Transcript United Nations Economic Commission for Europe Statistical Division UNECE Workshop on Consumer Price Indices Istanbul, Turkey,10-13 October 2011 Session 4: Calculation of elementary price indices Presentation.

United Nations Economic Commission for Europe
Statistical Division
UNECE Workshop on Consumer Price Indices
Istanbul, Turkey,10-13 October 2011
Session 4: Calculation of elementary
price indices
Presentation by Carsten Boldsen, UNECE
Overview
1.
2.
3.
4.
5.
Elementary aggregates
Calculation of elementary price indices
Use of weights in elementary indices
Chained or direct elementary indices?
Conclusion
UNECE Statistics Division
Slide 2
1. Elementary aggregates
The typical aggregation structure
Overall index
Higher-level indices
Elementary Indices
Individual price observations
UNECE Statistics Division
Value shares
(weights)
Slide 3
1. Elementary aggregates
Grouping of elementary aggregates:
Products – goods or services – that are as
similar as possible, i.e. homogeneous
Group products with similar price movements
to minimize expected dispersion of price
movements
In the absence of weights for the individual price
observations there are 3 main formulas for calculating
elementary indices …
UNECE Statistics Division
Slide 4
2. Calculation of elementary price indices
Carli index – the arithmetic mean of the price ratios
i


p
1
C
t
P0:t    i 
n
 p0 
Dutot index – the ratio of arithmetic mean prices
1
1
i
i
i
i
p
p
p

p



t
t
0
0
D
n
n
P0:t 

1
1
i
i
p
p


0
0
n
n
UNECE Statistics Division
Slide 5
2. Calculation of elementary price indices
Jevons index – the geometric mean of the price ratios
= the ratio of geometric mean prices
1n
i


p
J
t
P0:t    i 
 p0 
p 



 p 
UNECE Statistics Division
i
t
i
0
1n
1n
Slide 6
2. Calculation of elementary price indices
Example 1: Dutot, Carli and Jevons
January
February
Feb/Jan
A
10
8
0,8
B
14
14
1
C
20
24
1,2
Arithmetic mean
14,67
15,33
Geometric mean
14,09
13,90
Dutot
=
15,33/14,67 * 100
=
104,5
Carli
=
(0,8 + 1 + 1,2)/3 * 100
=
100,0
Jevons
=
13,90/14,09 * 100
=
98,6
=
(0,8 * 1 * 1,2) (1/3) * 100
=
98,6
UNECE Statistics Division
Slide 7
2. Calculation of elementary price indices
Example 2: Substitution effect in the Jevons index
May
June
June/May
Item A
10
12
1,20
Item B
10
8
0,80
Arithm. Mean
10,00
10,00
1,00
Geomean
10,00
9,80
0,98
Carli
Dutot
100,0
100,0
Jevons
98,00
Carli and Dutot keeps the implicit quantities constant
Jevons allows some substitution - households consume more of B
and less of A!
UNECE Statistics Division
Slide 8
2. Calculation of elementary price indices
Example 3: Upward bias in Carli
May
June
June/May
Item A
20
25
1,25
Item B
25
20
0,80
Arithm. Mean
22,50
22,50
1,00
Geomean
22,36
22,36
1,00
Carli
= (1,25 + 0,80)/2 * 100
=
102,5
Dutot
100,0
Jevons
100,0
Carli gives more weight to price increases than to decreases!
UNECE Statistics Division
Slide 9
2. Calculation of elementary price indices
Example 4: Dutot depends on the price level
December
January
Index
Product 1
25,53
16,06
62,9
Product 2
69,5
69,5
100,0
Product 3
201,67
221,67
109,9
98,9
102,4
Av. Price
Dutot index
102,4/98,9 * 100 =
103,5
Carli index
(62,9+100+109,9)/3 * 100 =
90,9
Price changes in Dutot are weighted according to the price in the reference period:
Price
Price weight
Product 1
25,53
0,09
Product 2
69,5
0,23
Product 3
201,67
0,68
296,7
1,00
Sum
Dutot index
62,9*0,09+100*0,23+109,9*0,68 =
UNECE Statistics Division
103,5
Slide 10
2. Calculation of elementary price indices
How to decide which formula to apply?
The economic approach - focuses on the
economic interpretation of the index
The axiomatic or test approach - focuses on the
statistical properties of the index
UNECE Statistics Division
Slide 11
2. Calculation of elementary price indices
The economic approach:
•
Assume utility maximizing households with perfect information.
The cost of living index is the ratio of the minimum
expenditures of keeping constant utility:
COLI 0:t 
•
•
•
=>
C  pti ,U 
C  p0i ,U 
The basket may change in response to consumer substitution
Usually, quantities are not available in practice
The assumptions are often not realistic
Difficult to calculated a COLI in practice
UNECE Statistics Division
Slide 12
2. Calculation of elementary price indices
The axiomatic approach:
Select a number of tests – axioms – that the index
should meet.
The more important tests:
Proportionality: If all prices change x%, the index should
also change by x%
Commensurability: The index should be invariant compared
to the unit in which prices are recorded
Time reversal: The index from period 0 to period t should
equal the reciprocal of the index from t to 0
Transitivity: The index from 0 to 1 multiplied (chained) by an
index from 1 to 2 should equal a direct index from 0 to 2.
UNECE Statistics Division
Slide 13
2. Calculation of elementary price indices
Carli
Dutot
Jevons
Proportionality
yes
yes
yes
Commensurability
yes
no
yes
Time reversal
no
yes
yes
Transitivity
no
yes
yes
•
Carli fails time reversal and transitivity
•
Dutot fails commensurability
•
Jevons passes all four
☞ Jevons recommended as the preferred index in
general
UNECE Statistics Division
Slide 14
3. Use of weights in elementary indices
Weighted averages
Laspeyres price index
La
0:t
P
pq 

