A MINIMUM DISTANCE AND THE GENERALISED EKS APPROACHES TO MULTILATERAL COMPARISONS OF PRICES AND REAL INCOMES

A MINIMUM DISTANCE AND THE GENERALISED EKS APPROACHES TO MULTILATERAL COMPARISONS OF PRICES AND REAL INCOMES D.S. Prasada Rao Sriram Shankar School of Economics The University of Queensland Australia Golamreza Hajarghasht Shiraz University, Iran Abstract The paper proposes a new approach: the minimum distance approach as an alternative to the minimum spanning tree (MST) approach; and revisits the weighted EKS (WGEKS) method for undertaking multilateral comparisons of prices and real incomes. These approaches are designed to make complete use of the information available and at the same time provide meaningful and fully operational methods for the computation of PPPs. Two measures of similarity proposed by Diewert (2009) are used in conjunction with the Paasche-Laspeyres spread in implementing the MST, WEKS and minimum distance approaches. The paper establishes analytical properties of the methods and also provides an empirical illustration using data generated from the ICP 2005 at the basic heading level. Robustness of the results to different methods and to alternative methods of similarity is examined. 6 April, 2010

1. Introduction There is on-going research on methods for international comparisons of prices, purchasing power parities (PPPs) and real incomes. Compilation of PPPs within the International Comparison Program (ICP) is usually undertaken in two stages. In the first stage, data on prices collected from different countries for products with specific characteristics are aggregated to yield PPPs at the basic headings (BH). Within the ICP there are 158 basic headings. The main feature of aggregation at this stage is that there are no commodity-specific weights available. The main aggregation method used at this stage is the countryproduct-dummy (CPD) method. In the OECD-Eurostat region, the EKS method is used. In the second stage, these BH parities are aggregated to yield PPPs for various aggregates such as Consumption, Investment, Government Expenditure and so on. Aggregation at this stage makes use of weights at the BH level available from the national accounts data from the participating countries. The adopted aggregation procedure in the second stage is the EKS method based on Fisher binary index numbers. As the Fisher binary indexes have axiomatic and economic theoretic properties and that Fisher indices are considered exact and superlative, the ICP uses transitive indices generated from the non-transitive binary Fisher indices using the EKS method. The EKS method has the least-squares property that it provides transitive indices that are the closest possible (in terms of logarithmic distance) to the binary Fisher indices. Additive methods such as the Geary-Khamis and Ikle indices are recommended if there is need for additively consistent international comparisons. The basic philosophy that underpins the EKS methodology is that the Fisher binary indices are the best way to make price comparisons for any given pair of countries and that imposing transitivity should respect this feature of the Fisher binary indices. However, there is now a growing realization that within the context of international comparisons involving countries from different regions and subregions and also at different levels of development. There are data related issues with respect to the comparability of the products within the basic heading levels and with respect to the similarity of the price and quantity structures when aggregation above the basic heading level is considered. It is generally recognized that some binary comparisons have more reliability or are considered more meaningful than some others. Without placing a finer point on it, for example we may consider a binary comparison between the United States and Germany may be more reliable than a comparison between United States and Nepal or Kenya. In this case, the basic premise that the binary comparisons should be preserved to the extent possible is no longer tenable. There have been a few developments in this direction in the recent years. Some of these are summarized in several contributions in Rao (2009a). Balk (2009a and 2009b) also provides an overview of the developments in this Field. Rao (2009b) pursues the EKS method further and suggests the use of the Generalised EKS and Weighted CPD methods of aggregation. The Generalised EKS method is pursued further in Section 2 of this paper. Robert Hill has been a strong proponent of the use of chained linking methods based on the use of Minimum Spanning Trees (MST) constructed using the PaascheLaspeyres spread as a measure of reliability of a given binary price of quantity comparison. Hill (2009) provides a good summary of the MST approach and discusses issues relating to the stability of the MST procedure. The MST procedure relies on the optimization criterion that the sum total of PLS measures involved in the chained comparisons is minimized over all possible spanning trees. While the MST procedure has an optimality property with respect to the overall spanning tree, there is no guarantee that any given binary comparison constructed using the chain of links implied by the MST is the best possible approach. If the main objective is to obtain the best possible comparison between any given pair of countries, then it is necessary to look beyond the MST method. In this paper, we pursue the minimum distance (MD) approach instead of the MST approach. Though Hill (2009) hints at a possible minimum distance approach, the idea for this goes further back to discussions when the Hill paper was first discussed at the CRIW conference in 1996 where the MST approach was first presented.

