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MAINTAINING THE RELEVANCE OF THE CPI
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 fixedbasket 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 fixedbase 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 fixedbase modified Laspeyres index formula (known as Lowe index(footnote 1) ) which keeps quantities fixed between weight updates but allows prices to vary. A Laspeyres (or in most cases a Laspeyrestype) 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 reweighted annually. HFCE data from the National Accounts is used to update the weights in interHES 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 reweight 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 macroeconomic statistics, and improved alignment with international standards. For more information on these changes, see the following information papers:
11.11 There is a family of indexes called superlative indexes. Superlative indexes make use of both beginningofperiod and endofperiod 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 available(footnote 2) . 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 Laspeyrestype 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 superlativetype index to provide an estimation of potential item (upper level) substitution bias in the fixedweight 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 Paaschetype 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 199899 and 200304, the second index was constructed on the 15th series CPI basis between 200304 and 200910, and the third constructed on the 16th series basis between 200910 and 201516. 11.18 Using the expenditure class weights at the weighted average of eight capital cities level, i) Laspeyres, ii) Paaschetype, and iii) superlative Fishertype indexes have been calculated at the All groups CPI level(footnote 3) . The indexes have all been calculated with the base period 199899 = 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 Paaschetype index is retrospectively modelled using the HES weights, and a linear model to derive weights for financial years in between the reweighting periods. The geometric mean of the Laspeyres and Paaschetype index approximates the Fisher. 11.21 The Laspeyres, Paaschetype and superlative Fishertype indexes were constructed using the same structure as the All groups CPI published at the time to allow for direct comparison. The indexes from 19981999 to 200304 were derived using the 14th series classification consisting of 88 expenditure classes. The index numbers from 200304 to 200910 were derived using the 15th series classification consisting of 90 expenditure classes, and the index numbers from 200910 to 201516 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 (Fishertype) 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 Fishertype index) was 5.6 percentage points after 17 years due to the inability of the fixedbase index to take account of the item substitution effect. The Laspeyres index increased by a total of 56.5% from 199899 to 201516. The retrospective superlative index, calculated using the Fishertype 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). Laspeyres_{CAGR}
= ((156.5/100.0) ^{(1/17)}  1) * 100 = 2.67% Fisher_{CAGR}
= ((150.9/100.0) ^{(1/17)}  1) * 100 = 2.45% 11.25 The average annual upper level substitution bias is calculated as Laspeyres_{CAGR}  Fisher_{CAGR} = 2.67%  2.45% = 0.22%. The CPI for the 199899 to 201516 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 reweights, 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 reweights. The reweighting periods in this analysis were 199899, 200304 and 200910.
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 sixyearly rather than threeyearly intervals.(footnote 4) 11.28 With the adoption of annual reweighting 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, July 2016 (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 reweighting, compared to 0.24% per annum under sixyearly 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.
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.(footnote 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 another(footnote 6) . 1 Consumer Price Indices; An ILO Manual, by Ralph Turvey et al (ILO, Geneva 1989). <back 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 nonalcoholic beverages group (see Use of transactions data in the Australian CPI for more details). <back 3 For a description of the indexes, refer to Price index theory of this manual. <back 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/cpiretrospectivesuperlativeindex2014.aspx <back 5 The RAP and APR formulas both give an index of 55.0 in this case. <back 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. <back Document Selection These documents will be presented in a new window.

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