 p


 w 
p
 p q 
i
t
i
0
i
0
i
0
i
t
i
0
i
0



Geometric Laspeyres price index
p

 pti 

  i  
wi
i
 p0 
p

 0
wi
P0:GLa
t
i
t
wi
UNECE Statistics Division
Slide 15
3. Use of weights in elementary indices
W
Dec
Jan
% change
A – Local calls
0.80
7
9
28,6
B – National calls
0.17
20
10
-50,0
C – International calls
0.03
28
12
-57,1
Weighted arithmetic mean of price ratios (Laspeyres)
(0,8*(9/7)+0,17*(10/20)+0,03*(12/28)) * 100 =
Index
112.64
Weighted geometric mean of price ratios (geometric Laspeyres)
((9/7)0.8 * (10/20)0.17 * (12/28)0.03) * 100 =
UNECE Statistics Division
105.95
Slide 16
4. Chained or direct elementary indices?

A direct index compares the prices of the current month with
those of a fixed reference month
P0:t  P0:t  p0 , pt 

A chained index compares month-on-month price changes
and multiplies the monthly indices into a long-term index
P0:t  P0:t  p0 , p1 , p2 ,..., pt 1 , pt   P0:1  P1:2  P2:3  ...  Pt 1:t

Direct = chained index when based on average prices,
and no replacements
pt
,
pt
pt
p1 p2 p3
P0:t 
   ... 

p0 p1 p2
pt 1 p0
UNECE Statistics Division
Slide 17
2. Calculation of elementary price indices
Example 5: A chained Carli index is biased upwards
Jan.
Feb.
March
Prices
A
40
45
44
B
60
55
66
Monthly price ratios
A
1,13
0,98
C
0,92
1,20
102,1
108,9
102,1
111,2
Monthly index
Chained monthly index
100
UNECE Statistics Division
Slide 18
5. Conclusion
Group homogenous products with similar expected
price movements into elementary aggregates
Carli and Jevons are independent of the price levels
– Dutot depends on the initial price levels
A chained Carli is upward biased and should not be
used
The Dutot index should only be used for
homogenous elementary aggregates
UNECE Statistics Division
Slide 19
5. Conclusion
Jevons is the generally recommended index
because of better statistical properties
Monthly chained indices appear to have some
practical advantages in the treatment of missing
prices and replacements
Explicit weights may be applied for the calculation of
elementary indices
Without explicit weights, there will still be implicit
weighting from the sampling!
UNECE Statistics Division
Slide 20