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CHAPTER 10 PPI AND ITPI CALCULATION IN PRACTICE
10.4 Each elementary aggregate, whether relating to the whole country, a region, or a group of providers, will typically contain a large number of individual products. In practise, only a small number of products can be selected for pricing. When selecting the products from period to period, the following considerations are taken into account:
THE AGGREGATION STRUCTURE 10.5 The aggregation structures for PPIs and ITPIs are discussed in Chapter 6, with an example provided in figure 6.1. Using different classifications of products and industries, the PPIs and ITPIs can be divided into broad divisions, subdivisions and groups, and then further refined into smaller classes. At the bottom of the standard classification structure, further disaggregation is made to reflect different products and different price behaviours. 10.6 Each component in the price index, from the root level (or top level or all products) of the aggregation structure down to each individual elementary aggregate, is associated with two distinct characteristics that allow future compilation of aggregate price index measures. These defining characteristics are:
10.7 Linking price indexes is discussed in more detail in Chapter 12. 10.8 In addition to these characteristics, the elementary aggregates have one additional feature. Elementary aggregates are the only components within the index structure to have price samples. From these price samples, it is possible to directly construct price indexes. A price index for an elementary aggregate should measure price change and correctly account for changes in quality and both new and disappearing products. This index is called an elementary aggregate Cindex. 10.9 Beginning with these two defining characteristics and the aggregation structure, price indexes are created by working upwards from the elementary aggregate Cindexes. All indexes above the elementary aggregate level are higher level indexes that can be calculated from the elementary price indexes using the elementary value aggregates as weights. The aggregation structure is consistent so that the weight (link period value aggregate) at each level above the elementary aggregate is always equal to the sum of its components. The price index at each higher level of aggregation can be calculated on the bases of the weights and price indexes for its components, i.e. the lower level or elementary indexes. The individual elementary price indexes are not necessarily sufficiently reliable to be published individually, but they remain the basic building blocks of all higher level indexes. THE COMPILATION OF ELEMENTARY PRICE INDEXES 10.10 Within the Australian PPIs and ITPIs, the elementary aggregate Cindexes are calculated using either the Laspeyres price index formula, the Lowe price index formula, or the Jevons (geometric mean) price index formula. The Lowe price index is common across most index calculations due the price reference period and the weight reference period being at different times. The Laspeyres price index is illustrated by means of a numerical example in Table 10.1. In the example, we have assumed that the following conditions apply:
10.11 This example has quite strong assumptions, because many of the problems encountered in practise are attributable to breaks in the continuity of the price series for the individual transactions for one reason or another(footnote 1) . 10.12 The calculation of the C index for each elementary aggregate begins through calculation of a weight for each price observation. For elementary aggregates that use the Jevons index formula, the weights are equal. The majority of elementary aggregates in the PPIs and ITPIs use the Laspeyres index formula, which is applied through the price relative form. In this form, as discussed in Chapter 4, price relatives are combined using weights that represent the value share in the reference period. These weights represent not only the value of the particular transactions included for pricing in the elementary aggregates each period but also the other transactions which these observations represent. The reference period value share is determined once for each observation and is only modified if the products in the elementary aggregate are changed; in which case the elementary aggregate undergoes sample maintenance, which is described in more detail in Chapter 11.
10.13 This example shows a price index of 111.0 in period 1, and 119.1 in period 2. The prices in the elementary aggregate have moved 11.0% in the first period, 7.3% in the second period and 19.1% since the reference period. IMPUTATION AND TEMPORARILY MISSING PRICE OBSERVATIONS WITHIN AN ELEMENTARY AGGREGATE 10.14 In any period an event may occur that makes it impossible to obtain a price measure for a particular product. For example, a product could be temporarily out of stock. 10.15 The Australian Bureau of Statistics (ABS) employs a number of imputation methods to address temporarily missing observations within price indexes. These include:
10.16 These options are known as imputation. Their purpose is to calculate a price for the temporarily missing product. The aim of imputation is to provide prices such that the resulting movement in the price index is the same as would have been calculated had all prices been observed. In achieving such a result, it is necessary to make an assumption regarding the price behaviour of the temporarily missing product. Imputation from other products in the price sample 10.17 The rationale for imputing a price movement from other products in the sample is that products are bought and sold in a competitive marketplace and in those cases where an individual product has not been observed it is assumed that its price behaviour is reflected by similar products in the sample. The design of elementary aggregates to contain products that are homogeneous in terms of price behaviour (as noted above) ensures that the assumption underlying this method of imputation is generally robust. 10.18 Imputing from other products in the sample is also mathematically equivalent to excluding the product, for which a price is unavailable in one period, from both periods involved in the index calculation. It strictly maintains the ‘matched sample’ concept. 10.19 In order to impute a movement resulting from excluding the product it is necessary to construct a measure of price change from the previous period to the current period for those products common to both periods. This calculation is dependent upon the price index formula used for the elementary aggregate. When the elementary aggregate is compiled using a Laspeyres formula, it is first necessary to derive the implicit quantity shares underlying the weights of the matched products. This can be achieved by dividing the weight for each product by its reference period price. 10.20 The resulting quantity shares for the matched products are then used to calculate the price change from the previous period to the current period. where S_{q, i} is the implicit quantity share in the reference period for matched product i, w_{i} is the weight for matched product i, p^{0}_{i}, p_{i}^{t1}, p^{t}_{i} are respectively the reference period price, previous period price, and current period price for matched product i (at time t), M^{t}_{t1} is the price movement between the previous and current period for the matched products, and is the imputed price for missing product j at time t. 10.21 An example of this calculation is shown in Table 10.2.
