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International Comparison Program
5th Technical Advisory Group Meeting April 18-19, 2011
Washington DC
[08.02]
Linking the Regions in the International
Comparisons Program at Basic Heading Level &
at Higher Levels of Aggregation
Robert J. Hill
Linking the Regions in the InternationalComparisons Program at Basic Heading Level and
at Higher Levels of Aggregation
Robert J. Hill∗
Department of Economics
University of Graz, Austria
April 10, 2011
Abstract:
The International Comparisons Program (ICP) compares the purchasing power of cur-
rencies and real output of almost all countries in the world. ICP is broken up into six
regions. Global results are then obtained by linking these regions together at both basic
heading level and the aggregate level in a way that satisfies within-region fixity (i.e.,
the relative parities of a pair of countries in the same region are the same in the global
comparison as in the within-region comparison). Standard multilateral methods (such
as CPD at basic heading level and GEKS above basic heading level) violate this within-
region fixity requirement and hence cannot be used to construct the global results. A
method is proposed here that resolves this problem in an optimal way by altering the
multilateral price indexes (calculated say using CPD or GEKS) by the minimum least-
squares amount necessary to ensure that within-region fixity is satisfied. The method’s
underlying rationale is similar to that of the GEKS method. In fact it can be viewed
as an extension of GEKS for achieving within-region fixity. The resulting global price
indexes are base country-invariant, and give equal weight to all regions. The method
can be applied equally well at basic heading level or at higher levels of aggregation, and
to either the ring country framework used in ICP 2005 or the core product framework
that will be used in ICP 2011. (JEL: C43, E31, O47)
Keywords: ICP; Price Index; Chaining; Within-Region Fixity; CPD; Multilateral Method;
GEKS; Base Country Invariance; Strong Factor Reversal Test
1 Introduction
The International Comparisons Program (ICP) is a huge undertaking coordinated by
the World Bank in collaboration with the OECD, Eurostat, IMF and UN. It compares
the purchasing power of currencies and real output of almost all countries in the world.
Its results underpin the Penn World Table (probably the most widely used data set in
the economics profession). The most recent comparison was made in 2005, and the next
is scheduled for 2011.
Both ICP 2005 and 2011 are broken up into six separate regional comparisons, each
of which has its own list of products for each basic heading.1 It is then necessary to link
the results across regions both at basic heading level and at the aggregate level. In ICP
2005, the regions were linked both at basic heading level and at the aggregate level using
an across-region variant proposed by Diewert (2008a) on the country-product-dummy
(CPD) method (see Summers 1973, Rao 2004, Diewert 2008b, and Hill and Syed 2010).
However, it later emerged that Diewert’s method of linking, as applied at the aggregate
level, was not invariant to the choice of within-region base countries (see Sergeev 2009
and Diewert 2010b). Hence, at least at the aggregate level, a new approach for linking
the regions will be required in ICP 2011.
An important complication that arises in the region-linking process is that the
global results must satisfy within-region fixity. In other words, the relative parities of a
pair of countries in the same region must be the same in the global comparison as in the
within-region comparison. Within-region fixity is required both at basic heading level
and at all higher levels of aggregation up to GDP. There are two reasons for imposing
within-region fixity. The first is essentially political. The European Union (EU) uses
the official EU results to calculate budget contributions and the disbursement of grants
and aid, and hence only wants one set of within-EU parities in the public domain.
More generally, in other regions there is also concern that the availability of multiple
1A basic heading is the lowest level of aggregation at which expenditure weights are available. A
basic heading consists of a group of similar products defined within a general product classification.
Food and non-alcoholic beverages account for 29 headings, alcoholic beverages, tobacco and narcotics
for 5 headings, clothing and footwear for 5 headings, etc. (see Blades 2007).
1
within-region parities could generate confusion. The second reason is that the within-
region comparisons are probably more reliable than the global comparisons. This partly
reflects the inherent regional structure of ICP. However, it is also the case that countries
within a region tend to have more similar levels of economic development, thus making
it easier to compare them.
In this paper a new and flexible method is proposed for linking the regions while
maintaining within-region fixity. It turns out the method is closely related to the
method currently used by Eurostat and the OECD at the aggregate level.2 Most of the
properties of the method developed here, though, were not previously known. In this
sense, one of the contributions of this paper is to provide firm theoretical foundations
for what Eurostat and OECD are already doing.
The method, which can be applied either at basic heading level or the aggregate
level, is optimal in the sense that it alters the multilateral price indexes (e.g., CPD
at basic heading level or GEKS at the aggregate level) by the minimum least-squares
amount necessary to ensure that within-region fixity is satisfied.3 The method’s un-
derlying rationale is similar to that of GEKS, which by comparison alters Fisher price
indexes by the minimum least-squares amount necessary to ensure transitivity. At the
aggregate level an equivalent least-squares problem can be solved for the quantity in-
dexes. It is shown that the least-squares price and quantity indexes, like GEKS, satisfy
the strong factor reversal test. Also, like GEKS, the method takes geometric means
of a number of chained comparisons between a pair of reference countries. In both
these senses, therefore, the method can be viewed as an extended version of GEKS
that achieves within-region fixity. The resulting global price indexes are base country-
invariant and give equal weight to all regions. The method can be applied equally well
at basic heading level or at the aggregate level, and to either the ring country framework
used in ICP 2005 or the core product framework (both frameworks are explained in the
next section) that will be used in ICP 2011.
