1 Consumer Price Indices; An ILO Manual, by Ralph Turvey et al (ILO, Geneva 1989).

2 Transactions data has access to timely quantity data, however this is currently only for a subset of the CPI basket, primarily the Food and non-alcoholic beverages group (see Use of transactions data in the Australian CPI for more details).

3 For a description of the indexes, refer to Price index theory of this manual.

4 Analytical retrospective superlative index based on New Zealand’s CPI: 2014, (Statistics New Zealand, 2014), available at http://archive.stats.govt.nz/browse_for_stats/economic_indicators/CPI_inflation/cpi-retrospective-superlative-index-2014.aspx

5 The RAP and APR formulas both give an index of 55.0 in this case.

6 The aggregation property of the Laspeyres and Paasche indexes allows them to be broken down into points contributions which is very useful for analysing the relative significance of items in the index, and their contributions to changes in the aggregate index. However, Diewert (2000) has a way to decompose superlative indexes.

# Maintaining the relevance of the CPI

## Introduction

11.1 At the expenditure class (published) level the ABS uses a fixed basket of goods and services in order to measure the price change in the Consumer Price Index (CPI) excluding any quality or quantity changes. However, as consumer expenditure patterns change over time in a dynamic economy, the fixed basket used in the CPI runs the risk of becoming unrepresentative and can lead to bias. There are a number of different types of bias that may affect price indexes, outlined in Price index theory of this manual. The ABS applies significant effort to address these biases. Some aspects, such as quality change, have been addressed in Price collection of this manual. Maintaining the relevance of the CPI includes the strategies the ABS uses to minimise the effect of substitution bias on the CPI and an estimation of one type of bias, the upper-level substitution basis.

11.2 Wherever the ABS refers to the use of a 'fixed basket', it is important to note that it is not the expenditure weights (price x quantity) that are fixed, but the quantities of the goods and services included in the basket. The price component of the expenditure weights are price updated each quarter (see Consumer Price Index calculation in practice of this manual for information on price updating). It is also important to note that the items within the CPI samples (what is actually priced) are regularly updated to reflect changes to consumer preferences. The term 'fixed basket' is used as an analogy to help users understand the concept of measuring pure price change.

## Limitations of fixed basket price indexes

11.3 The production of a price index by reference to a fixed basket of goods and services has several advantages. Firstly, the concept is easy to understand; price the same basket of goods and services at two different periods, and compare the total price of the basket. Secondly, by fixing both the goods and services within the basket and their quantities, the resulting values provide a measure of pure price change that is free from compositional change. In application, this process is more complex than the basket analogy would suggest. In practice, the transactions occurring in the market place are frequently changing. This observation reveals a dilemma, namely how can a price index use a fixed basket to measure pure price change and at the same time remain both contemporary and representative of the market?

## ABS strategy for reviewing and maintaining price indexes

11.4 The ABS has a policy of continual assessment of the samples of consumer goods and services that it uses in the CPI. Essentially there are three levels of maintaining representation of an index:

(i) Sample maintenance - ongoing updating and replacement of specifications, respondents, and weights for the prices collected in the CPI, which ensures that the structure of respondent samples and specifications remains relevant.

(ii) Sample review - a complete reassessment of the sample used to represent all products in the commodity classification; covering companies, products, pricing procedures and relative weights based on consumer expenditure. The end product of the sample review may be a new or revised sample (respondents, specifications and collection methods), the confirmation of the existing sample or a change to the index structure below the Expenditure Class (EC) level.

(iii) Index reviews - regular reviews of the overall index structure and the price collection methodology and updates to the weighting pattern. The ABS updates expenditure weights at the EC level annually in December quarters. The Household Expenditure Survey (HES) is used to re-weight the CPI in the years that it is available. In inter-HES years, the primary data source for updating the weights is Household Final Consumption Expenditure (HFCE) data from the National Accounts. Prior to the introduction of the 17th series CPI in the December quarter 2017, the EC level weights were updated six-yearly, in line with the release of the HES.

## Item substitution, index formulas and the frequency of CPI weight updates

11.5 Item substitution occurs when households react to changes in relative prices by choosing to reduce purchases of goods and services showing higher relative price change and instead buy more of those showing lower relative price change.

11.6 Under these circumstances, a fixed-basket Laspeyres index will overstate the price change of the whole basket as it does not take account of the substitutions that consumers make in response to relative price changes. For example, if the price of beef were to increase more than the price of chicken, one would expect consumers to purchase more chicken and less beef. As a fixed-base index would continue to price the original quantities of beef and chicken, the price change faced by consumers would be overstated.

