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Multilateral Sato–Vartia index for international comparisons of prices and real expenditures
Economics Letters 183 (2019) 108535
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Multilateral Sato–Vartia index for international comparisons of prices and real expenditures✩ ∗
Naohito Abe a , , D.S. Prasada Rao b,c a
Hitotsubashi University, Japan The University of Queensland, Australia c Hitotsubashi Institute for Advanced Study, Japan b
highlights • • • •
Prove existence of infinitely many log-change indices satisfying factor reversal test. A new log-change index satisfying factor reversal test is presented. Several transitive multilateral logarithmic indices are proposed. Present empirical results from different methods using World Bank Data.
article
info
Article history: Received 25 April 2019 Received in revised form 27 June 2019 Accepted 10 July 2019 Available online 30 July 2019 JEL classification: Codes C13 C83 E01 E31
a b s t r a c t The Sato–Vartia (SV) index for bilateral price comparisons has impressive analytical properties and is used intensively in recent international trade and macroeconomic analyses. We show that the SV index is only one of many logarithmic indices that satisfy the factor reversal test discussed in index number theory. In this paper we propose several transitive multilateral versions of the SV index but note that these do not satisfy factor reversal test. We derive closed form expressions for the transitive logarithmic indices and empirically implement the new indices for cross-country price comparisons using World Bank data from the 2011 International Comparison Program. © 2019 Elsevier B.V. All rights reserved.
Keywords: Sato–Vartia index Multilateral comparisons Transitivity Factor reversal test
1. Introduction The Sato (1976)–Vartia (1976) (SV) index is a logarithmic index to measure changes in prices and quantities which is specifically designed to satisfy factor reversal test. This index satisfies several other test properties including time reversal test, has the same properties as the Fisher (1922) ideal index. It is the first log-change index found to possess this property. Commonly
✩ Abe’s work was supported by JSPS KAKENHI, 19H01467, 16H06322, and 18H00864. Rao’s work was supported by the Australian Research Council Grant DP DP170103559. The authors wish to thank an anonymous referee for helpful comments. ∗ Correspondence to: Institute of Economic Research, Hitotsubashi University, Kunitachi Tokyo 186-8603, Japan. E-mail address: [email protected] (N. Abe). https://doi.org/10.1016/j.econlet.2019.108535 0165-1765/© 2019 Elsevier B.V. All rights reserved.
used Törnqvist–Theil and geometric-young indices fail the factor reversal test. The SV index has been shown to be exact cost-of-living (COLI) index for constant elasticity of substitution (CES) utility function (Sato, 1976; Feenstra, 1994). The SV index features prominently in trade and macroeconomic analyses (Feenstra, 1994; Redding and Weinstein, 2016; Rao et al., 2016). The SV index has not been used in multilateral international price comparisons as the index does not satisfy transitivity condition.1 The Gini–Eltető–Kö ves–Szulc (GEKS) approach is used to construct transitive multilateral index that is closest (in the least squares sense) to a desired binary index such as the Fisher Index. In this paper we show that the SV index offers a rich set of alternative methods of generating transitive comparisons. We prove several important results including non-uniqueness of the 1 See Balk (2008) for details of transitivity in multilateral comparisons.
2
N. Abe and D.S.P. Rao / Economics Letters 183 (2019) 108535
SV index except for the two-country two-commodity case. Empirical results based on data from the 2011 International Comparison Program (ICP) (World Bank, 2015) are presented and contrasted.
In the two commodity case (N = 2) it can be shown by simple algebra that (4) has unique solution given by:
2. Notation and the Sato–Vartia index k=1,2,...,M
Let {pik , qik }i=1,2,...,N respectively represent the price and quantity of ith commodity in country k. The price and quantity index numbers for country k with country j be denoted by Pjk and Qjk . The value or expenditure and the expenditure ∑N shares are denoted, respectively, by Ek ≡ i=1 pik qik and wik ≡ ∑N pik qik / i=1 pik qik (i = 1, 2, . . . , N). The value index is defined as Ejk ≡ Ek /Ej . An index number satisfies the factor reversal test if and only if
Pjk · Qjk = Ejk ⇔ ln Pjk + ln Qjk = ln Ejk ⇔ ln Ejk − ln Pjk − ln Qjk = 0 (1) Log-change index number formulae are essentially geometric averages of price and quantity changes which are additive in logarithmic form. Following Sato (1976), log-change price and quantity index numbers have the following generic form: N
ln Pjk =
∑
φijk ln
i=1
(
pik
)
pij
N
, ln Qjk =
∑
φijk ln
(
i=1
qik
)
qij
jk i
φ = 1, φ ≥ 0.
