EUROSTAT
Meeting of the Working Group on
Purchasing Power Parities
(LUX, May 2001)
Treatment of basic headings with negative nominal values within the aggregation procedures
S. Sergueev
„Statistik Austria“
The most popular aggregation procedures using within the ICP/ECP are the EKS method (the averaging of bilateral results) and the Geary-Khamis (GK) method (the block approach – averaging of national prices). Both multilateral methods are described in details in many reports[1].
However one particular problem should still be clarified. This is the treatment of so-called „Problem categories“ or „Balancing categories“, i.e. basic headings (BH) which are obtained as a balance of other categories and can have negative nominal values (and correspondingly - negative notional quantities). The following BH are regarded usually as „balancing categories“: „Change in stocks“, „Net exports“, „Net expenditures of residents abroad“[2] but, generally, practically all BHs from „Gross Fixed Capital Formation“ (GFCF) in the part „Machinery and Equipment“ and „Other products“ can have negative values. The main reason of this phenomenon is the export of „second hand“ equipment. So input data for the Eurostat 1997 comparison contained really several BH from GFCF with negative values (see para. III). Exactly this circumstance does not allow to carry out the recommendation of some authors – to distribute „Change in stocks“ and „Net exports“ between BH representing goods only to avoid the treatment of negative values.[3]
To guarantee meaningful results for the respective aggregates in general case, the „balancing“ categories need a special treatment. First of all, this was detected by the use of the GK method (see para.II) because negative values in input data led in some cases to meaningless overall GK results –negative international prices or even a negative global PPPs. However a special treatment for „balancing“ categories is needed, in principle, by all methods based on averaging of input data - even by very simple methods like Paasche and Laspeyres indices (see, para.I). An approach to include the „balancing“ categories in the EKS and the GK aggregation general procedures is proposed below.[4]
I. Treatment of „balancing“ BH within the EKS method[5]
The EKS method utilizes all available direct and indirect indices (usually of Fisher’s type). The EKS index between any two countries is the geometric average of the direct index between these countries and all possible indirect (via bridge-country) indices, in which the direct index is given twice the weight of each indirect index.
The basic point of the EKS method is the calculations of bilateral PPPs: Laspeyres and Paasche PPPs with further averaging by Fisher’s formula. Laspeyres-PPP can be interpreted as an arithmetic mean from particular PPPs with weights of base country and Paasche-PPPs can be interpreted as an harmonic mean from particular PPPs with weights of counterpart country. The Laspeyres and Paasche PPPs (and, in effect, EKS-PPPs) as correct averages can be calculated by correct averaging and this is based, first of all, on correct treatment of „Balancing categories“).
Strictly theoretically correct average price indices and quantity indices can be calculated on the basis of non-negative or even strictly positive (for price) input data (prices and quantities). The mechanical application of standard formulas violates the average test in the cases when headings with negative value have a significant share of the aggregate and can lead to fully meaningless results: obtaining of negative Laspeyres or Paasche PPPs (see para.III).
For example, the heading „Net exports“ was splitted in the first phases of the CIS comparisons (1994-1995) into two separate basic headings „Exports“ (positive nominal values) and „Imports“ (negative nominal values) with different Exchange Rates (specific PPPs) for them. Exports and imports were first treated separately because exchange rates of CIS countries were extremely volatile, and it was believed that the timing of foreign trade could have strong influence on results. The separation was done due to some objective reasons but nevertheless this led to the situation that many PPPs for „Net exports“ (1994 as well as 1995) were situated outside the diapason: PPP for „Exports“ - PPP for „Imports“. The most extreme case was PPP „Manat/1000 Russian Roubles“ for „Net exports“ for Turkmenistan in 1994 comparison:
PPP for „Exports“ was 6.87 M/RUR
PPP for „Imports“ was 7.86 M/RUR
but
F-PPP for „Net exports“ was 1.16 M/RUR only.
It was very difficult to interpret obtained results (gigantic differences between aggregated PPP and PPPs for particular headings) in the economic terms but fortunately it turned out that monthly exports and imports in CIS countries followed closely the same pattern in the next year and therefore the method could be abandoned in the further phases of the CIS comparison.
To avoid such cases it is possible to use a simple modification of the standard formulas of Laspeyres and Paasche PPPs: absolute nominal values should be used instead of actual nominal values of expenditure data (official present methodology) which are used as weights.
