systems of index numbers for international price comparisons based on the stochastic approach...
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![Page 1: Systems of Index Numbers for International Price Comparisons Based on the Stochastic Approach Gholamreza Hajargasht D.S. Prasada Rao Centre for Efficiency](https://reader030.vdocuments.net/reader030/viewer/2022032522/56649d615503460f94a422ac/html5/thumbnails/1.jpg)
Systems of Index Numbers for International Price Comparisons
Based on the Stochastic Approach
Gholamreza HajargashtGholamreza HajargashtD.S. Prasada RaoD.S. Prasada Rao
Centre for Efficiency and Productivity AnalysisCentre for Efficiency and Productivity AnalysisSchool of EconomicsSchool of Economics
University of QueenslandUniversity of QueenslandBrisbane, Australia Brisbane, Australia
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Standard formulae and the new index formula
Stochastic approach to index numbers
Derivation of index numbers using stochastic approach
Empirical application – OECD data
Concluding remarks
OutlineOutline
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NotationNotation
We have We have MM countries and countries and NN commodities commodities
observed price of the ith commodity in the jth countryobserved price of the ith commodity in the jth country
quantity of the ith commodity in the jth country quantity of the ith commodity in the jth country
purchasing power parity of jth countrypurchasing power parity of jth country
Average price for ith commodityAverage price for ith commodity
sharesshares
ijp
ijq
iP
1
ij ijij N
ij iji
p qw
p q
*
1
ijij M
ijj
ww
w
jPPP
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Commonly used methods like Laspeyres, Paasche, Fisher and Tornqvist methods are for bilateral comparisons – not transitive.
Geary-Khamis method – Geary (1958), Khamis (1970)
Elteto-Koves-Szulc (EKS) method – 1968
EKS method constructs transitive indexes from bilateral Fisher indexes
Weighted EKS index – Rao (2001)
Index number methods for international Index number methods for international comparisonscomparisons
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Variants of Geary-Khamis method – Ikle (1972), Rao (1990)
Methods based on stochastic approach:
Country-product-dummy (CPD) method – Summers (1973)
Weighted CPD method – Rao (1995)
Generating indexes using Weighted CPD method – Rao (2005), Diewert (2005)
Using CPD to compute standard errors – Rao (2004), Deaton (2005)
Index number methods - continuedIndex number methods - continued
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Geary-Khamis MethodGeary-Khamis Method Geary (1958) and Khamis (1970)
Based on twin concepts:
PPPs of currencies - PPPj’s
International averages of prices - Pi’s
Computations based on a simultaneous equation system:
1
1
( ) /M
ij ij jj
i M
ijj
p q PPP
Pq
1
1
N
ij iji
j N
i iji
p qPPP
P q
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Rao and Ikle variantsRao and Ikle variants
Rao Index Geometric Mean Rao Index Geometric Mean
Ikle Index Harmonic Mean Ikle Index Harmonic Mean
1
ijwNij
jii
pPPP
P
*
1
ijwMij
ijj
pP
PPP
1
1 Ni
ijj iji
Pw
PPP P
*
1
1 Mj
iji ijj
PPPw
P p
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The New IndexThe New Index
Why not an arithmetic meanWhy not an arithmetic mean
1
Nij
j ijii
pPPP w
P
*
1
Mij
i ijjj
pP w
PPP
Comments:
• For all these methods it is necessary to establish the existence of solutions for the simultaneous equations.
• Existence of Rao and Ikle indexes was established in earlier papers.
• Existence of the new index is considered in this paper.
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Relationship between the indices and Relationship between the indices and stochastic approachstochastic approach
The stochastic approach to multilateral indexes is based on the CPD model – discussed below.
The indexes described above are all based on the Geary-Khamis framework which has no stochastic framework.
Rao (2005) has shown that the Rao (1990) variant of the Geary-Khamis method can be derived using the CPD model.
Rest of the presentation focuses on the connection between the CPD model and the indexes described above.
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The Law of One PriceThe Law of One Price
Following Summers (1973), Rao (2005) and Following Summers (1973), Rao (2005) and Diewert (2005) we considerDiewert (2005) we consider
price of i-th commodity in j-th countryprice of i-th commodity in j-th country purchasing power paritypurchasing power parity
ij i j ijp P PPP u
World price of i-th commodity •random disturbance
The CPD model may be considered as a “hedonic regression model” where the only characteristics considered are the “country” and the “commodity”.
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CPD ModelCPD Model
Rao (2005) has shown that applying a weighted least Rao (2005) has shown that applying a weighted least square to the following equation results in Rao’s square to the following equation results in Rao’s
SystemSystem
where where are treated as parameters are treated as parameters The same result is obtained if we assume a log-normal The same result is obtained if we assume a log-normal
distribution for udistribution for uijij and use a weighted maximum and use a weighted maximum
likelihood estimation approach which is the same as the likelihood estimation approach which is the same as the weighted least squares estimator.weighted least squares estimator.
ln ln lnij i j ijp P PPP
ln and lni jP PPP
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Stochastic Approach to New IndexStochastic Approach to New Index
AgainAgain
This time assume This time assume
ij i j ijp PPPP u
~ ( , )iju Gamma r r
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Maximum LikelihoodMaximum Likelihood
It can be shown It can be shown
Taking logs we obtain Taking logs we obtain
1
( )( )
ij
i j
pr rr
PPPPijij r r
i j
prf p e
r P PPP
ln ln ( ) ( 1) ln ln ln ijij ij i j
i j
pLnL r r r r p r P r PPP r
PPPP
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Weights and M-EstimationWeights and M-Estimation
Define a weighted likelihood asDefine a weighted likelihood as
Then we haveThen we have
1 1 1 1 1 1
ln ( 1) ln ln lnn n n n n n
ij ij ij i ij ji j i j i j
WL r w p r w P r w PPP
1 1 1 1 1 1
ln ( ) ln ( )N M N M N M
ij ijij ij
i ji j i j i j
p wr r r w r w
PPPP
N
i
M
jij
ij LnLM
wLnWL
1 1
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First Order ConditionsFirst Order Conditions
The new Index independent of rThe new Index independent of r
The advantage of MLE is that we can calculate The advantage of MLE is that we can calculate standard errors for PPPsstandard errors for PPPs
1 1 1 1 1 1 1 1
1ln ( ) ln ( ln ln ln )
n n n n n n N Mij ij
ij ij ij i ij ji ji j i j i j i j
p wr r w p w P w PPP M
r M PPPP
*
1
1
0
0
Mij ij
ijj
Nij ij
jii
p wP
PPP
p wPPP
P
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Stochastic Approach to IkleStochastic Approach to Ikle
Now we consider the modelNow we consider the model
wherewhere
1 1ij
ij i j
up PPPP
~ ( , )iju Gamma r r
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First Order ConditionsFirst Order Conditions
Applying a weighted maximum likelihood we Applying a weighted maximum likelihood we obtain the following first order conditionsobtain the following first order conditions
i.e. Ikle’s Indexi.e. Ikle’s Index
1
1 Ni
ijj iji
Pw
PPP P
*
1
1 Mj
iji ijj
PPPw
P p
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Standard ErrorsStandard Errors
Computation of standard errors is an important Computation of standard errors is an important motivation for considering stochastic approach.motivation for considering stochastic approach.
An M-Estimator is defined as an estimator that An M-Estimator is defined as an estimator that maximizesmaximizes
It has the following asymptotic distributionIt has the following asymptotic distribution
wherewhere
1
1( ) ( , )
N
N i i ii
Q h yN
θ x ;θ
0 0 0 0ˆ( ) [ , ]dN N 1 1θ θ 0 A B A
0
01
1plim
Ni
i
h
N
2
θ
Aθ' θ
00
01
1plim
Ni i
i
h h
N
θ θ
Bθ θ'
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In practice, a consistent estimator can be obtained asIn practice, a consistent estimator can be obtained as
WhereWhere
In special cases like the standard maximum likelihood In special cases like the standard maximum likelihood
we havewe have thereforetherefore
1ˆ ˆ ˆˆN
1 1VAR(θ) A BA
1 ˆ
1ˆN
i
h
N
2
θ
Aθ' θ
ˆ1 ˆ
1ˆN
i i
i
h h
N
θ θ
Bθ θ'
1ˆ ˆ(N
1VAR θ) A
10
0A = -B
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Many software report this as their default Many software report this as their default variance estimatorvariance estimator
This Variance is not valid in our context This Variance is not valid in our context and the general formula should be usedand the general formula should be used
For example if we apply to a weighted For example if we apply to a weighted linear model (e.g. CPD)linear model (e.g. CPD)
But if we apply the general formulaBut if we apply the general formula
2ˆ( 1VAR θ) (X'ΩX)
2ˆ̂( ' ' 1 1VAR θ) (X'ΩX) (X Ω ΩX)(X'ΩX)
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Application to OECD countriesApplication to OECD countries
OECD data from 1996. OECD data from 1996. The price information was in the form of PPPs at The price information was in the form of PPPs at
the basic heading level for 158 basic headings, the basic heading level for 158 basic headings, with US dollar used as the numeraire currency. with US dollar used as the numeraire currency.
