

HANDBOOK OF THE INTERNATIONAL COMPARISON PROGRAMME
Annex II  METHODS OF AGGREGATION
 The present annex begins with the GearyKhamis method of aggregation,
explaining some of its advantages and disadvantages. It then compares
the GK results with several other methods of aggregation to illustrate
some of the differences.
A. The Geary system
 The valuation of a country's output in international prices can be
written as:
(1)
and where the p _{i}s are the international prices for
each of the basic headings and rgdp_{j} is GDP of country j
valued at those prices. The particular contribution of Geary was to
define the international prices in such a way that they would produce
an overall PPP for a country that was consistent with the prices. The
definition of the PPP in the ICP is:
(2)
where E_{ij} is the expenditure in national currency on basic
heading i by country j. That is, the purchasingpower parity over GDP
is the ratio of the GDP of a country in national currency to its GDP
in international prices.
 For Geary there were actual quantities and prices associated with
the agricultural output that he was concerned with valuing across countries.
The international prices would be in a numeraire currency, such as the
dollar, and the international prices would be so many dollars per unit
quantity, say, ton of rice. In the ICP, there are basic heading parities,
PPijS, that have been generated by EKS or CPD. These basic heading parities
have the dimension of units of currency of country j to the numeraire
currency for the basic heading.
 This means that the interpretation of quantity and price at the basic
heading level are not tons and rupees per ton. Rather, the quantity
in the GK technique as used in the ICP is what is termed a notional
quantity. It is defined as:
(3)
Each country's expenditure for a basic heading is converted to the
currency of the numeraire country; it is termed a notional quantity
because it serves the function of a quantity with its values at numeraire
country prices.
 One might ask why one cannot simply add up the notional quantities
for each basic heading for a country to get a GDP in a common currency.
The answer is that the result would use the relative prices between
each basic heading that prevailed in the numeraire country. This means
that the total would depend on which country was chosen as numeraire,
and the result would not be base country invariant.
 In the GK system, the international price for heading i is defined
as:
(4)
Equation (4) has been written as a weighted sum of the ratios of the
heading parities to the aggregate PPP. The weights used to obtain the
international prices typically are the notional quantities. Usually,
the expenditures (E_{ij}S) entering into equation (3) are the
total expenditures of a country, though alternative weights have been
used. a/ For each country this is a ratio that will centre on 1.0 because
in the Geary system the PPP_{j} is a weighted average of the
basic heading parities, where the weights are the notional quantities.
 An important feature of the GK system is illustrated in equation
(5), where the denominator of equation (4) is brought to the lefthand
side:
(5)
Each side of equation (5) is a measure of the contribution of output
of a basic heading to regional or world GDP. It is only in the GK system
that the valuation of quantities at international prices is consistent
with their basic heading parities and expenditures, as well as the overall
purchasingpower parity of each country.
 Equations (1) and (4) represent the complete GK system when PPP_{j}
and q_{ij} are defined as in equations (2) and (3). When m is
over 150 and n is over 60, this appears to be a large system to solve.
However, it turns out that the easiest way to solve the system is by
iteration; and it also turns out that the iterative procedure is itself
instructive, as the following discussion is intended to show.
 The basic data are the expenditures (E_{ij}S) and parities
(pp_{ij}S) at the basic heading levels, and from these the q_{ij}s
can be derived. Consider an iteration that begins by initially setting
each PPP_{j} equal to the exchange rate. For example, if the
