Table of Contents

Cover

Title Page

Preface

Organisation of the book
Additional material available online
Suggested routes through the book

Acknowledgements

Chapter 1: Introduction

1.1 What is an index number?
1.2 Example – the Consumer Prices Index
1.3 Example – FTSE 100
1.4 Example – Multidimensional Poverty Index
1.5 Example – Gender Inequality Index
1.6 Representing the world with index numbers
1.7 Chapter summary
References

Chapter 2: Index numbers and change

2.1 Calculating an index series from a data series
2.2 Calculating percentage change
2.3 Comparing data series with index numbers
2.4 Converting from an index series to a data series
2.5 Chapter summary
Exercise A

Chapter 3: Measuring inflation

3.1 What is inflation?
3.2 What are inflation measures used for and why are they important?
3.3 Chapter summary
References
Exercise B

Chapter 4: Introducing price and quantity

4.1 Measuring price change
4.2 Simple, un-weighted indices for price change
4.3 Price, quantity and value
4.4 Example – Retail Sales Index
4.5 Chapter summary
Exercise C

Chapter 5: Laspeyres and Paasche indices

5.1 The Laspeyres price index
5.2 The Paasche price index
5.3 Laspeyres and Paasche quantity indices
5.4 Laspeyres and Paasche: mind your Ps and Qs
5.5 Laspeyres, Paasche and the Index Number Problem
5.6 Laspeyres or Paasche?
5.7 A more practical alternative to a Laspeyres price index?
5.8 Chapter summary
References
Exercise D

Chapter 6: Domains and aggregation

6.1 Defining domains
6.2 Indices for domains
6.3 Aggregating domains
6.4 More complex aggregation structures
6.5 A note on aggregation structures in practice
6.6 Non-consistency in aggregation
6.7 Chapter summary
Exercise E

Chapter 7: Linking and chain-linking

7.1 Linking
7.2 Re-basing
7.3 Chain-linking
7.4 Chapter summary
Exercise F

Chapter 8: Constructing the consumer prices index

8.1 Specifying the index
8.2 The basket
8.3 Locations and outlets
8.4 Price collection
8.5 Weighting
8.6 Aggregation structure
8.7 Elementary aggregates
8.8 Linking
8.9 Owner occupier housing
8.10 Publication
8.11 Special procedures
8.12 Chapter summary
References
Exercise G

Chapter 9: Re-referencing a series

9.1 Effective comparisons with index numbers
9.2 Changing the index reference period
9.3 Why re-reference?
9.4 Re-basing
9.5 Chapter summary
References
Exercise H

Chapter 10: Deflation

10.1 Value at constant price
10.2 Volume measures in the national accounts
10.3 Chapter summary
Exercise I

Chapter 11: Price and quantity index numbers in practice

11.1 A big picture view of price indices
11.2 The harmonised index of consumer prices
11.3 UK measures of consumer price inflation
11.4 PPI and SPPI
11.5 PPPs and international comparison
11.6 Quantity indices
11.7 Gross domestic product
11.8 Index of Production
11.9 Index of services
11.10 Retail sales index
11.11 Chapter summary
11.12 Data links
References

Chapter 12: Further index formulae

12.1 Recap on price index formulae
12.2 Classifying price and quantity index formulae
12.3 Asymmetrically weighted price indices
12.4 Symmetric weighted price indices
12.5 Un-weighted price indices
12.6 Different formulae, different index numbers
12.7 Chapter summary
References
Exercise J

Chapter 13: The choice of index formula

13.1 The index number problem
13.2 The axiomatic approach
13.3 The economic approach
13.4 The sampling approach
13.5 The stochastic approach to index numbers
13.6 Which approach is used in practice?
13.7 Chapter summary
References
Exercise K

Chapter 14: Issues in index numbers

14.1 Cost-of-living versus cost-of-goods
14.2 Consumer behaviour and substitution
14.3 New and disappearing goods
14.4 Quality change
14.5 Difficult to measure items
14.6 Chapter summary
References

Chapter 15: Research topics in index numbers

15.1 The uses of scanner data
15.2 Variations on indices
15.3 Difficult items
15.4 Chaining
15.5 Some research questions
References

Appendix A: Mathematics for index numbers

A.1 Notation
A.2 Key results
A.3 Index Formula Styles

Appendix B: Choice of index formula

B.1 The axiomatic approach to index numbers
B.2 The economic approach to index numbers
References

Appendix C: Glossary of terms and formulas

C.1 Commonly used terms
C.2 Commonly used symbols
C.3 Unweighted indices (price versions only)
C.4 Weighted indices (price versions only)

Appendix D: Solutions to exercises

Exercise A (Chapter 2)
Exercise C (Chapter 4)
Exercise D (Chapter 5)
Exercise E (Chapter 6)
Exercise F (Chapter 7)
Exercise H (Chapter 9)
Exercise I (Chapter 10)
Exercise J (Chapter 12)

Appendix E: Further reading

E.1 Manuals
E.2 Books
E.3 Papers

Index

End User License Agreement

List of Illustrations

Chapter 1: Introduction

Figure 1.1 UK Consumer Prices Index and selected sectors, September 2011–2013.

