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

Producer Price Index Manual
tm , t+
1, m 0, m
(22.8) PAL
( p , p , s )
m = 1, 2,...12;
n∈S( m)
tm , t+
1, m 0, m
(22.9) PAP
( p , p , s )
m = 1, 2,...12;
( +
)
= ∑ s p p ;
0, m t 1, m t,
m
n n n
−1
−1
⎡
⎤
0, m t+
1, m t,
m
= ⎢ ∑ sn ( pn pn
) ⎥
⎢⎣n∈
S( m)
⎥
;
⎦
t, m t+
1, m 0, m 0, m
(22.10) PAF
( p , p , s , s )
tm , t+ 1, m 0, m tm , t+
1, m 0, m
PAL
( p , p , s ) P ( p , p , s )
≡ ;
m = 1, 2,...,12
−1
−1
⎡
⎤
tm , t+ 1, m tm , tm , t+
1, m tm ,
= ∑ sn ( pn pn ) ⎢ ∑ sn ( pn pn
) ⎥
n∈S( m)
⎢⎣n∈S ( m)
⎥⎦
.
22.23 The approximate Fisher year-over-year
monthly indices defined by equation (22.10) will
provide adequate approximations to their true
Fisher counterparts defined by equation (22.6)
only if the monthly revenue shares for the base
year 0 are not too different from their current year t
and t+1 counterparts. Thus, it will be useful to
construct the true Fisher indices on a delayed basis
in order to check the adequacy of the approximate
Fisher indices defined by equation (22.10).
22.24 The year-over-year monthly approximate
Fisher indices defined by equation (22.10) will
normally have a certain amount of upward bias,
since these indices cannot reflect long-term substitution
toward products that are becoming relatively
cheaper over time. This reinforces the case for
computing true year-over-year monthly Fisher indices
defined by equation (22.6) on a delayed basis,
so that this substitution bias can be estimated.
22.25 Note that the approximate year-over-year
monthly Laspeyres and Paasche indices, P AL and
P AP defined by equations (22.8) and (22.9), satisfy
the following inequalities:
tm , t+
1, m 0, m
(22.11) PAL
( p , p , s )
t+
1, m t, m 0, m
PAL
( p p s )
× , , ≥ 1;
m = 1, 2,...,12;
+
P p , p , s
tm , t 1, m 0, m
(22.12)
AP ( )
t+
1, m t, m 0, m
PAP
( p p s )
m = 1, 2,...,12;
× , , ≤ 1;
with strict inequalities if the monthly price vectors
p t,m and p t+1,m are not proportional to each other. 13
Equation (22.11) says that the approximate yearover-year
monthly Laspeyres index fails the time
reversal test with an upward bias while equation
(22.12) says that the approximate year-over-year
monthly Paasche index fails the time reversal test
with a downward bias. As a result, the fixedweights
approximate Laspeyres index P AL has a
built-in upward bias while the fixed-weights approximate
Paasche index P AP has a built-in downward
bias. Statistical agencies should avoid the use
of these formulas. However, they can be combined,
as in the approximate Fisher formula in equation
(22.10). The resulting index should be free from
any systematic formula bias, although some substitution
bias could still exist.
22.26 The year-over-year monthly indices defined
in this section are illustrated using the artificial
data set tabled in Section B. Although fixedbase
indices were not formally defined in this section,
these indices have similar formulas to the
year-over-year indices that were defined, with the
exception that the variable base year t is replaced
by the fixed-base year 0. The resulting 12 yearover-year
monthly fixed-base Laspeyres, Paasche,
and Fisher indices are listed in Tables 22.3 to 22.5.
22.27 Comparing the entries in Tables 22.3 and
22.4, it can be seen that the year-over-year
monthly fixed-base Laspeyres and Paasche price
indices do not differ substantially for the early
months of the year. However, there are substantial
differences between the indices for the last five
months of the year by the time the year 1973 is
reached. The largest percentage difference between
the Laspeyres and Paasche indices is 12.5 percent
for month 10 in 1973 (1.4060/1.2496 = 1.125).
13 See Hardy, Littlewood, and Pólya (1934, p. 26).
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