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

Producer Price Index Manual
= 1….m products, including the quality-adjusted
price p * t+
1
m
given on the left-hand side of equation
(7.13), must equal the average price change from
just using the overall mean of the rest of the i =
1….m – 1 products on the right-hand side of equation
(7.13). The discrepancy or bias from the
method is the balancing term Q. It is the implicit
adjustment that allows the method to work. The
arithmetic formulation is given here, although a
similar geometric one can be readily formulated.
The equation for one unavailable product is given
by
(7.13)
(7.14) Q
1 ⎡p
p
⎢
m ⎣ p p
* t+ 1 m−1
t+
1
m
i
+
t ∑ t
m i=
1 i
⎤
⎥
⎦
m 1 t + 1
⎡
−
1 p ⎤
i
= Q
t
( m 1)
∑ +
⎢⎣
− i=
1 pi
⎥
,
⎦
1 p
1 p
= − ∑ ,
m p m m − p
* t+ 1 m−1
t+
1
m
i
t
t
m ( 1)
i=
1 i
and for x unavailable products by
(7.15)
x t
m − x t
1 pm
x
pi
Q = −
t
m p m m − x p
* + 1 + 1
∑ ∑ .
( )
i= 1 m
i=
1
7.97 The relationships are readily visualized if
r
1
is defined as the arithmetic mean of price
changes of products that continue to be recorded
and r
2
is defined as the mean of quality-adjusted
unavailable products, that is, for the arithmetic
case where
m−
x
⎡ ⎤
1 ⎢∑
i i⎥
⎣ i=
1 ⎦
x
⎡ * t+
1 t⎤
r2
= ⎢∑ pi
/ pi
⎥
⎣ i=
1 ⎦
÷ x,
t+
1 t
(7.16) r = p / p ÷ ( m − x)
then the ratio of arithmetic mean biases from substituting
equation (7.16) into equation (7.15) is
t
i
x
(7.17) Q = ( r2 − r1)
,
m
which equals zero when r 1
= r 2
. The bias depends
on the ratio of unavailable values and the difference
between the mean of price changes for existing
products and the mean of quality-adjusted replacement
price changes. The bias decreases as either
( x/
m ) or the difference between r 1
and r 2
decreases. Furthermore, the method relies on a
comparison between price changes for existing
products and quality-adjusted price changes for the
replacement/ unavailable comparison. This is more
likely to be justified than a comparison without the
quality adjustment to prices. For example, let us
say there were m = 3 products, each with a price of
100 in period t. Let the t + 1 prices be 120 for two
products, but assume the third is unavailable, that
is, x = 1 and is replaced by a product with a price
of 140, of which 20 is the result of quality differences.
Then the arithmetic bias as given in equations
(7.16) and (7.17) where x = 1 and m = 3 is
( − + )
1 ⎡ 20 140 /100
3 ⎣
⎤⎦
− 1 ⎡( 120 + 120
) /2⎤
3 ⎣ 100 100 ⎦
= 0
Had the bias depended on the unadjusted price of
140 compared with 100, the imputation would be
prone to serious error. In this calculation, the direction
of the bias is given by ( r2 − r1)
and does not
depend on whether quality is improving or deterio-
A z > p t+
1
rating, that is, whether ( ) n
1
or A( z) < p t +
t 1
. If A( z) p +
n
>
n
, a quality improvement,
it is still possible that r 2
< r 1
and for
the bias to be negative, a point stressed by Triplett
(2002).
7.98 It is noted that the analysis here is framed
in terms of a short-run price change framework.
This means that the short-run price changes between
two consecutive periods are used for the imputation.
This is different from the long-run imputation,
where a base period price is compared with
prices in subsequent months and where the implicit
assumptions are more restrictive.
7.99 Table 7.3 provides an illustration whereby
the (mean) price change of products that continue
to exist, r 1
, is allowed to vary for values between
1.00 and 1.50: no price change and a 50 percent
increase. The (mean) price change of the qualityadjusted
new products compared with the products
162