# Comparison of PPP formulas

This page is a comparison of PPP formulas. A PPP formula is used to calculate price matrices.[1]

## General comparison

Index name Type First publication year Used in Notes
Laspeyres Bilateral
Paasche Bilateral
GEKS-Fisher
Geary–Khamis Multilateral 1958, 1972[3]:33 ICP 1975 (Kravis, Kenessey, Heston, and Summers)
Superlative method I think this is a class of methods defined by Walter Erwin Diewert; they are all the ones that satisfy a list of properties, see [1]
Gerardi[4] EUROSTAT[4]
Binary-Fisher[4]
Gini–Eltetö–Köves–Szulc (GEKS, sometimes just EKS)[4][3]:33 Multilateral OECD–Eurostat (a variant with representativity, sometimes called GEKS*)[5]:235
Walsh[4]
Van Yzeren[4] Multilateral European Coal and Steel Community (1960)[6]
Exchange rate[4]
Young[3]
Sidgwick–Bowley[3] Bilateral Arithmetic mean of Laspeyres and Paasche
Fisher ideal[3] (same as "Fisher"?) Bilateral NIPA 1999 in part of the contribution to percent change calculation[7]
Marshall–Edgeworth[3]:7
Carli[3]:7 Bilateral 1764
Jevons[3]:8 Bilateral 1865
Törnqvist[3]:9 Bilateral 1936
Konüs–Byushgens[3]:28 1926
Star method[3]:33 Multilateral
Democratic weights method[3]:33 Multilateral
Plutocratic weights method[3]:33 Multilateral
Own share method[3]:33 Multilateral
Country Product Dummy[3]:34 Multilateral 1973 ICP[6]:86
Unit-country weighting method[6]:78 Multilateral Common Market (1975) Developed by Dino Gerardi. Potentially the same as the Gerardi index.
Iklé method[8]:16 Multilateral 1972
Iklé–Dikhanov–Balk (IDB)[9]:3 1972[9]:9 Multilateral Africa region of ICP 2005[9]:3
Neary's GAIA system[9]:19 Multilateral 2004 This method "allows for nonhomothetic preferences on the part of final demanders" but "uses a single set of relative prices to value consumption or the gross domestic product (GDP) in all countries, no matter how different are the actual relative prices in each country".[9]:19
Minimum spanning tree (MST)[9]

Laspeyres No
Paasche No

## Properties of multilateral indices

Comparison style is one of:

• uses selective binary comparisons
• uses binary comparisons
• makes no use of any binary comparisons
Formula name Comparison style Reduces to in the case of two countries Base country invariant? Transitive? Matrix consistency? Factor-reversal test? Additive?
Van Yzeren Uses binary comparisons (Fisher) Fisher Yes
EKS Uses binary comparisons (Fisher) Fisher Yes
Walsh Yes
Geary–Khamis Yes Yes (strongly)[8]:13
Iklé–Dikhanov–Balk Yes[9]:3

## Notes/scratch work

Four multilateral methods are considered in detail: (1) Walsh, (2) EKS, (3) Van Yzeren, and (4) Geary-Khamis. Each method goes beyond the binary procedures of Chapter 4 by drawing upon price and quantity data of all countries simultaneously in aggregating up from the category level. They all are base country invariant, and have the transitivity property, and can be adapted to a form that gives additive consistency. The EKS method meets the factor-reversal test. The Geary-Khamis method also satisfies the test at the GDP level. Only in a purely definitional sense (that is, by deriving either the PPPs or the quantity index indirectly) can the Walsh and Van Yzeren methods and the Geary-Khamis subaggretates be said also to meet the test.[10]

What methods do the following use?

• OECD
• Eurostat
• World Bank
• IDB

chained/chain-linked vs fixed-base versions for each of the above? [2]:6 [3]:7

I think there's also bilateral vs multilateral versions of the above?

What parameters do each p (price) or q (quantity) variables take? I have seen time period (usually t or n), commodity/basic heading (c or i), country (j).

A lower bound on the number of price indices: "But Walsh (1901) and Fisher (1922) presented hundreds of functional forms for bilateral price indexes".[3] My current understanding is that only a handful are commonly used in practice, perhaps because the others fail to satisfy nice properties.

The multilateral indices can be divided into those that use binary comparisons (i.e. make use of some binary index) and ones that don't. Examples of the former are EKS and Van Yzeren. Examples of the latter are Walsh indices (apparently there are multiple).[6]:76, 77 This is probably why Walsh is listed as both a binary and a multilateral index: there is some binary formulation, and you can probably generalize it for multilateral comparison, but that multilateral index does not use the binary comparisons from the binary version. (My guess at what's going on.)

For multilateral indices, there is the question of what bilateral index it reduces to when the number of countries goes down to two.

A bilateral/binary index works just as well for locations as time periods. You just need as inputs two price vectors and two quantity vectors (where the coordinates are the different goods). The two price vectors can be two periods or two countries.

Balk lists the following multilateral methods: GEKS, Van IJzeren, Gerardi, WFBS (weighted Fisher blended system), WDOS (Diewert's Own Share system), YKS (Kurabayashi and Sakuma), Q-YKS, GK, EWGK (equally weighted GK method), KS-S (Kurabayashi and Sakuma), SRK (Sakuma, Rao, and Kurabayashi), Sergueev, Gerardi, Iklé, Rao.[8]

One taxonomy of multilateral indices is given by Hill.[11]