Comparison of PPP formulas: Difference between revisions

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This page is a '''comparison of PPP formulas'''. A [[PPP formula]] is used to calculate price matrices.<ref>{{cite web |url=https://en.wikipedia.org/wiki/Purchasing_power_parity#Measurement_issues |title=Purchasing power parity &sect; Measurement issues |accessdate=October 27, 2017 |publisher=English Wikipedia}}</ref>
This page is a '''comparison of PPP formulas'''. A [[PPP formula]] is used to calculate price matrices.<ref>{{cite web |url=https://en.wikipedia.org/wiki/Purchasing_power_parity#Measurement_issues |title=Purchasing power parity &sect; Measurement issues |accessdate=October 27, 2017 |publisher=English Wikipedia}}</ref>


==Comparison table==
==General comparison==


<ref>{{cite web |url=https://www.princeton.edu/~deaton/downloads/PPP_Exchange_rates_may04_relabeled_mar11.pdf |title=Purchasing power parity exchange rates from household survey data: India and Indonesia |date=May 2004 |first1=Angus |last1=Deaton |first2=Jed |last2=Friedman |first3=Vivi |last3=Alatas |accessdate=October 27, 2017 |quote=Originally developed by the International Price Comparison Project for the Penn World Table (PWT), there are now a number of different variants, most notably by the OECD, Eurostat and the World Bank. Although the formulas differ, all of these PPP estimates are based on prices and quantities for each country |publisher=[[wikipedia:Princeton University|Princeton University]]}}</ref>
<ref>{{cite web |url=https://www.princeton.edu/~deaton/downloads/PPP_Exchange_rates_may04_relabeled_mar11.pdf |title=Purchasing power parity exchange rates from household survey data: India and Indonesia |date=May 2004 |first1=Angus |last1=Deaton |first2=Jed |last2=Friedman |first3=Vivi |last3=Alatas |accessdate=October 27, 2017 |quote=Originally developed by the International Price Comparison Project for the Penn World Table (PWT), there are now a number of different variants, most notably by the OECD, Eurostat and the World Bank. Although the formulas differ, all of these PPP estimates are based on prices and quantities for each country |publisher=[[wikipedia:Princeton University|Princeton University]]}}</ref>
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{| class="sortable wikitable"
{| class="sortable wikitable"
|-
|-
! Formula name !! Type !! First publication year !! Used in !! Superlative? !! Additive?<ref name="un-gk" /> !! Notes
! Index name !! Type !! First publication year !! Used in !! Notes
|-
|-
| Laspeyres || Bilateral ||
| Laspeyres || Bilateral ||
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| Geary&ndash;Khamis || Multilateral || 1958, 1972<ref name="diewert-index-numbers" />{{rp|33}} || ICP 1975 (Kravis, Kenessey, Heston, and Summers)
| Geary&ndash;Khamis || Multilateral || 1958, 1972<ref name="diewert-index-numbers" />{{rp|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 [https://en.wikipedia.org/wiki/List_of_price_index_formulas#Superlative_indices]
| 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 [https://en.wikipedia.org/wiki/List_of_price_index_formulas#Superlative_indices]
|-
|-
| Gerardi<ref name="un-gk" /> || || || EUROSTAT<ref name="un-gk">{{cite web |url=https://unstats.un.org/unsd/methods/icp/ipc7_htm.htm |title=United Nations Statistics Division |accessdate=November 3, 2017}}</ref>
| Gerardi<ref name="un-gk" /> || || || EUROSTAT<ref name="un-gk">{{cite web |url=https://unstats.un.org/unsd/methods/icp/ipc7_htm.htm |title=United Nations Statistics Division |accessdate=November 3, 2017}}</ref>
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| Binary-Fisher<ref name="un-gk" /> || ||
| Binary-Fisher<ref name="un-gk" /> || ||
|-
|-
| EKS<ref name="un-gk" /> || Multilateral ||
| Gini–Eltetö–Köves–Szulc (GEKS, sometimes just EKS)<ref name="un-gk" /><ref name="diewert-index-numbers" />{{rp|33}} || Multilateral || || OECD–Eurostat (a variant with representativity, sometimes called GEKS*)<ref>{{cite web |url=https://www.oecd.org/std/prices-ppp/13-3012041ec015.pdf |title=Eurostat-OECD Methodological Manual on Purchasing Power Parities - Calculation and aggregation of PPPs |publisher=OECD/Eurostat |year=2012 |accessdate=November 10, 2017}}</ref>{{rp|235}}
|-
|-
| [[Walsh price index|Walsh]]<ref name="un-gk" /> || ||
| [[Walsh price index|Walsh]]<ref name="un-gk" /> || ||
|-
|-
