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

7. Treatment of Quality Change
given to each price comparison since some products
may account for much larger sales revenues
than others. The same consideration applies to
these hedonic indices. Diewert (2002e) has argued
that sales values should form the basis of the
weights over quantities. Two products may have
sales equal to the same quantity, but, if one is
priced higher than another, its price changes
should be weighted higher accordingly for the result
to be meaningful in an economic sense. In addition,
Diewert (2002e) has shown that it is value
shares that should form the weights, since values
will increase–over period t + 2, for example—with
prices, the residuals, and their variance thus being
higher in period t + 2 than in t. This heteroskedasticity
is an undesirable feature of a regression
model resulting in increased standard errors. Silver
(2002) has further shown that a WLS estimator
does not purely weight the observations by their
designated weights. The actual influence given is
also due to a combination of the residuals and the
leverage effect. The latter is higher since the characteristics
of the observations diverge from the average
characteristics of the data. He suggests that
observations with relatively high leverage and low
weights be deleted and the regression repeated.
G.2.2 Period-on-period hedonic indices
7.173 An alternative approach for a comparison
between periods t and t + 2 is to estimate a hedonic
regression for period t + 2 and insert the
values of the characteristics of each model existing
in period t into the period t + 2 regression to predict,
for each item, its price. This would generate
predictions of the prices of items existing in period
t based on their z t i
characteristics, at period t + 2
t+
2 t
shadow prices, pˆ i
( zi)
. These prices (or an average)
can be compared with the actual prices (or the
t t
average of prices) of models in period t, pi( zi)
as
a, for example, Jevons hedonic base period index:
(7.29a)
P
JHB
⎡
⎢
=
⎣
⎡
⎢
⎣
t
t
1/ N
N
⎤
t+
2 t
pˆ i
( zi)
⎥
i=
1 ⎦
t
t
1/ N
N
⎤
t t
∏ pi( zi)
⎥
i=
1
∏
⎦
t
t
t
1/ N
t
1/ N
N
N
t+ 2 t t+
2 t
pˆ
( ) ˆ
i
zi ∏pi ( zi)
i= 1 i=
1
≈
t
t
t 1/ N
t 1/ N
⎡
N
⎤ ⎡
N
⎤
t
t
⎢∏pˆ
i ⎥ ⎢∏pi
⎥
i= 1 i=
1
⎡ ⎤ ⎡ ⎤
⎢∏
⎥ ⎢ ⎥
≈
⎣ ⎦ ⎣ ⎦
⎣ ⎦ ⎣ ⎦
7.174 Alternatively, the characteristics of models
existing in period t + 2 can be inserted into a
regression for period t. Predicted prices of period t
+ 2 items generated at period t shadow prices,
t t 2
pi( z + i
), are the prices of items existing in period
t + 2 estimated at period t prices, and these prices
(or an average) can be compared with the actual
prices (or the average of prices) in period t +
t+ 2 t+
2
2, pi
( zi
); a Jevons hedonic current period index
is
(7.29b)
P
JHC
⎡
⎢
=
⎣
⎡
⎢
⎣
t+
2
N
∏
i=
1
t+
2
N
∏
i=
1
p
⎤
( z ) ⎥
⎦
t+ 2 t+
2
i i
p
t+
2
1/ N
t+
2
1/ N
⎤
t t+
2
i( zi
)
t+ 2 t+
2
t+ 2 1/ N
+ 2t
1/ N
⎡N
⎤ ⎡N
⎤
t+ 2 t+
2
⎢∏pˆ
i ⎥ ⎢∏pi
⎥
⎣ i= 1 ⎦ ⎣ i=
1 ⎦
t+ 2 t+
2
t+ 2 1/ N
t+
2
1/ N
N
N
t t+ 2 t t+
2
∏pi( zi ) ∏pi( zi
)
i= 1 i=
1
⎥
⎦
= =
⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
7.175 For a fixed base, bilateral comparison using
either equation (7.29a) or (7.29b), the hedonic
equation is only estimated for one period, the current
period t + 2 in equation (7.29a) and the base
period t in equation (7.29b). For reasons analogous
to those explained in Chapters 15, 16, and 17, a
symmetric average of these indices would have
some theoretical support. It would be useful as a
retrospective study to compare the results from
both approaches (7.29a) and (7.29b). If the discrepancy
is large, the results from either should be
treated with caution, similar to the way a large
Laspeyres and Paasche spread would cast doubt on
the use of either of these indices individually. It
would be evidence for the need to update the regressions
more often.
7.176 Note that a geometric mean of equations
(7.29a) and (7.29b) uses all of the data available in
each period, as does the hedonic index using a time
dummy variable in (7.29). If in (7.29) there is a
.
.
185