(2)
i=1 jk
where φi are weights which are usually functions of the observed expenditure shares in countries j and k. The Sato–Vartia log-change index, PjkSV , is defined as in (2) with the weights2 ;
φijk =
, i = 1, 2, . . . , N .
(3)
We establish the following result regarding uniqueness of the SV index. Proposition 1. Log-change index number satisfying the factor reversal test is not unique. It is unique up to a factor of proportionality if and only if the number of commodities is equal to two, in this case the log-change index is unique and equals the SV index.
( ) Proof. A log-change index with weights, φ jki : i = 1, 2, . . . , N satisfies factor reversal test if and only the following condition is satisfied:
( ln
Ek Ej
⇔
N
)
N ∑ i=1
∑
−
φijk ln
(
i=1 jk i ln
φ
(
wik wij
pik
N
)
pij
−
∑ i=1
φijk ln
(
qik qij
)
+ φ2jk ln
(
)
i∈N λi (wi2 − wi1 ) ⏐, κ = ⏐∑ + ⏐ ⏐ ⏐ i∈N− λi (wi2 − wi1 )⏐ ( ) w +w w where λi = ln wi2 · 2(wi2 −wi1 ) .
∑
i1
i2
i1
Also define
λ∗i = λi for i ∈ N+ and λ∗i = κ · λi for i ∈ N− . The following weights are nonnegative, add up to unity.
( wi2 +wi1 ) ( wi2 +wi1 ) i∈N+
2
κ· ( wi2 +wi1 )
2 ( ) for i ∈ N+ , ∑ i1 + i∈N− κ · wi2 +w 2 ( wi2 +wi1 )
( ) for i ∈ N− . (6) i1 κ · wi2 +w 2 ( ) ∑N w By construction i=1 φi∗ ln wi2 = 0, that is, a logarithmic index i1 with φi∗ as the weights passes the factor reversal test. Except for ∗ the case N = 2, φi are generally different from the SV weights. = ∑
i∈N+
2
+
2 ∑
i∈N−
Our Proposition is not inconsistent with Fattore (2010) who proves that the SV index is the only log-change index number that satisfies the factor reversal test when observed quantities follow a special structure. Otherwise, SV index is not unique. The choice of index satisfying (4) must depend on other properties. For example, SV index can be justified since it is exact for the CES utility function (Sato, 1976; Feenstra, 1994). 3. Sato–Vartia Index for multilateral comparisons and Transitivity Section 2 focused on SV index for bilateral comparison between two countries j and k. In the case of multilateral comparisons with M countries, we are interested in price and quantity comparisons between all pairs of countries represented by
{
Pjk , Qjk
}K =1,2,...,M j=1,2,...,M
. These indices satisfy transitivity if and only if
Pjl · Plk = Pjk or ln Pjl + ln Plk = ln Plk ∀j, k, l = 1, 2, . . . , M
=0
(5)
This proves the ‘‘if’’ part. To establish the only if part we provide a new logarithmic index which satisfies the factor reversal test and condition in (4). Let N+ and N− represent the items subsets for which (wi2 − wi1 )′ s are positive and negative respectively. Define
φi = ∑
,
jk i
wik −wij ln(wik )−ln(wij ) ∑N wik −wij i=1 ln(wik )−ln(wij )
(
∗
N
∑
w1k w1j
) w2k = 0; and φ1jk + φ2jk = 1 w2j (wi2 − wi1 ) /(ln wi2 − ln wi1 ) i = 1, 2 ⇒ φijk = ∑2 i=1 (wi2 − wi1 ) /(ln wi2 − ln wi1 )
φ1jk ln
(7)
The SV price index with weights in (3) does not satisfy transitivity property except in the case considered below.