| Van Yzeren<ref name="un-gk" /> || ||
| Van Yzeren<ref name="un-gk" /> || Multilateral || || European Coal and Steel Community (1960)<ref name="icrgp1982" /> ||
|-
|-
| Exchange rate<ref name="un-gk" /> || ||
| Exchange rate<ref name="un-gk" /> || ||
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| Young<ref name="diewert-index-numbers">{{cite web |url=https://www.business.unsw.edu.au/research-site/centreforappliedeconomicresearch-site/Documents/E.%20Diewert%20-%20Index%20Numbers.pdf |title=Index Numbers (working paper) |date=December 16, 2006 |journal=Journal of Economic Literature Classification Numbers |author=W. Erwin Diewert |publisher=Centre for Applied Economic Research (The University of New South Wales) |accessdate=November 4, 2017}}</ref> || ||
| Young<ref name="diewert-index-numbers">{{cite web |url=https://www.business.unsw.edu.au/research-site/centreforappliedeconomicresearch-site/Documents/E.%20Diewert%20-%20Index%20Numbers.pdf |title=Index Numbers (working paper) |date=December 16, 2006 |journal=Journal of Economic Literature Classification Numbers |author=W. Erwin Diewert |publisher=Centre for Applied Economic Research (The University of New South Wales) |accessdate=November 4, 2017}}</ref> || ||
|-
|-
| Sidgwick&ndash;Bowley<ref name="diewert-index-numbers" /> || Bilateral || || || || || Arithmetic mean of Laspeyres and Paasche
| Sidgwick&ndash;Bowley<ref name="diewert-index-numbers" /> || Bilateral || || || Arithmetic mean of Laspeyres and Paasche
|-
|-
| Fisher ideal<ref name="diewert-index-numbers" /> (same as "Fisher"?) || Bilateral ||
| Fisher ideal<ref name="diewert-index-numbers" /> (same as "Fisher"?) || Bilateral || || NIPA 1999 in part of the contribution to percent change calculation<ref>{{cite web |url=https://www.bea.gov/scb/pdf/national/nipa/1999/1099niw.pdf |title=A Preview of the 1999 Comprehensive Revision of the National Income and Product Accounts: Statistical Changes, October 1999 SCB - 1099niw.pdf |first1=Brent R. |last1=Moulton |first2=Eugene P. |last2=Seskin |date=October 1999 |publisher=Survey of Current Business |accessdate=November 5, 2017}}</ref>
|-
|-
| Marshall&ndash;Edgeworth<ref name="diewert-index-numbers" />{{rp|7}} || ||
| Marshall&ndash;Edgeworth<ref name="diewert-index-numbers" />{{rp|7}} || ||
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|-
|-
| Plutocratic weights method<ref name="diewert-index-numbers" />{{rp|33}} || Multilateral ||
| Plutocratic weights method<ref name="diewert-index-numbers" />{{rp|33}} || Multilateral ||
|-
| GEKS<ref name="diewert-index-numbers" />{{rp|33}} || Multilateral ||
|-
|-
| Own share method<ref name="diewert-index-numbers" />{{rp|33}} || Multilateral ||
| Own share method<ref name="diewert-index-numbers" />{{rp|33}} || Multilateral ||
|-
|-
| Average basket method<ref name="diewert-index-numbers" />{{rp|33}} || Multilateral ||
| Average basket method<ref name="diewert-index-numbers" />{{rp|33}} || Multilateral ||
|-
| Country Product Dummy<ref name="diewert-index-numbers" />{{rp|34}} || Multilateral || 1973 || ICP<ref name="icrgp1982" />{{rp|86}} ||
|-
| Unit-country weighting method<ref name="icrgp1982" />{{rp|78}} || Multilateral || || Common Market (1975) || Developed by Dino Gerardi. Potentially the same as the Gerardi index.
|-
| Iklé method<ref name="balk-2001" />{{rp|16}} || Multilateral || 1972
|-
| Iklé–Dikhanov–Balk (IDB)<ref name="diewert-2011-methods">{{cite web |url=http://siteresources.worldbank.org/ICPINT/Resources/270056-1255977254560/6483625-1291755426408/05_ICPBOOK_AggregationAboveBH_Final.pdf |title=Measuring the Size of the World Economy - Chapter 5: Methods of Aggregation above the Basic Heading Level within Regions |author=W. Erwin Diewert |year=2011 |archiveurl=https://web.archive.org/web/20171109004616/http://siteresources.worldbank.org/ICPINT/Resources/270056-1255977254560/6483625-1291755426408/05_ICPBOOK_AggregationAboveBH_Final.pdf |archivedate=November 9, 2017 |dead-url=no |accessdate=November 9, 2017}}</ref>{{rp|3}} || 1972<ref name="diewert-2011-methods" />{{rp|9}} || Multilateral || Africa region of ICP 2005<ref name="diewert-2011-methods" />{{rp|3}} ||
|-
| Neary's GAIA system<ref name="diewert-2011-methods" />{{rp|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".<ref name="diewert-2011-methods" />{{rp|19}}
|-
| Minimum spanning tree (MST)<ref name="diewert-2011-methods" /> ||
|}
|}