So according to the modified method the PPPs of Laspeyres-type and Paasche-type have to be calculated on the basis of absolute nominal values:
1) Laspeyres - Type:
S pj * |qk| S pppj/k * |wk|
(I.1) PPPLj/k = ------= ------
S pk * |qk| S |wk|
where
PPPLj/k - Laspeyres-PPP for the aggregate (Country „j“ to Country „k“),
pppj/k- PPPs for basic headings (Country „j“ to Country „k“),
wk - nominal values for basic headings in Country „k“.
2) Paasche - Type:
S pj * |qj| S |wj|
(I.2) PPPPj/k = ------= ------
S pk * |qj| S|wj| / pppj/k
where
PPPPj/k - Paasche -PPP for the aggregate (Country „j“ to Country „k“),
pppj/k- ppp for basic headings (Country „j“ to Country „k“),
wj - nominal value for basic headings in Country „J“.
The modified method guarantees the obtaining of positive meaningfull average PPPs. The absolute values are used for the calculation of bilateral PPPs (I.1 and I.2) only. Real values, etc. should be calculated on the basis of actual nominal values (with signs).
It is obvious from (I.1) and (I.2) that possible differences between the results obtained by official method (actual nominal values) and the modified method (absolute nominal values) are mainly depended on two factors: share of negative expenditure data and the variation of BH-PPPs. For example, if BH-PPPs are the same for all BH within an aggregate then the impact of BH with negative nominal values is eliminated. Some concrete numerical examples from 1997 Eurostat comparison are given in para.III.
II. Treatment of „balancing“ BH within the Geary-Khamis (GK) method
The main idea of the GK method proposed by R.Geary (1958) is the use of international prices which are calculated as average values weighted (the national quantities - physical or imaginary - are used as weights) from national prices revaluated with simultaneously calculating PPPs into a common currency (e.g. International Dollar, International Shilling, etc.). The average international prices and PPPs are interdependent being defined by an underlying set of simultaneous linear equations.
An average „International price“ of the ith item (denoted pi) is the quantity - weighted arithmetic average of the purchased-power-adjusted national prices of the ith item in the N participating countries. The global purchasing power parity = PPP (denoted fj) for the jth country for the aggregate in question is equal to the ratio of the total expenditure at international prices to the total expenditure at national prices.
These definitions lead to the system of (N+M) lineal equations in (N+M) unknowns (pi and fj) which can be modified to the reduced system with (N-1) unknown variables fj. The Gauss-method with the selection of main elements or an iterative method can be used for solving the reduced system of the equations.
As input the GK-method requires prices and physical quantities for the sets of products to be covered. In actual comparisons, input into GK are not quite those originally envisioned by R.Geary. Actually PPPs (‘National currency/Numeraire currency’) for primary groups are used as „notional“ (fictitious) prices and a set of „notional“ (fictitious) quantities, each obtained as ratio of nominal value (in national currency) to corresponding PPP.
S.H. Khamis (1970 and 1972) proved existence and uniqueness of a positive solution for the Geary system. He demonstrated that meaningful results can be guaranteed if non-negative input data („notional“ quantities) are used.
Therefore BH („problem categories“) which can have negative nominal values (and correspondingly- negative „notional“ quantities“) need a special treatment within the GK technique. „Problem categories“ are usually excluded from the GK-calculations and some special calculations (sometimes very complicated) are made for these categories after the main GK-calculation. Such procedure complicates the general algorithm of GK method and, speaking strictly, the separation of an aggregate into two (or several parts) leads to the non-invariant (relatively to the numeraire country) results.
