New Methodological Developments for the International Comparison

Program

Presentation at the Tinbergen Institute at Erasmus University, Rotterdam, October 3, 2008

W. Erwin Diewert

Department of Economics

University of British Columbia

Introduction

• The World Bank released the results for ICP 2005 in February of this year

• 146 countries in 6 regions participated in the comparisons of prices and volumes (or real outputs) for the year 2005

• Each of the 6 regions made up its own list of about 1000 narrowly defined products to be priced within the region

• These individual prices were aggregated into 155 Basic Headings

• Each participating country also provided a GDP breakdown of its expenditures on these 155 categories

• Thus if region r has C(r) countries, we have 2 matrices of size 155 by C(r)

• One matrix has the country price levels• The other matrix has the country

expenditures by 155 commodity classes• Now international comparisons of prices and

volumes within the region can be carried out using EKS or GK

• But how were the regions linked together?• Another commodity list was constructed; the

ring list and 18 countries across the regions priced out this list, enabling linking

• This is what led to new methodological developments; we now have a 2 stage procedure for linking the 146 countries

• Sections 2 and 3: how to link the 155 BH prices across (a) countries within a region and (b) across regions?

• Sections 4 and 5: how to construct aggregate price and volume comparisons across (a) within a region and (b) across regions?

2. Linking prices across countries within a region

• The Country Product Dummy (CPD) method (Summers (1973)) was used by African, Asian Pacific and West Asian regions

• The Extended (to include representativeness) CPD method (Cuthbert and Cuthbert (1988) was used by South America. Hill (2007) called this the CPRD method.

• The EKS* method was used by the OECD and CIS regions.

The CPD method with a balanced panel of price data works as follows:

(1) pcn = acbnucn ; c = 1,…,C; n = 1,…,N;

Taking logs of both sides of (1) leads to:

(2) ycn = c + n + cn ; c = 1,…,C; n = 1,…,N;

where ycn ln pcn, c ln ac, n ln bn and cn ln ucn.

• (2) is a linear regression model. The a’s are the country PPPs for the particular BH category under consideration and the b’s are product premiums that depend on the units of measurement

• The Basic CPDR model is:

(5) ycnu = c + n + u + cnu ; c = 1,…,C; n = 1,…,N;

u = 1,2

where the c are the log country PPP’s, the n are the log product price effects and the u are the two log

representativity effects and the cnu are independently

distributed random variables with mean zero and constant variances. In order to identify the parameters, we impose the following normalizations:

(6) 1 = 0 ; 1 = 0.

• This is another linear regression model. In principle, it should work better than the CPD method.

• The EKS* model is explained by Hill (2007)

3. Comparing Prices Across Regions• The model that was used to link BH price levels

across regions was the following generalization of the CPD model:

(7) prcn ar brc cn ; r = 1,…,5; c = 1,....,C(r); n = 1,...,N

(8) a1 = 1;

(9) br1 =1; r = 1,…,5where the above model pertains only to the price data

for the ring countries. There are C(r) ring countries in region r = 1,2,3,4,5, the a’s are interregional PPPs and (8) means that region 1 is chosen as the numeraire region, the b’s are the country PPPs for the countries in one of the 5 regions and (9) means that country 1 in each region is chosen as the numeraire country in that region and the c’s are commodity effects that depend on the units of measurement for the products.

• In order to respect the parities that were estimated by the regions, the following modification of the basic model above was run by the World Bank:

(13) ln prcn ln brc = ln ar+ln cn + rcn ; r = 1,…,5; c = 1,....,C(r); n = 1,...,N.

• The above model simplifies into:

(14) ln [prcn/brc] = r + n + rcn

which is a linear regression model. The r are the logs of the interregional PPPs and the n are the individual product effects for the products within the basic heading category of commodities which were price out by the ring countries.

4. Relative Prices and Volumes for Countries within a Region

• 5 of the 6 regions used the Gini (1924) (1931) EKS (1964) method to construct aggregate PPPs and relative volumes for the countries in their regions.