Chapter 2: Index numbers and change

Figure 2.1 Population on 1st January, European Union, 1995–2011.
Figure 2.2 Index of population on 1 January (2000 = 100), European Union, 1995–2011.

Chapter 5: Laspeyres and Paasche indices

Figure 5.1 Measures of price change for fruit between 2012 and 2013.

Chapter 6: Domains and aggregation

Figure 6.1 Example of COICOP structure.

Chapter 7: Linking and chain-linking

Figure 7.1 Price index of white goods, July 2011–June 2014.
Figure 7.2 Price index of white goods under old and new classification systems, July 2011–June 2014.
Figure 7.3 Options for linking price indices of white goods under old and new classification systems, July 2011–June 2014.
Figure 7.4 Linking price indices of white goods under old and new classification systems, July 2011–June 2014.
Figure 7.5 Combined alcohol index under two weighting patterns.
Figure 7.6 Index series with January base each year, 2008–2010.
Figure 7.7 Index series with January base each year and annually chain-linked, 2008–2010.

Chapter 9: Re-referencing a series

Figure 9.1 Beer and television prices, January–December.
Figure 9.2 Beer and television price indices, January–December.
Figure 9.3 Gross value added (GVA) chained volume measure at basic prices United Kingdom, by category of output.
Figure 9.4 Gross value added (GVA) chained volume measure at basic prices United Kingdom, by category of output.

Chapter 11: Price and quantity index numbers in practice

Figure 11.1 Index of GDP, UK, 1955–2013.
Figure 11.2 Index of production and main sub-indices, UK 1984–2013, rereferenced to 1948 = 100.
Figure 11.3 Index of services, UK 1997–2013.
Figure 11.4 Retail sales index adjusted and non adjusted for seasonal effects, UK 1996–2014.

Chapter 12: Further index formulae

Figure 12.1 Index numbers from different formulae with the same price and quantity data.

List of Tables

Chapter 1: Introduction

Table 1.1 Value of retail sales 2005–2008 for Great Britain.
Table 1.2 CPI values, 1- and 12-month inflation rates: September 2012–2013, United Kingdom.
Table 1.3 Weights for categories of goods and services in the CPI 2013, UK.
Table 1.4 Multidimensional Poverty Index dimensions, indicators and weights.
Table 1.5 Gender Inequality Index, dimensions and indicators.

Chapter 2: Index numbers and change

Table 2.1 Annual turnover for Company A, 2002–2013.
Table 2.2 Index of annual turnover for Company A, 2002–2013.
Table 2.3 Index of annual turnover for Company A, 2002–2013.
Table A.1 Number of employees 2005–2014.
Table A.2 Number of goals 2005/2006 to 2012/2013.
Table A.3 Index of salary 2009–2014.

Chapter 4: Introducing price and quantity

Table 4.1 Price of drinks on a particular day, 2012–2014.
Table 4.2 Price of drinks on a particular day, 2012–2013.
Table 4.3 Value of retail sales^a June 2013.
Table 4.4 Retail sales^a 2013.
Table C.1 Cake prices March-May.
Table C.2 Price and quantity of music lessons.
Table C.3 Price of wine 2013–2014.

Chapter 5: Laspeyres and Paasche indices

Table 5.1 Fruit prices.
Table 5.2 Selected fruit price indices.
Table 5.3 Fruit prices and quantities.
Table 5.4 Fruit prices and quantities.
Table 5.5 Price and quantity of tools, 2013 and 2014.
Table 5.6 Price and quantity relatives of tools, 2013–2014 with value shares.
Table D.1 Price and Quantity of clothing June to July.
Table D.2 Price and quantity for meat June and December.
Table D.3 Price and value data for a pharmacy.
Table D.4 Chicken and egg data, March and April.

