

HANDBOOK OF THE INTERNATIONAL COMPARISON PROGRAMME
 Purchasingpower parities for basic headings
 The EKS method
 The CPD approach
 Aggregation of the basic heading parities up to the
level of GDP
 The GK and EKS methods in brief
 Linking of regional results and the fixity question
 Extrapolation of benchmark estimates to other years
 PROCESSING OF THE BASIC DATA
 This chapter briefly sketches the methods of processing the price
and expenditure data provided by countries to the ICP coordinators.
Section A takes up the calculation of parities at the basic heading
level. Ube discussion in this section is fairly technical because it
is felt that a good understanding of estimation of the detailed heading
parities should make it possible for country statistical offices to
better appreciate the type of price information required by the ICP.
Considerable space is therefore given to the basic heading parities
with the view that this will help improve the overall quality of the
comparisons. Section B concerns aggregation from the basic heading level
to GDP. Many of the technical aspects of the methods are discussed in
Annex II. Section C of this chapter briefly discusses the question of
extrapolating benchmark estimates to years other than the benchmark
reference year, a task that country statistical offices may be called
upon to perform.
 By way of contrast, less space is devoted to aggregations above the
basic heading level. This is partly because these index number problems
have been the subject of a number of expert group meetings over the
past several years, and so there is ample material available discussing
the issues. 16/ Further, there is no unanimity about how the aggregations
should be made to build up either regional or world comparisons, so
the Handbook will simply sketch some of the issues and methods employed.
One general question is whether there should be symmetry in the methods
employed to obtain parities at the basic heading level and those used
to aggregate the basic headings. This issue was discussed in the expert
group meetings, but no consensus was reached; in the Handbook, the methods
of obtaining basic heading parities are treated independently from aggregations
of the basic headings.
A. Purchasingpower Parities for basic headings
 The parity at the basic heading level is an average of the individual
item price ratios of the specifications belonging to a given basic heading.
This section discusses two principal approaches to estimating these
basic heading parities, namely, the ÈltetöKövesSzulc
(EKS) and the countryproductdummy (CPD) methods. The principal differences
in the estimates generated by these two methods arise at the level of
the basic heading; as one moves to aggregations of the basic headings,
the overall results are unlikely to be affected by which method is chosen.
Both methods are described here because both have been extensively used
in ICP work.
 For purposes of illustration of how parities are obtained for basic
headings, a price tableau for a basic heading for four countries for
eight specifications will be used. In this illustration, no item weights
are provided but the countries have been able to indicate whether items
are important in their consumption by an asterisk (*). In this example,
country A will be taken as the numeraire, and price ratios between all
pairs of countries are given in rows (5) to (10), with those with respect
to country A given first. The six price ratios given will be termed
direct price ratios because they are formed directly by taking the prices
of the two countries, as in (B/A). An indirect price ratio derived from
two direct ratios, as the product of [(B/A) x (C/B)), will be denoted
(C/A)^.
Tableau of Prices and Price Ratios

Items


1

2

3

4

5

6

7

8

Countries

Prices

(1) A

2*

6*





10



1*

4

(2) B

12

35

3*

5

40*





18

(3) C

25

50

7

12*



10*





(4) D

150*

400*



100



70*

80



Country/Country

Price
ratios

(5) B/A

6

5.83





4





4.50

(6) C/A

12.5

8.33













(7) D/A

75

66.67









8.0



(8) C/B

2.083

1.429

2.333

2.40









(9) D/B

12.5

11.429



20.0









(10) D/C

6.0

8.0



8.333



7.0





 Binary comparisons at the basic
heading level are quite straightforward. Consider countries A and B
in the example above. The parity between A and B for the category is
taken as the geometric mean of the price ratios for the matching items
1, 2, 5, and 8, which, in the example, is 5.01 = (6 x 5.83 x 4 x 4.5
)^{1/4} . As noted, no item weights are given. However, where
the asterisked item system is used, importance of items is taken into
account in the following way: whenever items are marked with an asterisk
in one of the two countries, they are included in the calculation of
the parities. In the example above, only items 1, 2 and 5 would be included
in the comparison of A and B because these items have an asterisk (*)
in at least one of the two countries. The parity between A and B estimated
on the basis of asterisked items would be 5.19 = (6 x 5.83 x 4)^{1/3}.
 The price tableau in the illustration
has a number of items for which countries have not provided prices,
which is the usual situation. Suppose, however, that we consider a complete
tableau of only items (1) and (2), where each country has provided prices
for both items. In this case, the binary comparisons between each pair
of countries is transitive so that (C/A)^ = (C/A), that is, the direct
parity between C/A would be equal to the product of the parities B/A
and C/B. This can be seen below, where the price ratios are repeated
for items (1) and (2) from the price tableau above, and the geometric
mean of the price comparison is given for all possible binary comparisons:

B/A

C/A

D/A

C/B

D/B

D/C

Item 1

6.00

12.50

75.00

2.083

12.50

6.00

Item 2

5.833

8.333

66.67

1.428

11.43

8.00

Geometric mean

5.916

10.206

70.71

1.725

11.95

6.93

The geometric mean of B/A is
equal to (C/A)/(C/B), that is, 5.916 = 10.206/1.725, and so forth for
any other direct and indirect binary comparisons. Also, use of geometric
averages produces end results that remain base country invariant, i.e.
they are not influenced by which country played the role of numerator
and which of denominator. (With arithmetic averages, this would not
be the case, since the unweighted arithmetic average of the A/B ratios
is not the reciprocal of the unweighted arithmetic average of the B/A
ratios.)
 In chapter III a desirable property
of the geometric mean was mentioned in connection with timetotime
indexes. The property is that the ratio of the geometric mean of two
series is equal to the geometric mean of the product of the ratios of
the two series. In the above example, using only items 1 and 2, we may
note that the geometric mean of prices in B (12 x 35)^{1/2}'divided
by A (2 x 6)^{1/2} is 5.916 = 20.494/3.464. This leads us to
a discussion of the ÈltetöKövesSzulc (EKS) method,
which allows estimation of transitive multilateral parities based on
all possible binary comparisons.
1. The EKS
method
 When the price tableau is complete,
we have noted that the direct binary parity between B and A is equal
to the indirect binary derived through third countries such as C or
D. However, this is not the case when the price tableau is incomplete,
as can be seen from the geometric mean given below based on the full
price tableau:

B/A

C/A

D/A

C/B

D/B

D/C

Geometric mean

5.01

10.206

73.681

2.021

12.132

7.274

Geometric mean*

5.19

10.206

73.681

2.366

11.953

7.274

The first row above gives the
geometric mean of the price ratios between each possible pair of countries,
using all the prices in the price tableau, while the second row only
uses price ratios where the item has an asterisk (*) in at least one
of the countries. Considering (B/A), the direct ratio, it can be seen
that it does not equal the indirect ratio, (B/A)" = (C/A)/(C/B) in either
row. 17/ Or, put another way, transitivity is lost.
 The EKS method permits transitivity
to be restored by taking into account the indirect and direct comparisons
by the formula in the following equation:
(1)
The term "PP" is used to denote
a parity at the basic heading level. In EKS, the direct parities (PP_{ji},
where i = j) and (PP_{ki}, where i = k), are each counted, while
each indirect parity is counted once. In the above example with four
countries, the EKS calculation of the (C/A) parity from the geometric
mean for the asterisk (*) approach is:
C/A = [ (C/A) x (C/A) x ( (C/D)
x (D/A) ) x { (C/B) x (B/A) }] ¼, or
C/A = (10.206 x 10.206 x 10.130
x 12.280]1/4 = 10.670.
 All the EKS estimates, using
all prices, and the asterisk approach are given below:

B/A

C/A

D/A

C/B

D/B

D/C

EKS

5.267

10.167

70.352

1.930

13.357

6.920

EKS*

5.173

10.670

70.710

2.063

13.669

6.627

 An advantage of the EKS method
is that it produces transitivity and makes use of all the price information
available, including both direct price comparisons between each pair
of countries, and all indirect price relationships between each pair
of countries and the remaining countries. The EKS method is derived
from a minimization procedure that was basically mathematical in formulation,
though it can also be derived in a weighted form from some general considerations
of consumer behaviour. The CPD method that follows is derived from an
explicit model of how the price tableau is generated. 18/
2. The CPD
approach
 An alternative way of dealing
with an incomplete matrix of prices is the countryproductdummy (CPD)
procedure developed by Robert Summers (see Summers, 1973). It has been
employed in the ICP calculations for the initial studies, although in
recent years most regions have preferred to adopt the EKS method. CPD
is a multilateral method in which regression analysis is used to obtain
transitive parities for each basic heading. The prices are regressed
against two sets of dummyvariables: one set contains a dummy for each
specification and the second set a dummy for each country other than
the numeraire country. The transitive parities are derived from the
coefficients of the country dummies. The estimating equation is as follows:
(2) ln P_{j/k} = b_{1}X_{1}
+ b_{2}X_{2} + . . . + b_{n1}X_{n1}
+ Z_{1}Y_{1} + Z_{2}Y_{2} + . . . +
Z_{m}Y_{m} + u,
where n = number of countries,
m = number of items in a basic heading, j = 1,2,...,n1; k = 1,2 ....
m, and where ln P is the natural logarithm of the price of an item k
in country j. Each of the n1 countries being compared, other than the
numeraire country, is represented by an X dummy variable, and each of
the m items in the heading is represented by a Y dummy variable. The
country coefficients, the bs, are the natural logarithm of the estimated
country parity for the heading, and the item coefficients, the zs, are
the natural logarithms of the estimates of the item prices in the currency
of the numeraire country.
 The CPD estimates given below
are based on the price tableau, using all the observations in paragraph
213. The very high correlation is spurious since it basically results
from explaining the variance in the initial observations owing to different
currency units. Similarly, the size of the t statistics on the item
coefficients is of only limited application. However, the country and
item coefficients are of interest, particularly in column (3), where
they are given in their exponentiated form. The coefficients for each
country are pps in terms of currency unit of the country compared to
the numeraire country A, and the item prices are the estimated average
price of each item expressed in the currency unit of country A.
CPD regression example
Variable

Coefficient

t statistic

PPP
and item price estimates





Country B

1.574

17.55

4.83

Country C

2.315

22.44

10.12

Country D

4.296

43.88

73.44

Item 1

0.805

8.77

2.24

Item 2

1.766

19.24

5.85

Item 3

0.422

3.26

0.66

Item 4

0.171

1.51

1.19

Item 5

2.208

20.60

9.10

Item 6

0.030

0.23

0.97

Item 7

0.043

0.39

1.04

Item 8

1.351

12.60

3.86

adj R^{2} = 0.998
n = 21 df = 11 where n number of price observations and df the
degrees of freedom

