Linking the Regions in the International Comparisons Program

International Comparison Program [08.02] Linking the Regions in the International Comparisons Program at Basic Heading Level & at Higher Levels of Aggregation Robert J. Hill 5 th Technical Advisory Group Meeting April 18-19, 2011 Washington DC

Linking the Regions in the International Comparisons Program at Basic Heading Level and at Higher Levels of Aggregation Robert J. Hill Department of Economics University of Graz, Austria robert.hill@uni-graz.at April 10, 2011 Abstract: The International Comparisons Program (ICP) compares the purchasing power of currencies 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 withinregion 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 leastsquares 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 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 1 A 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 withinregion 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 underlying 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 indexes. 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 countryinvariant 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 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 2 This similarity was pointed out to me by Sergey Sergeev in e-mail correspondence. 3 GEKS is named after Gini (1931), Eltetö and Köves (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 products rather than ring countries. The products are products that are included in the product lists of every region. Every basic heading will include some products. In ICP 2011 therefore the products within each basic heading are divided into and non groups. The products are priced by all regions. The non products are region specific. The regions in the comparison are indexed here by A, B, C, etc. There are N A countries in region A, N B in region B, N C in region C, etc. denotes a within-region A price index for country, 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 and non). Similarly, denotes a within-region B price index for country with country B1 as the base, calculated over the full product list of region B, etc. P and P denote price indexes for countries and, 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., A1 B1 C1 1), there is only one base country in the comparison (i.e., P A1 in general P B1 1 and P C1 1). 1 while The indexes differ from the within-region indexes described above in two other important respects. First, they are only calculated over the 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 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 Fixity The objective here is to alter the price indexes by the minimum amount necessary to satisfy within region fixity. Before attempting to formulate this problem mathematically, 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 Eltetö and Köves 1964 and Szulc 1964). More precisely, GEKS solves the following problem: [ ( ) ] K K 2 Pk Min ln(pk /P j ) ln ln P F j,k j1 k1 P j. (1) That is, the solution for P k /P j 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 [ N A N C + ln c1 [ N A N B Min ln(λ),ln(µ),ln(ρ) global ) ln ln ( )] P 2 N B + P global ) ( )] P 2 ln P [ N C ln c1 global ) ln ( )] P 2 P, (2) 4 The method generalizes in a straightforward way to the case where there are four or more regions. 4

subject to the within-region fixity constraints: λ, µ, ρ, (3) where λ, µ, ρ are positive scalars, and, and denote arbitrary reference countries from regions A, B and C. The within-region fixity constraints can alternatively and perhaps more intuitively be written as follows: α, (4) β, (5) γ, (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: [ N A N B Min ln(λ),ln(µ),ln(ρ) [ N A N C + ln(µ) + ln + c1 [ N B N C c1 ln(ρ) + ln ln(λ) + ln region ) ( P region )] 2 ln P P region ) ln P ) region P ln region )] 2 P Solving (7), the following first order conditions are obtained: 5 [ N A N B 2N B ln(λ) + ln [ N C ln(µ) + ln c1 N A 2N C [ N B N C 2N C ln(ρ) + ln c1 region ) ln P region ) ln P region ) ln P region )] 2 P. (7) region )] 0. (8) P region )] 0. (9) P region )] 0. (10) P 5 The 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 N B region ) 1/NB / NA P region ) 1/NA. (11) P µ ρ N C c1 N C c1 region ) 1/NC / NA P region ) 1/NC / NB P region ) 1/NA. (12) P region ) 1/NB. (13) P Now substituting the solutions for λ, µ and ρ into (3), the following global price index formulas are obtained: λ µ ρ region ) ( NB ( NA ( NC c1 ( NA ( NC c1 NB P P P P P ) 1/NB ) 1/NA ) 1/NC ) 1/NA ) 1/NC ) 1/NB. (14). (15). (16) From (14), (15) and (16) it can be seen that the global price indexes are transitive. Hence the numerators and denominators can be separated into region specific components as follows: N A N B N C c1 ( ) P 1/NA, (17) ( ) P 1/NB, (18) ) 1/NC. (19) Again it should be noted that while each region has its own base country (i.e., A1 B1 1), there is only one base country in the comparison (i.e., P A1 1 while 6

in general P B1 1 and P C1 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 A1 1 by dividing through by N A (P / ) 1/N A. 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: Each term ( N A N B P P ( NB ( NA P P ) 1/NB ) 1/NA 1/(N A N B ) / ) (P /P ) ( / (20) ) in (20) can be thought of as a chained price index that compares countries and via countries and (i.e., the chaining path is ). / The overall global price index is the geometric average of the chained price indexes obtained by using all possible chain paths from to. 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 : A1 B2 B1 : A1 A2 B1 : ( ) p B1 p A1 ( p B2 p A1 region A2 A1 B1 B2 p B1 p A2 6 The global price indexes are invariant (after rescaling) to the choice of the regional base countries A1, B1, C1, and the base country in the comparison A1. ) ) 7

A1 A2 B2 B1 : region A2 p B2 p A2 B1 B2 ) A1 region A3 A1 A3 B1 : p B1 p A1 A3 region A3 A1 A3 B2 B1 : p B2 p ) B1 A1 A3 B2 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 price indexes are transitive. For example, as shown below the path reduces to : P P P P P P ) 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 Al reduces to : Al Al P P P P. An analogy can be drawn here with the GEKS formula, which transitivizes Fisher price indexes (denoted here by P F j,k where j and k denote countries) in a similar way: P GEKS 2 P GEKS 1 ( ) K P F 1/K k,2. (21) k1 Using the fact that Fisher price indexes satisfy the country reversal test, this formula can be rewritten as follows: P GEKS 2 P GEKS 1 [ P F k,1 K 1/K (P1,2) F 2 (P1,k F Pk,2)] F. k3 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 F 1,2, and K 2 chained price indexes obtained by chaining through all possible third countries, i.e., P F 1,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, withinregion 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 Countries With slight modifications the method can be applied to an ICP 2005 context in which ring countries were used instead of 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: N A N B N C γ1 ring ) 1/NA, (22) ring ) 1/NB, (23) ring ) 1/NC. (24) There are two changes here. First, the price indexes P have been replaced by ring price indexes P ring. Second, while a 1,..., N A 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 ) as follows: Az Az Az N A ( ) P 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 Az 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 comparison. Once the 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 price indexes that are used in (20), such as those with a more complete coverage of the products in the list 10

(while ensuring that there is enough representation from each region). Furthermore, the list of countries used in the comparison could vary from one basic heading to the next. The countries excluded from the 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 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 Fixity 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: [ N A N B Min ln(λ) subject to the within-region fixity constraint: where now and P unfixed ln global ) ( P unfixed )] 2 ln P unfixed (26) λ, (27) are both aggregate price indexes calculated using GEKS or some other multilateral formula such as Geary-Khamis, IDB or a minimum-spanning- 7 At 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 and P unfixed is that, in the case of, 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: ( ) NB P unfixed 1/NB ( ) 1/NA, (28) NA P unfixed which in turn up to a normalizing scalar implies that ( N A P unfixed ) 1/NA, (29) N B unfixed ) 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: [ ( N A N B Q global ) ( Q unfixed )] 2 Min ln(λ) ln Q global ln Q unfixed (31) subject to the within-region fixity constraint: Q global Q global λ Qregion Q region where Q global, Q region and Q unfixed denote quantity indexes. Solving this least-squares problem yields the solution: ( Q global NB Q unfixed Q global Qregion Q region Q region ( NA, (32) Q unfixed Q region ) 1/NB ) 1/NA, (33) 8 Geary-Khamis was developed by Geary (1958) and Khamis (1972), while minimum-spanning-tree methods were developed by Hill (1999) and Diewert (2010b). 9 In 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 Iklé (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 Q global Q global Q region Q region N A N B Combining (28) and (33) yields the following expression: N B unfixed Q global Q global Q unfixed Q region ( Q unfixed ) 1/NA Q region, (34) ( Q unfixed ) 1/NB Q region. (35) ) 1/NB / N A Q region Q region unfixed Q unfixed Q region ) 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 N B unfixed Q region Q region Q unfixed Q region ( Ii1 ) /( p,i q Ii1 ),i p,i q,i Ii1 p B1,i q Ii1 ; B1,i p A1,i q A1,i ) 1/NB N B. [( Ii1 p,i q,i Ii1 p A1,i q A1,i ) ( Ii1 p B1,i q B1,i Ii1 p,i q,i )] 1/NB ; N A unfixed Q unfixed Q region ) 1/NA N A [( Ii1 p,i q,i Ii1 p A1,i q A1,i ) ( Ii1 p A1,i q A1,i Ii1 p,i q,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. N B The product of the global price and quantity index can now be rewritten as follows: Q global Q global [( Ii1 ) /( p,i q Ii1 )],i p,i q,i Ii1 p B1,i q Ii1 B1,i p A1,i q A1,i [( Ii1 ) ( p,i q Ii1 )] 1/NB /,i p B1,i q NA B1,i Ii1 p A1,i q Ii1 A1,i p,i q,i [( Ii1 ) ( p,i q Ii1 )] 1/NA,i p A1,i q A1,i Ii1 p A1,i q Ii1 A1,i p,i q,i [( Ii1 ) /( p,i q Ii1 )] N,i p,i q B (,i Ii1 ) 1/NB p B1,i q B1,i Ii1 p B1,i q Ii1 B1,i p A1,i q Ii1 A1,i p A1,i q A1,i Ii1 p,i q,i Ii1 p,i q,i. 13