The actual implementation of either the generalised EKS or the MST approach requires adequate measures of price or quantity similarity. While Hill (1999, 2009) anchors the work on the well-known Paasche-Laspeyres spread, Diewert (2009) examines the construction of dissimilarity indices and proposes a number of measures that can be used in practice. In this paper, we consider the PLS measure and also two additional measures proposed in Diewert (2009), the weighted relative price dissimilarity measure and the weighted absolute quantity dissimilarity measure. The robustness of the price comparisons to the use of different dissimilarity measures is examined using international comparison data drawn from the 1996 OECD comparison. The outline of this paper is as follows. In section 2, we describe the generalised EKS (WEKS) procedure and implement with three different measures of dissimilarity and reliability. In Section3 we present and discuss three measures of dissimilarity used in the paper. Section 4 is devoted to a quick review of the MST approach. We focus on the minimum distance approach in Section 5. Empirical results and concluding remarks are provided in Section 6. 2. The Weighted-EKS procedure The following notation is used in this paper. Let pij and qij represent, respectively, the price and quantity of i th commodity in j th country. We consider a multilateral comparison with M countries and N commodities. The EKS1 method starts with a matrix of binary indexes computed using the Fisher formula. Let Fjk represent the Fisher price index (PPP) for country k with country j as the base. The Laspeyres (Ljk), Paasche (Pjk) and the Fisher index are given by. N L jk pik.qij i 1 N p.q i 1 ij N and Pjk ij p i 1 N ik.qik p.q i 1 ij 1/2 ; Fjk L jk.pjk (1) ik The matrix of Fisher binaries, Fjk, j,k = 1,2,,M, does not satisfy transitivity property. The EKS procedure essentially constructs a transitive set of comparisons from the Fisher binaries. The computational form for the EKS index, for a pair of countries j and k is given by EKS jk M F jl l 1 Flk 1M (2) where Fjk denotes the Fisher price index number for country k with country j as the base. The main 2 property of the EKS index is that it minimises (ln I jk ln F jk ) subject to the transitivity restriction: j k I jk I jl I lk j, k, l The main weakness of the EKS method is that it considers all Fisher binary indexes as equally meaningful or reliable. An EKS-based comparison between j and k given in equation (2) is a simple unweighted geometric mean of all the indirect comparisons computed using all the countries in the comparison. In this case, the EKS method can be seen to be giving the same weight irrespective of which 1 Even though the method is being referred to as Gini EKS or GEKS, in this paper we refer to it simply as EKS.