At the aggregate level, an alternative method for imposing within-region fixity, the
Heston-Dikhanov method (also referred to as the CAR method), is also considered. The
2This similarity was pointed out to me by Sergey Sergeev in e-mail correspondence.3GEKS is named after Gini (1931), Elteto and Koves (1964) and Szulc (1964).
2
underlying algrebraic structure of the two methods is compared. The Heston-Dikhanov
method is shown to have an arithmetic, as opposed to geometric, structure. As a result
it does not satisfy the strong factor reversal test. Empirically, however, the two methods
seem to generate quite similar results.
2 A GEKS-Type Method for Linking the Regions
at Basic-Heading Level while Retaining Within-
Region Fixity
2.1 Linking at Basic Heading Level Through Core Products
A key difference between ICP 2005 and 2011 is that in 2005 to facilitate the linking
of the regions an additional so-called ‘ring’ comparison was made between 18 countries
drawn from the six regions. This ring comparison had its own product list for each basic
heading. In 2011, the ring comparison has been dropped. The regions will be linked
through ‘core’ products rather than ring countries. The core products are products that
are included in the product lists of every region. Every basic heading will include some
core products.
In ICP 2011 therefore the products within each basic heading are divided into core
and noncore groups. The core products are priced by all regions. The noncore products
are region specific.
The regions in the comparison are indexed here by A,B,C, etc. There areNA coun-
tries in region A, NB in region B, NC in region C, etc. P regionAa denotes a within-region
A price index for country Aa, with country A1 as the base. The ‘region’ superscript
denotes the fact that the price index is calculated over the full product list of region A
(i.e., both core and noncore). Similarly, P regionBb denotes a within-region B price index
for country Bb with country B1 as the base, calculated over the full product list of
region B, etc.
P coreAa and P core
Bb denote price indexes for countries Aa and Bb, obtained from a
global comparison with country A1 as the base country. Hence, although there is a
3
base country for each region in the within-region comparisons (i.e., P regionA1 = P region
B1 =
P regionC1 = 1), there is only one base country in the core comparison (i.e., P core
A1 = 1 while
in general P coreB1 6= 1 and P core
C1 6= 1).
The core indexes differ from the within-region indexes described above in two other
important respects. First, they are only calculated over the core products within the
basic heading. Second, they are calculated over all countries in the world (that are
participating in ICP), and violate within-region fixity.
Both the within-region and core price indexes can be calculated using standard
formulas such as CPD or Eurostat Jevons-S (see Sergeev 2003 and Hill and Hill 2009).
2.2 A Least-Squares Method for Imposing Within-Region Fix-
ity
The objective here is to alter the core price indexes by the minimum amount necessary
to satisfy within region fixity. Before attempting to formulate this problem mathemati-
cally, it is illuminating first to consider a well-known least squares optimization problem
from the price index literature.
The GEKS method alters intransitive Fisher price indexes by the logarithmic least
squares amount required to obtain transitivity (see Elteto and Koves 1964 and Szulc
1964). More precisely, GEKS solves the following problem:
Minln(Pk/Pj)
K∑j=1
K∑k=1
[ln
(PkPj
)− lnP F
j,k
]2 . (1)
That is, the solution for Pk/Pj obtained from (1) is the GEKS formula stated in (21).
Mathematically, our least squares problem is slightly more complicated than this.
Supposing there are three regions A, B and C, it can be formulated as follows:4
Minln(λ),ln(µ),ln(ρ)
NA∑a=1
NB∑b=1
[ln
(P globalBb
P globalAa
)− ln
(P coreBb
P coreAa
)]2
+NA∑a=1
NC∑c=1
[ln
(P globalCc
P globalAa
)− ln
(P coreCc
P coreAa
)]2
+NB∑b=1
NC∑c=1
[ln
(P globalCc
P globalBb
)− ln
(P coreCc
P coreBb
)]2 , (2)
4The method generalizes in a straightforward way to the case where there are four or more regions.
4
subject to the within-region fixity constraints:
P globalBk
P globalAj
= λP regionBk
P regionAj
,P globalCl
P globalAj
= µP regionCl
P regionAj
,P globalCl
P globalBk
= ρP regionCl
P regionBk
, (3)
where λ, µ, ρ are positive scalars, and Aj, Bk and Cl denote arbitrary reference coun-
tries from regions A, B and C.
The within-region fixity constraints can alternatively and perhaps more intuitively
be written as follows:
P globalAj = αP region
Aj , (4)
P globalBk = βP region
Bk , (5)
P globalCl = γP region
Cl , (6)
where again α, β and γ denote positive scalars. Dividing (5) by (4), (6) by (4), and (6)
by (5), yield within-region fixity constraint of the form described in (3).