Imputation from another price sample 10.22 The second approach to imputation for the PPIs and ITPIs is to use the price movement from another related sample or comparable product. This approach is used in cases where price changes from a comparable product (or products) from a similar type of provider can be expected to be similar to the missing product. Carry forward imputation 10.23 The rationale for adopting a carry forward imputation is that failure to observe a price for a product reflects no transactions for the product, and hence there can be no price change. However, each product in the price sample represents similar products purchased and sold elsewhere in the marketplace, and such an assumption does not hold in most cases. Application of this method of imputation when transactions are actually occurring in a marketplace (but not observed by the sample) consistently biases the index towards zero (that is, biased downward when prices are rising and biased upward when prices are falling). It is for these reasons that the ABS applies this imputation mechanism only under specific conditions where it is known that failure to observe a transaction means that no transactions are occurring (such as where there is only one sale per year of a type of agricultural crop, for example, or where the price changes only once per year during annual price setting). PRICE INDEXES FOR HIGHER LEVEL COMPONENTS 10.24 Once a price movement for the elementary aggregate is determined, the resulting Cindex price movement is used to price update the value aggregate associated with the elementary aggregate. The resulting measure is known as the price updated value aggregate (or current period value aggregate). For a given elementary aggregate (EA): where VA^{t}_{EA} is the current period value aggregate for the elementary aggregate in period t, VA_{EA}^{t1} is the previous period value aggregate, and I^{t}_{C} and I_{C}^{t1}are respectively the current and previous period Cindexes for the elementary aggregate. 10.25 The price updated value aggregate is then used to determine the current period Pindex for the elementary aggregate. where I^{t}_{p} is the current period Pindex for the elementary aggregate in period t, VA_{EA}^{LINK}is the link period value aggregate for the elementary aggregate and I^{LINK}_{P,EA} is the link period Pindex for the elementary aggregate. 10.26 Once the current period value aggregates for all elementary aggregates are determined, the current period value aggregates for all higher level components of the index structure are calculated by summing the price updated value aggregates of their components. 10.27 Current period price indexes for any component in the aggregation structure are then calculated by price updating the link period Pindex for the component. That is, for any component, the current period Pindex is given by: where I^{t}_{p} is the current period Pindex in period t, VA^{t} is the current period value aggregate for the component, VA^{LINK} is the link period value aggregate for the component and I_{P}^{LINK} is the link period Pindex for the component of the index (or aggregation) structure. Points contribution and points change 10.28 Points contributions are also calculated using the value aggregates. In any period, the points contribution of a component to the top level is calculated by multiplying the root index number for the period by the value aggregate for the component in that period and dividing by the root value aggregate for that period. This can be stated algebraically as: where PC^{t}_{i} is the points contribution for component i in period t, I^{t}_{P,ROOT} is the Pindex for the root in period t,VA^{t}_{i} is the value aggregate for component i in period t and VA^{t}_{ROOT} is the value aggregate for the root of the index in period t. 10.29 Changes in points contribution for a component of a price index give an assessment of the component’s contribution to net price change. However, such a comparison is limited to periods between linking of price indexes. Comparisons of a component’s contribution to the index that cross a link period are comparing contributions on different weighting bases and therefore do not measure the contribution to net price change; any attempt at such comparison will confound change of weight with change of price. 10.30 Calculation of upper level price indexes is illustrated in Table 10.3. This table shows an input price index where products are classified by source (domestic and imported) and then by type of product. In Part 1 the Pindex for period 1 is calculated. Part 2 shows the calculation of the percentage movement in the elementary aggregate Cindex from period 1 to period 2. Part 3 shows how the current period value aggregates for period 2 are then derived and used to calculate the Pindex for period 2.
SECONDARY INDEXES 10.31 A key philosophy of price index design for the PPIs and ITPIs is to reuse components to maximise the utility of collected data (this philosophy is discussed in more detail in Chapter 8). One mechanism that helps achieve this aim is through the construction of secondary indexes. The preceding sections have described how elementary aggregate price indexes can be combined to produce higher level indexes. The particular combination of elementary aggregates is determined by the underlying classification of the price index (as noted in Chapter 5). 10.32 However, a given elementary aggregate may be classified in multiple ways. Reclassifying elementary aggregates according to a different aggregation structure results in a secondary index. The relationship between the original primary source index and the secondary index is marked by two important features. First, the elementary aggregates for the secondary index are the same as those in the source index, having the same Pindexes and value aggregate data. Second, the source index and the secondary index are identical at the root or top level of the index. The indexes only differ at the intermediate levels (between the root and the elementary aggregates), since a secondary index is defined through the different aggregation structure. 10.33 Frequent use of secondary indexes occurs within the ITPIs, with classification by both Standard International Trade Classification (SITC) and Broad Economic Categories (BEC) (further details of these classifications are provided in Chapter 5). 10.34 An example of a secondary index is provided in Table 10.4. This example uses a reclassification of the elementary aggregates presented in Table 10.3, with emphasis on type of product rather than the domestic or imported split.
10.35 The key feature of secondary indexes is that they rearrange the existing basic building blocks of the price index along a different compilation structure, and in doing so retain both the price movements and underlying value aggregates of the elementary aggregates. TERTIARY INDEXES 10.36 It is also possible to construct tertiary indexes, where price movements are retained but an entirely new weighting pattern is applied. In this case the resulting tertiary index has consistent price movements at the elementary aggregate level, but results in a different price movement at the top or root of the index. This device is a powerful analytical tool that allows further reuse of price samples. Footnote 1 Departures from these assumptions are discussed separately: introduction of new providers is discussed in Chapter 4 (Price index theory), the treatment of new products is described in Chapter 9 (Quality change and new products), imputation for missing prices is discussed in this chapter and changes to the types of products in the marketplace are discussed in Chapter 11 (Maintaining relevance). <back Document Selection These documents will be presented in a new window.

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