country is used as the link country. For example, in a comparison between the USA and Germany the EKS method gives the same weight to the indirect comparisons derived using UK as the link and another using India as the link. It is clear that the indirect comparison through UK would be more reliable as it involves comparisons among similar countries compared to a comparison through India. Rao (2009b) proposed a method of addressing this problem associated with the EKS method. In the first instance, Rao has shown that the EKS indices can be derived as: EKS jk exp( ˆ k ) exp( ˆk ˆ j ) exp( ˆ j ) (3) where s are the ordinary least squares estimators of s (which are the best linear unbiased estimators) in the following model specification ln F jk k j u jk with E (u jk ) 0 and (4) v(u jk ) 2 In equation (4), we can see that the regression model postulates equal variance for all the binary Fisher indices. Once this assumption of homoscedasticity is relaxed and if variance of each disturbance is treated as different, we can derive weighted EKS (WEKS) indices. The corresponding specification is: ln F jk k j u jk with E (u jk ) 0 and v(u jk ) 2 w jk (5) The specification in equation (5) implies that each Fjk has a different variance and therefore reliability attached to each binary comparison is different. The generalised least squares estimator of π s can then be used in finding WEKS indices.2 3. Measures of Reliability or Dissimilarity In order to implement the WEKS method, we need to specify our measure of reliability. We may use measures of dissimilarity as a measure of reliability. Hill (1999) uses the Paasche Laspeyres spread as it reflects variability in the price and quantity ratios as well as the strength of the correlation between the price and quantity ratios over time or across countries. The dissimilarity between two countries j and k (djk) is measured for all j and k by L jk d jk ln Pjk (6) where Ljk and Pjk respectively refer to Laspeyres and Paasche index numbers. It can be noted that the distance measure in equation (6) is the same whether the price index numbers or quantity index numbers are used. Since a large value of djk represents a larger spread between the Laspeyres and Paasche indices, and therefore the corresponding binary comparison is considered less reliable. 2 See Rao (2009b?) for further details of the implementation of the WEKS method WGEKS mehod?.

Recognising a major issue with the Paasche-Laspeyres spread which may take a value equal to zero (suggesting perfect similarity) even when the price vectors in countries j and k are quite dissimilar, Diewert (2009) offers an axiomatic treatment of the construction of price and quantity dissimilarity measures.3 Without revisiting the technical material presented by Diewert4, we select the following two measures, one based on quantity dissimilarity and another based on price dissimilarity, for further application. Weighted asymptotically quadratic index of absolute quantity dissimilarity (WAQD) This index is defined as: 2 2 q qij 1 ik DWAQD ( p, p, q, q ) d jk sij sik 1 1 qij qik i 1 2 j k j k N (7) where sij is the expenditure share of i-th commodity in j-th country. Implementation of equation (7) requires all the quantities are positive. In practice, quantities can be zero and in that case the ratios are well-defined. In the empirical implementation, we replaced zero quantities by a small non-zero quantity of 0.5. Weighted asymptotically quadratic index of relative price dissimilarity (WRPD) 2 2 P( p j, p k, q j, q k ). pij p ik DWPRD d jk 1/ 2 sij sik 1 1 pik P( p j, p k, q j, q k ). pij i 1 N (8) The relative price dissimilarity index requires the use of a suitable binary index to reduce the price data into a comparable form. In our empirical implementation we make use of the Fisher binary price index for this purpose. However, given the premise that the Fisher binary price index5 may not be reliable in all the cases, we may consider the use of Fisher index as a first stage approximation. One may then revise the index in the light of an improved binary index computed using the weighted EKS or some other method. 4. The Minimum Spanning Trees The minimum spanning tree is a graph theoretic concept. First we start with a graph with M vertices, with each vertex representing a country. In an international comparison context, we are interested in making price and real income comparisons between countries. For any binary comparison, we can use the Fisher index. For multilateral comparisons, Hill (1999) proposes the construction of a spanning tree. Main Features: 1. Spanning tree is a graph that covers all the vertices (countries) with the property that between any two countries there is only one path that links the two countries. Therefore, there are no cycles in the graph. 3 As is the case with Diewert, we also opt to use dissimilarity measure in preference to similarity measures. Diewert states a number of axioms including: continuity; identity; positivity; symmetry; invariance to changes in units of measurement; montonicity; invariance to ordering of commodities, etc. 5 In fact this would be true for any other price index like the Tornqvist index. 4