Substituting (3) into (2) reduces the optimization problem to the following:
Minln(λ),ln(µ),ln(ρ)
NA∑a=1
NB∑b=1
[ln(λ) + ln
(P regionBb
P coreBb
)− ln
(P regionAa
P coreAa
)]2
+NA∑a=1
NC∑c=1
[ln(µ) + ln
(P regionCc
P coreCc
)− ln
(P regionAa
P coreAa
)]2
+NB∑b=1
NC∑c=1
[ln(ρ) + ln
(P regionCc
P coreCc
)− ln
(P regionBb
P coreBb
)]2 . (7)
Solving (7), the following first order conditions are obtained:5
2NB
NA∑a=1
NB∑b=1
[ln(λ) + ln
(P regionBb
P coreBb
)− ln
(P regionAa
P coreAa
)]= 0. (8)
2NC
NA∑a=1
NC∑c=1
[ln(µ) + ln
(P regionCc
P coreCc
)− ln
(P regionAa
P coreAa
)]= 0. (9)
2NC
NB∑b=1
NC∑c=1
[ln(ρ) + ln
(P regionCc
P coreCc
)− ln
(P regionBb
P coreBb
)]= 0. (10)
5The second derivatives of (7) with respect to each of λ, µ and ρ are positive, thus ensuring that
the solution found is a minimum.
5
which on rearranging simplify to
λ =NB∏b=1
(P regionBb
P coreBb
)1/NB/
NA∏a=1
(P regionAa
P coreAa
)1/NA
. (11)
µ =NC∏c=1
(P regionCc
P coreCc
)1/NC/
NA∏a=1
(P regionAa
P coreAa
)1/NA
. (12)
ρ =NC∏c=1
(P regionCc
P coreCc
)1/NC/
NB∏b=1
(P regionBb
P coreBb
)1/NB
. (13)
Now substituting the solutions for λ, µ and ρ into (3), the following global price
index formulas are obtained:
P globalBk
P globalAj
= λP regionBk
P regionAj
=
P regionBk
P regionAj
∏NBb=1
(P core
Bb
P regionBb
)1/NB
∏NAa=1
(P core
Aa
P regionAa
)1/NA
. (14)
P globalCl
P globalAj
= µP regionCl
P regionAj
=
P regionCl
P regionAj
∏NCc=1
(P core
Cc
P regionCc
)1/NC
∏NAa=1
(P core
Aa
P regionAa
)1/NA
. (15)
P globalCl
P globalBk
= ρP regionCl
P regionBk
=
(P regionCl
P regionBk
)∏NCc=1
(P core
Cc
P regionCc
)1/NC
∏NBb=1
(P core
Bb
P regionBb
)1/NB
. (16)
From (14), (15) and (16) it can be seen that the global price indexes are tran-
sitive. Hence the numerators and denominators can be separated into region specific
components as follows:
P globalAj = P region
Aj
NA∏a=1
(P coreAa
P regionAa
)1/NA , (17)
P globalBk = P region
Bk
NB∏b=1
(P coreBb
P regionBb
)1/NB , (18)
P globalCl = P region
Cl
NC∏c=1
(P coreCc
P regionCc
)1/NC . (19)
Again it should be noted that while each region has its own base country (i.e., P regionA1 =
P regionB1 = 1), there is only one base country in the core comparison (i.e., P core
A1 = 1 while
6
in general P coreB1 6= 1 and P core
C1 6= 1).6 By contrast, at is stands, none of the global
indexes (17), (18) and (19) are normalized to 1. However, they can easily be rescaled
to achieve whatever normalization is desired. For example, they can be normalized so
that P globalA1 = 1 by dividing through by
∏NAa=1(P
coreAa /P region
Aa )1/NA .
The global price indexes derived above satisfy within-region fixity, while at the
same time giving equal weight to all regions.
2.3 An Alternative Perspective on the Least Squares Method
The global price index formula in (14) can be rewritten as follows:
P globalBk
P globalAj
=
P regionBk
P regionAj
∏NBb=1
(P core
Bb
P regionBb
)1/NB
∏NAa=1
(P core
Aa
P regionAa
)1/NA
=NA∏a=1
NB∏b=1
P regionAa
P regionAj
× P coreBb
P coreAa
× P regionBk
P regionBb
1/(NA×NB)
(20)
Each term (P regionAa /P region
Aj )×(P coreBb /P core
Aa )×(P regionBk /P region
Bb ) in (20) can be thought
of as a chained price index that compares countries Aj and Bk via countries Aa and
Bb (i.e., the chaining path is Aj − Aa − Bb − Bk). The overall global price index
P globalBk /P global
Aj is the geometric average of the chained price indexes obtained by using
all possible chain paths from Aj to Bk. For example, suppose there are three countries
in region A, denoted by A1, A2, and A3, and two countries in region B, denoted by B1
and B2. There are then a total of six possible paths from say A1 to B1. These and
their corresponding chained price indexes are listed below:
A1−B1 :
(pcoreB1
pcoreA1
)
A1−B2−B1 :
(pcoreB2
pcoreA1
× P regionB1
P regionB2
)
A1− A2−B1 :
(P regionA2
P regionA1
× pcoreB1
pcoreA2
)6The global price indexes are invariant (after rescaling) to the choice of the regional base countries
A1, B1, C1, and the base country in the core comparison A1.
