User:Poetlister/Elementary aggregates

Elementary aggregates: this page assumes knowledge of Averages.

Elementary aggregates are an important concept in the construction of index numbers, especially retail or consumer price indices (RPI, CPI). An index number may cover many different items and is then generally calculated by constructing separate indices for each item and producing a weighted average. At the lowest level, there may be no suitable data to calculate weights; their combination is then called an elementary aggregate.

Application to price indices

Suppose you have an elementary aggregate of prices at two different times and you want to calculate the inflation between those two times. (An elementary aggregate in this context is a collection of prices for the same type of item in different shops, often restricted to shops of the same type - e.g. supermarkets - or in the same region or both.) For each pair of prices of exactly the same item in the same shop, we may compute the price relative: the price at the later date divided by that at the earlier date. We then combine these by an elementary aggregation formula.

Properties of elementary aggregation formulae

An elementary aggregation formula must have the following properties, very similar to the properties of averages:

1. If all the price relatives are equal, the price change should also equal this value: EA(x, x, x, ...) = x.
2. The price change must not exceed the maximum of the price relatives nor be less than their minimum. Unlike averages, we would normally wish to be strict and say that if not all the price relatives are equal, it must be greater than the minimum and less than the maximum.
3. The formula must be multiplicatively linear, i.e. if all price relatives are multiplied by the same constant k the price change will be multiplied by the same number: EA(kx, ky) = k.EA(x, y).
4. The price change must be order invariant; if we permute the numbers, it will not change their average: EA(y, x) = EA(x, y). This rules out just picking the first number, or the average of the first and last, or other weighted averages.
5. The price change must be strictly monotonic; if any one number increases (the others being unchanged), the price change must not decrease, and vice versa. Again, we would normally wish to be strict and say that if any price relative increases, so must the price change.

Common formulae

There are two types of formula:

a. some type of average of the price relatives, and

b. some type of average of the prices in the later period divided by the same type of average of the prices in the earlier period.

Traditionally, the commonest formula has been the Average of Relatives (AR) or Carli, the arithmetic mean of the price relatives.

Increasingly, the Geometric Mean (GM) of the price relatives or Jevons is being used. Obviously, this will give a lower rate of inflation (or a higher rate of deflation for items with falling prices) than the Carli.

Another widely used method is to divide the arithmetic mean price at the later date by the mean at the earlier date. This is known as the Ratio of Averages (RA) or Dutot. Although it is of type b above, it can also be formulated as a weighted mean of the price relatives so it is also of type a.

The ratio of geometric means does not give a new formula, as it always equals the geometric mean of the price relatives. {why?} Thus again it is of both types a and b. (In general, however, a formula will not be of both types.)

The AR and RA are the only elementary aggregation formulae used in the United Kingdom RPI. By European agreement, the AR is not used in the CPI produced for comparison with other members of the European Union. Indeed, very few countries anywhere outside the UK now use it in their retail or consumer price indices. Instead, the CPI and many other price indices use the GM.

These three formulae are the only ones currently in use in any official RPI or CPI. Other methods that have been advocated include the GAHM of the price relatives, known as the Carruthers-Ward-Sellwood-Dalén or CWSD, and the Power plus 1 mean with s = -1/2, known as the Walsh-Balk (WB) or Balk-Walsh (BW). The CWSD was at one time used in the Swedish price index.

Inequalities

How do the results from these formulae compare? If all the price relatives are equal, all these formulae will always give the same result. Otherwise, GM is always less than AR, so it gives a smaller price increase or a bigger decrease, and the difference is roughly proportional to the variance of the price relatives. (See Quantifying the arithmetic-geometric inequality.) This difference makes GM very popular with finance ministries. But of course there is no theoretical reason to prefer the GM just because it gives a lower result.

