6461.0 - Consumer Price Index: Concepts, Sources and Methods, 2009  
ARCHIVED ISSUE Released at 11:30 AM (CANBERRA TIME) 17/12/2009   
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11.1 As discussed in detail in Chapter 4, there are a number of different types of bias that may affect price indexes. The ABS applies significant effort to address these biases. Some aspects, such as quality change, have been addressed elsewhere. This chapter looks at the way the ABS deals with two other aspects of bias:

  • the way the ABS maintains the sample to ensure that it remains representative of goods and services being acquired by contemporary households; and
  • the selection of the appropriate index number formula.


11.2 The development of a price index by reference to a fixed basket of goods and services has several advantages. First, 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, and perhaps more importantly, by fixing both the items within the basket and their quantities, the resulting values provide a measure of pure price change that is free from compositional changes. Of course, in application this process is more complex than the basket analogy would suggest. So, although the ABS uses a fixed basket to construct its consumer price index, in practice it finds that the transactions occurring in the market place are frequently changing. This observation results in 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 as a whole.


11.3 The ABS has a policy of continually assessing the samples of consumer goods and services that it uses in the CPI. Sample maintenance is indicated where minor problems are identified such as the need to find replacements for items or respondents. Sample maintenance is conducted regularly throughout the year. Where the sample assessment indicates that changes to the index structure or the distribution of the weights are required, a sample review is undertaken. Essentially, a sample review involves selecting a component of the CPI (it could be one or more expenditure classes, or part of an expenditure class) and subjecting it to detailed examination. The review determines what changes should be made to the items priced, the outlets they are sourced from and the weights to be applied to the commodities and outlets.


11.4 As different index number formulae 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 the so-called axiomatic approach. Although it is certainly useful, a few practical issues need to be considered as well, 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 is the index likely to be to the best measure.

11.5 The range of tests developed for index numbers has expanded over the years. Diewert (1992) describes twenty tests for weighted index formulae, and Diewert (1995) provides seventeen tests for equally weighted (or elementary) index formulae, and attributes the tests to their authors. It is beyond the scope of this chapter to describe all the tests, but several important ones are shown below. Many of the tests apply to both types of formulae.
  • 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 multiperiod test (essentially a test of chaining). It requires that the product 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.
  • Permutation or price bouncing This test requires that, if the order of the prices in either period 0 or period 1 (or both) is changed, but not the individual prices, the index number should not change. This test is appropriate in situations where there is considerable volatility in prices; for example, because of seasonal factors or sales competition.
  • 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.
  • Factor reversal test This test is not appropriate for the elementary index formulae. It requires that the product of the price index number for any period, and an index of quantity obtained from the formula by interchanging the price and quantity terms, should equal the ratio of expenditure in that period to the base period expenditure.(footnote 1)

11.6 The Fisher Ideal index formula passes the tests on time reversal, circularity, commensurability, and factor reversal; whereas the Laspeyres and Paasche only pass the test of commensurability.

11.7 Regarding the three equally weighted price index formulas discussed in Chapter 4, the APR fails the first three tests, the RAP fails the commensurability test, but the GM approach passes all tests. Of Diewert's seventeen tests for elementary index formulas, the RAP passes fifteen tests and the GM sixteen tests.

11.8 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 per cent. Because the GM maintains equal expenditure shares in each period, it effectively gives a larger weight to lower prices.(footnote 2)

11.9 Only in recent years has the GM formula become more widely accepted in official circles for compiling consumer price indexes. For example, Canada switched to using geometric means in the late 1980s; the United States introduced the GM formula for items making up about 61 per cent of its CPI in January 1999; and Australia began introducing the formula in the December quarter 1998. (However, where there is a likelihood of zero occurring in the price sample the geometric mean is inappropriate, and the ABS generally uses the relative of average prices (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). There appear to be two reasons for the slow take-up of the GM approach. One is that before the use of computers in compiling official indexes, calculation of geometric means was a very laborious task. A second reason is the perceived difficulty in explaining the measure to users of the statistics.

11.10 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. Diewert (1978) shows that the fixed weighted Laspeyres and Paasche indexes may be aggregated consistently, and the Fisher and Törnqvist indexes are (very) closely consistent.(footnote 3)

1 Diewert (1992, p. 222) notes that some researchers have objected to this test, and he does not include it in his list of twenty tests. <back
2 The RAP and APR formulae both give an index of 55.0 in this case. <back
3 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