2. With M countries, we can construct many spanning trees. The maximum number of spanning trees possible is MM-2. Out of all the possible spanning trees, we wish to select one that is optimal in some sense. 3. Hill proposes uses the Paasche-Laspeyres (PLS) spread as a measure of distance between two countries. That means, for each j and k we have d(j,k) as a weight attached to the edge connected j and k. The LPS is given by L jk d jk ln Pjk 4. 5. 6. where Ljk and Pjk are the Laspeyres and Paasche index numbers Hill suggests that the Minimum Spanning Tree is selected for use it is that tree that minimizes the sum of distances attached to the edges of the tree. If all the distances are different, then the MST is unique. The MST can be constructed using Kruskal s algorithm. In this paper we implement the MST approach using the relative price similarity and absolute quantity similarity indices discussed in Section 3. We make use of data from the 1996 OECD comparison with basic heading PPPs and implicit quantities for 158 basic haedings. 5. Comparisons based on minimum distance paths While the MST has some optimality properties with respect to the tree as a whole, there is no guarantee that the comparison between countries j and k derived using the MST is necessary best for the pair of countries. In order to guarantee this, we suggest that we identify an optimal path of countries to link country j and k based on the distances defined in Section 3 above. Suppose we start with a similarity measure denoted by d(xj,xk) or simply djk. Suppose a subset of (distinct) countries {i1, i2,,ip }, a subset of {1,2,,M}, defines a path between countries j and k. Then the distance associated with the path that links these two countries is simply the sum of the distances between the links. Thus the distance associated with a path is: P 1 d path ( x j, xk ) d ( x j, xi1 ) d ( xil, xil 1 ) d ( xip, xk ) l 1 In a complete graph where each pair of countries are directly compared using an index similar to the fisher index, there are numerous paths that link countries j and k. Out of all such paths, we choose the path that has the minimum distance. P 1 d Min ( x j, xk ) min path d path ( x j, xk ) d ( x j, xi1 ) d ( xil, xil 1 ) d ( xip, xk ) l 1 min path Sum of w ' s over all paths connecting x and y Once the minimum distance path is identified, then the price comparison between two countries j and k with a minimum path defined by countries with labels {i1, i2,,ip } is defined as: P 1 MD jk ( Fisher ) Fj,i1. Fil,il 1.FiP,k l 1 (9) It is possible to use any other binary index number formula as long as it satisfies the country-reversal test.

Properties of the minimum distance approach: 1. For all pairs of countries, j and k, we have d MD ( x j, xk ) d MST ( x j, xk ) for all j and k This means that the minimum distance approach comparisons between j and k are always at least as good as those derived using MST. 2. d MD ( x j, xk ) provides Pareto optimal comparisons relative to the direct binary comparisons. We have d MD ( x j, xk ) d ( x j, xk ) j, k Equality holds for all x and y if and only d ( x j, xk ) satisfies the triangular inequality. 3. d MD ( x j, xk ) is a proper distance metric that satisfies the basic distance axioms including triangular inequality. These can be seen from the definitions. 4. The MDFisherjk based on the links generated by d MD ( x j, xk ) is generally not transitive but provides the best possible comparison (even when compared to the direct comparison). One may use the EKS transformation on this to generate transitive comparisons. We denote the EKS generated using MDFisher as the MDEKSjk index. As the work is motivated by the fact that the Fisher binary indices may not be the best binary comparisons and the fact that the MD approach generates the best possible comparison, the idea of preserving the MD binaries in generating transitive comparisons is consistent with the general purpose of the EKS procedure. The MDEKS procedure is implemented and the results presented here. The process of identifying minimum path from a given country (source country) to a destination country is based on Dijkstra s algorithm drawn from Jungnickel (2008) entitled Graphs, Networks and Algorithms. Dijkstra s algorithm is coded in MATLAB and is based on the general greedy algorithm. The Disjkstra s algorithm finds the shortest paths from a given source country to all the countries in the comparison at the same time. We have also coded the Kruskal s algorithm in MATLAB for constructing the MSTs. 6. Empirical Results In this section we present estimated PPPs derived using the WEKS, MST and MD approaches described in the paper derived using the 1996 OECD data. The price information that we have is in the form of PPPs at the basic heading level for 158 basic headings, with US dollar used as the numeraire currency. In addition we have expenditure, in national currency units, for each basic heading in all the OECD countries. These nominal expenditures provide the expenditure share data used in deriving the weighted maximum likelihood estimators under alternative stochastic specification of the disturbances. In Table 1 we present PPPs derived using the WEKS procedure and the three distance measures, PLS, WRPD and WAQD measures of dissimilarity. For purposes of comparison, we also present the Fisher and EKS PPPs. The results are quite interesting. When US is used as the reference currency, the PLS based EKS are above those derived using WRPD and WAQD measures of dissimilarity. While the percentage differences are small for the high income countries like France and Germany the differences are around 5 to 6% for countries like Portugal and Spain. We also find the standard EKS and the weighted EKS based on the weighted relative price dissimilarity measure are almost equal. These results indicate that even in the case of OECD the use of different dissimilarity measure has the potential to generate