On rearrangement this becomes Ii1 / p,i q,i Q global Ii1 p,i q,i Q global P global The direct global price indexes / are identical to the implicit global price global global indexes P / P derived from the direct global quantity indexes via the factorreversal 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: [ ( N A N B P global ) ( P unfixed )] 2 Min ln(λ) ln ln P unfixed, (36) subject to the fixity constraints: The constraints in (37) and (38) can be combined as follows: λ. (37) P unfixed. (38) λ P unfixed. (39) Substituting (39) into (36) reduces the optimization problem to the following: [ ( N A N B P region )] 2 Min ln(λ) ln(λ) ln P unfixed. (40) 14

Solving (40) yields the following first order condition: which on rearranging simplifies to [ ( N A P unfixed )] 2N B ln(λ) ln 0. (41) λ N A unfixed ) 1/NA. (42) Now substituting the solution for λ from (42) into (39) yields the following global price index formulas: or alternatively, P unfixed N A N A region ) 1/NA P unfixed, (43) unfixed ) 1/NA, (44) P unfixed. (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 properties 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 A 1 and A 2 and that countries a 1,..., j lie in the first subregion while countries a j + 1,..., N A lie in the second subregion. Optimal within-region price indexes that satisfy within-subregion fixity are now obtained as follows: P subregion A 1 a j P reg unfixed 1/j A 1 a P subregion, for a 1,..., j, A 1 a 15

where P subregion A 1 a P subregion A 2 a and A 2 and P reg unfixed A 1 a of region A. N A aj+1 P reg unfixed A 2 a P subregion A 2 a 1/(N A j 1, for a j + 1,..., N A, and P subregion A 2 a denote the within-subregion price indexes of regions A 1 and P reg unfixed A 2 a denote the unfixed within-region price indexes The overall global results for region A are now obtained as follows: P subregion A 2 a P subregion A 1 a N A aj+1 A 1 a N A j P reg unfixed 1/j NA A 1 a P subregion A 1 a P reg unfixed A 2 a P subregion A 2 a A 2 a N A 1/(N A j 1) NA unfixed ) 1/NA unfixed ) 1/NA, for a 1,..., j, unfixed ) 1/NA unfixed ) 1/NA for a j + 1,..., N A. 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: s global s unfixed A s region, (46) where the value shares are essentially rescaled quantity indexes. s global Q global Nn1 Q global n is the share of country in total world output in the global comparison. s unfixed A NA Q unfixed Nn1 Q unfixed n,, 16

is the share of region A in total world output obtained from the unfixed global comparison (where within-region fixity is not satisfied). s region Q region NA Q region is the share of country in the total output of region A obtained from the within-region A comparison. Converting (46) into quantity indexes yields the following expression: Q global Nn1 Q global n ( NA Q unfixed Nn1 Q unfixed n, ) Qregion NA Q region An equivalent expression for another country is as follows: Q global Nn1 Q global n ( NB Qunfixed Nn1 Q unfixed n ) ( Q region NB Qregion Now dividing the latter by the former, and rearranging yields the following: Q global ( Q global Qregion NB ) /( Qunfixed NA Q unfixed ) Q region NB Qregion NA Q region. (47) It is easily verified that the global quantity indexes derived from (47) satisfy withinregion fixity. Replacing with Ak, the term on the righthand side of (47) collapses to Q region Ak /Q region. 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 indexes 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 Eurostat/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 10 The percentage differences are calculated as follows: 100 [max(q EOH k, Q HD k 1]. The same percentage differences are obtained if the price indexes are used. )/min(q EOH k, Q HD k ) 18

