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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CHAPTER 10 CONSUMER PRICE INDEX CALCULATION IN PRACTICE


INTRODUCTION

10.1 The CPI has been described as a basket of goods and services which is notionally purchased each quarter. As prices change from one quarter to the next so too will the total cost (or price) of the basket. Of the various ways in which a CPI could be described, this description conforms closely with the procedures actually followed.

10.2 Using this description, the CPI can be thought of as being constructed in five major steps:

(i) subdividing the total expenditure into individual items for which price samples can be selected;

(ii) collecting price data;

(iii) estimating price movements for individual items;

(iv) calculating the current period cost of the basket; and

(v) calculating index numbers and points contribution.

10.3 This chapter provides a stylised account of the steps above. It also indicates how analytical indexes are calculated, and describes the ABS rounding practices.


SUBDIVIDING THE BASKET

Expenditure aggregates

10.4 Based mainly on the results of the ABS Household Expenditure Survey (HES), estimates are obtained for total annual expenditure of private households in each capital city for each of the ninety expenditure classes in the CPI. As these estimates are for the expenditure of households in aggregate, they are referred to as expenditure aggregates.

10.5 Expenditure aggregates are derived for well defined categories of household expenditure (e.g. bread), but are still too broad to be of direct use in selecting samples of products for pricing. For this purpose, expenditure aggregates need to be subdivided into as fine a level of commodity detail as possible. As the HES is generally not designed to provide such fine level estimates, it is necessary to supplement the HES data with information from other sources such as other official data collections and industrial data. The processes involved are illustrated below using a hypothetical example for the Bread expenditure class of the CPI.

10.6 Suppose that, based on information reported in the HES, the annual expenditure on bread by all private households in a particular city is estimated at $8 million. Further, suppose that some industrial data exist on the market shares of various types of bread. In combination, these two data sources can be used to derive expenditure aggregates at a much finer level of detail than that available from the HES alone. The hypothetical results are shown in Table 10.1 below.

10.7 The next stage in the process involves determining the types of bread for which price samples should be constructed. This is not a simple exercise, and relies on the judgement of the prices statisticians. In reaching decisions about precisely which items to include in price samples, a balance needs to be struck between the cost of data collection (and processing) and the accuracy of the index. Factors taken into account include the significance of individual items, the extent to which different items are likely to exhibit similar price behaviour, and any practical problems with measuring prices to constant quality.

10.1 DISAGGREGATION OF EXPENDITURE DATA

Market Share
HES data
Derived expenditure aggregates
Type of bread
%
$'000
$'000

1 White, sandwich, sliced
30
-
2 400
2 White, sandwich, unsliced
2
-
160
3 White high fibre
20
-
1 600
4 White high top
3
-
240
5 Wholemeal
10
-
800
6 Multigrain
15
-
1 200
7 Bread rolls
15
-
1 200
8 Specialty
5
-
400
Total Bread
100
8 000
8 000



10.8 In this example, a reasonable outcome would be to decide to construct pricing samples for varieties 1, 3, 5 and 6. Separate price samples would not be constructed for items 2 and 4 because of their small market share relative to the others. Pricing samples would also not be constructed for bread rolls and specialty breads (items 7 and 8) as they would prove difficult to price to constant quality because these items are usually sold by number and not by weight.


Elementary aggregates must have a price sample

10.9 When no more information is available to disaggregate the expenditure values any further, the resulting product definitions are called elementary aggregates. Each elementary aggregate has its own price sample. Ideally, all the products in an elementary aggregate (and there should only be a few) would be homogeneous goods or services, and would be substitutes for each other. In the Australian CPI, there are approximately 1,000 elementary aggregates for each of the eight capital cities. This gives around 8,000 price samples nationally. The expenditure aggregates for the items that are not explicitly priced are reallocated across the elementary aggregates of closely related goods or services under the assumption that the price movements for these products are similar.

10.10 In the bread example, the reallocation is carried out in two stages. First, the expenditure aggregate for unsliced white sandwich loaves is added to sliced white sandwich loaves resulting in an elementary aggregate for white sandwich loaves (as being white bread and sandwich loaves makes them likely to experience similar price movements). White high top loaves would be treated similarly. In the second stage, the expenditure aggregates for bread rolls and specialty breads, which have no closely matching characteristics with any of the other types of bread, would be allocated proportionally across the remaining elementary aggregates under the assumption that the average movement in prices for all other bread types is the best estimate. The outcome of this process is presented in Table 10.2.

10.11 In summary, the rationale for this allocation is as follows. Price behaviour of item 2 (white, sandwich, unsliced) is likely to be best represented by the price behaviour of item 1 (white, sandwich, sliced). Items 4 (white high top) and 3 (white high fibre) are treated similarly. The price behaviour for items 7 (bread rolls) and 8 (specialty bread) is likely to be best represented by the average price behaviour of all other breads.