Chapter 6: Domains and aggregation

Table 6.1 UK consumer prices index, percentage change on a year earlier selected CPI divisions, May 2014.
Table 6.2 Sales information from Joe's bike shop 2005–2010.
Table 6.3 Joe's bike shop turnover shares and price relatives.
Table 6.4 Joe's bike shop turnover and turnover shares.
Table E.1 Price and value data for a bar, 2013–2014.
Table E.2 Value shares for drink categories.
Table E.3 Price and quantity data for newspapers and magazines.
Table E.4 Items sold by a gift shop and example price relatives.

Chapter 7: Linking and chain-linking

Table 7.1 Quarterly index of turnover for company A, 2012–2013.
Table 7.2 Turnover share for Lager and Cider.
Table 7.3 Data for personal grooming services – 1.
Table 7.4 Data for personal grooming services – 2.
Table 7.5 Data for personal grooming services – 3.
Table F.1 Index series, 2012–2013.
Table F.2 Index numbers with previous period base.
Table F.3 Index series, 2001–2013.

Chapter 9: Re-referencing a series

Table 9.1 Beer and television prices, January–December.
Table 9.2 Beer and television price indices, January–December (January = 100).
Table 9.3 Short index series (2011 = 100).
Table 9.4 Short index series (2011 = 100 and 2009 = 100).
Table 9.5 A simple series.
Table H.1 Transport costs 2007–2014.

Chapter 10: Deflation

Table 10.1 Change in price and value of cakes sold by a bakery in 2012.
Table I.1 Value of sales 2007–2012.
Table I.2 CYP and Laspeyres quantity index series, 2004–2009.

Chapter 12: Further index formulae

Table 12.1 Price and quantity data for sales of tea.
Table 12.2 Index numbers for sales of tea using a variety of index formulae.
Table 12.3 Price and quantity data.

Chapter 13: The choice of index formula

Table 13.1 Fruit prices and quantities.

This edition first published 2015

Registered office

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom

For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.

The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.

Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought.

Library of Congress Cataloging-in-Publication Data

Ralph, Jeff.

A practical introduction to index numbers / Jeff Ralph, Rob O'Neill, Joe Winton.

pages cm

Includes bibliographical references and index.

ISBN 978-1-118-97781-1 (paperback)

1. Index numbers (Economics) I. O'Neill, Rob. II. Winton, Joe. III. Title.

HB225.R35 2015

330.01′5195– dc23

2015019379

A catalogue record for this book is available from the British Library.

Preface

The inspiration for writing an introduction to index numbers arose from our work in delivering index number training. We provide introductory and advanced training courses for the Government Statistical Service, for the Official Statistics module of the Operational Research, Applied Statistics and Risk MSc course at Cardiff University and for students sitting the index number components of the Royal Statistical Society (RSS) Ordinary and Higher Certificates. A number of students have asked us to recommend an introductory text to supplement the training, and we have struggled to find a suitable modern book to recommend. While other statistical topics are well supported by introductory books, this is not the case for index numbers.

There are, of course, a number of very good books on index numbers; however, they are directed towards more advanced study. We wanted to write a short, practical book that would introduce the subject and be suitable for students and others who would like a general introduction to the subject. We have therefore based the content partly on the syllabuses for the two RSS Certificates and partly on the content of the workshops we have developed at the Office for National Statistics.

Index numbers is a topic whose concepts are applied very widely. In the economic sphere, many of the most prominent of our economic indicators are presented as index numbers, including the Consumer Prices Index, Gross Domestic Product and the Retail Sales Index. Readers of most newspapers will find regular mentions of these indices and many others too. However, it is not just in the economic sphere that index number concepts appear. Increasingly, they are finding application in the social sphere through composite measures such as those aiming to quantify complex concepts such as poverty and prosperity.

Despite finding widespread application, few statistics or economics books contain even a mention of index numbers, let alone any detailed description; a curious student might well wonder which academic subject they fall under. The answer is that the topic tends to fall between economics and statistics. Where it does appear, it is usually studied as part of a specialist topic, like measurement economics.

At a basic level, the subject may appear simple to the beginning student, or casual academic observer; however, in our experience this is not the case. There are many subtle and complex aspects that have gradually developed over many years and are still the subject of much discussion and academic research.¹ Indeed, the practical implementation of a price index like the Consumer Prices Index is a massive undertaking that requires a considerable amount of effort and expertise on the part of National Statistics Institutes around the world. It is the intention of this book to highlight some of these aspects, hopefully resulting in an enhanced understanding of index numbers as commonly used in Official Statistics.