7
A1− A2−B2−B1 :
(P regionA2
P regionA1
× pcoreB2
pcoreA2
× P regionB1
P regionB2
)
A1− A3−B1 :
(P regionA3
P regionA1
× pcoreB1
pcoreA3
)
A1− A3−B2−B1 :
(P regionA3
P regionA1
× pcoreB2
pcoreA3
× P regionB1
P regionB2
)The global price index between A1 and B1 in this case is obtained by taking the
geometric mean of these six chained price indexes.
Countries from third regions, such as C, drop out if they are included in the chain
path. This is because the core price indexes are transitive. For example, as shown
below the path Aj − Cl −Bk reduces to Aj −Bk:
P regionAa
P regionAj
× P coreCl
P coreAa
× P coreBb
P coreCl
× P regionBk
P regionBb
=P regionAa
P regionAj
× P coreBb
P coreAa
× P regionBk
P regionBb
.
Similarly, additional countries from either regions A or B when included drop out of
the chain path since the within-region price indexes are also transitive. For example,
as shown below, the path Aj − Al − Aa−Bb−Bk reduces to Aj − Aa−Bb−Bk:
P regionAl
P regionAj
× P regionAa
P regionAl
× P coreBb
P coreAa
× P regionBk
P regionBb
=P regionAa
P regionAj
× P coreBb
P coreAa
× P regionBk
P regionBb
.
An analogy can be drawn here with the GEKS formula, which transitivizes Fisher
price indexes (denoted here by P Fj,k where j and k denote countries) in a similar way:
PGEKS2
PGEKS1
=K∏k=1
(P Fk,2
P Fk,1
)1/K
. (21)
Using the fact that Fisher price indexes satisfy the country reversal test, this formula
can be rewritten as follows:
PGEKS2
PGEKS1
=
[(P F
1,2)2K∏k=3
(P F1,k × P F
k,2)
]1/K
.
Written in this way, it can be seen that GEKS can itself be interpreted as taking the
geometric mean of a direct price index (given double the weight), i.e., P F1,2, and K − 2
chained price indexes obtained by chaining through all possible third countries, i.e.,
P F1,k × P F
k,2. In this sense, the method in (20) is again analogous to GEKS (except that
it does not give twice the weight to the direct comparison).
8
The representation of the between-region links of the least-squares method as a
geometric mean of all possible paths between pairs of countries drawn one from each
region is also useful for demonstrating that the method gives equal weight to all regions.
Suppose now we have two countries in region A (denoted by A1 and A2), and four
countries in region B, denoted by B1, B2, B3, and B4. Using A1 and B1 as the regional
bases, this means we take a geometric mean of the price indexes obtained by chaining
along the following eight paths from A1 to B1: (i) A1-B1, (ii) A1-A2-B1, (iii) A1-B2-B1,
(iv) A1-A2-B2-B1, (v) A1-B3-B1, (vi) A1-A2-B3-B1, (vii) A1-B4-B1, (viii) A1-A2-B4-
B1. The geometric mean of these eight paths contains more within-region price indexes
from region B than from region A. More specifically, paths (ii), (iv), (vi) and (viii)
contain a within-region price index from region A, while paths (iii), (iv), (v), (vi), (vii)
and (viii) contain a within-region price index from region B. In other words, within-
region A price indexes feature in only four of the eight chain paths, while within-region
B price indexes feature in six of the eight chain paths. This does not imply though that
region B is exerting a ”greater influence” on the overall results. What matters here is
the link countries between regions. A1 is the link country in region A in paths (i). (iii),
(v) and (vi). A2 is the link country for the other four paths. Similarly, B1 is the link
country for region B in paths (i) and (ii), B2 is the link country in paths (iii) and (iv),
etc. This means that each country in region A is the link country in four of the paths,
while each country in region B is the link country in only two paths. In other words,
each country in region A has twice the weight in the comparison as compared with each
country in region B. However, there are twice as many countries in region B. Hence,
overall the weight allocation for the two regions is the same. The within-region price
indexes should be viewed as simply rebasing a particular chain path’s comparison back
into units of the numeraire of that region, rather than as exerting weight in the overall
comparison.
9
2.4 Linking at the Basic Heading Level Through Ring Coun-
tries
With slight modifications the method can be applied to an ICP 2005 context in which
ring countries were used instead of core products to link the regions together. The ring
comparison in ICP 2005 had its own product list and involved 18 countries drawn from
the regions.
In a ring country context, the formulas in (17), (18) and (19) are modified slightly
as follows:
P globalAj = P region
Aj
NA∏a=1
(P ringAa
P regionAa
)1/NA , (22)
P globalBk = P region
Bk
NB∏b=1
(P ringBb
P regionBb
)1/NB , (23)
P globalCl = P region
Cl
NC∏γ=1
(P ringCc
P regionCc
)1/NC . (24)
There are two changes here. First, the core price indexes P coreAa have been replaced
by ring price indexes P ringAa . Second, while a = 1, . . . , NA in (17) indexes all the countries
in region A, in (22) it indexes only the ring countries in region A. The global price
index for a non-ring country (say Az) is then obtained by linking it to the global price
index of one of the ring countries from its region (say Aj) as follows:
P globalAz =
P regionAz
P regionAj
× P globalAj = P region
Az
NA∏a=1
(P coreAa
P regionAa
)1/NA . (25)
It can be seen from (25) that for a region containing more than one ring country (as all
regions did in ICP 2005), it does not matter which is used as the link for the non-ring
countries in that region. The linking ring country (here Ak) drops out of the P globalAz
formula.
More generally, moving beyond ICP 2005, there is no reason why in ICP 2011
every country from every region must be included in the core comparison. Once the
core products have been decided, it is still possible to select only a sample of countries
from each region to participate in the calculation of the core price indexes that are used
in (20), such as those with a more complete coverage of the products in the core list
10
(while ensuring that there is enough representation from each region). Furthermore,
the list of countries used in the core comparison could vary from one basic heading to
the next. The countries excluded from the core comparison can then be linked back in
using (25). Such an approach might prove useful if some countries are unable to price
any of the core products in a few basic headings.
3 Linking the Regions at the Aggregate Level while
Retaining Within-Region Fixity
3.1 A Least Squares Method for Imposing Within-Region Fix-
ity at the Aggregate Level
ICP 2005 and 2011 require within-region fixity to be satisfied at all levels of aggregation
from basic heading level up to GDP. This means that, once the basic heading price
indexes have been linked across regions, it is not enough to simply use a standard
multilateral method to construct global price indexes at the aggregate level since this
would lead to a violation of within-region fixity.7
A least-squares formulation of the problem of imposing within-region fixity at the
aggregate level exists similar to that in (2). To simplify matters, consider the case where
there are only two regions A and B. The least-squares problem can then be written as
follows:
Minln(λ)
NA∑a=1
NB∑b=1
[ln
(P globalBb
P globalAa
)− ln
(P unfixedBb
P unfixedAa
)]2 (26)
subject to the within-region fixity constraint:
P globalBk
P globalAj
= λP regionBk
P regionAj
, (27)
where now P regionAj and P unfixed
Aj are both aggregate price indexes calculated using GEKS
or some other multilateral formula such as Geary-Khamis, IDB or a minimum-spanning-
7At a conceptual level, such an approach might also be undesirable since while the basic headings
may match from one region to the next, the underlying products from which the basic heading price
indexes are constructed do not.
11
tree.8 The difference between P regionAj and P unfixed
Aj is that, in the case of P regionAj , the
GEKS formula is only applied to the countries in region A, while the unfixed price
indexes are obtained by applying the GEKS formula globally. These price indexes have
the superscript ‘unfixed’ since they do not satisfy within-region fixity.9
Solving this least-squares problem yields the following solution:
P globalBk
P globalAj
=
P regionBk
P regionAj
∏NBb=1
(Punfixed
Bb
P regionBb
)1/NB
∏NAa=1
(Punfixed
Aa
P regionAa
)1/NA
, (28)
which in turn up to a normalizing scalar implies that
P globalAj = P region
Aj
NA∏a=1
(P unfixedAa
P regionAa
)1/NA , (29)
P globalBk = P region
Bk
NB∏b=1
(P unfixedBb
P regionBb
)1/NB . (30)
One important difference here, however, is that one has the option at the aggregate
level of minimizing the deviations between global and unfixed price indexes or global and
unfixed quantity indexes. By contrast below basic heading level there are no quantities
or expenditure shares. A quantity index version of the least squares problem at the
aggregate level takes the following form:
Minln(λ)
NA∑a=1
NB∑b=1
[ln
(QglobalBb
QglobalAa
)− ln
(QunfixedBb
QunfixedAa
)]2 (31)
subject to the within-region fixity constraint:
QglobalBk
QglobalAj
= λQregionBk
QregionAj
, (32)
where QglobalAa , Qregion
Aj and QunfixedAa denote quantity indexes.
Solving this least-squares problem yields the solution:
QglobalBk
QglobalAj
=
QregionBk
QregionAj
∏NBb=1
(Qunfixed
Bb
QregionBb
)1/NB
∏NAa=1
(Qunfixed
Aa
QregionAa
)1/NA
, (33)
8Geary-Khamis was developed by Geary (1958) and Khamis (1972), while minimum-spanning-tree
methods were developed by Hill (1999) and Diewert (2010b).9In ICP 2005, the within-region price indexes at the aggregate level were calculated using GEKS
for all regions except Africa which used IDB. IDB was proposed by Ikle (1972), Dikhanov (1997), and
Balk (1996) (see also Diewert 2010a for a detailed analysis of the properties of IDB).
12
which in turn up to a normalizing scalar implies that
QglobalAj = Qregion
Aj
NA∏a=1
(QunfixedAa
QregionAa
)1/NA , (34)
QglobalBk = Qregion
Bk
NB∏b=1
(QunfixedBb
QregionBb
)1/NB . (35)
Combining (28) and (33) yields the following expression:
P globalBk ×Qglobal
Bk
P globalAj ×Qglobal
Aj
=
P regionBk ×Qregion
Bk
P regionAj ×Qregion
Aj
×
NB∏b=1
(P unfixedBb ×Qunfixed
Bb
P regionBb ×Qregion
Bb
)1/NB/NA∏
a=1
(P unfixedAa ×Qunfixed
Aa
P regionAa ×Qregion
Aa
)1/NA .
