1 Reliability of Ideal Indexes Gholamr eza Hajargasht

2025-04-30 0 0 754.67KB 36 页 10玖币

侵权投诉

Reliability of Ideal Indexes

Gholamreza Hajargasht

The University of Queensland

Email: g.hajargasht@uq.edu.au

Abstract

The Fisher and Gini-Eltetö-Köves-Szulc (GEKS) are celebrated as ideal

bilateral and multilateral indexes due to their superior axiomatic and econ-

theoretic properties. The Fisher index is the main index used for constructing

CPI by statistical agencies and the GEKS is the index used for compiling

PPPs in World Bank’s International Comparison Program (ICP). Despite

such a high status and the importance attached to these indexes, the

stochastic approach to these indexes is not well-developed and no measures

of reliability exist for these important economic statistics. The main objective

of this paper is to fill this gap. We show how appropriate reliability measures

for the Fisher and GEKS indexes, and other ideal price index numbers can be

derived and how they should be interpreted and used in practice. In an

application to 2017 ICP data on a sample of 173 countries, we estimate the

Fisher and GEKS indexes along with their reliability measures, make

comparisons with several other notable indexes and discuss the implications.

JEL Codes: C43; C13; C18; C80

Key words: Stochastic Approach; Purchasing Power Parities (PPP); ICP; Product-

Dummy (PD) method; Bootstrap; Reliability Measures; Walsh; Fisher

The author is indebted to D. S. Prasada Rao for helpful suggestions and comments on

earlier drafts of this paper. Financial support from the Australian Research Council

(ARC) Discovery Project DP170103559 is gratefully acknowledged.

1- Introduction

National statistical agencies compile and publish consumer (and producer) price

index numbers on a monthly or quarterly basis. Estimates of gross domestic product

at constant prices are also published as a part of the national accounts data.

International agencies such as the World Bank, OECD and Eurostat produce estimates

of purchasing power parities (PPPs) and real expenditures through the International

Comparison Program (ICP). A distinguishing feature of the national publications on

CPI and international publications on PPPs is the lack of any indication of reliability

of the published figures. For example, in the case of international comparisons, one

may have a higher level of confidence in price comparisons between countries at a

comparable level of development (e.g. between USA and Germany) than a

comparison between two “dissimilar” countries (e.g. USA and Kenya). The main

objective of this paper is to fill this gap by providing a framework for computing and

understanding reliability measures for Fisher, GEKS and other notable index

numbers.

Fisher (1922) in his momentous book, The Making of Index Numbers, listed 126 index

number formulae. Additional formulae have been proposed since then. Among them,

the Fisher index has been crowned as the ideal price index for bilateral comparisons

due to its superior axiomatic properties and being superlative (see e.g., Consumer

Price Index Theory, 2020). It is the index used in constructing consumer price indexes

(CPI) by many national and international agencies (also recommended by the

Consumer Price Index Manual: Concepts and Methods

, 2020) and in constructing

Hereafter we refer to this book as the Manual.

purchasing power parities (PPPs) by the World Bank’s International Comparison

Program (World Bank 2020). The Fisher index is also a building block in constructing

multilateral price indexes for temporal and spatial comparisons. Fisher along with

Törnqvist are also considered ideal for computing productivity measures both in

theory (see e.g., Diewert, 1992) and in practice (e.g., by US Bureau of Labor Statistics).

Despite the popularity of the Fisher index, the stochastic approach to it is not well-

developed. In particular, no formula for its variance exists.

This paper seeks to fill this gap by developing a modern stochastic approach to the

Fisher index and provide a simple formula for its variance. We also discuss measures

of reliability for other important indexes including Törnqvist, Sato-Vartia and Walsh

and compare them to the Fisher index. Empirical results reported here make use of

data from the 2017 Round of the International Comparison Program (ICP) covering

173 countries. Other contributions of the paper on bilateral indexes include: (i) a

stochastic approach and a formula for the variance of the Walsh index; (ii) interpreting

variance of log-Fisher and other indexes as price dissimilarity measures; and (iii)

demonstrating that the two main stochastic approaches to index numbers produce the

same index and incredibly the same variance.

