A Practical Introduction to Index Numbers
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Dr Jeff Ralph, Head of Index Number Methodology, Office for National Statistics, Cardiff, UK
Mr Joe Winton, Statistical Training Unit, Office for National Statistics, Cardiff, UK
Dr Robert O'Neill, Lecturer in Economics, University of Huddersfield, UK
Content
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
3.2.7 Index-linked debt 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 Linking and chain-linking 67
7.1 Linking 68
7.2 Re-basing 71
7.3 Chain-linking 74
7.4 Chapter summary 75
Exercise F 76
8 Constructing the consumer prices index 79
8.1 Specifying the index 79
8.2 The basket 80
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
8.8 Linking 84
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
11.12 Data links 115
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
14.4.3 Option 3: linking 147
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 Further reading 211
E.1 Manuals 211
E.2 Books 211
E.3 Papers 212
Index 213