Assuming the methods used to compute the within-region and global unfixed price and
quantity indexes satisfy the weak factor reversal test, then it follows that
P regionBk ×Qregion
Bk
P regionAj ×Qregion
Aj
=
(∑Ii=1 pBk,iqBk,i∑Ii=1 pB1,iqB1,i
)/(∑Ii=1 pAj,iqAj,i∑Ii=1 pA1,iqA1,i
);
NB∏b=1
(P unfixedBb ×Qunfixed
Bb
P regionBb ×Qregion
Bb
)1/NB
=NB∏b=1
[(∑Ii=1 pBb,iqBb,i∑Ii=1 pA1,iqA1,i
)(∑Ii=1 pB1,iqB1,i∑Ii=1 pBb,iqBb,i
)]1/NB
;
NA∏a=1
(P unfixedAa ×Qunfixed
Aa
P regionAa ×Qregion
Aa
)1/NA
=NA∏a=1
[(∑Ii=1 pAa,iqAa,i∑Ii=1 pA1,iqA1,i
)(∑Ii=1 pA1,iqA1,i∑Ii=1 pAa,iqAa,i
)]1/NA
;
where A1 and B1 are the base countries in regions A and B, and A1 is the base country
in the global unfixed comparison.
The product of the global price and quantity index can now be rewritten as follows:
P globalBk ×Qglobal
Bk
P globalAj ×Qglobal
Aj
=
[(∑Ii=1 pBk,iqBk,i∑Ii=1 pB1,iqB1,i
)/(∑Ii=1 pAj,iqAj,i∑Ii=1 pA1,iqA1,i
)]
×NB∏b=1
[(∑Ii=1 pBb,iqBb,i∑Ii=1 pA1,iqA1,i
)(∑Ii=1 pB1,iqB1,i∑Ii=1 pBb,iqBb,i
)]1/NB/
NA∏a=1
[(∑Ii=1 pAa,iqAa,i∑Ii=1 pA1,iqA1,i
)(∑Ii=1 pA1,iqA1,i∑Ii=1 pAa,iqAa,i
)]1/NA
=
[(∑Ii=1 pBk,iqBk,i∑Ii=1 pB1,iqB1,i
)/(∑Ii=1 pAj,iqAj,i∑Ii=1 pA1,iqA1,i
)]×
NB∏b=1
(∑Ii=1 pB1,iqB1,i∑Ii=1 pA1,iqA1,i
)1/NB
=
∑Ii=1 pBk,iqBk,i∑Ii=1 pAj,iqAj,i
.
13
On rearrangement this becomes
P globalBk
P globalAj
=
∑Ii=1 pBk,iqBk,i∑Ii=1 pAj,iqAj,i
/QglobalBk
QglobalAj
=P globalBk
P globalAj
.
The direct global price indexes P globalBk /P global
Aj are identical to the implicit global price
indexes P globalBk /P global
Aj derived from the direct global quantity indexes via the factor-
reversal equation. The direct and implicit global quantity indexes are likewise identical.
In other words, the global linking method as defined in (28) and (33) satisfies the strong
factor reversal test.
This means that solving the least squares optimization problem for the global
quantity indexes in (31) implies also solving it for the global price indexes in (26).
Hence the method is well-founded irrespective of whether the main focus of interest is
the price or quantity indexes.
3.2 Asymmetric Fixity, Subregion Fixity and the OECD Method
Suppose now the objective is to alter the unfixed global price indexes by the minimum
amount necessary to satisfy fixity in one region (say region A). I refer to this scenario
as asymmetric fixity, since it treats the regions asymmetrically. In this case, without
loss of generality, all the other regions can be collected into a second region B.
This can be formulated as a least squares problem as follows:
Minln(λ)
NA∑a=1
NB∑b=1
[ln
(P globalBb
P globalAa
)− ln
(P unfixedBb
P unfixedAa
)]2 , (36)
subject to the fixity constraints:
P globalAa = λP region
Aa . (37)
P globalBb = P unfixed
Bb . (38)
The constraints in (37) and (38) can be combined as follows:
P globalBk
P globalAj
= λP unfixedBk
P regionAj
. (39)
Substituting (39) into (36) reduces the optimization problem to the following:
Min− ln(λ)
NA∑a=1
NB∑b=1
[ln(λ)− ln
(P regionAa
P unfixedAa
)]2 . (40)
14
Solving (40) yields the following first order condition:
2NB
NA∑a=1
[ln(λ)− ln
(P unfixedAa
P regionAa
)]= 0. (41)
which on rearranging simplifies to
λ =NA∏a=1
(P unfixedAa
P regionAa
)1/NA
. (42)
Now substituting the solution for λ from (42) into (39) yields the following global
price index formulas:
P globalBb
P globalAa
=P unfixedBb
P regionAa
NA∏a=1
(P regionAa
P unfixedAa
)1/NA
, (43)
or alternatively,
P globalAa = P region
Aa
NA∏a=1
(P unfixedAa
P regionAa
)1/NA
, (44)
P globalBb = P unfixed
Bb . (45)
Comparing (44) and (45) with (29) and (34) it can be seen that the asymmetric method
proposed here is a special case of the symmetric method, where the globalization formula
is applied only to countries in region A.