) =0
(4) jk
Any index number formula with weights φi satisfying (4) is a log-change index number with factor reversal test property. Eq. (4) is one linear homogeneous equation in N − 1 unknown weights since the weights add up to unity. Therefore, there are infinitely many log-change index numbers that satisfy the factor reversal test. Sato–Vartia index is only one from this class. 2 The SV quantity index is similarly represented. Weights are defined using logarithmic averages of expenditure shares in countries j and k.
Proposition 2. The SV log-change index number satisfies transitivity jk if and only if the weights satisfy the condition: φi = φi ∀ j, k = 1, 2, . . . , M ; i = 1, 2, . . . , N. Proof. See Appendix When the condition in Proposition 2 holds, the resulting index is: Pjk =
]φ N [ ∏ pik i i=1
pij
for all j, k = 1, 2, . . . , M
(8)
N. Abe and D.S.P. Rao / Economics Letters 183 (2019) 108535
This result is consistent with an important theorem by Funke et al. (1979) which proves that the index in (8) is the only index that satisfies identity, commensurability and transitivity properties. SV Based Gini–Eltető–Köves–Szulc (GEKS) Index The GEKS method is a technique to generate transitive indices that are closest to the binary comparisons that lack transitivity property.
}k=1,2,...,M
Proposition 3. Let PjkSV , PjkSV −GEKS denote the SV and SVj=1,2,....M based GEKS indices, then solution to the following minimization problem
{
M M ∑ ∑ [
min Pjk
− ln Pjk
]2
˜ φi =
1
N where
φ αi =
+ φ αi − φ α for all i
M M 1 ∑∑
M
αijk · φijk for all i; φ α =
j=1 k=1
is : PjlSV · PlkSV
N 1 ∑
N
φ αi
(14)
i=1
Proof. { }i=N Follows from first order conditions. The optimal weights ˜ φi i=1 in (14) are non-negative and add up to 1.
Eq. (13) simplify to
subject to Pjl · Plk = Pjl ∀ j, k, l = 1, 2, . . . , M
M ∏ [
(13)
{
αijk
}i=N ; j,k=M i=1;j,k=1
, in Proposition 5 satisfy the
{ }i=N
condition α1jk = α2jk = · · · = αNjk ; ∀j, k, then weights ˜ φi
j=1 k=1
PjkSV −GEKS =
is given by
Corollary. If weights, ln PjkS −V
3
]1/M
(9)
min φi
The GEKS approach is the only way to generate transitive indices from Fisher binary indices. In the case of SV indices, we propose three additional methods to generate transitive indices that retain the general character of the SV index. Proposition 2 implies that we need to generate a set of weights {φi }Ni=1 . We propose three different criteria for generating transitive weights, φi ’s, that are as close as possible (maintain charac-
{
jk
teristicity) to the observed non-transitive SV weights φi
}j,k=M j,k=1
.
}i=N φ αi i=1
i=1
in
in Eq. (14).
The third approach is to construct optimal shares φi which minimize the magnitude by which the factor reversal test is not satisfied. This is equivalent to the following quadratic optimization minimization problem:
l=1
Derivation of (9) can be found in Rao and Banerjee (1985).
{
[ N M M ∑ ∑ ∑ j=1 k=1
i=1
)]2
wik φi ln wil
(
subject to 0 < φi < 1 and
N ∑
φi = 1.
(15)
i=1
No closed form solutions similar to (9), (11) and (13) are available and hence not considered further. Our multilateral SV indices of the form (8) with weights in (11) and (14) have the additional advantage that they satisfy the strong identity test (see Balk, 2008 for details) which is not satisfied by the Fisher and SV-based GEKS indices.
jk
Proposition 4. Suppose we have observed weights, φi ; ∀i and j, k, which are non-negative and add up to unity {for} any j and k,
∑N
i=1
jk i
φ = 1. Then the optimum set of weights, φˆ i M
M
)2 1 ∑ ∑ ∑ ( jk φi − φi min φi
2
that
i=1
N
N
s.t .
∑
φi = 1
(10)
i=1
j=1 k=1 i=1
is given by
∑M ∑M ( φˆ i = φ i =
j=1
k=1
φijk
4. Empirical results
i=N
We make use of Household Consumption data drawn from ICP 2011.3 We compute: (i) GEKS based on Fisher index; (ii) GEKS based on bilateral SV index (Eq. (9)); (iii) SV index with optimal weights in (11) associated with the unweighted objective function (Eq. (10)); (iv) SV index with optimal weights in (13) with three different specification of weights:
)
M2
1
for all i
(11)
Proof is straight forward.
