¹ Departures from these assumptions are discussed separately: introduction of new providers is discussed in the Price index theory section , the treatment of new products is described in the Quality change and new products section, imputation for missing prices is discussed in this section and changes to the types of products in the marketplace are discussed in the Maintaining relevance section.

# The aggregation structure

The development of the aggregation structures for Producer and International Trade Price Indexes follow the technical methodology for weighting explored above.

Using different classifications of products and industries, the Producer and International Trade Price Indexes can be divided into broad divisions, sub-divisions, and groups, and then further refined into smaller sub-groups/classes depending on the classification structure adopted for the indexes (see below for further information on the classifications used for each Producer and Trade Price Index). At the bottom of the standard classification structure, further disaggregation is made to reflect different products and different price behaviours.

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, as defined in the Index Review methodology section above. These defining characteristics are:

**The link period value aggregate:**The value aggregate defined at the period when the index weighting structures commence; this measure effectively determines the underlying quantity weights of the price index**The link period P-index:**The price index number at the period when the index weighting structures commence; it measures the price change for the component that occurred between the link period and the price index reference period; in the case where the link period and the index reference period are the same, the link period P-index takes a value of 100.0.

In addition to these characteristics, the elementary aggregates have one additional feature in that they 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.

The construct of an elementary aggregate index is also known as an elementary aggregate C-index.

Beginning with these two defining characteristics and the aggregation structure, price indexes are created by working upwards from the elementary aggregate C-indexes. 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

Within the Australian PPIs and ITPIs, the elementary aggregate C-indexes 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 4.1.

In the example, we have assumed that the following conditions apply:

- prices are collected for four representative products within an elementary aggregate
- the quality of each product remains unchanged over time so that the period-to-period changes compare like with like
- a set of weights is available for use in the Laspeyres index formula
- prices are collected for all four products in every period covered so that there is a complete set of prices
- there are no disappearing products, no missing prices and no replacement products.

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 another¹.

The calculation of the elementary aggregate C- index 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 Producer and International Trade Price Indexes use the Laspeyres index formula, which is applied through the price relative form. In this form, 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 the Maintaining relevance section below.

Reference period | Period 1 | Period 2 | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Reference period value share | Price ($) | Price Relative | Weight x relative | Price ($) | Price Relative | Weight x relative | Price ($) | Price Relative | Weight x relative | |

Product A | 30 | 5 | 1.000 | 30.000 | 6 | 1.200 | 36.000 | 7 | 1.400 | 42.000 |

Product B | 20 | 7 | 1.000 | 20.000 | 7 | 1.000 | 20.000 | 6 | 0.857 | 17.143 |

Product C | 10 | 2 | 1.000 | 10.000 | 3 | 1.500 | 15.000 | 4 | 2.000 | 20.000 |

Product D | 40 | 5 | 1.000 | 40.000 | 5 | 1.000 | 40.000 | 5 | 1.000 | 40.000 |

Laspeyres price index | 100.0 | 111.0 | 119.1 | |||||||

Percentage change from previous period | 11.0% | 7.3% |

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.