A bilateral index is suitable for comparing two time periods or two countries. When

the comparison is extended to three or more, the index needs to become multilateral.

Transitivity is an internal consistency requirement for a multilateral index. None of

the widely used index number methods, including Fisher, Tornqvist and Walsh

indices, satisfy transitivity. Several approaches exist for obtaining transitive

multilateral price indexes (see e.g., Balk 1996; Hill 1997; or Diewert 1999). However, a

method due to Gini (1931), Eltetö and Köves (1964) and Szulc (1964), known as the

GEKS, is the recommended method for international price comparisons. Since 2005,

ICP uses GEKS to compare prices and real incomes across countries (see e.g., Diewert

2013). GEKS is used by the OECD in its triennial comparisons and by Eurostat in its

annual comparisons (see Eurostat 2012). GEKS has also been suggested as a good

method to construct temporal chained indexes (Diewert and Fox, 2020). Again, despite

the importance attached to GEKS, the stochastic approach to GEKS is not well-

developed. In this paper we propose an appropriate stochastic approach and use the

2017 ICP data to show that it produces sensible reliability measures and discuss the

implications.

The structure of the paper is as follows: Section 2 introduces the new stochastic

approach and Section 3 applies it to the Fisher ideal index. Section 4 presents the

stochastic approach to several other notable bilateral indexes. In Section 5 we discuss

alternative interpretations of the reliability measures. Section 6 develops the stochastic

approach to GEKS. In Section 7, we estimate the Fisher and GEKS indexes using 2017

ICP data on a sample of 173 countries and compare them with some of the other

notable indexes. Two appendices complement the materials in the text.

2- Stochastic Approaches to Price Index Numbers

Suppose there are N commodities indexed with

1,....,nN=

is the price of n-th item

in the j-th location (or time period)

and

is expenditure on the n-th item in j-th

location/period. Quantity is defined as

nj nj nj

q e p=

, expenditure shares as

- In our notation, j indicates a location (e.g. a country) in the former and a time period in the latter case. We use

location/period to include both.

nj nj nj

s e e

=

and

denotes the price index of the j-th relative to the k-th

location/period.

The stochastic approach to index numbers (see e.g. Consumer Price Index Theory,

2020, Chapter 4) can be conducted in two basic ways. One approach relies on

modelling price ratios, the alternative approach models prices according to the so-

called law of one price. In the analysis below we use the former approach, but in

Appendix-A, we show that the two approaches are equivalent i.e. they produce the

same formula for an index and its variance. The stochastic approach to index numbers

also requires a way of incorporating weights and an appropriate estimation

procedure. There have been two methods to incorporate weights in the stochastic

approach. Here we briefly describe these two methods and propose a new method

that we use in this paper.

Weights through heteroskedasticity: A stochastic approach to index numbers starts with

a specification such as

,1,...,

n jk jk n

P v n N



= + =

(1)

with

( ) 0

Ev =

and where

,n jk



’s are price relatives (e.g.

,n jk nj nk



) or log of price

relatives and

is a bilateral or log of a bilateral price index

. Under the early

stochastic approach of 1980s and 1990s the weights (

s) are incorporated through the

variance of the error term by specifying

()

Var v w



and the model is estimated

using generalized least squares (GLS). This heteroskedastic specification has been

criticized (see e.g. Diewert 1995) since variability in price ratios may not have a simple

Alternative choices of weights lead to alternative indexes such as Laspeyres, Paasche, Tornqvist, etc.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

1ReliabilityofIdealIndexesGholamrezaHajargashtTheUniversityofQueenslandEmail:g.hajargasht@uq.edu.auAbstractTheFisherandGini-Eltetö-Köves-Szulc(GEKS)arecelebratedasidealbilateralandmultilateralindexesduetotheirsuperioraxiomaticandecon-theoreticproperties.TheFisherindexisthemainindexusedforconstructin...

展开>> 收起<<

1 Reliability of Ideal Indexes Gholamr eza Hajargasht.pdf

共36页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

1 Reliability of Ideal Indexes Gholamr eza Hajargasht

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: