Calculation of the results
of the Eurostat GDP comparison
with the use of fixity
S. Sergeev
Statistik Austria
May 2005
The Eurostat comparison comprises not only the EU Members but also some other countries (EFTA countries and CC). However, the results of the Eurostat comparison are calculated with the principle of fixity. The procedure of fixity keeps the relationships between the EU countries (obtained in the pure EU comparison) in each wider comparison. It means that data of nonEU Members has no impact on the results of the EU Members. The fixity is used mainly due to the political reasons but also due to the fact the results are used for some practical purposes within the EU like the contributions / receipts of the EU Members to / from the Structural Funds.
The procedure of fixity applied by Eurostat was never described in detail. The recent printed Eurostat publication „PPPs and related economic indicators. Results for 1998“ (Eurostat, 2000) contains only one sentence about this point (p. 28): „... the results of the first calculation (EU15  remark of author) replace the results of the second calculation (EFTA18  remark of author) for the European Union countries“.
This very generalized description can be interpreted in different ways those lead to different results. Which concrete results of the 1st calculation replaced the results of the 2nd calculation? The subjects of fixity can be the PPPs, the shares of real values (volumes)^{1}, etc. and this point is necessary to clarify. This note refers to the possible procedures of fixity of results within the Eurostat comparisons^{2}. It is important for a better understanding of the applied methodology and the interpretation of the results^{3}.
I. General description of the application of fixity principle
The practical impact has the fixity for the EU Members only and Eurostat uses from 1999 onwards only one level of fixity – for the EU Members (EU15 for 19992003 and EU25 – from 2004 onwards). However it is better to describe this principle for a more general case because the fixity can be used not only for the EU Members but also for several hierarchical sets of the countries. The description of the Eurostat 1998 comparison is done below for the demonstration. Eurostat used in 1998 (and in preceding years) the fixity for the following sequent subsets of countries^{4}:
1) EU15 (fifteen member states of the EU)
2) EU  EFTA18 (EU15, CH, IS, NO)
3) Eurostat20 (EFTA18, Poland, Cyprus).
It means that 3 calculations of the EKS results were carried out step by step:
1) Firstly, for EU15 countries only,
2) Secondly, for EUEFTA18 countries,
3) Thirdly, for all 20 countries participating in the Eurostat 1998 comparison.
In accordance with fixity principle the following combination of set results were given as the final output. The inside results for the EU countries are the results from EU15 calculation; for 3 EFTA countries (CH, IS, NO): inside results  from the EUEFTA18 calculation, outside results  a combination from inside EUEFTA18 results and the results from EU15, and for other countries (Poland, Cyprus)  the results obtained during the calculation for all 20 participating countries in combination with the results of EU15 and EUEFTA18 (for 3 EFTA countries). It means that the final results between EU countries are the same as the results of EU15 comparison (but the results between 15 EU countries and 3 EFTA countries are not the same as within EUEFTA18 countries), i.e. 3 EFTA countries, Poland and Cyprus had no impact on the results of the EU countries (but not vice versa).
II. Presentation of the PPPs in the Eurostat comparison
To better understand all operations during the procedures of fixity, the specific numeraire used by Eurostat by the presentation of the PPPs should be explained.
The PPPs within the Eurostat comparisons are calculated from 1990 by the EKS method. It was agreed to express the EKS parities with an average of all Member countries (EU15  till 2004 and EU25 – from 2004) as the base rather than an individual country. The EKSPPPs derived in this way are named as the standardized EKSPPPs. A more detail description of the procedure of the standardization of the PPPs is given below.
A square matrix of transitive PPPs „Country A / Country B“ (for each pair of countries) is obtained within the EKSprocedure for each heading^{5}. Due to the transitivity of EKSPPPs it is possible to use just one row or one column only for practical purposes, i.e. EKSPPPs to any country selected as base. The selection of base country has no impact on the final results (it is necessary only as an intermediate step) because this operation does not change relations between countries due to the transitivity of the EKSPPPs. However, it is not always desirable to use a concrete country as base due to bureaucratic (political) and also practical reasons (e.g. reference PPPs for missing PPPs for some countries). Therefore the original EKSPPPs are transformed into a standardized form with the base „Whole region (EU) = 1“, i.e. an imaginary (fictious) "average" currency of all EU members is selected as a numéraire. This artificial unit (currency) was named as PPS = Purchasing Power Standard.^{6}
