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

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

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

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

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

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

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

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.

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.

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”.

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

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

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

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

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

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

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

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'

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

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)

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.

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

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

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.