Index Calculation

2.11.1 Computation of lowest level indices (elementary aggregation)

There are several methods for combining prices to give elementary aggregates (lowest level indices):

• the ratio of averages (RA) – the average price of a sample of observations in the current period is compared to the average price of the same sample in the previous (or base) period, and in order to obtain a correct result, it is vital that the samples of varieties are the same in both periods (matched samples). If a price is missing in the current period and no action is taken the index will be biased. RA is not suitable if the spread of prices being aggregated is so large that the prices are heterogeneous;

• the average of relatives (AR) – is based on comparing each individual price with its corresponding price in the previous (or base) period to give a price relative for each observation. These price relatives are then weighted together, using either the standard or the modified Laspeyres formulae. AR assumes an elasticity of substitution between varieties of zero, and therefore suffers from upward drift when this assumption is invalid and prices are rising. If a price is missing in the current period, then its price relative cannot be calculated (possibly causing computer error). If this price relative has a specific weight, then omitting the relative from the next level of aggregation may cause errors if the weights of the other price relatives are not adjusted accordingly;

• the geometric mean (GM) – is being introduced by more and more countries (as can be seen in Table 7). The approach is to calculate a GM of prices in both periods and then derive the price relative or, alternatively, calculate a geometric average of the price relatives - both calculations will yield the same results. The GM assumes an elasticity of substitution between varieties of one. However, the problem of missing observations is still the same as in the previous two cases. The sample of observations used from each period must have the same number of observations for computing the geometric mean of prices in order to avoid a biased measure of price change. In the case where a weighted geometric mean is used, the weights for missing observations would also need to be distributed to the remaining observations to avoid any bias.

2.11.2 Index aggregation

Elementary aggregation may directly give local, regional or national indices, i.e., relatively few prices may be combined to give local item indices or a wider spread of prices may be combined to give regional indices or, if regional indices are not published and prices are sufficiently homogeneous, then all prices for an item may be combined at this first stage (weighted if possible) to give a national index.

The elementary aggregates obtained above are combined using some kind of index number formula and weights based on expenditure (or population, but this is less desirable). In the case of CPIs, all countries use a Laspeyres formulation – either in the standard or modified form.

Table 7: Consumer prices: Index aggregation

Elementary aggregation Higher level aggregation

Canada GM + some RA Modified Laspeyres

Mexico Weighted AR Standard Laspeyres

United States GM + weighted AR Modified Laspeyres

Australia GM + some RA Modified Laspeyres

Japan RA Standard Laspeyres

Korea RA Standard Laspeyres

New-Zealand RA Standard Laspeyres

Austria AR + some RA Standard Laspeyres

Belgium RA Standard Laspeyres

Czech Republic RA Modified Laspeyres

Denmark GM Standard Laspeyres

Finland GM Standard Laspeyres

France GM + RA for food Chained Laspeyres

Germany .. Standard Laspeyres

Greece .. Standard Laspeyres

Hungary RA Chained Laspeyres

Iceland GM Standard Laspeyres

Ireland Weighted RA Modified Laspeyres

Italy GM Chained Laspeyres

Luxembourg .. Standard Laspeyres

Table 7: Consumer prices: Index aggregation (continued)

Elementary aggregation Higher level aggregation

Norway GM Chained Laspeyres

Poland GM Chained Laspeyres

Portugal GM+ weighted AR Standard Laspeyres

Slovak Republic Weighted RA Standard Laspeyres

Spain AR Standard Laspeyres

Sweden GM Chained linked index

Switzerland AR Standard Laspeyres

Turkey RA Standard Laspeyres

United Kingdom AR + RA Standard Laspeyres

GM: geometric mean; RA: ratio of averages; AR: average of relatives ..: metadata are not available

2.11.3 Alignment of expenditure and price reference base

In practice, data generally are not available as required by the Laspeyres formula. Expenditure weights are calculated using average prices for the base period (preferably a year), and price relatives are calculated using as convenient price reference base period, for example, to correspond with the price reference base of the overall index - commonly either a whole year or a single month35. Thus the base period for the weights is different to that of the price relatives. In order to use price relatives based on a different period to their corresponding weights, link factors (or adjustment coefficients) are needed.

2.11.4 Chaining re-weighted indices

A chain index consists of a series of successive indices, each linked (spliced) to its predecessor. Linking consists of multiplying the values of the successor index by the value of its predecessor in an overlap period (linking coefficients are calculated using values in the overlap period), so that the index base period of the successor becomes the same as for the predecessor index, i.e. the indices have a common reference base. Linked indices can be produced at any level, i.e. item, product, group, total CPI. However, it should be remembered that aggregating linked sub-indices will give a different result to linking aggregated indices. The recommendation is that aggregation should always be done before chain-linking, i.e. linking should always be the last stage in the process.

2.12 CPI Problem Areas