10.2 OUTCOME OF ELEMENTARY AGGREGATE RATIONALISATION

Bread Type
Initial
Stage 1
Stage 2
Elementary aggregate

1
2,400
2,560
3,200
White sandwich
2
160
-
-
3
1,600
1,840
2,300
White high fibre
4
240
-
-
5
800
800
1,000
Wholemeal
6
1,200
1,200
1,500
Multigrain
7
1,200
1,200
-
8
400
400
-
Total
8,000
8,000
8,000




Determining outlet types

10.12 The next step is to determine the outlet types (respondents) from which the prices will be collected. In order to accurately reflect changes in prices paid by households for bread, prices need to be collected from the types of outlets from which households normally purchase bread. Data are unlikely to be available on the expenditures at the individual elementary aggregate level by type of outlet. It is more likely that data will be available for expenditure on bread in total by type of outlet. Suppose industrial data indicate that supermarkets account for about 80 per cent of bread sales, and bakery outlets the remainder. A simple way to construct a pricing sample for each elementary aggregate that is representative of household shopping patterns is to have a ratio of four supermarkets for every bakery.


COLLECTING PRICE DATA

Selecting respondents

10.13 When the pricing samples are worked out, ABS field staff decide from which individual outlets the prices will be collected. The respondents are chosen to be representative of the types of outlets (in the example above, supermarkets and bakeries) taking into account the demographic characteristics of the city, and the numbers required for the sample. Prices are collected from any particular respondent on the same day in each collection period (e.g. the first Monday of each month).


Selecting items to price

10.14 When a pricing sample contains respondent standard specifications, the field staff will decide which specific items are most representative of the required type of product. Usually they do this by consulting with the manager of the outlet. Using the bread example above, at one outlet they might decide that a 680g sliced white sandwich loaf best represents white sandwich bread, but at another outlet it might be a 700g white sandwich loaf. Once selected, the same item will be priced at that respondent so long as it remains the most representative example of the product.

10.15 An important part of the price collection process is the continual monitoring of the items for quality change. In the bread example, quality change could occur with (say) a change in the size (weight) of the loaf of bread. In this case, the price movement attributable to the change in loaf size would be removed to derive a pure price movement for the loaf.


ESTIMATING PRICE MOVEMENTS FOR ELEMENTARY AGGREGATES

10.16 Price relatives are calculated for each price in the sample, and mostly the geometric mean of these is used in the calculations. The ratio of the current period’s geometric mean of price relatives to the previous period’s geometric mean of price relatives provides the change in the average price for the elementary aggregate. Using the hypothetical bread example, Table 10.3 shows price relatives being used to estimate the price movement for bread. These estimates of price movements are used to revalue the expenditure aggregates to current period prices by applying the period to period price movement to the previous period's expenditure aggregate for each elementary aggregate. The updated expenditure aggregate provides an estimate of the cost of acquiring the base period quantity of the elementary aggregate’s products in the current period.

10.3 ESTIMATING PRICE MOVEMENT FOR AN ELEMENTARY AGGREGATE

Price relative in
Period 1
Period 2
Price
movement
%

White sandwich loaf
Supermarket A
1.025
1.030
0.5
Supermarket B
1.030
0.950
-7.8
Supermarket C
1.040
1.065
2.4
Supermarket D
0.980
1.100
12.2
Bakery
1.100
1.250
13.6
Geometric mean
1.034
1.075
4.0




CALCULATING THE CURRENT COST OF THE BASKET

10.17 The price updated expenditure aggregates for the elementary aggregates are then summed to derive the current cost of the basket of goods and services (or any portion of the basket) . Index numbers are calculated from the expenditure aggregates at every level of the index. The table below shows the calculation of the expenditure value for the total of bread (an expenditure class in this example).

10.4 AGGREGATION OF EXPENDITURE AGGREGATES FOR EXPENDITURE CLASS

Expenditure aggregate $'000
Percentage
change
%
Expenditure aggregate $'000

Elementary aggregate (Description)
Period 1
Period 1 to Period 2
Period 2
White sandwich
3200
4.0
3328
White high fibre
2300
3.5
2381
Wholemeal
1000
0.0
1000
Multigrain
1500
1.7
1526
Total
8000
2.9
8235




CALCULATING INDEX NUMBERS AND POINTS CONTRIBUTIONS

10.18 Table 10.5 shows the calculation of index numbers and points contribution. It is assumed that index numbers already exist for the link period (June quarter 2005 for the 15th series CPI) and period 1. Assume the expenditure aggregate for Cereals has been calculated using the same method as that for Bread so that the two can be added and a movement calculated for Bread and Cereals. Similarly, assume the expenditure aggregates for period 2 have been calculated for Other foods and Non-food so that expenditure aggregates can be calculated for Food and All groups.