• But Africa used a new additive method due to Doris Iklé (1972) and Yuri Dikhanov (1994), who made her method intelligible. Bert Balk (1996) provided the first existence proof for the method so we will call the method the IDB system.

• We will explain these two methods in the next few slides along with the Geary Khamis (GK) method

• Both methods are implemented at the basic heading level where we have price and quantity data available for each country for the 155 basic headings.

4. Relative Prices and Volumes for Countries within a Region

4.1 The Gini EKS Method (GEKS)Define country vectors of BH prices as pk [p1

k,...,pNk], country

vectors of BH quantities as yk [y1k,...,yN

k], country expenditure vectors as ek [e1

k,...,eNk] and country

expenditure share vectors as sk [s1k,...,sN

k] for k = 1,...,K.

(17) PF(pk,pj,yk,yj) [pjyj pjyk/pkyj pkyk]1/2 j = 1,...,K ; k = 1,...,K.

The aggregate PPP for country j, Pj, is defined as follows:

(18) Pj k=1K [PF(pk,pj,yk,yj)]1/K ; j = 1,...,K.

GEKS (continued)GEKS country real outputs or volumes Yj can be defined as the

country expenditures pjyj in the reference year divided by the corresponding GEKS purchasing power parity Pj:

(19) Yj pjyj/Pj ; j = 1,...,K.

The GEKS country shares of world product are defined as follows:

(20) Sk Yk/j=1K Yj ; k = 1,...,K.

Aside on exact and superlative indexes and the role of the Fisher indexes; consistent with perfect substitutability and no substitution at all (Leontief preferences) but also consistent with flexible functional forms in the case of homothetic preferences.

4.2 The Geary Khamis Method (GK)The GK system of equations involves K country price levels

or PPPs, P1,...,PK, and N international commodity reference prices, 1,...,N. The equations which determine these unknowns (up to a scalar multiple) are the following ones:

(21) n = k=1K [yn

k/j=1K yn

j][pnk/Pk] ; n = 1,...,N ;

(22) Pk = pkyk/yk ; k = 1,...,K.

(24) Yk = pkyk/Pk ; k = 1,...,K

= yk using (22).

Problem: Big countries get “undue” weight in the n .

4.3 The Ikle Dikhanov Balk Method (IDB)

Dikhanov’s (1994; 9-12) equations that are the counterparts to the GK equations (21) and (22) are the following ones:

(27) n = [k=1K sn

k [pnk/Pk]1/j=1

K snj]1 ; n = 1,...,N

(28) Pk = [n=1N sn

k [pnk/n]1]1 k = 1,...,K.

Note the use of share weighted harmonic means in (27) and (28). The use of share weights gives the IDB parities a more “democratic” flavour. Equations (24) are still used to define the country volumes Yk. Thus both GK and IDB are termed additive methods since both methods use a common set of international prices to value output components across countries.

5. Aggregate price and Volume Comparisons Across Regions

• Reorganize the countries into 5 regions (we regard the OECD/Eurostat/CIS countries as forming one region).

• Consider region r which has C(r) countries in it. Let pnrc denote the

within region PPP for basic heading class n and country c in region r and let en

rc denote the corresponding expenditure in local currency. • The total regional expenditure on commodity group n in currency

units of country 1 in each region, Enr, is defined as follows:

(31) Enr pn

r1 c=1C(r) en

rc/pnrc ; r = 1,...,5 ; n = 1,...,155.

• The corresponding regional PPPs by region and commodity, Pnr, are

defined to be the world BH parities for the numeraire country in each region:

(32) Pnr pn

r ; r = 1,...,5 ; n = 1,...,155.

• Now each region can be treated as if it were a single supercountry with supercountry expenditures and basic heading PPPs defined by (31) and (32) respectively for the 5 supercountries. The EKS method was used to link these supercountries.

• Once the interregional price and volumes have been determined, the regional price and volume aggregates can be used to provide world wide price and volume comparisons for each individual country. This method necessarily preserves all regional relative parities.

• Hill (2007e) shows that the overall procedure does not depend on the choice of numeraire countries, either within regions or between regions; i.e., the relative country parities will be the same no matter what the choices are for the numeraire countries.

