2 t 2 t 2 t
p + i
≡ p + i
− β + k
z +
ik
−zik
i=
1
ˆ exp[ ∑ ( )] .
7.185 In equation (7.33a), the two bounds on
their respective theoretical indices have been
shown to be brought together under an assumption
of homothetic preference (see Chapter 17). The
calculation of such indices is no small task. For
examples of its application see Silver and Heravi
(2001a and 2003) for comparisons over time and
Kokoski, Moulton, and Zieschang (1999) for price
comparisons across areas of a country.
7.186 Note that unlike the hedonic indices in
Sections G.2.1 and G.2.2, the indices in equations
(7.30), (7.31) and (7.33) need not be based on
matched data. Kokoski, Moulton, and Zieschang
(1999) used a sample from a replacement universe
of otherwise matched data from the U.S. Bureau of
Labor Statistics CPI, although the sample benefited
from rotation. Silver and Heravi (2001a and
2003) used scanner data for the universe of transactions
via a two-stage procedure whereby first,
cells were defined according to major pricedetermining
features much like strata; such feature
included all combinations of brand, outlet type,
and screen size (for television sets. There may be a
gain in the efficiency of the final estimate, since
the adjustment is for within-strata variation, much
in the way that stratified random sampling improves
on simple random sampling. The average
price in each matched cell could then be used for
the price comparisons using equations (7.30a),
(7.31a) or (7.33a), except that to ensure that the
quality differences in each coming cell from characteristics
other than these major ones did not influence
the price comparison, adjustments were
made for quality changes using equations (7.30b),
t
si
187