An usual question from the users is the following – How is calculated the PPS? This question is incorrect. The PPS is an imaginary currency (measuring unit) and this can’t be calculated (this is similar with national currencies which are also not calculated but are determinated conventionally). However the PPPs between national currencies and this artificial currency PPS can be calculated.
Concretely the PPP between the currency of the given EU country J and a standard of reference (PPS) is calculated as the nonweighted geometric average of all PPPs for given country J with all EU countries: PPP „J/Country 1“, PPP „J/Country 2“, ... , PPP „J/Country N“ (incl. PPP „J/J“ = 1), i.e. the following formula is used:
N
(1) PPP „J/PPS“ = (PPP^{J/i})^{1/N}; j =1, 2, ..., N
i = 1
where
PPP „J / PPS“^{7}  purchasing power parity between currency of EU country J and PPS,
PPP^{J/i}  purchasing power parity between currencies of countries J and i,
N  no. of EU countries (at the moment  25).
The product of all standardized PPPs „J / PPS“ for the EU countries obtained by formula (1) is equal to 1.
In practice, Eurostat uses a slightly modified procedure. The modification concerns only the sequence of calculation but not the results as such. The following steps are carried out actually:
a) row of EKSPPPs for selected base country (usually Germany, i.e. PPP „Country K/EuroDE“) is the starting point;
b) calculation of an unweighted geometric mean from PPPs for all EU countries „Country K/ EuroDE: PPPs „EuroBE/EuroDE“, „FREuro/ EuroDE“, „ITEuro/ EuroDE“, etc.)“ in this row (incl. PPP „ EuroDE/ EuroDE“ = 1), i.e. a PPP „Average imaginary currency (PPS)/ EuroDE“
c) calculation of standardized PPPs by formula:
(2) PPP „K/PPS“ = PPP „K/EuroDE“ : PPP „PPS/ EuroDE“.
Obviously, the results calculated by formula (2) are identical to the results calculated by formula (1).
The standardized PPPs for nonEU countries are calculated as a product of an original PPP „Country  Base EU country (=DE)“ and a standardized PPP for this Member country selected as base (DE):
(3) PPP „Country J/PPS“ = PPP „J/ EuroDE“ * PPP „ EuroDE / PPS“.
All further considerations concerning the procedures of fixity are given in the terms of standardized PPPs.
III. Possible procedures of fixity
As it was mentioned earlier, the subjects of fixity can be different: the PPPs as well as the country’s shares of real values. Respectively, two different possible procedures of fixity are described below:
I) Fixity of PPPs within the given set of countries,
II) Fixity of shares of countries in the total real value for the given set.
These procedures are illustrated on the example with 4 imaginary countries (C.1, C.2, C.3, C.4) where the countries C.1 and C.2 are a zone of fixity. All calculations are presented in the Annex 1.
III.1 Fixity of PPPs
The fixity of PPPs for the fixity purposes is the simplest way. The following steps are carried out during this procedure:
1) Calculation of PPPs for fixity zone with the standardization (Zone = 1)
Let the original PPPs (to Cou.1) are:
Original PPPsI
PPP „Cou.1 / Cou.1“ = 1
PPP „Cou.2 / Cou.1“ = 2
***************************************
Geometric Mean = (1*2)^0.5= 1.41
Standardized PPPs are calculated by formula (2):
Standardized PPPsI
PPP „Cou.1 / PPS“ = 0.71
PPP „Cou.2 / PPS“ = 1.41
2) Calculation of PPPs for all four countries
Let the original PPPs (to Cou.1) are:
Original PPPsII
PPP „Cou.1 / Cou.1“ = 1
PPP „Cou.2 / Cou.1“ = 3
PPP „Cou.3 / Cou.1“ = 5
PPP „Cou.4 / Cou.1“ = 10
2a) Standardization of PPPs
Geometric Mean of PPPs for the Cou.1 and Cou.2 = (1*3)^0.5 = 1.73
Standardized PPPs are calculated by formula (2) for countries 1 & 2 and by formula (3) for countries 3 & 4:
Standardized PPPsII
PPP „Cou.1 / PPS“ = 0.58
PPP „Cou.2 / PPS“ = 1.73
PPP „Cou.1 / PPS“ = 2.89
PPP „Cou.2 / PPS“ = 5.77
3) Replacement of standardized PPPsII for fixity zone (countries 1 & 2) by standardized PPPsI obtained during the 1st calculation:
Final standardized PPPs Final PPPs to Cou.1
PPP „Cou.1 / PPS“ = 0.71 PPP „Cou.1 / Cou.1“ = 1.00
PPP „Cou.2 / PPS“ = 1.41 PPP „Cou.2 / Cou.1“ = 2.00
PPP „Cou.1 / PPS“ = 2.89 PPP „Cou.3 / Cou.1“ = 4.08
PPP „Cou.2 / PPS“ = 5.77 PPP „Cou.4 / Cou.1“ = 8.16
It is visible that the results inside the subsets (for the countries 1 & 2 and for the countries 3 & 4) are unchanged relatively the original calculations but the external results, i.e. the results between countries from different subsets, are different from the original calculation (see original PPPsII and final PPPs to the base country 1). This approach can be regarded as the fixity of PPPs without the use of the country’s expenditure data.
III.2 Fixity of shares of the countries in the total real value for the given subsets
The approach described above is looked as very natural and unique. However the use of fixity of shares of the countries in the total real value for the given subsets (this approach can be regarded also as the fixity of PPPs with the use of the country’s expenditure data) is also possible and, as it will be explained in the next section, can be more preferable for some purposes.
The following steps are carried out to keep the fixed Volume shares of the countries:
1) Calculation of PPPs for zone of fixity
1a) Standardization of PPPs (Zone = 1).
This is the same operation as in the III.I) Fixity of PPPs
1b) Calculation of real values (RV) in the term of standardized PPS
RV(j) = NV(j) / PPP“J / PPS“,
where
RV(j)  real value for country J measured in PPS
NV(j)  nominal value for country J measured in national currency
1c) Calculation of shares (RSH) of countries in the total real value for zone
RSH(j) = RV(j) / SUM(RV(j)),
where
RSH(j)  share of country J in the total RV for zone measured in PPS
2) Calculation of PPPs for all countries
2a) Standardization of PPPs
This is the same operation as in the III.I) Fixity of PPPs
2b) Calculation of real values (RV) in the terms of standardized PPS for the countries and for the zones
2c) Calculation of shares (RSH) of countries in the total real value for zones
3) Distribution of Total Real Value for fixity zone obtained during the 2nd calculation between the countries 1 & 2 according to the Real Values shares obtained during the 1st calculation.
The obtained real values can be named as fixed real values (FRV):
FRV(j) = Total RVII * SHR(j)I.
4) Calculation of new PPPs (FPPPs) for the countries from the fixity zone:
FPPPs (j) = NV(j) / FRV(j).
It is obviously that the results inside the subsets (for the countries 1 & 2 and for the countries 3 & 4) are unchanged relatively the original calculations, but external results, i.e. the results between countries from different subsets are different from the original calculation (see original PPPsII and final PPPs to the base Country 1).
Further it is visible that the final PPPs obtained by this approach are dependent on the expenditure data and therefore they are some different from the fixity PPPs obtained by the 1st approach.
IV. Recommendations for the practical use and the realization in the computer computational programs
Taking into account the general accounting identity of the PPP exercise:
Iq = Ie / Ip,
where
Ie  expenditure ratio,
Ip  price ratio,
Iq – quantity (Volume) ratio
it is clear that the choice of the indicator for fixity (PPPs or Shares of Volumes) has an impact only on results for the nonEU countries (outside the zone of fixity); the results for the EU Members will be the same in both versions. Additionally, as showed experimental calculations, even for the nonEU members the differences in the results are insignificant – see, for example, the results for Eurostat 2002 exercise in Annex 2. Nevertheless it is desirable to include both approaches in the forthcoming new (ORACLE based) software for PPP calculations because the theoretical methodological preferability can be different at different stages of the calculations.
The fixity of PPPs is inevitable during the calculation at the basic heading level because the BHPPPs are calculated earlier than BH expenditure data is available. The fixity of PPPs is also reasonable within the EKS aggregation procedure during the calculation of reference PPPs because of two circumstances: (1) each reference PPP is an estimation / proxy and therefore the simplest method should be used and (2) the expenditure data used for the calculation of PPPs and the expenditure data for which this reference PPPs should be used are different in this case.
The present Eurostat practice is based on the use of fixity for PPPs in all cases (for BHPPPs, for the reference PPPs as well as for the PPPs for analytical categories), although, as it was mentioned earlier, there is no any official indication on this in the methodological documents. It seems that this was done from the point of view of simplicity for the calculations and understanding. Nevertheless the main output of international GDP comparisons is the Volume indicators. Therefore, the fixity for the Shares of Volumes at the level of the analytical categories would be more consistent with this general aim. We do not propose to change the present practice but we insist that the forthcoming comprehensive PPP software should be flexible in this context:

It should be possible to keep fixity for the several hierarchical (without overlap) sets of countries with the possibility of their indication by the users,

It should be possible to keep the fixity for the PPPs as well as for the Shares of Volumes (a parameter given by a user for a concrete calculation),

Fixity should refer not only to the results for the analytical categories but also to hierarchical and nonhierarchical reference PPPs should be calculated also with the fixity (without the use of expenditure data).
The present VBA PPP software prepared by the author of this notice included all features listed above and therefore it can be as a prototype for the respective part of the forthcoming Eurostat PPP (ORACLE based) package. Obviously some “environmental” improvements can be done to increase the friendliness of computer programs for the users and to do the output of the results more analytical.
Annex 1
Imaginary example for the clarification of the "fixity" principle
(with expenditure – fixity of Volume Shares and without expenditure – fixity of PPPs)

Countries 1 and 2 are a zone for fixity










NV

Origin. PPPsI

PPSI stand.

RVI (in st. PPS)

ShareI of RV

Origin. PPPsII

PPSII stand.

RVII
(in st. PPS)


ShareII of RV

PPSII fix. with exp.data

Final RV (in st. PPS)


PPSII fix.(C.1=1)

Stand.PPSII fix. w/o exp. data

PPSII fix. w/o exp. data


C.1

10

1

0.7

14.1

0.400

1

0.58

17.3


0.303

0.722

13.9


1.000

0.707

1.000


C.2

30

2

1.4

21.2

0.600

3

1.73

17.3

RV for Fix. Gr.

0.303

1.443

20.8

RV for Fix.gr.

2.000

1.414

2.000


C.3

40





5

2.89

13.9

34.6

0.242

2.887

13.9

34.6

4.000

2.887

4.082


C.4

50





10

5.77

8.7


0.152

5.774

8.7


8.000

5.774

8.165






35.4

1.000



57.2


1.000


57.2












RVi/RVii (1&2)

1.020621





























FIXITY  on the basis of Shares of Real Values (Volumes)










NV


PPSI

RVI (in st. PPS)

ShareI of RV


PPSII stand.

RVII
(in st. PPS)


ShareII of RV

PPSII fix. with exp.data

Final RV (in st. PPS)


PPSII fix.(C.1=1)

Stand.PPSII fix. w/o exp. data

PPSII fix. w/o exp. data

C.1

10

1

0.7

14.1

0.286

1

0.58

17.3


0.210

0.758

13.2


1.000

0.707

1.000

C.2

50

2

1.4

35.4

0.714

3

1.73

28.9

RV for Fix. Gr.

0.350

1.516

33.0

RV for Fix.gr.

2.000

1.414

2.000

C.3

80





5

2.89

27.7

46.2

0.336

2.887

27.7

46.2

3.810

2.887

4.082

C.4

50





10

5.77

8.7


0.105

5.774

8.7


7.619

5.774

8.165





49.5

1.000



82.6


1.000


82.6











RVi/RVii (1&2)

1.071652









Annex 2
Eurostat 2002 comparison^{8}:
PPP for GDP by different fixity approaches
Countries

Fixity of PPPs

Ffixity of Shares
of Volumes

GERMANY

1.06797

1.06797

BELGIUM

0.98241

0.98241

DENMARK

9.36701

9.36701

GREECE

0.75734

0.75734

SPAIN

0.82904

0.82904

FRANCE

0.98722

0.98722

IRELAND

1.09658

1.09658

ITALY

0.91751

0.91751

LUXEMBOURG

1.10472

1.10472

NETHERLANDS

1.02686

1.02686

AUSTRIA

1.01583

1.01583

PORTUGAL

0.73290

0.73290

FINLAND

1.09752

1.09752

SWEDEN

10.4355

10.4355

UNITED KINGDOM

0.68180

0.68180

ICELAND

103.188

103.142

NORWAY

10.2147

10.2101

SWITZERLAND

2.06796

2.06704

BULGARIA

0.64619

0.64591

CYPRUS

0.47163

0.47142

CZECH REPUBLIC

15.8627

15.8556

ESTONIA

8.23014

8.22647

HUNGARY

128.328

128.271

LATVIA

0.26505

0.26494

LITHUANIA

1.55447

1.55377

MALTA

0.27419

0.27407

POLAND

2.03705

2.03615

ROMANIA

10 913.7

10 908.9

SLOVAKIA

17.9934

17.9854

SLOVENIA

159.274

159.203

TURKEY

668 848

668 550

Annex 3
OECD practice with the fixity for the results of the Eurostat comparison^{9}
EurostatOECD PPP exercise is the common Program undertaken by Eurostat and OECD. The overall EurostatOECD comparison covers presently covers 42 countries: 31EUR + 7 NonEuropean OECD countries + ISR + RUS + MKD + HRV). Usually each international organisation wants to keep the results of its own comparisons as fixed within a broader comparison. However EurostatOECD cvomparison has a complicated overlapped structure. The EU comparison (presently 25 countries) as well as Eurostat comparison (presently 31 countries) cover the 23 OECD members as well as 8 nonOECD Members (SVN, CYP, MLT, BGR, ROM, 3 Baltic countries). The OECD (presently 30 countries) comprises 19 EU members as well as 11 nonEU members. Therefore it is impossible to keep the fixity of the results simultaneously for the EU countries and for the OECD countries.
In effect, it was agreed that the results of Eurostat comparison for all 31 countries are fixed within the overall EurostatOECD comparison and the results for nonEuropean OECD countries are fixed within the OECD30 comparison. To satisfy this requirement, the OECD receives the input data and the results from the Eurostat comparison and carries out two EKS calculations and following procedures:
1) EKSPPP calculation for OECD30  this calculation is used to obtain the fixed PPPs between nonEuropean OECD countries on the basis of data from the OECD countries only
2) EKSPPP calculation for all 42 countries
3) GM for EUR31 in the calculation (2) are redistributed in accordance with the ratios to GM31 obtained within the EUR31 comparison and GM for 7 NonEuropean OECD in the calculation (2) are redistributed in accordance with the ratios to GM7NonEuropean OECD obtained within (1)
4) The redistributed results from (3) are scaled to OECD30 = 1 by XR to USD