This asymmetric method for imposing fixity has been used by the OECD in its
within-region comparisons since 1990 (see Sergeev 2005), although without the prop-
erties of the method being fully derived or appreciated. A need for asymmetric fixity
arises in the OECD region due to Eurostat’s requirement of within-region fixity for the
European Union subregion. It can be seen that the method used by OECD is in fact
an optimal solution to the problem posed.
More generally, it is possible that some regions may require sub-region fixity in ICP
2011. Focusing specifically on region A, suppose it has two subregions denoted here
by A1 and A2 and that countries a = 1, . . . , j lie in the first subregion while countries
a = j+ 1, . . . , NA lie in the second subregion. Optimal within-region price indexes that
satisfy within-subregion fixity are now obtained as follows:
P regionAa = P subregion
A1a
j∏a=1
P reg−unfixedA1a
P subregionA1a
1/j
, for a = 1, . . . , j,
15
P regionAa = P subregion
A2a
NA∏a=j+1
P reg−unfixedA2a
P subregionA2a
1/(NA−j−1
, for a = j + 1, . . . , NA,
where P subregionA1a
and P subregionA2a
denote the within-subregion price indexes of regions A1
and A2 and P reg−unfixedA1a
and P reg−unfixedA2a
denote the unfixed within-region price indexes
of region A.
The overall global results for region A are now obtained as follows:
P globalAa = P region
A1a
NA∏a=1
(P unfixedAa
P regionAa
)1/NA
= P subregionA1a
j∏a=1
P reg−unfixedA1a
P subregionA1a
1/jNA∏a=1
(P unfixedAa
P regionAa
)1/NA
, for a = 1, . . . , j,
P globalAa = P region
A2a
NA∏a=1
(P unfixedAa
P regionAa
)1/NA
= P subregionA2a
NA∏a=j+1
P reg−unfixedA2a
P subregionA2a
1/(NA−j−1)NA∏a=1
(P unfixedAa
P regionAa
)1/NA
for a = j + 1, . . . , NA.
3.3 The Heston/Dikhanov Method for Imposing Within-Region
Fixity
Kravis, Heston and Summers (1982), Heston (1986) and Dikhanov (2007) suggest an
alternative method for imposing within-region fixity on the global aggregate results.
The method, which Heston also refers to as the CAR method, is typically expressed in
terms of value shares as follows:
sglobalAj = sunfixedA × sregionAj , (46)
where the value shares are essentially rescaled quantity indexes.
sglobalAj =QglobalAj∑N
n=1Qglobaln
,
is the share of country Aj in total world output in the global comparison.
sunfixedA =
∑NAa=1Q
unfixedAa∑N
n=1Qunfixedn
,
16
is the share of region A in total world output obtained from the unfixed global compar-
ison (where within-region fixity is not satisfied).
sregionAj =QregionAj∑NA
a=1QregionAa
,
is the share of country Aj in the total output of region A obtained from the within-region
A comparison. Converting (46) into quantity indexes yields the following expression:
QglobalAj∑N
n=1Qglobaln
=
(∑NAa=1Q
unfixedAa∑N
n=1Qunfixedn
) QregionAj∑NA
a=1QregionAa
.An equivalent expression for another country Bk is as follows:
QglobalBk∑N
n=1Qglobaln
=
(∑NBb=1Q
unfixedBb∑N
n=1Qunfixedn
)(QregionBk∑NB
b=1QregionBb
).
Now dividing the latter by the former, and rearranging yields the following:
QglobalBk
QglobalAj
=
QregionBk
QregionAj
(∑NBb=1Q
unfixedBb∑NB
b=1QregionBb
)/(∑NAa=1Q
unfixedAa∑NA
a=1QregionAa
). (47)
It is easily verified that the global quantity indexes derived from (47) satisfy within-
region fixity. Replacing Bk with Ak, the term on the righthand side of (47) collapses
to QregionAk /Qregion
Aj .
Also, it is interesting to compare (47) with the global quantity index formula
derived earlier in (31). The key difference between the two formulas is that the method
described in section 3.1, and given its similarity to the Eurostat-OECD method is
henceforth referred to as the Eurostat-OECD/Hill (EOH) method, takes geometric
means of the unfixed and within-region quantity indexes while the Heston-Dikhanov
method takes arithmetic means. As a consequence the former satisfies the strong factor
reversal test while the latter does not.
3.4 An Illustrative Example Using ICP 2005 Data
How much difference does it make empirically whether the EOH or Heston-Dikhanov
method is used? To help answer this question, an example is provided below using the
ICP 2005 basic heading data for six countries in the Asia-Pacific region, four countries
from South America and five countries from the Eurostat-OECD region. The countries
17
included are the following:
Asia-Pacific:
Hong Kong, Macao, Singapore, Taiwan, Brunei and Bangladesh
South America:
Argentina, Bolivia, Brazil, Chile
Eurostat-OECD:
Austria, Portugal, Finland, Sweden, United Kingdom
The aggregate within-region, global unfixed, and global fixed price and quantity in-
dexes are shown in Table 1. The within-region and global unfixed indexes are calculated
using GEKS. Two sets of global fixed results are provided. The first set is calculated
using the geometric averaging method of section 3.1, (i.e., the Eurostat-OECD/Hill
(EOH) method) while the second set is calculated using the Heston-Dikhanov (HD)
method.