The estimates of PPPs based on the new index, The estimates of PPPs based on the new index, Ikle’s and the weighted CPD for 24 OECD Ikle’s and the weighted CPD for 24 OECD countries along with their standard errors are countries along with their standard errors are presented in the following table.presented in the following table.
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Country MLE Estimates
New Index CPD Ikle
PPP S.E PPP S.E PPP S.E.
GER 1.887 0.136 2.034 0.144 2.187 0.147
FRA 6.092 0.429 6.554 0.455 7.035 0.466
ITA 1425.96 109.727 1504.02 115.509 1584.381 119.196
NLD 1.921 0.150 2.056 0.155 2.205 0.156
BEL 35.491 2.577 37.890 2.698 40.450 2.728
LUX 33.578 2.488 35.816 2.618 38.191 2.700
UK 0.603 0.043 0.642 0.044 0.682 0.045
IRE 0.637 0.051 0.669 0.055 0.696 0.060
DNK 8.525 0.586 9.131 0.615 9.762 0.631
GRC 180.470 13.452 188.482 13.891 196.640 14.005
SPA 112.414 8.304 118.546 8.606 124.799 8.738
PRT 126.043 10.400 129.037 10.994 130.317 12.002
AUT 12.770 0.881 13.730 0.928 14.728 0.948
SUI 2.050 0.168 2.183 0.177 2.320 0.180
SWE 9.424 0.686 10.075 0.720 10.758 0.742
FIN 6.159 0.432 6.598 0.453 7.070 0.462
ICE 86.828 7.000 89.541 6.975 92.329 6.810
NOR 8.807 0.684 9.238 0.736 9.642 0.764
TUR 6304.23 579.128 6321.42 544.907 6357.003 506.991
AUS 1.264 0.099 1.333 0.103 1.407 0.104
NZL 1.464 0.111 1.530 0.113 1.596 0.115
JAP 182.031 13.622 187.429 14.282 192.392 14.780
CAN 1.168 0.090 1.229 0.094 1.295 0.096
USA 1.00 1.00 1.00
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ConclusionConclusion
A new system for international price comparison A new system for international price comparison is proposed. Existence and uniqueness is proposed. Existence and uniqueness establishedestablished
A stochastic framework for generating the Rao, A stochastic framework for generating the Rao, Ikle and the new index has been established.Ikle and the new index has been established.
Using the framework of M-estimators, standard Using the framework of M-estimators, standard errors are obtained for the PPPs from each of errors are obtained for the PPPs from each of the methods. the methods.
Empirical application of the new approach to Empirical application of the new approach to OECD data shows the feasibility of the approachOECD data shows the feasibility of the approach
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Future researchFuture research
A more complete existence theoremA more complete existence theorem Statistical Comparison of the three Indices – Statistical Comparison of the three Indices –
choosing between the three distribution choosing between the three distribution specificationsspecifications
Bayesian Estimation and Inference seems to be Bayesian Estimation and Inference seems to be promising in this contextpromising in this context
Finding a suitable model to generate the Geary-Finding a suitable model to generate the Geary-Khamis methodKhamis method
Using residuals from the regression framework Using residuals from the regression framework to generate standard errors for more standard to generate standard errors for more standard index number formulae.index number formulae.