 Turning first to the country
estimates, the value of country A is 1.0, since it is the numeraire.
If these coefficients are compared to the EKS estimates in paragraph
219, the largest difference is for B, about 9 per cent. As was mentioned
earlier, if there are no holes in the price matrix, then the CPD and
EKS estimates are identical and all direct and indirect binary parities
between countries are transitive. The more prices missing from a given
price matrix, the less reliable are either the EKS or the CPD estimates
compared to a direct countrytocountry comparison using the geometric
average of price ratios; and the more prices missing, the larger will
be the differences between the EKS and the CPD estimates. It is not
possible to say which method is closer to the truth; both are approximations.
 The CPD method estimates a common
item price in the currency unit of the numeraire country that, together
with the heading parity, in effect produces a full price matrix. The
item prices given above are a part of the estimation procedure of the
CPD that are of considerable interest in themselves since they are an
estimate of the average price for each specification in the currency
of the numeraire country across the group of countries. In regional
comparisons, for example, a byproduct of application of the CPD method
in the ESCAP region in 1985 was a set of Asian item prices. These CPD
prices provide a basis of comparison for any country in a region of
their specification prices with the average. They also have applications
where it may be desired to link a country that did not participate in
the benchmark comparison to an existing ICP study. For example, for
a country whose prices were available at a later date than the initial
CPD for a region, it is possible to link the country to a regional or
world comparison in the following way: first, the item prices within
a basic heading for such a country would be divided by the CPD estimates
of the same item prices in a numeraire currency; then the geometric
mean of these price ratios would be calculated to provide the basic
heading parity that would allow the country to be linked to the regional
and world comparison. This exercise was carried out as a nonofficial
research exercise for Taiwan Province of China, based on the CPD average
ESCAP prices for 1985.
 The EKS and CPD systems can
also be used with weights for the individual items, or by use of the
asterisk (*) system. In phase IV, for example, the CPD system was used
for 20 core countries that were used to link the various world regions.
Some core countries had items marked with an asterisk (*) and these
were given larger weight than nonasterisk items within any basic heading.
 The Handbook considers the CPD
and the EKS as two alternative methods for the multilateral parity calculations
at the basic heading level, without trying to give a clear preference
to one or the other. The actual conditions in a given region as well
as the preferences of regional experts and organizers should determine
which of the two methods is applied.
B. Aggregation
of the basic heading Parities up to the level of GDP
 Once parities are obtained for
each basic heading the aggregated results have to satisfy the basic
requirements of international comparisons, commensurably. Expenditure
data must be converted by these parities from national currency to the
currency of the numeraire country or to an international currency unit.
When expenditure data are converted by parities into a common currency
and unit of account at the basic heading level, they are then comparable
across countries. Thus, by dividing the expenditure of country A by
the expenditure of country B in the same currency, interspatial quantity
indexes can be obtained for each basic heading. At the basic heading
level the quantity estimates vary in reliability and the volume of data
is large so they usually are not published. However, the basic heading
data are the building blocks necessary to obtain the converted aggregates
for both the summary categories and GDP.
1. The GK
and EKS methods in brief
 To aggregate the basic heading
parities and expenditures, a method adopted from the suggestion of Geary
has usually been employed in comparisons at the regional as well as
the world level. 19/ This method is known as the GearyKhamis or GK
method and produces transitive comparisons between all countries. The
EKS and other methods proposed for aggregation also produce transitive
results at the level of GDP.
 A principal advantage of the
GK method is that it produces additive results that have the property
of matrix consistency, where the results can be compared down the basic
headings and across countries for any basic heading or aggregation.
There are strong arguments that gross domestic product should retain
this property even after conversion to another currency since it is
in accord with standard national accounting practice. Such additive
consistency is advantageous not only because it permits an easier analysis
of the structure of the aggregates (e.g., it enables the calculation
of distribution percentages), but also because it allows comparison
across countries.
 Not all index formulas provide
additive results. Neither the Fisher ideal formula (the geometric mean
of the Laspeyres and Paasche formulas), nor any method based on the
Fisher formula (such as the EKS) will meet the additivity requirement.
Nor do chain indexes, where different weights are used in the different
defined composite elements (in the various bilateral comparisons), meet
this requirement.
 Any method of aggregation uses
some implicit or explicit set of weights for the importance of each
country in the comparison. In the usual GK application, countries are
accorded the weight of their own total GDP in the aggregation. 20/ This
accords with standard national accounts methodology, where prices embedded
in national accounts are an average weighted by the quantities produced
in each region.
 Most other methods of aggregation
use a weighting system that accords the same importance to each country.
For example, the implicit weighting system of EKS type systems gives
the same importance to, say, Luxembourg, as to France, even though France's
economy is over 50 times larger than that of Luxembourg. As a consequence,
the different aggregation methods produce different results at the aggregate
level. These questions are discussed in more detail in annex II, but
as an empirical generalization it can be said that systems such as EKS
tend to produce somewhat larger differences between per capita incomes
between rich and poor countries than the GK method. All of the aggregation
systems proposed produce results much closer to one another than to
the nominal results obtained by converting by exchange rates.
 While aggregation methods such
as the EKS and GK systems move us towards better measures or real output
between countries, there is not yet agreement on criteria that would
allow one to say that one system was to be always used. To summarize,
the main claim for the GK method is that it follows the conventions
of national income accounting and produces additive results. As discussed
in annex II, the EKS system may have more basis in consumer theory than
the GK. In GK, the quantity of each item of a country is the weight,
while in EKS each country is given equal weight. There remain differences
among the experts on which system should be adopted, and in phase VI
it is likely that the results of both methods will be presented. 21/
