### XMPI Manual Chp. 9

```United Nations Economic Commission for Europe
Statistical Division
Calculation of the consumer price index Recommendations of the CPI Manual
Joint EFTA/UNECE/SSCU Seminar
July 2007
Presentation by Carsten B. Hansen, UNECE
Overview
1. Overview of the CPI Manual
2. Calculation of the CPI
3. The size and distribution of the sampled
prices
4. E-commerce
July 2007
UNECE Statistical Division
Slide 2
Overview of the CPI Manual
1.
2.
3.
4.
5.
6.
7.
8.
Introduction
Uses of CPIs
Concepts and scope
Expenditure weights
Sampling
Price collection
Item substitution and new
products
9. Calculation of the CPI in
practice
10. Special cases
11. Errors and bias
12. Organization & management
July 2007
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
Publication and dissemination
The system of price indices
Basic index number theory
Axiomatic and stochastic
approaches
The economic approach, I
The economic approach, II
Price index using an artificial
data set
Elementary indices
Quality changes and hedonics
Seasonal products
Durables and user costs
UNECE Statistical Division
Slide 3
Calculation of the CPI
The CPI calculated in 2 stages:
1. Elementary aggregate indices
Calculated on basis of a sample of prices for
individual products – and perhaps individual
price weights
2. Higher-level indices
Calculated as weighted averages of elementary
aggregate indices, using the expenditure shares as
weights
July 2007
UNECE Statistical Division
Slide 4
Calculation of the CPI
Construction of Elementary aggregates
• Groups of goods or services that are as similar
•
July 2007
as possible, and preferably fairly
homogeneous.
Should consist of products with similar
expected price movements; try to minimize
the dispersion of price movements
UNECE Statistical Division
Slide 5
Calculation of the CPI
Calculation of elementary aggregate price indices:
•
The arithmetic mean of the price ratios – Carli index
1  pti 
P   i 
n  p0 
C
0:t
•
The ratio of arithmetic mean prices – Dutot index
1
i
i
i
i
p

t
p
p

p


t
0
0
D
n
P0:t 

i
1
p
i
0
p0

n
July 2007
UNECE Statistical Division
Slide 6
Calculation of the CPI
•
The geometric mean of the price ratios = the ratio of
geometric mean prices – Jevons index
1n
p 
P   
p 
J
0:t
i
t
i
0
p



 p 
i
t
i
0
1n
1n
How to decide which formula to apply?
•
•
July 2007
The economic approach
The axiomatic or test approach
UNECE Statistical Division
Slide 7
Calculation of the CPI
The economic approach:
•
Assume utility maximizing households with perfect information.
Derive the cost of living index as the ratio of the minimum
expenditures of keeping constant utility:
COLI 0:t 
•
•
•
=>
C  pti ,U 
C  p0i ,U 
The basket is allowed to vary in response to consumer substitution
Usually, quantities are not available in practice
The assumptions are often not realistic
Difficult or impossible to calculated a COLI in practice
July 2007
UNECE Statistical Division
Slide 8
Calculation of the CPI
The axiomatic approach:
Select a number of tests – or axioms – that that the index
should meet. Important tests are:
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.
July 2007
UNECE Statistical Division
Slide 9
Calculation of the CPI
Carli
Dutot
Jevons
Proportionality
yes
yes
yes
Commensurability
yes
no
yes
Time reversal
no
yes
yes
Transitivity
no
yes
yes
•
•
•
=>
July 2007
Carli fails last two – time reversal and transitivity
Dutot fails commensurability
Jevons passes all four
Jevons recommended as the preferred index in general
UNECE Statistical Division
Slide 10
Calculation of the CPI
Example 1: Substitution effect in the Jevons index
May
June
June/May
Item A
10,00
11,00
1,10
Item B
10,00
9,00
0,90
Arithm. Mean
10,00
10,00
1,00
Geomean
10,00
9,95
0,99
Carli
100,00
Dutot
100,00
Jevons
99,50
Jevons allows the households to consume more of B and less of A.
Carli and Dutot keeps the implicit quantities constant
July 2007
UNECE Statistical Division
Slide 11
Calculation of the CPI
Example 2: The Dutot index depends on initial price level
May
June
June/May
Item A
10,00
11,00
1,10
Item B
20,00
18,00
0,90
Arithm. mean
15,00
14,50
0,97
Geomean
14,14
14,07
0,99
Carli
100,00
Dutot
96,67
Jevons
99,50
Jevons and Carli are independent of the price levels
Dutot weights the price changes after the initial price level
July 2007
UNECE Statistical Division
Slide 12
Calculation of the CPI
Example 3: Upward bias in the chained Carli
May
June
June/May
Item A
20,00
25,00
1,25
Item B
25,00
20,00
0,80
Arithm. mean
22,50
22,50
1,00
Geomean
22,36
22,36
1,00
Carli
102,50
Dutot
100,00
Jevons
100,00
Carli gives more weight to price increases than to decreases.
A chained Carli is upward biased and should not be used.
If in July prices return to the prices of May, a chained Carli
gives 102,50 * 102,5/100 = 105,06 !
July 2007
UNECE Statistical Division
Slide 13
Calculation of the CPI
Chained or direct elementary aggregate indices?


•
•
•
A direct index compares the prices of the current month
with those of a fixed reference month
A chained index compares month-to-month price changes
and multiplies the monthly indices into long-term price
indices
Chained and direct index give same results for Dutot and
Jevons
Carli is not transitive and upward biased.
Monthly chained indices appear to have some practical
advantages in the treatment of missing prices and imputations
July 2007
UNECE Statistical Division
Slide 14
Calculation of the CPI
The Calculation of Higher Level Indices
Target indices

Economic index: Fisher, Walsh or Törnqvist

Basket index: Lowe or Laspeyres (?) – or Walsh
The situation in practice
Weight
reference
period
b
July 2007
Price
reference
period
0
Current
period
t
UNECE Statistical Division
End of
T
Slide 15
Calculation of the CPI
In practice, higher-level indices are calculated as the
expenditure share weighted arithmetic average of the
elementary price indices:
I 0:t   wib I i0:t
•
•
It is up to the statistical office to decide whether to priceupdate the weights from b to 0, or not
Caution with automatic price-updating for items with
unusual price development, e.g. pc’s, high-tech.
July 2007
UNECE Statistical Division
Slide 16
The size and distribution of the sampled prices
•
•
•
Outlets and products may be selected on the basis of a
(stratified) probability sample, cut-off sampling, or other
measures.
From time to time the sample should be examined and
updated to ensure its representativity.
Optimize the sample – avoid over-sampling and save
resources and reduce the response burden
July 2007
UNECE Statistical Division
Slide 17
Numbers of price observations
Population
(in mio.)
Total no. of
observations
No. of prices per
mio. inhabitants
Luxembourg
0,4
6.656
15.162
Ireland
3,8
42.379
11.056
Finland
5,2
45.870
8.853
Denmark
5,3
25.000
4.674
Austria
8,0
53.475
6.667
Sweden
8,9
29.899
3.366
Portugal
10,3
103.691
10.110
Belgium
10,3
91.980
8.962
Greece
10,9
33.687
3.082
Netherlands
16,0
115.522
7.226
Spain (E)
40,5
119.143
2.943
Italy (I)
57,0
288.553
5.066
United Kingdom (UK)
59,0
130.981
2.220
France (F)
60,9
171.088
2.809
Germany (D)
82,3
326.615
3.971
Ukraine
47,3
275.000
5.814
EU-15
378,8
1.584.539
4.184
D, E, F, I, UK
299,6
1.036.380
3.459
July 2007
UNECE Statistical Division
Slide 18
The size and distribution of the sampled prices
A practical way of assessing the distribution of prices:
1)
2)
3)
Calculate the average percentage contribution of each
elementary index on the 12-months rate of change of the total
CPI for a period of a year or more
Compare the average relative importance of the elementary
indices with the relative distribution of prices
The distribution of price observations should, roughly,
correspond to the importance of the elementary indices
This is a general measure only – there are exceptions, e.g. for goods
and services with very few suppliers.
July 2007
UNECE Statistical Division
Slide 19
E-commerce
What is E-commerce?
•
•
•
Traditional goods/services purchased from Internet sites
- books, CDs, IT-equipment, clothing,
Goods or services sold exclusively on internet sites
- special models/brands of existing products
- Internet-banking
- e-newspapers
Consumption of Internet provided services
- connection to Internet
- e-mail
- games
- Internet TV and telephone services (Skype etc.)
July 2007
UNECE Statistical Division
Slide 20
E-commerce
•
•
•
The growing importance of E-commerce and the
competition between E-commerce and traditional outlets
is likely to lead to different price developments
As E-commerce grows, it therefore becomes more
important to include it in the CPI
Inclusion of E-commerce may influence both the weights
of the CPI and the price changes
Data sources for the weights
•
The household budget survey
•
IT related medias and journals, research articles, surveys
July 2007
UNECE Statistical Division
Slide 21
E-commerce
Steps to be taken:
•
•
•
•
•
•
Estimate the importance (weight) of E-commerce
Compare the development of E-prices with that of normal
prices
Decide whether to include or exclude E-commerce
If included, select the goods or services to be priced
Decide which Internet web pages/E-outlets from where to
collect prices
In principle the recorded prices should be net of transport
and delivery costs (should be recorded under transport
services). However, in practice it is not always possible to
disentangle delivery costs
July 2007
UNECE Statistical Division
Slide 22

Consumer price index Manual. Theory and practice. ILO
(2004). Electronic version available on:
www.ilo.org/public/english/bureau/stat/guides/cpi/index.htm

The Ottawa Group on Price Statistics. Webpage:
www.ottawagroup.org

The Voorburg Group on Services Statistics. Webpage:
http://www4.statcan.ca/english/voorburg/

Papers from joint UNECE/ILO meetings on CPI are
available on: www.unece.org/stats/archive/docs.date.e.htm
July 2007
UNECE Statistical Division
Slide 23
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