significant differences. However, these differences could be much larger when the countries in the comparison are at different levels of development which is the starting point for the purpose of WEKS. We present the MSTs generated using the three measures of dissimilarity described in Section 3. Figure 1 shows the PLS-based MST. As is generally the case with MSTs, there are a number of counter intuitive paths. For example, Spain and Greece are connected through Portugal, Denmark, USA, UK, Germany, Switzerland, Austria, Sweden, Italy. Similarly Australia and New Zealand are connected through the UK, Germany, Switzerland and Austria. Now we turn to Figure 2 where MST based on relative price distance measure is provided. The links in WRPD based MST are a lot more intuitive and are consistent with the notion of price similarity of the countries. For example, Spain, Italy, Portugal, Greece and Turkey are all connected directly, USA-Canada has a direct link so is the pair Ireland-United Kingdom. Countries like Sweden, Finland, Iceland, Norway and Denmark are all connected together. The main conclusion emerging from Figure 2 is that the WRPD is a better measure of price similarity than the PLS used in the standard MST applications. In Figure 3 we have the MST generated using the weighted quantity dissimilarity measure which shows quite different paths linking pairs of countries compared to those in Figure 2. As absolute quantities drive this measure, we have similar size countries bunched up together. For example, Australia and Canada, Spain and Italy, USA and Germany and Luxembourg and Belgium are all directly connected. As mentioned in our discussion of the quantity dissimilarity measure, we encountered a problem as there are several basic headings with zero quantities in several countries. The WAQD measure in Diewert (2009) assumes positive quantities. So, in order to implement the WAQD measure, we arbitrarily replaced zero quantities with 0.5 which still retains the main flavor of the absolute quantity dissimilarity measure. In the next step we set out to make a comparison of the distances between pairs of countries under the MST and the MD approach. We have computed these matrices of percentage differences for all the three measures of dissimilarity. In Table 2, we present a comparison of the percentage differences and the differences in the paths for a few selected pairs of source and destination countries, all computed using the PLS spread as the measure of dissimilarity. For example, Greece and Spain are connected through a long chain and the MST distance (sum of the PLS measures) is 0.431. In comparison the MD approach uses a direct comparison with a distance of only 0.133, roughly a third of the MST distance. Similarly in the case of Italy and France, the MST distance is 0.177 compared to the MD distance of 0.79, less than half of the MST distance. We also find similar difference between the MST and MD distances when we consider the absolute quantity dissimilarity measure. But an interesting feature is that when we use the weighted relative price dissimilarity measure (WRPD), there is considerable agreement between the MST and MD distances. The main conclusion we draw from the MST and MDs identified for different measures is that the choice of dissimilarity measure is likely to affect the MST links to a larger degree. However, the use of MDs are likely to produce more robust set of links as the minimization is undertaken separately for each pair of countries. Finally, we present in Table 3, PPPs from the MST, MD and MDEKS methods are presented along with the Fisher and EKS PPPs. As expected PPs from the MST and MD approaches based on PLS differ for some pairs of countries. We find the resulting differences between MST and MD parities when the relative price dissimilarity is used to be counter intuitive. We are currently rechecking the calculations. However, when it comes to the use of absolute quantity dissimilarity measure, the results from the MST and MD approaches are quite similar.