"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 Gery-Khamis subaggretates be said also to meet the test."<ref>{{cite web |url=http://documents.worldbank.org/curated/en/199981467988893189/pdf/multi-page.pdf |title=World Bank Document - multi-page.pdf |accessdate=November 3, 2017}}</ref>
==Properties of bilateral indices==
 
{| class="sortable wikitable"
|-
! Formula name !!  Superlative? !! Additive?<ref name="un-gk" /> !! Transitive/circular?
|-
| 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
 
{| class="sortable wikitable"
|-
! 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&ndash;Khamis || || || Yes || || || || Yes (strongly)<ref name="balk-2001">{{cite web |url=https://www.oecd.org/std/prices-ppp/2424931.pdf |first=Bert M. |last=Balk |date=June 15, 2001 |title=Aggregation Methods in International Comparisons: What Have We Learned? |archiveurl=https://web.archive.org/web/20171108203336/https://www.oecd.org/std/prices-ppp/2424931.pdf |archivedate=November 8, 2017 |dead-url=no |accessdate=November 8, 2017}}</ref>{{rp|13}}
|-
| Iklé–Dikhanov–Balk || || || || || || || Yes<ref name="diewert-2011-methods" />{{rp|3}}
|}
 
==Notes/scratch work==
 
{{quotation|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.<ref>{{cite web |url=http://documents.worldbank.org/curated/en/199981467988893189/pdf/multi-page.pdf |title=World Bank Document - multi-page.pdf |accessdate=November 3, 2017}}</ref>}}


What methods do the following use?
What methods do the following use?
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A lower bound on the number of price indices: "But Walsh (1901) and Fisher (1922) presented hundreds of functional forms for bilateral price indexes".<ref name="diewert-index-numbers" /> My current understanding is that only a handful are commonly used in practice, perhaps because the others fail to satisfy nice properties.
A lower bound on the number of price indices: "But Walsh (1901) and Fisher (1922) presented hundreds of functional forms for bilateral price indexes".<ref name="diewert-index-numbers" /> 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).<ref name="icrgp1982">{{cite web |url=http://documents.worldbank.org/curated/en/974171468766774952/pdf/multi-page.pdf |title=World Product and Income: International Comparisons of Real Gross Product (United Nations International Comparison Project Phase III) |year=1982 |first1=Irving B. |last1=Kravis |first2=Alan |last2=Heston |first3=Robert |last3=Summers |publisher=Johns Hopkins University Press |accessdate=November 5, 2017}}</ref>{{rp|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.<ref name="balk-2001" />
One taxonomy of multilateral indices is given by Hill.<ref>{{cite web |url=http://www.roiw.org/1997/49.pdf |author=Robert J. Hill |title=A Taxonomy of Multilateral Methods for Making International Comparisons of Prices and Quantities |journal=Review of Income and Wealth |date=March 1997 |accessdate=November 8, 2017}}</ref>


==See also==
==See also==
* [[Intertemporal versus interspatial comparison]]


==External links==
==External links==

Latest revision as of 18:13, 12 November 2017

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

General comparison

[2]

Index name Type First publication year Used in Notes
Laspeyres Bilateral
Paasche Bilateral
GEKS-Fisher
Geary–Khamis Multilateral 1958, 1972[3]Template:Rp 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]Template:Rp Multilateral OECD–Eurostat (a variant with representativity, sometimes called GEKS*)[5]Template:Rp
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]Template:Rp
Carli[3]Template:Rp Bilateral 1764
Jevons[3]Template:Rp Bilateral 1865
Törnqvist[3]Template:Rp Bilateral 1936
Konüs–Byushgens[3]Template:Rp 1926
Star method[3]Template:Rp Multilateral
Democratic weights method[3]Template:Rp Multilateral
Plutocratic weights method[3]Template:Rp Multilateral
Own share method[3]Template:Rp Multilateral
Average basket method[3]Template:Rp Multilateral
Country Product Dummy[3]Template:Rp Multilateral 1973 ICP[6]Template:Rp
Unit-country weighting method[6]Template:Rp Multilateral Common Market (1975) Developed by Dino Gerardi. Potentially the same as the Gerardi index.
Iklé method[8]Template:Rp Multilateral 1972
Iklé–Dikhanov–Balk (IDB)[9]Template:Rp 1972[9]Template:Rp Multilateral Africa region of ICP 2005[9]Template:Rp
Neary's GAIA system[9]Template:Rp 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]Template:Rp
Minimum spanning tree (MST)[9]

Properties of bilateral indices

Formula name Superlative? Additive?[4] Transitive/circular?
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]Template:Rp
Iklé–Dikhanov–Balk Yes[9]Template:Rp

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]Template:Rp [3]Template:Rp

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]Template:Rp 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]

See also

External links

References

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  4. ↑ 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Script error: No such module "citation/CS1".
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  7. ↑ Script error: No such module "citation/CS1".
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  9. ↑ 9.0 9.1 9.2 9.3 9.4 9.5 9.6 Script error: No such module "citation/CS1".
  10. ↑ Script error: No such module "citation/CS1".
  11. ↑ Script error: No such module "citation/CS1".

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