6429.0 - Producer and International Trade Price Indexes: Concepts, Sources and Methods, 2014  
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CHAPTER 10 PPI AND ITPI CALCULATION IN PRACTICE


INTRODUCTION

10.1 The purpose of this chapter is to describe the way in which Australian Producer Price Indexes (PPIs) and International Trade Price Indexes (ITPIs) are calculated in practice.

10.2 The calculation of price indexes proceeds in two stages. First, price indexes are estimated for the elementary aggregates from a sample of prices. These elementary price indexes are then averaged to obtain higher level indexes using the relative values of the value aggregates for the elementary aggregates as weights. This chapter starts by explaining how the elementary aggregates are constructed and what economic and statistical criteria need to be taken into consideration in defining the aggregates. The different ways of imputing for missing prices are also explained. This chapter then explains how price indexes for higher level components are constructed and the way components are re-used to maximise the utility of collected data.


THE COMPOSITION OF THE ELEMENTARY AGGREGATES

10.3 Elementary aggregates are constructed by grouping homogeneous individual products and transactions. Groups may be formed from products in various regions of the country or from the country as a whole. Likewise, elementary aggregates may be formed from different types of providers or from various sub-groups of products. The key points in constructing an elementary aggregate are:

  • Elementary aggregates consist of groups of products that are as similar as possible in terms of price determining characteristics (i.e. the group of products is homogeneous).
  • They consist of products that may be expected to have similar price movements. The objective should be to try to minimise the dispersion of price movements within the aggregate.

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 transactions selected are ones whose price movements are believed to be representative of all the products represented by the elementary aggregate.
  • The number of transactions within each elementary aggregate for which prices are collected should be large enough for the estimated price index to be statistically reliable. The minimum number required will vary between elementary aggregates depending on the nature of the products and their price behaviour.
  • The objective is to try to track the price of the same product over time for as long as possible, or as long as the product continues to be representative. Therefore the products selected are ones that are expected to remain on the market for some time so that like can be compared with like.


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, sub-divisions 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:
  • 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.

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 C-index.

10.9 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

10.10 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 10.1. 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.

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.

TABLE 10.1 EXAMPLE OF ELEMENTARY AGGREGATE PRICE INDEX USING THE LASPEYRES PRICE RELATIVE APPROACH

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%



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:
  • impute a movement for the product based on the price movement for all other products in the sample
  • use the price movement from another price sample
  • repeat the previous period’s price of the product (also called carry forward or show no change).

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.

Formula

where Sq, i is the implicit quantity share in the reference period for matched product i, wi is the weight for matched product i, p0i, pit-1, pti are respectively the reference period price, previous period price, and current period price for matched product i (at time t), Mtt-1 is the price movement between the previous and current period for the matched products, and Formula is the imputed price for missing product j at time t.

10.21 An example of this calculation is shown in Table 10.2.

TABLE 10.2 EXAMPLE OF IMPUTATION FROM OTHER PRODUCTS IN THE PRICE SAMPLE

Reference Period Value Share
Reference Period Price ($)
Previous Period Price
Current Period Price

Product A
30
5
8
12
Product B
60
10
16
20
Product C
10
2
4
n.a.
Implicit quantities
Implicit quantity share
Share x Previous Period Price
Share x Current Period Price
Product A
6
0.5
4.0
6.0
Product B
6
0.5
8.0
10.0
Total
12.0
16.0
Movement
1
Current period Price after impute
Price relative after impute
Weight x relative
Product A
12
2.4
72
Product B
20
2
120
Product C
5.333
2.666667
26.66667
Laspeyres price index
218.6667




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 C-index 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):

Formula

where VAtEA is the current period value aggregate for the elementary aggregate in period t, VAEAt-1 is the previous period value aggregate, and ItC and ICt-1are respectively the current and previous period C-indexes for the elementary aggregate.

10.25 The price updated value aggregate is then used to determine the current period P-index for the elementary aggregate.

Formula

where Itp is the current period P-index for the elementary aggregate in period t, VAEALINKis the link period value aggregate for the elementary aggregate and ILINKP,EA is the link period P-index 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 P-index for the component. That is, for any component, the current period P-index is given by:

Formula

where Itp is the current period P-index in period t, VAt is the current period value aggregate for the component, VALINK is the link period value aggregate for the component and IPLINK is the link period P-index 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:

Formula

where PCti is the points contribution for component i in period t, ItP,ROOT is the P-index for the root in period t,VAti is the value aggregate for component i in period t and VAtROOT 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 P-index for period 1 is calculated. Part 2 shows the calculation of the percentage movement in the elementary aggregate C-index 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 P-index for period 2.

TABLE 10.3 EXAMPLE OF AGGREGATION OF A PRICE INDEX

Value aggregates

P-Index

Elementary Aggregate C-Index

% movement
Price Updated Value aggregate
(Period 2)
P-Index
(period 2)
Link Period
Period 1
Link Period
Period 1
Period 1
Period 2

Total inputs
105,479
133,610
105.6
133.7
152625
152.7
Imports
41,198
44,909
110.0
119.9
47989
128.1
Textile, clothing, footwear
5,682
5,750
109.3
110.6
110.6
109.7
-0.8%
5704
109.7
Wood and paper products
4,654
4,753
100.3
102.4
102.4
106.3
3.8%
4934
106.3
Chemicals, plastic, rubber
11,127
10,742
97.1
93.7
93.7
96.2
2.7%
11029
96.2
Fabricated products
16,099
17,885
107.8
119.7
119.7
120.7
0.8%
18035
120.7
Agricultural products
562
548
119.9
116.9
116.9
121.8
4.2%
571
121.8
Mining products
3,074
5,230
103.2
175.6
175.6
259.1
47.%6
7717
259.1
Domestic
64,281
88,701
104.7
144.5
104635
170.5
Agricultural products
28,036
38,530
107.9
148.3
148.3
148.1
-0.1%
38478
148.1
Electricity and gas
11,169
12,289
110.0
121.0
121.0
125.6
3.8%
12756
125.6
Forestry and logging
1,472
1,738
113.0
133.4
133.4
142.4
6.7%
1856
142.4
Mining products
23,604
36,144
102.6
157.0
157.0
223.9
42.6%
51546
224.0




SECONDARY INDEXES

10.31 A key philosophy of price index design for the PPIs and ITPIs is to re-use 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 P-indexes 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.

TABLE 10.4 EXAMPLE OF A SECONDARY PRICE INDEX

Value aggregates

P-Index

Link Period
Period 2
Link Period
Period 2

Materials used
105,479
152,625
105.6
152.7
Agricultural products
28,597
39,048
108.1
147.6
Domestic
28,036
38,478
107.9
148.1
Imported
562
571
119.9
121.8
Chemicals, plastic, rubber
11,127
11,029
97.1
96.2
Imported
11,127
11,029
97.1
96.2
Electricity and gas
11,169
12,756
110.0
125.6
Domestic
11,169
12,756
110.0
125.6
Fabricated products
16,099
18,035
107.8
120.7
Imported
16,099
18,035
107.8
120.7
Forestry and logging
1,472
1,856
113.0
142.4
Domestic
1,472
1,856
113.0
142.4
Mining Products
26,679
59,263
102.6
228.0
Domestic
23,604
51,546
102.6
224.0
Imported
3,074
7,717
103.2
259.1
Textile, clothing, footwear
5,682
5,704
109.3
109.7
Imported
5,682
5,704
109.3
109.7
Wood and paper products
4,654
4,934
100.3
106.3
Imported
4,654
4,934
100.3
106.3



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 re-use 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