⏐ ; (b) αijk = (a) αijk = ⏐ ⏐wik − wij ⏐ (c ) αijk =
The log-change price index with weights in (11) is an alternative to the SV-GEKS index in (9). The weights in (11) are obtained when discrepancies for all commodities are considered to be of equal importance. It is possible to accord differential weights reflecting relative importance of items through weighted optimization below. jk
Proposition 5. Suppose we have observed weights, φi ; ∀i and j, k, which are non-negative and add up to unity for any j and k,
{ }i=N ; j,k=M φijk = 1. Let αijk be a set of non-negative set of 1;j,k=1 ∑M ∑M i=∑ jk N weights with α = 1. Then the optimum set of { }i=N j=1 k=1 i=1 i weights, ˜ φi i=1 that minimizes ∑N
2
j=1 k=1 i=1
( )2 αijk φijk − φi
s. t .
N ∑ i=1
φi = 1
(12)
⏐
2
; (16)
⏐,
N × ⏐PLjk − PPjk ⏐
where PLjk and PPjk are bilateral Laspeyres and Paasche indices, respectively. Complete set of transitive multilateral price indices with USA as base are in Appendix Tables. We find that the Fisher-based GEKS and SV-based GEKS are similar in magnitude as expected. The four transitive indices proposed in the paper are also close to the Fisher and SV-based GEKS. The relative performance of various transitive SV indices can be examined using the following two distance measures,
i=1
M M N 1 ∑∑∑
PLjk
( ) wik + wij
∆1 = SQRT
⎧ ⎨
1
⎩ M (M −1) 2
M M ∑ ∑ [ (
ln
j=1,j̸ =k k=1
PjkSV −T
)
− ln
PjkSV
(
⎫ )]⎬ ⎭
(17)
3 The authors gratefully acknowledge the data made available by the Global Office of the ICP.
4
N. Abe and D.S.P. Rao / Economics Letters 183 (2019) 108535
Appendix A. Supplementary data
Table 1 Relative performances of four transitive Sato–Vartia indices. Weight
Unweighted
(a)
(b)
(c)
Distance from the binary S-V Distance from the factor reversal
0.0471 4.9742
0.0481 4.8890
0.0476 4.8889
0.0464 4.2187
Supplementary material related to this article can be found online at https://doi.org/10.1016/j.econlet.2019.108535. References
M M ∑ ∑ [ (
1
∆2 = SQRT{ M (M −1) 2
− ln
QjkSV −T
(
)]
ln Ejk − ln PjkSV −T
)
(
)
j=1,j̸ =k k=1
}
(18)
∆1 and ∆2 show the distance from the binary SV index and factor reversal, respectively.4 Table 1 summarizes the results. The index with weights (3) based on the Laspeyres–Paashce gap (column c) is the best performing index. Correlations between the six transitive logged price indices (four transitive SV and two GEKS) are greater than 0.999. 5. Conclusions In this paper we construct several transitive Sato–Vartia multilateral price and quantity indexes. We provide a new logarithmic index that satisfies factor reversal test. Based on our analysis, we recommend the use of index of the form (8) with optimal weights (c) in Eq. (16). Our proposed index preserves the core philosophy of the SV index in generating transitive multilateral indices and unlike Fisher and SV-based GEKS indices it satisfies the strong identity test.
4 To calculate ∆ , we need to construct quantity index that requires strictly 2 positive quantity. Thus, we drop observations with zero nominal expenditures when calculating ∆2 .
Balk, B.M., 2008. Price and Quantity Index Numbers: Models for Measuring Aggregate Change and Difference. Cambridge University Press. Fattore, M., 2010. Axiomatic properties of geo-logarithmic price indices. J. Econometrics 156 (2), 344–353. Feenstra, R.C., 1994. New product varieties and the measurement of inter-national prices. Amer. Econ. Rev. 84 (1), 157–177. Fisher, I., 1922. The Making of Index Numbers. Houghton-Mifflin, Boston, MA. Funke, H., Hacker, G., Voeller, J., 1979. Fisher’s circular test reconsidered. Swiss J. Econ. Stat. 115 (IV), 677–688. Rao, D.S.P., Banerjee, K.S., 1985. A multilateral index number system based on the factorial approach. Statist. Papers 27 (1), 297–313. Rao, D.S.P., Rambaldi, A.N., Balk, B.M., 2016. On Measuring Regional or Global Growth and Inflation, Discussion Papers Series 552, School of Economics, University of Queensland, Australia. Redding, S.J., Weinstein, D.E., 2016. Measuring Aggregate Price Indexes with Demand Shocks: Theory and Evidence for CES Preferences, NBER Working Papers 22479. Sato, K., 1976. The ideal log-change index number. Rev. Econ. Stat. 58 (2), 223–228. Vartia, Y.O., 1976. Ideal log-change index numbers. Scand. J. Stat. 3, 121–126. World Bank, 2015. Purchasing Power Parities and the Real Size of World Economies. The World Bank.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Multilateral Sato–Vartia index for international comparisons of prices and real expenditures✩ ∗
Naohito Abe a , , D.S. Prasada Rao b,c a
Hitotsubashi University, Japan The University of Queensland, Australia c Hitotsubashi Institute for Advanced Study, Japan b
highlights • • • •
Prove existence of infinitely many log-change indices satisfying factor reversal test. A new log-change index satisfying factor reversal test is presented. Several transitive multilateral logarithmic indices are proposed. Present empirical results from different methods using World Bank Data.
article
info
Article history: Received 25 April 2019 Received in revised form 27 June 2019 Accepted 10 July 2019 Available online 30 July 2019 JEL classification: Codes C13 C83 E01 E31
a b s t r a c t The Sato–Vartia (SV) index for bilateral price comparisons has impressive analytical properties and is used intensively in recent international trade and macroeconomic analyses. We show that the SV index is only one of many logarithmic indices that satisfy the factor reversal test discussed in index number theory. In this paper we propose several transitive multilateral versions of the SV index but note that these do not satisfy factor reversal test. We derive closed form expressions for the transitive logarithmic indices and empirically implement the new indices for cross-country price comparisons using World Bank data from the 2011 International Comparison Program. © 2019 Elsevier B.V. All rights reserved.
Keywords: Sato–Vartia index Multilateral comparisons Transitivity Factor reversal test
1. Introduction The Sato (1976)–Vartia (1976) (SV) index is a logarithmic index to measure changes in prices and quantities which is specifically designed to satisfy factor reversal test. This index satisfies several other test properties including time reversal test, has the same properties as the Fisher (1922) ideal index. It is the first log-change index found to possess this property. Commonly
✩ Abe’s work was supported by JSPS KAKENHI, 19H01467, 16H06322, and 18H00864. Rao’s work was supported by the Australian Research Council Grant DP DP170103559. The authors wish to thank an anonymous referee for helpful comments. ∗ Correspondence to: Institute of Economic Research, Hitotsubashi University, Kunitachi Tokyo 186-8603, Japan. E-mail address: [email protected] (N. Abe). https://doi.org/10.1016/j.econlet.2019.108535 0165-1765/© 2019 Elsevier B.V. All rights reserved.
used Törnqvist–Theil and geometric-young indices fail the factor reversal test. The SV index has been shown to be exact cost-of-living (COLI) index for constant elasticity of substitution (CES) utility function (Sato, 1976; Feenstra, 1994). The SV index features prominently in trade and macroeconomic analyses (Feenstra, 1994; Redding and Weinstein, 2016; Rao et al., 2016). The SV index has not been used in multilateral international price comparisons as the index does not satisfy transitivity condition.1 The Gini–Eltető–Kö ves–Szulc (GEKS) approach is used to construct transitive multilateral index that is closest (in the least squares sense) to a desired binary index such as the Fisher Index. In this paper we show that the SV index offers a rich set of alternative methods of generating transitive comparisons. We prove several important results including non-uniqueness of the 1 See Balk (2008) for details of transitivity in multilateral comparisons.
2
N. Abe and D.S.P. Rao / Economics Letters 183 (2019) 108535
SV index except for the two-country two-commodity case. Empirical results based on data from the 2011 International Comparison Program (ICP) (World Bank, 2015) are presented and contrasted.