6. Problem Areas and Future Research

• The problem of pricing exports and imports. At present, exchange rates are taken as the price of exports and imports.

• Inaccurate expenditure weights can cause grave difficulties.

• Methodological difficulties with hard to measure areas of the accounts. There are particular problems with housing, financial services and nonmarket production. These are problem areas for regular country accounts as well due to the lack of consensus on an appropriate methodology.

• The fact that current System of National Accounts conventions do not allow an imputed interest charge for capital that is used in the nonmarket sector tends to understate the contribution of this sector and the degree of understatement will not be constant across rich and poor countries.

The lack of matching of products. The same problem occurs in the time series context due to the introduction of new products and the disappearance of “old” products but the lack of matching is much worse in the international context due to differences in tastes and big differences in the levels of development across countries, leading to very different consumption patterns.

However, Structured Product Descriptions were introduced in the current ICP round and this does open up the possibility for undertaking hedonic regression exercises in the next round in order to improve the matching process.

There are many problems to be addressed however, and it would be wise to undertake experimental hedonic studies well in advance of the next round.

• The fact that the ring list of commodities to be priced was almost entirely different from the regional lists means that there is the possibility of anomalies in the final results; i.e., if entirely different products are priced in the ring list, we cannot be sure the relative ring price levels really match up with the relative prices within the regions.

• Thus in the next ICP round, there should be at least some coordination in the determination of the ring product list with the regional product lists so that within each basic heading level, one or more products are on all of the lists.

• It would be advisable to undertake some studies on alternative methods of aggregation at the higher levels of aggregation. In particular, the program of making comparisons based on the degree of similarity of the price and quantity data being compared that was initiated by Robert Hill seems to be sensible but users have not embraced it, perhaps due to the instability of the method. In any case, the World Bank now has a considerable data set based on the current ICP round that could be used to experiment with alternative methods of aggregation.

• Looking ahead into the more distant future, it would be desirable to integrate the ICP with the EU KLEMS project, which is assembling data on the producer side of the economy as opposed to the final demand side, which is the focus of the ICP. Producer data are required in order to calculate relative productivity levels across economies, a topic of great interest to policy makers.

• The data disclosure problem.

7. Conclusion

• The regions liked the idea that they could define their own list of products for international pricing and this improved the quality of the data.

• The new methodology to link prices across the regions using ring countries also seems to be a clear improvement over previous rounds.

• The use of hand held computers and the structured product description methodology led to improvements in the production of national price statistics in many cases.

• Overall, ICP 2005 was a major success!

Appendix: Numerical ExamplesExample 1 from Diewert 1999This was a three country, two commodity example. (A84) p1 [1,1]; p2 [10, 1/10]; p3 [1/10,10] ; y1 [1,2]; y2

[1,100]; y3 [1000,10].Note that the geometric average of the prices in each country

is 1, so that average price levels are roughly comparable across countries, except that the price of commodity 1 is very high and the price of commodity 2 is very low in country 2 and vice versa for country 3. As a result of these price differences, consumption of commodity 1 is relatively low and consumption of commodity 2 is relatively high in country 2 and vice versa in country 3. Country 1 can be regarded as a tiny country, with total expenditure (in national currency units) equal to 3, country 2 is a medium country with total expenditure equal to 20 and country 3 is a large country with expenditure equal to 200.

Example 1 (continued)

Table 1: Fisher Star, GEKS, GK and IDB Relative Volumes for Three Countries Fisher 1 Fisher 2 Fisher 3 GEKS GK IDB

Y1 1.00 1.00 1.00 1.00 1.00 1.00Y2 8.12 8.12 5.79 7.26 47.42 33.67Y3 57.88 81.25 57.88 64.81 57.35 336.67

It can be seen that the GK parity for Y3/Y1, 57.35, is reasonable but the parity for Y2/Y1, 47.42, is too large. The cause of this unreasonable estimate for Y2 is the fact that the GK international price vector, [1,2], is equal to [1, 9.00] so that these relative prices are closest to the structure of relative prices in country 3, the large country.

