# A Practical Introduction to Index Numbers

## Books

This book provides an introduction to index numbers for statisticians, economists and numerate members of the public. It covers the essential basics, mixing theoretical aspects with practical techniques to give a balanced and accessible introduction to the subject. The concepts are illustrated by exploring the construction and use of the Consumer Prices Index which is arguably the most important of all official statistics in the UK. The book also considers current issues and developments in the field including the use of large-scale price transaction data.

A Practical Introduction to Index Numbers will be the ideal accompaniment for students taking the index number components of the Royal Statistical Society Ordinary and Higher Certificate exams; it provides suggested routes through the book for students, and sets of exercises with solutions.

Preface xi

Acknowledgements xv

1 Introduction 1

1.1 What is an index number? 1

1.2 Example – the Consumer Prices Index 2

1.3 Example – FTSE 100 5

1.4 Example – Multidimensional Poverty Index 6

1.5 Example – Gender Inequality Index 6

1.6 Representing the world with index numbers 7

1.7 Chapter summary 8

References 8

2 Index numbers and change 9

2.1 Calculating an index series from a data series 9

2.2 Calculating percentage change 11

2.3 Comparing data series with index numbers 13

2.4 Converting from an index series to a data series 14

2.5 Chapter summary 16

Exercise A 17

3 Measuring inflation 19

3.1 What is inflation? 19

3.2 What are inflation measures used for and why are they important? 20

3.2.1 Determination of monetary policy by a central bank 21

3.2.2 Changing of provisions for private pensions 21

3.2.3 Changes in amounts paid over long-term contracts 21

3.2.4 Changes in rail fares and other goods 22

3.2.5 Evaluating investment decisions 22

3.2.6 Inputs to economic research and analysis 23

3.2.8 Tax allowances 23

3.2.9 Targets for stability of the economy in an international context 23

3.3 Chapter summary 24

References 24

Exercise B 25

4 Introducing price and quantity 27

4.1 Measuring price change 27

4.2 Simple, un-weighted indices for price change 30

4.2.1 Simple price indices 30

4.2.2 Simple quantity indices 33

4.3 Price, quantity and value 34

4.4 Example – Retail Sales Index 35

4.5 Chapter summary 36

Exercise C 37

5 Laspeyres and Paasche indices 39

5.1 The Laspeyres price index 40

5.2 The Paasche price index 41

5.3 Laspeyres and Paasche quantity indices 43

5.4 Laspeyres and Paasche: mind your Ps and Qs 45

5.4.1 Laspeyres price index as a weighted sum of price relatives 45

5.4.2 Laspeyres quantity index as a weighted sum of quantity relatives 46

5.4.3 Paasche price index as a weighted harmonic mean of price relatives 46

5.4.4 Paasche quantity index as a weighted harmonic mean of quantity relatives 46

5.5 Laspeyres, Paasche and the Index Number Problem 48

5.6 Laspeyres or Paasche? 49

5.7 A more practical alternative to a Laspeyres price index? 51

5.8 Chapter summary 51

References 52

Exercise D 53

6 Domains and aggregation 55

6.1 Defining domains 55

6.2 Indices for domains 57

6.3 Aggregating domains 58

6.4 More complex aggregation structures 62

6.5 A note on aggregation structures in practice 62

6.6 Non-consistency in aggregation 63

6.7 Chapter summary 63

Exercise E 64

7.2 Re-basing 71

7.4 Chapter summary 75

Exercise F 76

8 Constructing the consumer prices index 79

8.1 Specifying the index 79

8.3 Locations and outlets 81

8.4 Price collection 81

8.5 Weighting 81

8.6 Aggregation structure 82

8.7 Elementary aggregates 83

8.9 Owner occupier housing 85

8.10 Publication 85

8.11 Special procedures 86

8.12 Chapter summary 86

References 86

Exercise G 88

9 Re-referencing a series 89