In the two commodity case (N = 2) it can be shown by simple algebra that (4) has unique solution given by:
2. Notation and the Sato–Vartia index k=1,2,...,M
Let {pik , qik }i=1,2,...,N respectively represent the price and quantity of ith commodity in country k. The price and quantity index numbers for country k with country j be denoted by Pjk and Qjk . The value or expenditure and the expenditure ∑N shares are denoted, respectively, by Ek ≡ i=1 pik qik and wik ≡ ∑N pik qik / i=1 pik qik (i = 1, 2, . . . , N). The value index is defined as Ejk ≡ Ek /Ej . An index number satisfies the factor reversal test if and only if
Pjk · Qjk = Ejk ⇔ ln Pjk + ln Qjk = ln Ejk ⇔ ln Ejk − ln Pjk − ln Qjk = 0 (1) Log-change index number formulae are essentially geometric averages of price and quantity changes which are additive in logarithmic form. Following Sato (1976), log-change price and quantity index numbers have the following generic form: N
ln Pjk =
∑
φijk ln
i=1
(
pik
)
pij
N
, ln Qjk =
∑
φijk ln
(
i=1
qik
)
qij
jk i
φ = 1, φ ≥ 0.
(2)
i=1 jk
where φi are weights which are usually functions of the observed expenditure shares in countries j and k. The Sato–Vartia log-change index, PjkSV , is defined as in (2) with the weights2 ;
φijk =
, i = 1, 2, . . . , N .
(3)
We establish the following result regarding uniqueness of the SV index. Proposition 1. Log-change index number satisfying the factor reversal test is not unique. It is unique up to a factor of proportionality if and only if the number of commodities is equal to two, in this case the log-change index is unique and equals the SV index.
( ) Proof. A log-change index with weights, φ jki : i = 1, 2, . . . , N satisfies factor reversal test if and only the following condition is satisfied:
( ln
Ek Ej
⇔
N
)
N ∑ i=1
∑
−
φijk ln
(
i=1 jk i ln
φ
(
wik wij
pik
N
)
pij
−
∑ i=1
φijk ln
(
qik qij
)
+ φ2jk ln
(
)
i∈N λi (wi2 − wi1 ) ⏐, κ = ⏐∑ + ⏐ ⏐ ⏐ i∈N− λi (wi2 − wi1 )⏐ ( ) w +w w where λi = ln wi2 · 2(wi2 −wi1 ) .
∑
i1
i2
i1
Also define
λ∗i = λi for i ∈ N+ and λ∗i = κ · λi for i ∈ N− . The following weights are nonnegative, add up to unity.
( wi2 +wi1 ) ( wi2 +wi1 ) i∈N+
2
κ· ( wi2 +wi1 )
2 ( ) for i ∈ N+ , ∑ i1 + i∈N− κ · wi2 +w 2 ( wi2 +wi1 )
( ) for i ∈ N− . (6) i1 κ · wi2 +w 2 ( ) ∑N w By construction i=1 φi∗ ln wi2 = 0, that is, a logarithmic index i1 with φi∗ as the weights passes the factor reversal test. Except for ∗ the case N = 2, φi are generally different from the SV weights. = ∑
i∈N+
2
+
2 ∑
i∈N−
Our Proposition is not inconsistent with Fattore (2010) who proves that the SV index is the only log-change index number that satisfies the factor reversal test when observed quantities follow a special structure. Otherwise, SV index is not unique. The choice of index satisfying (4) must depend on other properties. For example, SV index can be justified since it is exact for the CES utility function (Sato, 1976; Feenstra, 1994). 3. Sato–Vartia Index for multilateral comparisons and Transitivity Section 2 focused on SV index for bilateral comparison between two countries j and k. In the case of multilateral comparisons with M countries, we are interested in price and quantity comparisons between all pairs of countries represented by
{
Pjk , Qjk
}K =1,2,...,M j=1,2,...,M
. These indices satisfy transitivity if and only if
Pjl · Plk = Pjk or ln Pjl + ln Plk = ln Plk ∀j, k, l = 1, 2, . . . , M
=0
(5)
This proves the ‘‘if’’ part. To establish the only if part we provide a new logarithmic index which satisfies the factor reversal test and condition in (4). Let N+ and N− represent the items subsets for which (wi2 − wi1 )′ s are positive and negative respectively. Define
φi = ∑
,
jk i
wik −wij ln(wik )−ln(wij ) ∑N wik −wij i=1 ln(wik )−ln(wij )
(
∗
N
∑
w1k w1j
) w2k = 0; and φ1jk + φ2jk = 1 w2j (wi2 − wi1 ) /(ln wi2 − ln wi1 ) i = 1, 2 ⇒ φijk = ∑2 i=1 (wi2 − wi1 ) /(ln wi2 − ln wi1 )
φ1jk ln
(7)
The SV price index with weights in (3) does not satisfy transitivity property except in the case considered below.