Example 2 • Yuri Dikhanov rightly objected to the previous example,

noting that the amount of price variation across countries was too extreme compared to the actual amounts. He was nice enough to give me data (from the 2005 ICP) on 5 consumption components for 8 countries.

• The 8 countries are: 1=Hong Kong, 2=Bangladesh; 3=India, 4=Indonesia; 5=Brazil; 6=Japan; 7=Canada and 8=US.

• The 5 commodity groups are: 1=durables; 2=food, alcohol and tobacco; 3=other nondurables excluding food, alcohol, tobacco and energy, 4=energy and 5=services

• The expenditure data (converted to US dollars) and the quantity data for the 8 countries are on the next slide.

Example 2 (continued)Expenditures by commodity (row) and country (column)14320 1963 23207 8234 52722 307547 94121 967374

10562 24835 176782 83882 105527 448995 82056 778665

14951 5100 60748 15158 60798 272875 69461 992761

2619 3094 42126 17573 39933 125835 43342 524288

62124 11627 166826 61248 273669 1736977 379629 5559458

Quantities by commodity (row) and country (column)

15523 2312 30189 9781 46146 280001 81021 967374

9164 47509 356756 138273 163868 251846 63689 778665

17564 10588 180964 29879 65274 200614 58261 992761

1095 3033 38377 22084 23963 59439 35714 524288

81148 47611 786182 223588 541236 1695136 417210 5559458

• We use the above data to compute various indexes.

Example 2 (continued)The GEKS volumes turned out to be:HK BGD INDIA INDO BRA JPN CAN US0.01315    0.01332    0.15317    0.04966    0.09128    0.26556    0.07357    1.00

The Market exchange rate volumes are:0.01185    0.00528    0.05324    0.02109    0.06037    0.32782    0.07578    1.00

The GK volumes turned out to be:0.01386    0.01357    0.16258    0.05057    0.09613    0.27814    0.07431    1.00

The IDB volumes turned out to be:0.01346    0.01392    0.16187    0.05143    0.09441    0.27076    0.07417    1.00

Vector of percentage differences, (GK/GEKS) – 1: 0.05413    0.01898    0.06147    0.01841    0.05306    0.04737    0.01015    0.00

Vector of percentage differences, (IDB/GEKS) – 1:0.02332    0.04497    0.05685    0.03568    0.03429    0.01957    0.00823    0.00

Conclusion: IDB no better than GK relative to EKS

Example 2 (continued)A final method: Robert Hill’s spatial linking method:• Need a measure of similarity in the structure of relative

prices across two countries; see Diewert (2002); is basically a share weighted average of log price ratios, where the prices of one country are deflated by the Fisher price index between the two countries to eliminate the effects of absolute differences in price levels

The Hill volumes turned out to be:

HK BGD INDIA INDO BRA JPN CAN US0.01349    0.01310    0.14720    0.04779    0.09214    0.27596    0.07429    1.00

• The 8 countries grouped themselves into two groups that had similar price structures: rich countries HK, JPN, CAN and US and poorer countries: BGD, INDIA, INDO and BRA. The linking between the two groups took place via HK and BRAZIL.

Example 2 (continued)• My preferred method is the Hill spatial linking method

• The differences between GEKS, GK and IBD are as follows:

HK BGD INDIA INDO BRA JPN CAN US

(GEKS/HILL) – 1:-0.02544    0.01713    0.04054    0.03907   -0.00934   -0.03768   -0.00980    0.0

(GK/HILL) – 1:0.02732    0.03643    0.10450    0.05820    0.04323    0.00790    0.00025    0.0

(IDB/HILL) – 1:-0.00271    0.06287    0.09969    0.07614    0.02463   -0.01885   -0.00165    0.0

• For India, both GK and IDB overstate Hill by 10.0%, for Bangladesh, IDB overstates by 6.3% and GK overstates by 3.6%; for Indonesia, IDB overstates by 7.6% and GK by 5.8%; for Brazil, IDB overstates by 2.5% and GK by 4.3%. These are very substantial differences.

• Conclusion: the choice of multilateral method matters!