9.1 Effective comparisons with index numbers 89

9.2 Changing the index reference period 92

9.3 Why re-reference? 94

9.4 Re-basing 95

9.5 Chapter summary 96

References 96

Exercise H 97

10 Deflation 99

10.1 Value at constant price 101

10.2 Volume measures in the national accounts 102

10.3 Chapter summary 103

Exercise I 104

11 Price and quantity index numbers in practice 105

11.1 A big picture view of price indices 105

11.2 The harmonised index of consumer prices 106

11.3 UK measures of consumer price inflation 107

11.4 PPI and SPPI 108

11.5 PPPs and international comparison 109

11.6 Quantity indices 109

11.7 Gross domestic product 110

11.8 Index of Production 111

11.9 Index of services 112

11.10 Retail sales index 113

11.11 Chapter summary 114

References 115

12 Further index formulae 119

12.1 Recap on price index formulae 119

12.2 Classifying price and quantity index formulae 120

12.3 Asymmetrically weighted price indices 120

12.4 Symmetric weighted price indices 123

12.5 Un-weighted price indices 124

12.6 Different formulae, different index numbers 126

12.7 Chapter summary 127

References 127

Exercise J 129

13 The choice of index formula 131

13.1 The index number problem 131

13.2 The axiomatic approach 133

13.3 The economic approach 134

13.4 The sampling approach 135

13.5 The stochastic approach to index numbers 136

13.6 Which approach is used in practice? 137

13.7 Chapter summary 138

References 138

Exercise K 140

14 Issues in index numbers 141

14.1 Cost-of-living versus cost-of-goods 141

14.2 Consumer behaviour and substitution 143

14.3 New and disappearing goods 144

14.4 Quality change 145

14.4.1 Option 1: do nothing – pure price change 146

14.4.2 Option 2: automatic linking – pure quality change 146

14.4.4 Option 4: imputation 147

14.4.5 Option 5: hedonics 147

14.5 Difficult to measure items 148

14.6 Chapter summary 149

References 149

15 Research topics in index numbers 151

15.1 The uses of scanner data 151

15.1.1 Improvements at the lowest level of aggregation 152

15.1.2 Understanding consumer behaviour 152

15.1.3 Alternative measurement schemes 153

15.1.4 Frequency of indices 153

15.2 Variations on indices 154

15.2.1 Regional indices 154

15.2.2 Variation by socio-economic group or income quantile 154

15.3 Difficult items 155

15.3.1 Clothing 155

15.3.2 New and disappearing goods 156

15.3.3 Hedonics 157

15.4 Chaining 157

15.5 Some research questions 158

References 158

A Mathematics for index numbers 161

A.1 Notation 161

A.1.1 Summation notation 161

A.1.2 An alternative representation 163

A.1.3 Geometric indices 164

A.1.4 Harmonic indices 164

A.2 Key results 165

A.2.1 The value ratio decomposition 165

A.2.2 Converting between the two forms of price and quantity indices 166

A.2.3 Other examples of the price-relative/weights 167

A.2.4 The value ratio as a product of Fisher indices 167

A.3 Index Formula Styles 168

B Choice of index formula 169

B.1 The axiomatic approach to index numbers 169

B.1.1 An introduction to the axiomatic approach 169

B.1.2 Some axioms 170

B.1.3 Choosing an index based on the axiomatic approach 173

B.1.4 Conclusions 174

B.2 The economic approach to index numbers 174

B.2.1 The economic approach to index numbers 174

B.2.2 A result on expenditure indices 177

B.2.3 Example 1: Cobb-Douglas and the Jevons index 179

B.2.4 Example 2: CES and the Lloyd-Moulton index 181

B.2.5 Issues with the economic approach 183

References 184

C Glossary of terms and formulas 185

C.1 Commonly used terms 185

C.2 Commonly used symbols 189

C.3 Unweighted indices (price versions only) 190

C.4 Weighted indices (price versions only) 191

D Solutions to exercises 193

E.1 Manuals 211

E.2 Books 211

E.3 Papers 212

Index 213

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