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

15. Basic Index Number Theory
401
Appendix 15.3: Relationship Between
Young Index and its Time
Antithesis
15.105 Recall that the direct Young index,
P Y (p 0 ,p t ,s b ), was defined by equation (15.48) and
its time antithesis, P Y *(p 0 ,p t ,s b ), was defined by
equation (15.52). Define the ith relative price between
months 0 and t as
t 0
(A15.3.1) r ≡ p / p ; i = 1,...,n ,
i i i
and define the weighted average (using the baseyear
weights s i b ) of the r i as
(A15.3.2) r
∗
n
≡ ∑ s r ,
i=
b
i i
which turns out to equal the direct Young index,
P Y (p 0 ,p t ,s b ). Define the deviation e i of r i from their
weighted average r* using the following equation:
∗
(A15.3.3) r = r (1 + e ); i = 1,...,n.
i
i
If equation (A15.3.3) is substituted into equation
(A15.3.2), the following equations are obtained:
∗ b ∗
(A15.3.4) r ≡ s r (1 + e )
(A15.3.5) e
n
∑
i=
1
n
i
r ∗ ∗
= + r ∑ s e since
∗
n
∑
i=
1
i=
1
b
i i
b
≡ s e = 0.
i
i
i
n
∑
i=
1
s
b
i
= 1
Thus, the weighted mean e* of the deviations e i
equals 0.
15.106 The direct Young index, P Y (p 0 ,p t ,s b ), and
its time antithesis, P Y *(p 0 ,p t ,s b ), can be written as
functions of r*, the weights s i b and the deviations
of the price relatives e i as follows:
(A15.3.6)
P p p s r ∗
0 t b
Y
( , , ) = ;
n
∗ 0 t b ⎡ b ∗
Y
= ⎢∑
i
⎣ i=
1
+
i
n
−1
∗⎡
b −1⎤
⎢∑
i
(1
i) ⎥ .
i=
1
(A15.3.7) P ( p , p , s ) s { r (1 e )}
= r s + e
⎣
⎦
−1
−1
⎤
⎥
⎦
15.107 Now, regard P Y *(p 0 ,p t ,s b ) as a function of
the vector of deviations, e ≡ [e 1 ,…,e n ], say P Y *(e).
The second-order Taylor series approximation to
P Y *(e) around the point e = 0 n is given by the following
expression: 81
(A15.3.8)
n n n n
b b b b
P ∗ () e ≈ r ∗ + r ∗ s e + r ∗ s s ee −r ∗ s e
∑ ∑∑ ∑
[ ] 2
Y i i i j i j i i
i= 1 i= 1 j= 1 i=
1
n n n
∗ ∗ ∗ b
⎡
b
⎤
b ∗
= r + r 0 + r ∑ si ⎢ ∑ sjej⎥ei −r*
∑ s ⎡
i ⎣ei
−e
⎤
⎦
i= 1 ⎣ j= 1 ⎦
i=
1
using equation (A15.3.5)
n
n
∗ ∗ b
∗ b ∗
∑ i [ 0]
i ∑ ⎡
i ⎣ i
i= 1 i=
1
= r + r s e −r s e −e
⎤
⎦
using equation (A15.3.5)
n
0 t b 0 t b b
Y( , , )
Y( , , ) ∑ ⎡
i ⎣ i
i=
1
= P p p s −P p p s s e −e ∗ ⎤
⎦
using equation (A15.3.6)
= P p p s − P p p s Var e
0 t b 0 t b
Y( , , )
Y( , , )
where the weighted sample variance of the vector e
of price deviations is defined as
b
(A15.3.9) Var e ≡ s ⎡
i ⎣ei
−e ∗ ⎤
⎦ .
n
∑
i=
1
15.108 Rearranging equation (A15.3.8) gives the
following approximate relationship between the direct
Young index P Y (p 0 ,p t ,s b ) and its time antithesis
P Y *(p 0 ,p t ,s b ), to the accuracy of a second-order
Taylor series approximation about a price point
where the month t price vector is proportional to
the month 0 price vector:
(A15.3.10)
0 t b ∗ 0 t b 0 t b
P ( p , p , s ) ≈ P ( p , p , s ) + P ( p , p , s ) Var e.
Y Y Y
81 This type of second-order approximation is credited to
Dalén (1992, p. 143) for the case r* = 1 and to Diewert
(1995, p. 29) for the case of a general r*.
2
2
2
2