) =0
(4) jk
Any index number formula with weights φi satisfying (4) is a log-change index number with factor reversal test property. Eq. (4) is one linear homogeneous equation in N − 1 unknown weights since the weights add up to unity. Therefore, there are infinitely many log-change index numbers that satisfy the factor reversal test. Sato–Vartia index is only one from this class. 2 The SV quantity index is similarly represented. Weights are defined using logarithmic averages of expenditure shares in countries j and k.
Proposition 2. The SV log-change index number satisfies transitivity jk if and only if the weights satisfy the condition: φi = φi ∀ j, k = 1, 2, . . . , M ; i = 1, 2, . . . , N. Proof. See Appendix When the condition in Proposition 2 holds, the resulting index is: Pjk =
]φ N [ ∏ pik i i=1
pij
for all j, k = 1, 2, . . . , M
(8)
N. Abe and D.S.P. Rao / Economics Letters 183 (2019) 108535
This result is consistent with an important theorem by Funke et al. (1979) which proves that the index in (8) is the only index that satisfies identity, commensurability and transitivity properties. SV Based Gini–Eltető–Köves–Szulc (GEKS) Index The GEKS method is a technique to generate transitive indices that are closest to the binary comparisons that lack transitivity property.
}k=1,2,...,M
Proposition 3. Let PjkSV , PjkSV −GEKS denote the SV and SVj=1,2,....M based GEKS indices, then solution to the following minimization problem
{
M M ∑ ∑ [
min Pjk
− ln Pjk
]2
˜ φi =
1
N where
φ αi =
+ φ αi − φ α for all i
M M 1 ∑∑
M
αijk · φijk for all i; φ α =
j=1 k=1
is : PjlSV · PlkSV
N 1 ∑
N
φ αi
(14)
i=1
Proof. { }i=N Follows from first order conditions. The optimal weights ˜ φi i=1 in (14) are non-negative and add up to 1.
Eq. (13) simplify to
subject to Pjl · Plk = Pjl ∀ j, k, l = 1, 2, . . . , M
M ∏ [
(13)
{
αijk
}i=N ; j,k=M i=1;j,k=1
, in Proposition 5 satisfy the
{ }i=N
condition α1jk = α2jk = · · · = αNjk ; ∀j, k, then weights ˜ φi
j=1 k=1
PjkSV −GEKS =
is given by
Corollary. If weights, ln PjkS −V
3
]1/M
(9)
min φi
The GEKS approach is the only way to generate transitive indices from Fisher binary indices. In the case of SV indices, we propose three additional methods to generate transitive indices that retain the general character of the SV index. Proposition 2 implies that we need to generate a set of weights {φi }Ni=1 . We propose three different criteria for generating transitive weights, φi ’s, that are as close as possible (maintain charac-
{
jk
teristicity) to the observed non-transitive SV weights φi
}j,k=M j,k=1
.
}i=N φ αi i=1
i=1
in
in Eq. (14).
The third approach is to construct optimal shares φi which minimize the magnitude by which the factor reversal test is not satisfied. This is equivalent to the following quadratic optimization minimization problem:
l=1
Derivation of (9) can be found in Rao and Banerjee (1985).
{
[ N M M ∑ ∑ ∑ j=1 k=1
i=1
)]2
wik φi ln wil
(
subject to 0 < φi < 1 and
N ∑
φi = 1.
(15)
i=1
No closed form solutions similar to (9), (11) and (13) are available and hence not considered further. Our multilateral SV indices of the form (8) with weights in (11) and (14) have the additional advantage that they satisfy the strong identity test (see Balk, 2008 for details) which is not satisfied by the Fisher and SV-based GEKS indices.
jk
Proposition 4. Suppose we have observed weights, φi ; ∀i and j, k, which are non-negative and add up to unity {for} any j and k,
∑N
i=1
jk i
φ = 1. Then the optimum set of weights, φˆ i M
M
)2 1 ∑ ∑ ∑ ( jk φi − φi min φi
2
that
i=1
N
N
s.t .