United States were the base country, then its initial PPP would be 1.0
and the initial PPPs for the other countries would be their exchange
rate relative to the dollar. Then a set of international prices can
be estimated using equation (4). These p _{i}s can then
be plugged into equation (1) and then equation (2) to estimate a set
of PPP_{j}S. The process can then be repeated beginning with
the new PPP_{i}S. The iteration will be complete when the difference
between the initial set of PPP_{j}S and the end set is very
small. Typically, in eight iterations the differences will only be observed
at the fourth decimal place. It is unlikely that when the last iteration
is complete the new PPP for the United States will equal 1.0. The system
is then normalized so that each new PPP is adjusted so that the United
States value will be 1.0, and the p _{i}s appropriately
scaled so that, for the United States, gdp and rgdp as obtained from
equation (1) are equal.
 While one can begin the iteration with any set of values, there is
another way to begin that is also instructive. Consider setting each
of the initial international prices (p _{i}s) equal to
1.0. The same loop can then be followed, estimating the PPPjs when the
p _{i}s are all 1.0, and work back through the system to
obtain a new set of international prices, and a new set of PPPs and
so on. A normalization as described in paragraph 9 would also be carried
out to make the PPP of the base country 1.0. Beginning with all international
prices equal to 1.0 is equivalent to using the relative price structure
of the numeraire country. The fact that the final set of international
prices will differ substantially from 1.0, no matter which country is
numeraire, again illustrates why one cannot simply sum up the notional
quantities given in equation (3).
 This discussion should also make clear that the international prices
of the ICP centre around 1.0 and are used to value a quantity that has
no natural dimension, such as a kilogram, but has a notional character
depending on the numeraire currency. b/ The iteration procedure also
illustrates how the Geary system achieves additivity across countries
and basic headings to achieve matrix consistency.
 As discussed in the text the major advantage of the Geary system is
that the international prices are analogous to the prices used to generate
the national accounts of an individual country. In the Geary formulation,
large rich countries receive more weight in determining international
prices used to value quantities in each country. This means that the
structure of international prices will tend to be closer to those of
rich countries. There is also usually an inverse relationship between
price and quantity across countries, so that items that are expensive
in poor countries, for example, will be consumed in relatively small
quantities and vice versa. The GK price structure will tend to value
the large quantities of relatively inexpensive items in poor countries,
such as services, at higher prices. Conversely, those items that are
relatively cheap in rich countries, such as transport equipment, will
be valued at international prices closer to their national value. This
effect is present in all of the aggregation systems since it is part
of the world economic structure that the ICP is attempting to represent.
 However, the international price systems that are explicit or implicit
in other systems are usually closer to middleincome countries because
the weights used are not in proportion to country GDP. As a consequence,
the GK system tends to lower the income of rich countries relative
to poor countries more than the other aggregation methods. Some regard
this as a desired result stemming from the national accounts basis of
the GK system, while others regard it as a drawback. c/
B. Other aggregation methods
1. Additive systems
 One type of aggregation system, devised by D. Gerardi, that was used
by EUROSTAT was based on international prices used to evaluate notional
quantities, as in equation (1) above. The Gerardi system was compared
with the GK system by Hill (1982, pp. 5159), and that discussion will
not be repeated here. The objective of both the Gerardi system and other