11.7 Item substitution bias is due to changes in the pattern of household consumption which takes place over time as a result of both demand and supply changes. The longer the time period between weight updates, the more time there is for consumers to substitute towards or away from goods and services in reaction to relative price changes and as a result of changes in income. Similarly, supply conditions (and therefore the availability, or otherwise, of certain goods and services) can change substantially over the period in which the weights are fixed.

11.8 Like most CPIs, the Australian CPI is calculated using a fixed-base modified Laspeyres index formula (known as Lowe index¹) which keeps quantities fixed between weight updates but allows prices to vary. A Laspeyres (or in most cases a Laspeyres-type) index measures the change in the cost of purchasing the same basket of goods and services in the current period as was purchased in a specified base period. The weights reflect expenditures from a historical period, the base period. See Price index theory of this manual for more detail.

11.9 The ABS has previously updated expenditure weights at the EC level every six years using data from the HES. Beginning with the introduction of the 17th series CPI in the December quarter 2017, the Australian CPI is re-weighted annually. HFCE data from the National Accounts is used to update the weights in inter-HES years. The use of expenditure aggregates from the National Accounts is an internationally recommended approach in situations where the time interval between household surveys is large.

11.10 The ability to re-weight the Australian CPI more frequently has significant benefits to the user community. These include more accurately reflecting consumer spending patterns, addressing stakeholder concerns following the 16th Series CPI review, coherence across macro-economic statistics, and improved alignment with international standards. For more information on these changes, see the following information papers:

- Information paper: Increasing the Frequency of CPI Expenditure Class Weight Updates (cat. no. 6401.0.60.002).
- Information paper: An Implementation Plan to Annually Re-weight the Australian CPI (cat. no. 6401.0.60.005)

11.11 There is a family of indexes called superlative indexes. Superlative indexes make use of both beginning-of-period and end-of-period information on both prices and quantities (expenditures), thereby accounting for substitution across items. However, in order to construct a superlative index price and quantity (expenditure) data are required for both periods under consideration.

11.12 Superlative indexes for the entire CPI basket of goods and services can only be produced retrospectively once the required weighting data are available². Given that current period expenditure data for households is not available on a sufficiently timely basis (generally not available until 12 months after the reference period), a superlative formula cannot be used in the routine production of the CPI, which is why statistical agencies rely on fixed baskets. Most, if not all, statistical agencies use a Laspeyres-type index. The requirement for current period quantity data in real time is the reason a superlative index is an impractical option for statistical offices for the compilation of the CPI.

## Estimation of the upper level substitution bias

11.13 The ABS has constructed a retrospective superlative-type index to provide an estimation of potential item (upper level) substitution bias in the fixed-weight Australian CPI. While there are five main sources of bias in CPIs (described further in Price index theory), this analysis focuses on one type only - upper level item substitution bias - and therefore the results in the analysis should not be taken to equate to the total bias in the CPI, which will be the cumulative impact of all sources of bias. This analysis can only be conducted retrospectively, when new expenditure data are available.

11.14 Superlative indexes allow for substitution as they make use of weights for both the earlier and later periods under consideration (basically averaging across historical and current expenditures to derive a ‘representative’ set of weights for the period) whereas the Laspeyres index uses only base period weights.

11.15 The estimate of upper level substitution bias has been made at relatively high levels of aggregation. The analysis is calculated based on the amount of consumer substitution between expenditure classes as this is the lowest level for which reliable weighting information (from the HES and other alternative data sources) is available and the level at which the underlying quantity weights remain fixed between CPI reviews. Thus, the analysis captures substitution from one expenditure class to another, e.g. from beef and veal to poultry, but not within a given expenditure class, e.g. from beef to veal. The substitution within an expenditure class is called lower level substitution bias which is minimised through regular sample maintenance, sample reviews and choice of index formulas. In the December quarter 2017, the ABS implemented new methods, known as multilateral methods, to compile 28 ECs in the CPI. These methods utilise a census of products available in big datasets, and use expenditure data to weight products, mitigating the risk of lower level substitution bias.

11.16 The ABS enhanced the method to estimate upper level substitution bias as part of the 17th series review. This approach calculates financial year Laspeyres and Paasche-type indexes, using the HES weights for each series of the CPI and financial year estimates of price change.