2. Linking
of regional results and the fixity question
 There is one more problem that
arises when passing from the regional comparisons to the world comparison.
If the GK or another aggregation system is applied at the regional
level, results will be expressed at regional average prices, which vary
from region to region. How should the results of the world comparison
be expressed? If world average prices are used as weights, the results
obtained in the world comparison between any two countries belonging
to the same region may be different from those obtained originally in
the regional comparison.
 Many users and producers of
ICP results would like to avoid having two different relationships between,
say, France and Italy, depending on whether the result was obtained
from an EC comparison or a world comparison where different international
prices prevail. This view is especially strong in those regions where
ICP results are also used for administrative purposes, as in the European
Community. It explains why, in phases IV and V, the ICP organizers accepted
the socalled "fixity principle", which requires that results obtained
in a regional comparison remain unchanged in a comparison covering a
larger number of countries.
 The price to be paid for complying
with the fixity requirement is relatively high. Essentially the matrix
consistency of the GK method must be given up at the world level if
the fixity principle is applied. 22/ However, this limitation is only
observed in official publications. For purposes of research concerned
with the structure of the world economy across regions, individual researchers
or research organizations may aggregate the basic heading data in other
ways that may be more suitable for the analysis of the economic structure
of countries.
C. Extrapolation
of benchmark estimates to other years
 Typically, benchmark estimates
are obtained every five years. However, since benchmark estimates are
not available until at least two or three years after the benchmark
year, this means that the latest available benchmark estimate for a
participating country may be from two to eight years prior to the current
year. This is one reason why countries often need to approximate estimates
between benchmarks. In the case of the OECD countries, these extrapolations
are regularly published with estimates of real GDP and the implicit
PPPs moved backwards and forwards from the latest benchmark estimate.
The European Community has gone further in this direction, moving towards
annual benchmarks. For EC, this partly reflects the fact that operational
uses of real output numbers often require very current estimates. 23/
 The general method of extrapolation
is quite straightforward. OECD, for example, can take a benchmark GDP
estimate of each country for, say, 1985 and extrapolate it forward and
backwards by the national growth rate of GDP for each country. The benchmark
estimate is in 1985 dollars and the entire series for nonbenchmark
years will also be in 1985 dollars. One can easily obtain an implicit
purchasing power parity from this type of extrapolation. 24/ This discussion
has been framed in terms of national growth rates as the basis for extrapolation
and it should be noted that one could also have extrapolated PPP for
GDP using the implicit deflator.
 The same method can be used
for any subaggregate of GDP for which national growth rates (implicit
deflators) are available. Further, if one extrapolates, say, the main
components of GDP from a benchmark year to a later year, one could simply
add up these components to obtain an estimate of GDP. This estimate
of GDP would not be the same as that using the national growth rate
of GDP. The reason for this is that the national growth rates of components
are in one case being weighted by the shares of GDP in international
prices and in the other by shares at national prices. The case for extrapolating
components at international prices to obtain GDP growth is that it most
closely replicates what a new benchmark estimate will produce. The case
for using the national growth rate of GDP, and perhaps distributing
components so as to preserve additivity, is that it does preserve the
national growth rate. At present, there is not a recommended practice,
and the method used is likely to depend on the specific purpose for
which the extrapolation is being carried out.
16/ Meetings were held at EUROSTAT
in June 1989 and at OECD in June 1990. These expert group meetings were
jointly sponsored by EUROSTAT, OECD and the Statistical Office of the
United Nations Secretariat. Reports of the meetings are available from
any of the secretariats.
17/ That is, the direct value of
(B/A) in the first row is 5.01, while (B/A )A is 10.206 (C/A)/2.021 (C/B)
= 5.05; in the second row the direct value is 5.19 and the indirect is
4.31.
18/ In the EKS method, the importance
assigned to individual price observations is variable and is not selfevident.
Even when the asterisk system is used, the importance assigned to individual
prices will depend on the number of observations and on whether they are
marked with an asterisk (*). Further, the weight given to indirect price
ratios will normally not depend on the number of price observations involved
unless explicit weighting is employed. This last problem is also present
in the CPD method, where it has been explicitly treated by assigning the
same weight to each country so that each price for a country will receive
a weight inversely related to the total number of prices for a basic heading
for the country. While this could be done in EKS, in applications of EKS
it has not been carried out.
19/ Citations to Geary and early
phases of the ICP are provided in a volume reviewing aggregation methods
(Hill, 1982).
20/ Other systems of weighting
are discussed in annex II.
21/ In the phase IIII reports,
the results of seven different aggregation methods are reported at the
GDP level. These results allow one to gain an impression of how sensitive
the results are to the method of aggregation used. See, for example, Kravis,
Heston and Summers (1982), pp. 9697.
22/ If countries within a region
retain their relationship at the GDP level obtained from a regional aggregation,
then when they are linked to the world comparison some compromise must
be adopted. There are several ways to carry out such a linking, two of
which are discussed in United Nations, Economic Commission for Europe
(1985) and United Nations and EUROSTAT (1986).
23/ Another reason that EC is moving
towards annual estimates is because successive benchmark estimates do
not necessarily give results that are consistent with the deflated growth
estimates of the countries. In effect, by moving to annual estimates,
EC will be generating purchasingpower parity estimates that are more
consistent with the national deflation practices of the countries.
24/ There are several ways one
could do this. One method would be to take the extrapolated value of GDP
for a country in a particular year, say Italy in 1990, as a ratio to the
value for the United States, the OECD numeraire. Similarly, calculate
the same ratio converting Italy's lira GDP in 1990 in current prices at
exchange rates relative to current United States GDP. The ratio of the
GDPs at exchange rates to the ratio at real 1985 dollars for 1990 gives
an estimate of the comparative price level of Italy for 1990, which when
multiplied by the exchange rate yields the estimated PPP of Italy for
1990.