international price systems with which EUROSTAT has experimented has
been to retain an additive system that does not use international prices
close to those of larger countries. Another way of putting this is to
say that there are those who want matrixconsistent comparisons, but
do not want to use a set of prices that are quantity weighted as in
national accounts. Gerardi's international prices, for example, were
initially based on equal weights to the pyqs of each country.
 Another motivating factor for those seeking alternatives to the GK
system that are inherently additive is that GK is a simultaneous system
that requires all information from all countries before it can be calculated.
A price change in one basic heading can, in principle, change the estimates
of other basic headings. d/ Also, results of the GK system can change
as the number of countries included in the aggregation changes, though
this is also true for most other aggregation systems.
2. EKS and related systems
 Erwin Diewert has made an extensive review of indexes that might be
used in international comparisons, and has come up with a class of what
he terms superlative indexes (Diewert, 1978). What he finds is that
indexes built up from Fishertype comparisons between two countries
have a number of desirable properties that flow from the theory of consumer
choice. From this it follows that a multilateral index based on Fisher
binary indexes, such as the EKS system, appears to have more theoretical
rationale than the GK system.
 While Diewert's arguments provide some support for EKS, the issue
is not so easily resolved. First, 30 to 40 per cent of expenditures
on GDP are typically not chosen on the basis of relative prices. That
is, most government expenditures and much of investment is not allocated
on the basis of the principles underlying consumer choice. It is not
claimed that EKS, GK or any other system is necessarily better for
comparing these expenditures, but that the theory of consumer choice
is applicable to a portion of GDP only.
 The second point relates to additivity. There are several systems
that have been used that, like EKS, produce an overall comparison for
all the basic headings entering the aggregation. One of these systems,
the van Yzeren system, was proposed for the European Coal and Steel
Community and another, the Walsh or expenditure weight system, has been
used in Latin American comparisons. e/ The EKS system, as well as the
Walsh and van Yzeren systems, provide a PPP over GDP or whatever aggregate
for which they have been computed. However, they do not have an implicit
system of international prices, so there is no explicit allocation of
the expenditures within the aggregate and no inherent additivity. It
is simple enough to impose additivity by, say, distributing the expenditures
on GDP obtained by EKS across the categories according to the distribution
of those expenditures in national currencies. The disadvantage of this
is that the method is arbitrary and no information about the price structure
in other countries is used in comparing the structure of expenditures
in one country with those in another.
 One further point is that for some purposes the only number sought
is for an aggregate such as consumption. One might, for example, want
to use the PPP for consumption to compare real wages across countries.
In this case, an EKS aggregation may be preferred to the GK method
for two reasons. First, since additivity is not needed in this example,
one drawback to using EKS is removed. Secondly, the implicit weighting
involved in EKS is equal among countries so that for converting wages
across countries it may make more sense to think of using a PPP that
assigns the same importance to the market basket of each country. (The
latter weighting system can also be achieved by GK.)
 To give some impression of what differences are involved in the various
methods, results are given below from the phase III report for six countries,
spanning the range of per capita incomes in the world. The entries give
the per capita income of each country relative to the United States
as 100 for each country.
Country per capita GDP, 1975 (US = 100)
Method