In common with other topics in statistics and economics, some observers may consider index numbers a dry subject, distant from the general population and perhaps not very interesting. While we can understand why this view might be held, we see the subject very differently. It is a complex and absorbing subject that provides a good return on investment in study. It should also be pointed out that through index number measures such as the Consumer Prices Index, all of our lives are affected in one way or another by the application of the subject.

This is not a book that will examine and explain all of the complex aspects of major index number outputs. However, there are facets of the Consumer Prices Index that are instructive for the student to consider, and we have included some of them in this book. We hope they will also prove interesting and will give the student a better understanding of this key indicator when watching the news, listening to the radio or reading a newspaper.

One of the significant advantages of developing and delivering training courses over a number of years is the opportunity to continuously improve the teaching material based on feedback from students. Two particular aspects of the training have proved popular with students. Firstly, including exercises that require students to calculate index numbers from a modest amount of data, and secondly, mixing material on the theoretical aspects of the subject with sections on the practical aspects of price statistics. We have continued this approach in this book.

Organisation of the book

This book has 15 chapters and includes 11 sets of exercises. Chapter 1 provides a general introduction to the concepts of an index and an index number; it identifies a range of applications from both the economic and social spheres. Chapter 2 introduces the reader to the procedure for converting a simple data series into a series of index numbers and back again. It shows how to calculate percentage change from an index series and identifies some benefits of working with data in index number form. The first exercise gives the reader the opportunity to develop the basic skills with some simple data.

One of the major applications of index numbers is the measurement of inflation; it is arguably the most important output any National Statistics Institute produces and it is the subject of Chapter 3. We explain briefly what is meant by inflation and why it is important. Chapters 4 and 5 set out the basic material to enable the student to understand how inflation is calculated in practice. Chapter 4 introduces price, quantity and value, and the application of simple unweighted price indices. An exercise follows in which the reader is invited to calculate price index numbers for goods on sale in a shop. Chapter 5 describes value change for a basket of goods between two time periods and introduces the iconic price (and quantity) indices of Étienne Laspeyres and Hermann Paasche. It explains the difference between the two indices and shows how value change can be decomposed into measures of pure price change and pure quantity change; it also introduces the index number problem. An exercise follows.

Chapters 6 and 7 explore two aspects of the subject that are important for the practical application of index numbers – domains and aggregation, and linking and chain-linking. How the Consumer Prices Index is constructed is the subject of Chapter 8; it uses the knowledge gained from Chapters 2 to 7. Two more practical topics are covered in Chapters 9 and 10 – re-referencing and rebasing, and deflation; both chapters are followed by exercises. A brief description of the major price and quantity indices produced by National Statistics Institutes is given in Chapter 11.

Chapter 12 returns to the theoretical development of the subject, considering further index formulae including the Fisher, Walsh and Törnqvist indices; an exercise includes both algebra and calculation based questions. Chapter 13 gives an overview of the various approaches that have been made to identify the ‘best’ formula to use. More advanced topics that are much discussed and studied in the index number research field are the subject of Chapter 14. They include: consumer substitution behaviour, new goods and disappearing goods, and hard to measure goods like housing. These are all challenging topics that producers of price indices need to address and we seek only to introduce them in this chapter. Current research topics, such as the role of big data in price indices are explored in Chapter 15.

Six appendices provide supporting material:

A: Mathematics for index numbers
Index numbers is a mathematical topic – though the level of mathematical knowledge needed is relatively modest at the introductory level that this book covers. However, we do not want readers to be inhibited by the mathematical requirements, so we have explained the mathematical notation and knowledge required in each chapter as it arises. Appendix A provides further support by providing a fuller explanation of the key mathematical concepts and index number results.
B: Choice of index formula
This appendix provides a more detailed look at two of the approaches to index numbers that were introduced in Chapter 13 – the axiomatic and economic. This appendix is more advanced than any of the other material in the book.
C: Glossary of terms and formulae
D: Solutions to most of the exercises
E: Further reading

Additional material available online

The companion website, www.wiley.com/go/ralph/index_numbers, hosts additional content for this book. This includes presentations, example data and R code to run index number calculations. There is also some additional help for students who attempt the exercises.

Suggested routes through the book

While it is hoped that many readers will work through all chapters in the book, there are options for students taking the Royal Statistical Society exams² who want to focus on the specific material they need. The following chapters are particularly relevant:

Ordinary certificate: 1–5 and some parts of 7 and 12
Higher certificate: 1–13 and Appendix A.

For a reader who would like to learn about the concept of index numbers, their use and some of the challenging aspects, but without wanting to develop skills in calculation, the following is a suggested subset of the book: Chapters 1–3, 8, 11, 14 and 15.