It seems that there is a simple approach to include all categories in general GK-calculation. We propose to use the absolute values of indicators wij and q ij (without sign) instead of their actual values (with signs). In effect, we have the GK system which will produce always positive international prices and PPPs:
(II.1) pi = / ; i = 1,2,...,M
(II.2) fj = / ; j = 1,2,...,N
where
Pij is „notional“ price of ith item in the jth country
qij is „notional“ quantity (weight) for ith item in the jth country;
Qi = - total quantity of the ith basic heading (sum of absolute quantities),
fj is the global purchasing power parity[6] „International currency/National currency“ of given aggregate (GDP) for the jth country
wij = pij * |qij| - modified nominal value for ith item in the jth country (in national currency);
Wj = - modified total value of the aggregate in question for country j at national prices,
N - number of participating countries;
M - number of basic headings (primary groups),
It can believe this approach is not only practicable but also correct from a theoretical point of view of the calculation of „true“ averages. So, international prices are average weighted values from recalculated national prices by help of Global PPPs with qij/Qi as the weights and global PPPs are average weighted values (from individual PPPs „International price/National price“) with the values wij/Wj as the weights. Any correct average (which is between maximal and minimal values) can be calculated if the weights are non-negative. In our case the using of absolute values of indicators qij and wij in (II.1) and (II.2) gives us the possibility for the calculations of correct average values.
The absolute quantities are used within the GK-method for the calculations of international prices only. The actual quantities (values) based upon fact (with sign) and the international prices calculated by formula (II.1) should be used for calculations of real values and respective volume indices, etc. Consequently the global purchasing power parities fj (II.2) are used for the calculation of the average international prices only. The „standard“ PPP „International currency/National currency“ (Fj) which should be used for the calculation of Real Values (Volumes) are calculated as the ratio of Real GDPs at international prices to Nominal GDPs at national prices. Of course, this procedure needs an additional explanation for users but the general „gain“ seems to be more than some „new“ problems.
III. Some examples and experiments within EUROSTAT 1997 comparison
The last EU/OECD comparisons focused mainly on the results obtained by EKS method. Therefore several numerical examples and experiments will be done on the basis of data from Eurostat 1997 comparison.
As it was mentioned above the following BH are regarded traditionally as „balancing categories“: „Other products“, „Changes in stocks“, „Net exports“, „Net expenditures of residents abroad“. Surprisingly, two BH from the aggregate „Machinery and Equipment“ (GFCF) had negative values (a possible reason of this phenomenon can be the export of „second hand“ equipment) within the 1997 Eurostat comparison:
- 14130211 „Boats, etc.“ – for Sweden,
- 14130231 „Aircrafts, etc.“ – for Sweden and Iceland.
In effect, Sweden and Iceland had negative expenditure data for Heading 141302 „Other Transport equipment“ as a whole. This led in the initial versions of the 1997 calculations to several negative binary Laspeyres-PPP and to impossibility to calculate the respective Fisher-PPPs; many of other binary PPPs were meaningless for Sweden and Iceland. The detailed analysis of these curiosities are given in the Table 4 and Table 5.
The software used earlier by Eurostat for the aggregation calculation produced the results for 54/53 analytical categories[7] only and therefore the problem of „balancing“ categories was hidden. A new VBA program[8] used now by Eurostat produced the results for each Heading in the GDP classification (more than 400 headings) and therefore the problem with Heading 141302 „Other transport equipment“ was detected immediately. The official methodology said nothing about the treatment of „balancing“ categories and therefore a practicable simplification was used: Fisher-PPPs were obtained as geometric means from the product of absolute Laspeyres and Paasche PPPs. Obviously this was not a scientific solution but a practicable „trick“ and it was agreed to investigate this problem in detail and to find an appropriate solution for the future comparisons.
The theoretical considerations and proposal were given in para.I. The results of some numerical experiments on the basis of data from 1997 Eurostat comparison are given below. The PPPs for several analytical categories were calculated by official methodology (the use of actual nominal values) as well as by modified methodology (the use of absolute nominal values). It was explained above that the differences between the results are depended mainly on two factors: share of negative expenditure data and the variation of BH-PPPs.
The following analytical categories contained BH with negative values were included in the experiments[9]: GDP, Private Final Consumption (national), Gross Fixed Capital Formation, Machinery and Equipment, Transport equipment.
First of all, the results at the GDP level were examined because they are interesting for all users. As showed the experiments, the impact of BH with negative values was remarkable for some countries even at the GDP level: the differences between the results by official methodology and by experimental methodology were about 2.5 - 6.5% for the countries with a relatively high share of „negative“ BH and a high variation of BH-PPPs (last factor at the GDP level concerns mainly the countries with the high differences between Exchange Rate(XR)[10] and BH-PPPs). The summary of the GDP results for all 20 countries participated in the 1997 Eurostat comparison are given in Table 1 (the differences higher than 1% are given in red script).