Since both the EOH and HD methods satisfy within-region fixity, it follows that
their results are identical for the base region (here the Asia-Pacific). For the other
regions, the EOH and HD results differ, although empirically the differences seem to
be relatively small, equalling 0.4 percent for Latin America and 0.7 percent for Euro-
stat/OECD.10 It would be useful to repeat this example using all the countries that
participated in ICP 2005.
Insert Table 1 Here
4 Conclusion
The method proposed here for linking regions in an ICP context is very flexible. It can
be applied at either basic heading level or at the aggregate level, and can be combined
with any multilateral method. It is algebraically quite simple and intuitive, and is
optimal in a least-squares sense in the same way as GEKS is in a multilateral context.
It remains still to apply the method to real ICP data. It may be problematic for some
10The percentage differences are calculated as follows: 100×[max(QEOHk , QHD
k )/min(QEOHk , QHD
k )−
1]. The same percentage differences are obtained if the price indexes are used.
18
basic headings to compute CPD price indexes over the core products for all participating
countries (of which there will be about 150 in ICP 2011). If some countries fail to price
any of the core products in a particular basic heading, then as mentioned earlier these
countries can simply be excluded from the core comparison and then linked back in
later using the formula in (25) (with ‘ring’ replaced by ‘core’). If, however, across all
countries there are simply not enough core product price quotes in a particular basic
heading to estimate the 150 country dummy variables with sufficient accuracy, then it
may be preferable at least for this heading to stick with the regional variant on the
CPD method proposed by Diewert and used in ICP 2005, and which although it lacks
the least-squares optimality property of my method has the advantage that it requires
the estimation of only region (of which there are only six) rather than country dummy
variables. The practical relevance of this problem can only be determined empirically
using real ICP data.
At the aggregate level, however, a new method will definitely be required in ICP
2011 due to the failure of the Diewert region-CPD method in this context to satisfy
base-country invariance. The aggregate level version of the method proposed here in
section 3.1, the EOH method, looks like a strong candidate for this role, particularly
since at the aggregate level there is no need to make a global CPD-type comparison
(and hence the criticisms above are no longer relevant).
Also, it seems probable that at the aggregate level GEKS will be used to make the
global unfixed comparison and most of the within-region comparisons. As a matter of
internal consistency therefore it is desirable that the regions should be linked in a way
that maintains the underlying GEKS structure. EOH is the most natural generalization
of GEKS for imposing within-region fixity. Four illustrations of this are provided below.
First, EOH solves a generalization of the GEKS least squares optimization problem.
Second, equations (31) and (47) provide the EOH and Heston/Dikhanov quantity index
formulas (with the latter expressed in terms of quantity indexes rather than value
shares). It can be seen that EOH quantity indexes are constructed from geometric
means of the unfixed and within-region indexes while the Heston/Dikhanov indexes
are constructed from arithmetic means of these same indexes. In other words, EOH
maintains the geometric averaging structure of GEKS while Heston/Dikhanov does not.
19
Third, section 2.3 explains how the EOH between-region links can be derived as the
geometric means of all possible paths between a pair of regions. This again is a natural
extension of the GEKS ethos of taking the geometric mean of all possible paths between
a pair of countries. Finally, like GEKS, EOH satisfies the strong factor reversal test.
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22
Table 1. A Comparison of EOH and HD Price and Qauntity Indexes Using ICP 2005 Data
HKG MAC SGP TWN BRN BGD ARG BOL BRA CHL AUT PRT FIN SWE GBR
Within-Region Quantity 1.000 0.070 0.712 2.443 0.068 0.708 1.000 0.082 3.764 0.473 1.000 0.759 0.564 1.021 6.797
Unfixed Quantity 1.000 0.071 0.702 2.543 0.069 0.744 1.626 0.132 6.188 0.770 1.183 0.865 0.677 1.231 8.070
Global Quantity (EOH) 1.000 0.070 0.712 2.443 0.068 0.708 1.603 0.131 6.035 0.759 1.164 0.884 0.656 1.188 7.913
Global Quantity (HD) 1.000 0.070 0.712 2.443 0.068 0.708 1.598 0.130 6.014 0.756 1.156 0.878 0.652 1.180 7.860
Within-Region Price 1.000 0.957 0.197 3.382 0.168 4.021 1.000 1.756 1.073 264.58 1.000 0.802 1.137 10.67 0.740
Unfixed Price 1.000 0.947 0.200 3.248 0.166 3.825 0.237 0.417 0.251 62.524 0.150 0.125 0.168 1.569 0.111
Global Price (EOH) 1.000 0.957 0.197 3.382 0.168 4.021 0.240 0.421 0.257 63.492 0.152 0.122 0.173 1.626 0.113
Global Price (HD) 1.000 0.957 0.197 3.382 0.168 4.021 0.241 0.423 0.258 63.706 0.153 0.123 0.174 1.637 0.114
Percentage difference 0.000 0.000 0.000 0.000 0.000 0.000 0.338 0.338 0.338 0.338 0.685 0.685 0.685 0.685 0.685