India

Kenya

Colombia

Republic of Korea

Japan

France

1. BinaryFisher

6.0

5.8

19.7

17.2

67.5

80.2

2. GearyKhamis

6.6

6.5

22.6

19.9

68.6

81.9

3. EKS

5.7

5.4

19.9

17.8

65.3

81.1

4. Walsh

6.4

4.8

19.5

17.6

66.1

80.0

5. Van Yzeren

5.7

5.4

19.9

17.7

65.3

81.0

6. Gerardi

5.7

5.8

20.4

18.5

66.6

77.8

7. Exchange rate

2.0

3.4

7.9

8.1

62.3

89.6

Source: Kravis, Heston and Summers, 1982, pp.
9697.
 The differences between the
first six rows for any one country are less than 5 per cent for Japan
and France, and less than 15 per cent for the remaining countries. A
seventh row is also provided for the exchange rate conversions, indicating
that all of the other methods are much closer to one another than to
use of the exchange rate, and for Japan and France, the deviations are
over 10 per cent and in opposite directions. Thus, while results of
the different methods can vary from one another, their general orders
of magnitude for each country and variation across countries tell a
fairly consistent story.
 It would be nice if there were
a simple conclusion to be drawn from this discussion, but that would
imply that somehow the ICP had solved the index number problem, which
it assuredly has not. One longs for one measure because that would be
simple to explain to users, especially users providing resources for
the work. It is also tidier to have only one result. However, because
there are a variety of uses for which the ICP results are desired, for
the present, more than one result will be produced, though in official
publications the differences will be minimized.
C. Some loose ends
 Some expenditure categories
can be negative, such as change in stocks or the net foreign balance.
These categories do not make much sense in any method using international
prices because the Geary system, for example, is based on positive quantities
and prices. Therefore, in the GK system, the actual solution is carried
out over the nonnegative basic headings. The parities assigned to the
net foreign balance and the net expenditures of residents abroad is
the exchange rate. (A different treatment is made for countries with
a large amount of tourist expenditures, such as Austria, where net expenditures
of residents abroad may be distributed among the important headings,
and no expenditure is retained in that heading.) In phases I to III
of the ICP, the international price for these two headings was defined
as in equation (4), but has since been assumed to be 1.0.
 For change in stocks, a parity
is calculated from the GK result based on those basic headings that
are commodities. That parity is assigned to the change in stocks. The
international price for the category is then calculated from equation
(4) above. Any normalization to make the PPP of the base country 1.0
is then carried out involving the international prices of all basic
headings. In the comparisons of methods given above, the actual comparison
is over the nonnegative categories, since this appeared to put all
methods on the most comparable basis.
a/ For the world comparisons
in phases IIV, countries were assigned additional weight to reflect the
importance of countries not included in the benchmark comparisons. The
total expenditures of a country were termed its supercountry weight, and
the sum of all supercountry expenditures would be world GDP. One reason
for using supercountry weights was to estimate the international prices
that were implicit in world GDP. Since the GK result does depend on the
number of countries in the calculation, the use of supercountry weights
was designed to approximate the international prices if all countries
in the world were participating in the ICP. This in turn should, in principle,
make the results from earlier benchmark ICP comparisons, when relatively
few countries participated, better approximate later comparisons involving
more benchmark countries.
In the Geary system, it is also possible to use per capita expenditure
weights or other weighting systems, For example, one could assign equal
weights to each country over all expenditures and in effect use the percentage
expenditure for each basic heading as the country weight. The discussion
in this annex assumes that the overall weight for each country is their
GDP, or supercountry GDP.
b/ The international prices
will depend on the numeraire country chosen. This point is discussed in
Kravis, Heston and Summers (1982, pp. 9495). Two other technical points
may be noted. First, some regions have chosen to use a numeraire currency
outside the region, as, for example, Africa. In the African comparisons,
all prices and expenditures are initially converted into United States
dollars at exchange rates. In the African study, no single country is
used as the base, but rather the average of all countries is used. While
the results of the African study are presented in dollars, this does not
make them comparable with other countries, such as the United States,
because dollar conversions have only been carried out at exchange rates.
A second point is that when an average of a group of countries is used,
as in Africa or the European Communities, there will still be a set of
international prices implicit in the calculation. In the African case,
the system would be normalized to make the sum of expenditures of all
headings and all countries converted at exchange rates equal to the sum
of all notional quantities valued at international prices. For any particular
basic heading, this equality would not hold, and the ratio of the sum
across all countries of the basic heading notional quantities valued at
international prices to their value at exchange rates would be the international
price for that basic heading.
c/ Usually, the GK results
are criticized because they depart from Fisher binary results, being closer
to the Laspeyres than the Paasche estimate for poor countries. However,
the binary comparisons being used as a reference weight each country the
same. The EKS system, which is an indirect least squares type of estimate
from the binaries, naturally comes closer to the Fisher result than does
GK. However, Prasada Rao has shown that, if a binary is done using the
GDP weights of the GK system, then the multilateral GK is a direct least
squares estimate based on the binaries and, of course, comes much closer
to the GK binaries than does EKS. The point, then, is that it is really
the weighting system that produces more difference between methods than
other factors (see Prasada Rao (1972)).
d/ This can readily be seen
by examining equation (5). A price change affects a PPij and that may
affect the PPPjS and work itself through the entire system. Any other
system that was matrix consistent would also be affected. Systems like
EKS would be affected in the aggregate but because there is no explicit
estimates of basic heading quantities in EKS, there is no visible effect
at the detailed level.
e/ These systems are discussed
in Kravis, Kenessey, Heston and Summers (1975), pp. 6668.
United Nations Statistics Division  Time Use