11.17 Three superlative indexes have been constructed and linked together to form one continuous series. The first index was constructed on the 14th series CPI basis between 1998-99 and 2003-04, the second index was constructed on the 15th series CPI basis between 2003-04 and 2009-10, and the third constructed on the 16th series basis between 2009-10 and 2015-16.

11.18 Using the expenditure class weights at the weighted average of eight capital cities level, i) Laspeyres, ii) Paasche-type, and iii) superlative Fisher-type indexes have been calculated at the All groups CPI level³. The indexes have all been calculated with the base period 1998-99 = 100.0. The Fisher index is regarded as the best practical approximation of a 'true' (or 'ideal') price index, being the geometric average of the Laspeyres and Paasche indexes.

11.19 Under this approach, the Laspeyres index is a true Laspeyres, rather than a Lowe index as in the case of the published All groups CPI.

11.20 The Paasche-type index is retrospectively modelled using the HES weights, and a linear model to derive weights for financial years in between the re-weighting periods. The geometric mean of the Laspeyres and Paasche-type index approximates the Fisher.

11.21 The Laspeyres, Paasche-type and superlative Fisher-type indexes were constructed using the same structure as the All groups CPI published at the time to allow for direct comparison. The indexes from 1998-1999 to 2003-04 were derived using the 14th series classification consisting of 88 expenditure classes. The index numbers from 2003-04 to 2009-10 were derived using the 15th series classification consisting of 90 expenditure classes, and the index numbers from 2009-10 to 2015-16 were derived using the 16th series classification consisting of 87 expenditure classes.

11.22 Using these indexes, an estimate of upper level substitution bias in the CPI was obtained by subtracting the superlative (Fisher-type) index from the Laspeyres.

## Analysis of the upper level substitution bias

11.23 The analysis found the total upper level substitution bias of the All groups CPI (as measured by the difference between the Laspeyres index and the Fisher-type index) was 5.6 percentage points after 17 years due to the inability of the fixed-base index to take account of the item substitution effect. The Laspeyres index increased by a total of 56.5% from 1998-99 to 2015-16. The retrospective superlative index, calculated using the Fisher-type index, increased by 50.9% over the same period.

11.24 To estimate the average annual upper level substitution bias, the indexes can be expressed as Compound Annual Growth Rates (CAGR).

\(\begin{array}{l}\text{Laspeyres}_{\text{CAGR}}\\{=\left(\left(\text{I}_{\mathrm{I,2015-16}} / \text{I}_{\mathrm{I,1998-99}}\right)^{(1 / 17)}-1\right) \times 100}\\{=\left((156.5 / 100.0)^{(1 / 17)}-1\right) \times 100}\\{=2.67\%} \\ \\\text{Fisher}_{\text{CAGR}}\\{=\left(\left(\text{I}_{\mathrm{F,2015-16}} / \text{I}_{\mathrm{F,1998-99}}\right)^{(1 / 17)}-1\right) \times 100}\\{=\left((150.9 / 100.0)^{(1 / 17)}-1\right) \times 100}\\{=2.45\%}\end{array}\)

11.25 The average annual upper level substitution bias is calculated as \(\text{Laspeyres}_{\text{CAGR}}\) - \(\text{Fisher}_{\text{CAGR}}\) = 2.67% - 2.45% = 0.22%. The CPI for the 1998-99 to 2015-16 period was potentially upwardly biased by 0.22 of a percentage point per year on average due to the inability to take account of the upper level item substitution effect. These results are consistent with previous analysis and studies by other national statistical agencies.

11.26 The results show that the longer the period between re-weights, the larger the potential upper level item substitution bias effect on the index. Table 11.1 illustrates that the average annual substitution bias increases at a faster rate the longer the period between re-weights. The re-weighting periods in this analysis were 1998-99, 2003-04 and 2009-10.

Time since re-weight | LaspeyresCAGR - FisherCAGR |
---|---|

1 year | 0.11 |

2 years | 0.17 |

3 years | 0.17 |

4 years | 0.18 |

5 years | 0.21 |

(b)6 years | 0.20 |

- This takes the average of the annual substitution bias for the 1998-99 to 2003-04, 2003-04 to 2009-10 and 2009-10 to 2015-16 periods.
- The six year average annual substitution bias is only based on the 2003-04 to 2009-10 and 2009-10 to 2015-16 periods.