∑
φi = 1
(10)
i=1
j=1 k=1 i=1
is given by
∑M ∑M ( φˆ i = φ i =
j=1
k=1
φijk
4. Empirical results
i=N
We make use of Household Consumption data drawn from ICP 2011.3 We compute: (i) GEKS based on Fisher index; (ii) GEKS based on bilateral SV index (Eq. (9)); (iii) SV index with optimal weights in (11) associated with the unweighted objective function (Eq. (10)); (iv) SV index with optimal weights in (13) with three different specification of weights:
)
M2
1
for all i
(11)
Proof is straight forward.
⏐ ; (b) αijk = (a) αijk = ⏐ ⏐wik − wij ⏐ (c ) αijk =
The log-change price index with weights in (11) is an alternative to the SV-GEKS index in (9). The weights in (11) are obtained when discrepancies for all commodities are considered to be of equal importance. It is possible to accord differential weights reflecting relative importance of items through weighted optimization below. jk
Proposition 5. Suppose we have observed weights, φi ; ∀i and j, k, which are non-negative and add up to unity for any j and k,
{ }i=N ; j,k=M φijk = 1. Let αijk be a set of non-negative set of 1;j,k=1 ∑M ∑M i=∑ jk N weights with α = 1. Then the optimum set of { }i=N j=1 k=1 i=1 i weights, ˜ φi i=1 that minimizes ∑N
2
j=1 k=1 i=1
( )2 αijk φijk − φi
s. t .
N ∑ i=1
φi = 1
(12)
⏐
2
; (16)
⏐,
N × ⏐PLjk − PPjk ⏐
where PLjk and PPjk are bilateral Laspeyres and Paasche indices, respectively. Complete set of transitive multilateral price indices with USA as base are in Appendix Tables. We find that the Fisher-based GEKS and SV-based GEKS are similar in magnitude as expected. The four transitive indices proposed in the paper are also close to the Fisher and SV-based GEKS. The relative performance of various transitive SV indices can be examined using the following two distance measures,
i=1
M M N 1 ∑∑∑
PLjk
( ) wik + wij
∆1 = SQRT
⎧ ⎨
1
⎩ M (M −1) 2
M M ∑ ∑ [ (
ln
j=1,j̸ =k k=1
PjkSV −T
)
− ln
PjkSV
(
⎫ )]⎬ ⎭
(17)
3 The authors gratefully acknowledge the data made available by the Global Office of the ICP.
4
N. Abe and D.S.P. Rao / Economics Letters 183 (2019) 108535
Appendix A. Supplementary data
Table 1 Relative performances of four transitive Sato–Vartia indices. Weight
Unweighted
(a)
(b)
(c)
Distance from the binary S-V Distance from the factor reversal
0.0471 4.9742
0.0481 4.8890
0.0476 4.8889
0.0464 4.2187
Supplementary material related to this article can be found online at https://doi.org/10.1016/j.econlet.2019.108535. References
M M ∑ ∑ [ (
1
∆2 = SQRT{ M (M −1) 2
− ln
QjkSV −T
(
)]
ln Ejk − ln PjkSV −T
)
(
)
j=1,j̸ =k k=1
}
(18)
∆1 and ∆2 show the distance from the binary SV index and factor reversal, respectively.4 Table 1 summarizes the results. The index with weights (3) based on the Laspeyres–Paashce gap (column c) is the best performing index. Correlations between the six transitive logged price indices (four transitive SV and two GEKS) are greater than 0.999. 5. Conclusions In this paper we construct several transitive Sato–Vartia multilateral price and quantity indexes. We provide a new logarithmic index that satisfies factor reversal test. Based on our analysis, we recommend the use of index of the form (8) with optimal weights (c) in Eq. (16). Our proposed index preserves the core philosophy of the SV index in generating transitive multilateral indices and unlike Fisher and SV-based GEKS indices it satisfies the strong identity test.
4 To calculate ∆ , we need to construct quantity index that requires strictly 2 positive quantity. Thus, we drop observations with zero nominal expenditures when calculating ∆2 .
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