7. Conclusions In this paper we revisit the use of EKS and MST approaches to the computation of PPPs. We make use of the newly proposed dissimilarity measures by Diewert (2009), the WEKS approach in Rao (2009) and using the minimum distance approach proposed here. The feasibility of the MD approach and the likely superiority of this approach are demonstrated using the 1996 OECD data set. The results confirm the need for further research and empirical applications of the MD approach involving a more dissimilar set of countries. Currently work is underway to examine the 2005 basic heading level data set to see how these approaches influence the PPP estimates. References Diewert, W.E. (2009), Similarity Indexes and Criteria for Spatial Linking, pp. 183-216 in Purchasing Power Parities of Currencies: Recent Advances in Methods and Applications, D.S. Prasada Rao (ed.), Cheltenham UK: Edward Elgar. Jungnickel, D. (2008), Graphs, Networks and Algorithms (3rd Ed.), Springer, Berlin. Hill, R.J. (1999a), Comparing Price Levels across Countries Using Minimum Spanning Trees, The Review of Economics and Statistics 81, 135-142. Hill, R.J. (2009), Comparing Per Capita Income Levels Across Countries Using Spanning Trees: Robustness, Prior Restrictions, Hybrids and Hierarchies, pp. 217-244 in Purchasing Power Parities of Currencies: Recent Advances in Methods and Applications, D.S. Prasada Rao (ed.), Cheltenham UK: Edward Elgar. Prasada Rao, D.S., (2009) "Generalised Elteto-koves-Szulc (EKS) and Country-Product-Dummy (CPD) Methods for International Comparisons" in Prasada Rao (ed.) Purchasing Power Parities: Recent Advances In Methods And Applications, Edward Elgar Publishing Company, 86-120.

Country Table 1: Purchasing Power Parities of Currencies OECD Countries, 1996 (USA = 1.0) WEKS Indices PLS WRPD WAQD Fisher Indices EKS Indices GER 2.158 2.140 2.071 2.048 2.137 FRA 6.959 6.908 6.702 6.629 6.903 ITA 1576.305 1565.195 1513.925 1593.428 1564.760 NLD 2.204 2.190 2.125 2.083 2.192 BEL 40.369 39.977 38.462 38.830 39.898 LUX 39.332 38.375 36.004 38.279 38.051 UK 0.677 0.669 0.650 0.700 0.671 IRE 0.697 0.694 0.674 0.698 0.694 DNK 9.673 9.586 9.492 9.721 9.616 GRC 201.952 200.380 192.796 203.451 200.067 SPA 126.453 124.873 120.131 130.897 125.410 PRT 131.987 130.943 126.299 133.820 131.096 AUT 14.603 14.430 13.878 14.898 14.438 SUI 2.255 2.248 2.169 2.184 2.251 SWE 10.682 10.568 10.432 10.865 10.598 FIN 6.978 6.914 6.794 7.057 6.927 ICE 93.762 92.959 85.241 95.095 93.121 NOR 9.944 9.871 9.704 9.681 9.832 TUR 6735.835 6685.458 6495.195 6851.691 6680.602 AUS 1.417 1.407 1.361 1.411 1.404 NZL 1.605 1.591 1.551 1.616 1.595 JAP 202.648 201.148 196.881 185.180 201.488 CAN 1.297 1.299 1.245 1.284 1.299

Source Des. MST Dist MD Table 2: The MST and MD Paths for Selected Countries Using the Paasche-Laspeyres Spread % diff b/w MST and MD MST path Min Dist path GRC SPA 0.431 0.133 225.175 {GRC,ITA,SWE,AUT,SUI,GER,UK,US,DNK,PRT,SPA} {GRC,SPA} FIN UK 0.112 0.059 87.664 {FIN,SWE,AUT,SUI,GER,UK} {FIN,UK} ITA FRA 0.177 0.079 124.139 {ITA,SWE,AUT,SUI,GER,FRA} {ITA,FRA} TUR IRE 0.364 0.191 90.581 {TUR,IPRT,DNK,USA,UK,GER,SUI,IRE} {TUR,IRE} DNK AUT 0.089 0.033 168.538 {DNK,USA,UK,GER,SUI,AUT} {DNK,AUT}