basic headings to compute CPD price indexes over the products for all participating countries (of which there will be about 150 in ICP 2011). If some countries fail to price any of the products in a particular basic heading, then as mentioned earlier these countries can simply be excluded from the comparison and then linked back in later using the formula in (25) (with ring replaced by ). If, however, across all countries there are simply not enough 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. References Balk, Bert M. (1996), A Comparison of Ten Methods for Multilateral International Price and Volume Comparisons, Journal of Official Statistics 12, 199-222. Blades, Derek (2007) GDP and Main Expenditure Aggregates. 2003-2006 Handbook. Washington DC: The World Bank. Chapter 3 in ICP Diewert, W. Erwin (2008a), New Methodology for Linking the Regions, Discussion Paper 08-07, Department of Economics, University of British Columbia, Vancouver, Canada. Diewert, W. Erwin (2008b), New Methodological Developments for the International Comparison Program, Discussion Paper 08-08, Department of Economics, University of British Columbia, Vancouver, Canada. Diewert, W. Erwin (2010a), Methods of Aggregation above the Basic Heading Level within Regions, Chapter 7 in Rao D. S. and F. Vogel (eds.), Measuring the Size of the World Economy: A Framework, Methodology and Results from the International Comparison Program (ICP), forthcoming. Diewert, W. Erwin (2010b), Methods of Aggregation above the Basic Heading Level: Linking the Regions, Chapter 8 in Rao D. S. and F. Vogel (eds.), Measuring the Size of the World Economy: A Framework, Methodology and Results from the International Comparison Program (ICP), forthcoming. Dikhanov, Yuri (1997), Sensitivity of PPP-Based Income Estimates to Choice of Aggregation Procedures, Development Data Group, International Economics Department, The World Bank, Washington D.C., January. Dikhanov, Yuri (2007), Two Stage Global Linking with Fixity: Method 1 (EKS), 20

World Bank, Mimeo. Eltetö, Oded and Pal Köves (1964), On a Problem of Index Number Computation Relating to International Comparison, Statisztikai Szemle 42, 507-518. Geary, Roy G. (1958), A Note on Comparisons of Exchange Rates and Purchasing Power between Countries, Journal of the Royal Statistical Society, Series A 121, 97-99. Gini, Corrado (1931), On the Circular Test of Index Numbers, International Review of Statistics, Vol.9, No.2, 3-25. Heston, Alan (1986), World Comparisons of Purchasing Power and Real Product for 1980. United Nations and Eurostat, New York: United Nations. Hill, Robert J. (1999), Comparing Price Levels Across Countries using Minimum Spanning Trees, Review of Economics and Statistics 81(1), February 1999, 135-142. Hill, Robert J. and T. Peter Hill (2009), Recent Developments in the International Comparison of Prices and Real Output, Macroeconomic Dynamics 13 (supplement no. 2), 194-217. Hill, Robert J. and Iqbal Syed (2010), Improving International Comparisons of Real Output: The ICP 2005 Benchmark and its Implications for China. Report prepared for World Bank. Iklé, Doris M. (1972), A New Approach to the Index Number Problem, Quarterly Journal of Economics 86(2), 188-211. Khamis, Salem H. (1972), A New System of Index Numbers for National and International Purposes, Journal of the Royal Statistical Society, Series A 135, 96-121. Kravis, Irving B., Alan Heston, and Robert Summers (1982), World Product and Income: International Comparisons of Real Gross Product. Published for the World Bank by Johns Hopkins University Press: Baltimore. Rao, D. S. Prasada (2004), The Country-Product-Dummy Method: A Stochastic Approach to the Computation of Purchasing Power Parities in the ICP, Paper 21

presented at the SSHRC Conference on Index Numbers and Productivity Measurement, Vancouver, Canada, 30 June-3 July, 2004. Sergeev, Sergey (2003), Equi-representativity and Some Modifications of the EKS Method as the Basic Heading Level. Paper presented at the Joint Consultation on the European Comparison Programme, ECE, Geneva, 31 March-2 April 2003. Sergeev, Sergey (2005), Calculation of the Results of the Eurostat GDP Comparison with the Use of Fixity. Paper prepared for ICP Technical Advisory Group. Sergeev, Sergey (2009), The Evaluation of the Approaches Used for the Linking of the Regions in the ICP 2005, Statistics Austria, Mimeo. Summers, Robert (1973), International Price Comparisons Based upon Incomplete Data, Review of Income and Wealth 19(1), 1-16. Szulc, Bohdan J. (1964), Indices for Multiregional Comparisons, Przeglad Statystyczny 3, Statistical Review 3, 239-254. 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