11.27 This finding is consistent with the Statistics New Zealand (SNZ) analysis which showed that item substitution bias is considerably greater when NZ CPI weights are updated at six-yearly rather than three-yearly intervals.⁴

11.28 With the adoption of annual re-weighting from the December quarter 2017, consumer substitution effects are more effectively captured. Empirical analysis published in Information Paper: Increasing the Frequency of CPI Expenditure Class Weight Updates, (cat. no. 6401.0.60.002) showed that the average annual substitution bias for the CPI over the September 2005 to September 2011 period would have been 0.09% under annual re-weighting, compared to 0.24% per annum under six-yearly weights updates.

11.29 While there are five main sources of bias in CPIs, this analysis focuses on one type only - upper level item substitution bias - and therefore the results in the analysis should not be taken to equate to the total bias in the CPI, which will be the cumulative impact of all sources of bias.

11.30 The use of transactions data and implementation of multilateral index methods in the Australian CPI from December quarter 2017 mitigates the risk of lower level substitution bias and bias from the failure to introduce new goods when they first appear in the market. These methods are discussed in more detail in Use of transactions data in the Australian CPI of this manual.

## Choosing an index number formula

11.31 As different index number formulas produce different results, the ABS has to decide which formula to use. The usual way is to evaluate the performance of a formula against a set of desirable mathematical properties or tests. This is called the axiomatic approach. This approach is certainly useful however a few practical issues need to be considered, such as: the relevance of the tests for the application at hand; the importance of a particular test (some tests are more important than others); and even if an index formula fails a test, how close in practice will the index likely be to the best measure?

11.32 The range of tests developed for index numbers has expanded over the years. Diewert (1992) describes twenty tests for weighted index formulas, and Diewert (1995) provides seventeen tests for equally weighted (or elementary) index formulas, and attributes the tests to their authors. It is beyond the scope of this discussion to describe all the tests, but several important ones are outlined below. Many of the tests apply to both types of formulas.

- Time reversal. This test requires the index formula to produce consistent results whether it is calculated from period 0 to period 1 or from period 1 to period 0. More specifically, if the price observations for period 0 and period 1 are changed around then the resulting price index should be the reciprocal of the original index.
- Circularity (often called transitivity). This is a multi-period test (essentially a test of chaining). It requires that the multiplication of the price index obtained by going from period 0 to period 1 and from period 1 to 2 is the same as going directly from period 0 to period 2.
- Commensurability. This test requires that if the units of measurement of the item are changed (e.g. from kilograms to tonnes), then the price index should not change.

11.33 The Fisher Ideal index formula passes the tests on time reversal and commensurability; whereas the Laspeyres and Paasche only pass the test of commensurability. The Lowe index used in the Australian CPI passes the circularity and commensurability tests.

11.34 Regarding the three equally weighted price index formulas discussed in Price index theory, the arithmetic mean of price relatives (APR) fails the time reversal and circularity tests, the relative of average prices (RAP) fails the commensurability test, but the geometric mean (GM) approach passes all tests.

11.35 Although the equally weighted GM appears to have considerable appeal as an elementary index formula, there are some situations in which it produces an undesirable result. The GM cannot handle zero prices which might occur, for example, if the government introduced a policy to subsidise fully a particular good or service. In addition, the GM may not produce acceptable movements when a price falls sharply. For example, consider a price sample of two items, each selling for $10 in one period, with the price of one of the items falling to $1 in the second period. The GM produces an index of 31.6 for the second period (assuming it was 100 in the first period), a fall of around 68%. Because the GM (implicitly) maintains equal expenditure shares in each period, it effectively gives a larger weight to lower prices.⁵

11.36 The GM formula has become more widely accepted in official circles for compiling consumer price indexes. For example, Canada switched to using GMs in the late 1980s; the United States introduced the GM formula for items making up about 61% of its CPI in January 1999; and Australia began introducing the formula in the December quarter 1998. (However, where there is a likelihood of a zero price occurring in the sample or there are barriers to item substitution, it is inappropriate to use the GM, then the ABS generally uses the RAP formula instead.) Furthermore, the GM formula is prescribed by the European Union for calculation of price sample means in its Harmonised Indices of Consumer Prices (HICP).

11.37 There is another aspect to indexes that is worth considering, although it is not rated as a test in the literature. In most countries the CPI is produced at various levels of aggregation. Typically there are three or more levels between the lowest published level, and the total of all goods and services. In practice, it is desirable that the same result is obtained whether the total index is compiled directly from the lowest level or in a staged way using progressively higher levels of aggregation. This property is known as 'additivity'. Diewert (1978) shows that the fixed weighted Laspeyres and Paasche indexes may be aggregated consistently, and the Fisher and Törnqvist indexes are very close approximations of one another⁶.