Index calculation

The index formula chosen for the elementary aggregates⁵² can vary as a function of the sampling method, information available, specific nature of industry and resource restrictions. A more extensive description of widely used elementary aggregate formulas and their justification can be found in the PPI Manual, Chapter 20. When aggregating sample price relatives⁵³ within elementary aggregates, in many cases the geometric average may be recommended as the best micro index formula particularly when information on weights is not available. An advantage of using geometric averaging is also that outliers have less influence on results.

However weights may often be required for price relatives due to the sampling methodology used (see 3.7.2)

The index number formula used in the aggregation of elementary aggregates should be selected as a function of the indicator’s purpose.

Individual countries may have different priorities and the formulas may therefore differ from country to country. It is recommended that the EU member states should use a Laspeyres-type index formula that is in line with the EU STS regulation.

3.7.2 Weights and aggregation of an index

As discussed in sections 1.3 and 1.9, there are two common methods for aggregating data collected from enterprises to form a particular SPPI, usually at the 4-digit industry level. By strict definition, in the product SPPI the aggregation is based on service products provided by any industry either as principal or secondary production, and in the

52 An elementary aggregate is generally the lowest level within an index structure (i.e. family tree) for which reliable weights exist for the population of interest (e.g. relatively homogeneous group of service products), generally obtained from some source not directly related to the collection of prices (e.g.

independent survey). Samples of prices are collected within each elementary aggregate.

53 Sampled price relatives are the micro indices of sampled prices contributing to the elementary aggregate

industry SPPI establishments/LKAUs (in practice often enterprises) are used first to compile a price index for the unit which are then aggregated to cover the whole service industry. Note that in both these cases the weights given to particular production units (i.e. establishments/LKAUs, enterprises) should reflect the inverse of their probability of selection, or at least be such that the weight of a selected production unit takes into account other similar production units it represents in the index. These weights are generally regarded as first stage weights. Second stage weights refer to the size of the production unit (e.g. total employment or preferably total turnover).

1) Product SPPI: aggregation through elementary aggregates or a ‘family tree’ of service products

If a product classification can be determined within a service industry, then this classification can be used to form an elementary aggregate or

‘family tree’ structure for an industry. The produced classification might originate from an existing classification (rare) or through a specific survey that has been undertaken (more common) or through advice from an industry association.

Each of these elementary aggregate components is weighted according to the estimated proportion of production for these products within the industry.

Using this approach, surveyed businesses will contribute one or more sample price relatives⁵⁴ to the elementary aggregate structure⁵⁵. The weight of a sample price relative will be equal to the production units’ first stage weight multiplied by

54 In this context, following from the previous footnote, a price relative refers to a service product surveyed within the enterprise for which a micro index has been estimated. This micro index could have been made up of one or several price quotations for similar service products provided by the enterprise. In the case of several service products contributing to the micro level index, equal weights would generally be used in forming the micro level index at the firm level unless the firm had very detailed information available to allow different percentage weights to be used at this micro index level.

55 Note, a business will only contribute a specification to a component of the elementary aggregate structure if it has significant production for the particular service product in question. Thus one does not expect a business to contribute specifications to each component of the indices’ elementary aggregate structure.

a proportion of its second stage weight. This proportion should equal the estimated proportion of its total production on service products within the elementary aggregate component that the sample price relative is contributing to. See Box 16 below as an example of this weighting methodology.

An advantage of this form of index aggregation is that sub-indices for service products (i.e. which form the elementary aggregate structure) are produced which may be a useful output if deemed to be of suitable quality. A potential disadvantage of this method is that it relies on the relevance of the product classification forming the family tree structure. If this phase of the index development has not been done well then the family tree structure may not represent important product groups within the industry and the index could be of low quality.

2) Industry SPPI: aggregation of firm level indices

Another possibility is to first form specific indices for each production unit (e.g. establishment) in the survey. These specific price indices are then aggregated together to form the service industry index, where the relevant weights are the multiplication of the enterprises’ first and second

stage weights as described above. The advantage of this method is that the production units can use their own familiar service classification, so that the potential classification bias can be reduced.

Within a production unit, sample price relatives may or may not be weighted depending on the diversity and relative production values of the different service products produced by the unit.

As pointed out in section 1.3, product SPPI can in general be regarded as preferable. A drawback of industry SPPI is that the production unit may produce services that are officially classified for some other industry than the one that is their main activity. However this could possibly be corrected to arrive at product SPPI by moving certain price relatives to contribute to the aggregation of the appropriate 4-digit industry. Another serious shortcoming of industry SPPI is that the method is unable to provide information about price development of individual services (e.g. as provided through the elementary aggregate structure described in method 1 above). This is especially problematic if the service industry in question includes a large number of different service groups.

56 Probability proportional to size (PPS) sampling is recommended in the PPI Manual, see Chapter 5, pages 113-114, of this manual for a worked example. In simple terms, the probability of an establishment being selected in the sample is proportional to a measure of size for the establishment obtained from the sampling frame (e.g. turnover). Consequently, the larger the measure of size of the establishment is, the higher the probability that it will be included in the sample.

57 The first stage weight for an establishment is equal to the inverse of its probability of selection.

Having produced SPPI at a 4-digit industry classification, there may be a desire to aggregate these to higher levels (e.g. ISIC division level) to form macro level indicators, which may be of particular interest to central banks or treasuries monitoring inflationary pressures. The most appropriate source of weights for combining 4-digit industry level indices to higher aggregates within the industry classification would be industry based statistics on value added from the national accounts (often input-output tables).

Box 16. Weighting example for product SPPI of 4-digit level industry xxxx

An industry survey showed that the industry xxxx produces 4 main product groups y1, y2, y3 and y4 with approximate shares 20%, 30%, 15% and 35%. This defines the elementary aggregate structure for the SPPI.

5 establishments are chosen in the sample for industry xxxx using PPS sampling⁵⁶. They have the following characteristics:

Establishment Probability

selection Turnover Shares of y1, y2, y3

and y4 on turnover, % Price quotations within product groups taken for each establishment

A 1/5 100 (0.25, 0.25, 0.5, 0) (1, 2, 3, 0)

B 1/8 70 (0.7, 0, 0, 0.3) (4, 0, 0, 2)

C 1/10 40 (0, 0.4, 0, 0.6) (0, 3, 0, 3)

D 1/50 10 (0, 0.5, 0.5, 0) (0, 3, 3, 0)

E 1/100 4 (0, 0, 0, 1) (0, 0, 0, 4)

For each establishment, price quotations (specifications) within a product group have equal weights. Price relative for each specifications are aggregated geometrically to form an enterprise level price relative for the product group. These price relatives (micro indices) must then be weighted together with other enterprises' price relatives to form the elementary aggregate (i.e. product group) indices.

Establishment 1st stage

weight⁵⁷ 2nd stage

weight Total weight

(1st x 2nd) Elementary index weight for enterprise price relative

Product group y1 A 5 100 x 0.25 = 25 125 125/(125+392) = 0.24

B 8 70 x 0.7 = 49 392 0.76

Product group y2 A 5 100 x 0.25 = 25 125 0.23

C 10 40 x 0.4 = 16 160 0.30

D 50 10 x 0.5 = 5 250 0.47

Product group y3 A 5 100 x 0.5 = 50 250 0.5

D 50 10 x 0.5 = 5 250 0.5

Product group y4 B 8 70 x 0.3 = 21 168 0.21

C 10 40 x 0.6 = 24 240 0.30

E 100 4 x 1 = 4 400 0.49