10.19 When a price index has not been linked, indexes for any component can be calculated simply by dividing the current period expenditure aggregate by its expenditure aggregate in the reference period (when the index is set to 100.0). However, the CPI has been linked several times since its reference base (1989-90) and the index numbers must be calculated from

Equation: Chp10_10.1(10.1)

where ILP is the index number in the link period (June quarter 2005 for the 15th series CPI), and VCP and VLP are the expenditure aggregates in the current period and link periods respectively. Thus the index number for Bread in period 2 is given by 108.0 x 8235 / 6500 = 136.8.

Points contributions are also calculated using the expenditure aggregates. In any period, the points contribution of a component to the All groups index number is calculated by multiplying the All groups index number for the period by the expenditure aggregate for the component in that period, and dividing by the All groups expenditure aggregate for that period. This can be stated algebraically as

Equation: chp10_10.2(10.2)

Where ItAG is the index for All groups in period t, ItiEquation: chp10_10.2ais the expenditure aggregate for component i in period t and is VtAG the expenditure aggregate for All groups in period t.

10.20 In the example in Table 10.5 below, the points contribution for Bread in period 2 is calculated as 141.3 * (8235 / 144268) = 8.07.

10.21 The change in index points contribution for a component between any two periods is found by simply subtracting the points contribution for the previous period from the points contribution for the current period. For example, the change in index points contribution for Bread between periods 1 and 2 is 8.07 - 7.84 = 0.23.

10.22 The CPI publication does not show the expenditure aggregates, but rather the index numbers derived from the expenditure aggregates. Expenditure aggregates vary considerably in size, and showing them would make the publication difficult to read and interpret. Index numbers and points contributions are a better way to present the information.

10.5 AGGREGATION OF EXPENDITURE AGGREGATES FOR ENTIRE INDEX

Link period
Period 1
Period 2

Expenditure aggregates ($)

All groups
122,500
138,100
144,268
Food
32,500
40,100
41,368
Bread and cereals
12,500
15,000
15,515
Bread
6,500
8,000
8,235
Cereals
6,000
7,000
7,280
Other foods
20,000
25,100
25,853
Non-food
90,000
98,000
102,00

Movement in expenditure aggregates (period 1 to period 2)

All groups
1.045
Food
1.032
Bread and cereals
1.034
Bread
1.029
Cereals
1.04
Other foods
1.03
Non-food
1.05

Index numbers

All groups
120.0
135.3
141.3
Food
115.0
141.9
146.4
Bread and cereals
110.0
132.0
136.5
Bread
108.0
132.9
136.8
Cereals
113.0
131.8
137.1
Other foods
117.0
146.8
151.2
Non-food
125.0
136.1
142.9

Points contribution

All groups
120.0
135.3
141.3
Food
31.84
39.29
40.52
Bread and cereals
12.24
14.7
15.2
Bread
6.37
7.84
8.07
Cereals
5.88
6.86
7.13
Other foods
19.59
24.59
25.32
Non-food
88.16
96.01
100.78



Note: It is assumed the reference base period precedes period 1.


SECONDARY INDEXES

10.23 A range of analytical indexes are also published reusing data from the CPI. Examples of these are the All groups excluding (each of the groups in turn), Goods and Services, Tradables, Non-tradables, and Market goods and services (with exclusions). These are called secondary indexes as they use the same weights (or expenditure aggregates) as the CPI, and are compiled by summing the appropriate value aggregates. For example, in the table above, the starting point for compiling an index for All groups excluding Bread and cereals would be to add up the value aggregates for Other foods and Non-food and then calculate index values as described previously.


TERTIARY INDEXES

10.24 A further range of analytical indexes are compiled from the price samples collected for the CPI. Price indexes compiled under the outlays approach are published annually for four population subgroups: employees; age pensioners; self funded retirees; and recipients of other government transfer. These indexes, unlike the secondary indexes, have their own weighting patterns. For each component in the population subgroup indexes, the movement in the corresponding CPI index is used to update the expenditure aggregate and index number for the population subgroup. The purpose of the population subgroup indexes is to show any differences in the price changes faced by each of the four demographic groups arising from their differing expenditure patterns.


CONSUMER PRICE INDEX ROUNDING CONVENTIONS

10.25 To ensure consistency from one publication to the next, the ABS uses a set of rounding conventions or rules for calculating and presenting the results. These conventions strike a balance between maximising the usefulness of the information for analytical purposes, and retaining a sense of the underlying precision of the estimates. These conventions need to be taken into account when CPI data is used for analytical or other special purposes.

10.26 Index numbers are always published relative to a base of 100.0. Index numbers and percentage changes are always published to one decimal place, and the percentage changes are calculated from the rounded index numbers. Index numbers for periods longer than a single quarter (e.g. for financial years) are calculated as the simple arithmetic average of the rounded quarterly index numbers.

10.27 Points contributions are published to two decimal places. Change in points contributions is calculated from the rounded points contributions. Rounding differences can arise in the points contributions where different levels of precision are used.