Table 3: Purchasing Power Parities of Currencies OECD Countries, 1996 (USA = 1.0) Country MST MD MD EKS Fisher EKS PLS WRPD WAQD PLS WRPD WAQD PLS WRPD WAQD GER 2.246 2.086 2.048 2.246 2.048 2.048 2.217 2.158 2.048 2.048 2.137 FRA 7.149 6.797 6.678 7.149 7.138 6.678 7.097 7.001 6.673 6.629 6.903 ITA 1623.425 1532.273 1495.174 1635.923 1553.357 1495.174 1618.472 1583.150 1495.174 1593.428 1564.760 NLD 2.270 2.108 2.091 2.288 2.252 2.091 2.245 2.221 2.092 2.083 2.192 BEL 42.420 39.398 35.475 42.420 38.688 35.475 41.725 40.365 35.462 38.830 39.898 LUX 42.298 38.614 34.769 41.142 39.101 34.769 41.350 38.982 34.756 38.279 38.051 UK 0.700 0.627 0.640 0.700 0.700 0.640 0.692 0.675 0.640 0.700 0.671 IRE 0.727 0.647 0.667 0.707 0.726 0.667 0.711 0.707 0.667 0.698 0.694 DNK 9.721 9.101 9.460 9.721 9.721 9.460 9.890 9.711 9.456 9.721 9.616 GRC 207.804 196.216 193.066 210.709 210.709 191.465 207.967 203.580 189.787 203.451 200.067 SPA 123.662 121.444 118.504 130.897 128.502 118.504 128.607 126.410 118.665 130.897 125.410 PRT 128.999 126.685 126.616 128.999 137.839 126.616 133.716 133.551 126.441 133.820 131.096 AUT 15.317 14.080 13.685 14.898 14.898 13.685 14.962 14.582 13.680 14.898 14.438 SUI 2.362 2.194 2.189 2.298 2.184 2.189 2.309 2.263 2.188 2.184 2.251 SWE 11.029 10.179 10.581 10.865 10.865 10.581 10.942 10.660 10.577 10.865 10.598 FIN 7.141 6.591 6.771 7.057 7.035 6.771 7.123 6.991 6.763 7.057 6.927 ICE 99.814 89.124 82.047 95.095 90.281 82.047 96.879 94.012 82.015 95.095 93.121 NOR 10.422 9.462 9.721 10.137 10.283 9.721 10.196 10.092 9.709 9.681 9.832 TUR 6726.770 6508.843 6647.638 6899.820 6899.820 6647.638 6909.919 6780.494 6645.102 6851.691 6680.602 AUS 1.472 1.355 1.342 1.472 1.411 1.342 1.442 1.425 1.351 1.411 1.404 NZL 1.677 1.542 1.535 1.616 1.616 1.535 1.638 1.601 1.535 1.616 1.595 JAP 215.444 197.047 195.689 209.556 193.490 195.689 209.460 203.096 197.090 185.180 201.488 CAN 1.284 1.284 1.221 1.284 1.284 1.221 1.286 1.316 1.222 1.284 1.299

Figure 1: MST with PLS distance measure IRE JAP BEL NLD NOR SUI LUX ICE FRA GER AUT SWE ITA GRC NZL FIN AUS UK USA CAN DNK TUR PRT SPA

Figure 2: MST with weighted relative price distance measure ICE IRE UK DNK SWE FIN NOR AUS NLD SUI NZL JAP LUX BEL GER AUT FRA USA CAN SPA GRC TUR PRT ITA

Figure 3: MST with weighted absolute quantity distance measure JAP NOR GRC NLD CAN AUS FIN ICE FRA UK AUT IRE LUX BEL SPA ITA PRT DNK NZL USA GER TUR SWE SUI