RA is nearly always less than AR, but not always. RA is less than AR if the price relatives are negatively correlated with the base prices, or in other words if the brands, varieties or shops with higher prices in the base period tend to have smaller price increases in subsequent months than those with lower prices. That is what usually happens in practice. However, if the price relatives are positively correlated with the base prices, or in other words if the brands, varieties or shops with higher prices in the base period tend to have larger price increases than those with lower prices, RA will exceed AR. This is very unlikely to happen if the prices relate to brands, varieties or shops that are fairly homogeneous so that consumers will readily move to cheaper alternatives. It may happen if they are quite inhomogeneous.

In practice, RA and GM are usually fairly close. RA is less than GM if the variance of prices divided by the square of the arithmetic mean of the prices in the current month is less than in the base month, and greater if the variance divided by the square of the mean is greater. In fact, we have approximately

GM/RA = 1 + ½(var(B)-var(C))

where var means variance, B is the set of base prices divided by their mean and C is the set of current prices divided by their mean.

Both the WB and CWSD typically give results close to the GM, especially the WB. In fact, the WB will always lie between the GM and CWSD. If the distribution of the logarithms of the price relatives has a positive skew, they will be higher than the GM, and if it has a negative skew, they will be lower than the GM. The effect of skew is larger for the CWSD than for the WB.

Price bounce

This is a problem with the AR that occurs if the index is chained, and is one of the main reasons why AR is nowadays deprecated. Chaining is necessary to allow the weights at the higher levels of aggregation to be updated, and it is considered good practice to chain price indices annually.

If chaining is done say each January, if prices change between say January 2010 and January 2011, and then in January 2012 revert to their values in January 2010, you might expect that the net inflation over the two years would be zero. This is the case with all of the formulae considered above except for the AR, which will always show an apparent price increase. In general, if prices bounce up and down the AR will show what may be considered a spurious upward drift in prices.

If there is no chaining, price bounce cannot occur even with the AR, but it is then impossible to modify the higher level weights.

A stronger requirement is transitivity, which means that chaining never affects the measured inflation. Transitivity is only guaranteed if the aggregation formula can be expressed as a ratio of average prices,as the RA and the GM can be. AR always yields a greater price increase if there is chaining; with CWSD, the difference may go either way.

In practice, elementary aggregates are not chained, only higher level aggregates that do involve weights. Since such weights may change every year, then even if prices in January 2012 revert to their values in January 2010, the net inflation over the two years will not in general be zero, whichever elementary aggregate method is used. Obviously, transitivity will not work either.

Fixed weights

The AR and RA can both be regarded as fixed weight indices. In other words, the market shares by volume of different shops, or of different brands or varieties of an item, do not change during the year even if relative prices change. The textbooks say that for the AR, the weights of the price relatives are equal, implicitly implying that each shop, brand or variety has an equal share of expenditure in the base month. For the RA, the weights of the price relatives are proportional to the price in the base month, implicitly implying that each shop, brand or variety has an equal share of sales volume regardless of price. Neither of those assumptions is very plausible, especially regarding different shops.

However, it must be stressed when considering implicit shares that prices are only recorded in a sample of shops. Each shop used for price collection represents not just itself, but many other shops and the implicit shares refer to all the shops a given sampled shop represents. Shops may be selected by simple random sample so that each shop has an equal chance of selection, but more commonly it is done by probability proportional to size, so larger shops are more likely to be selected. The effect of this is that a small shop represents more shops than a large one, so it is fairly realistic to assume that all shops have an equal share.

If the set of prices within an item is very variable, so that some prices are many times higher than others, the high prices will dominate the price change measured by RA. This is considered undesirable, so RA should only be used for more homogeneous items that do not have large price ranges. The homogeneity can be improved by splitting up the item into smaller items; for example, silk ties could be distinguished from other ties. The disadvantage is that either more prices must be collected (increasing costs), or there will be fewer price quotes per item. In the latter case, it may be necessary to abandon regional or shop type stratification. Homogeneity is not an issue for AR or GM.

GM weights

By contrast, the GM cannot be formulated as a fixed weight index. It assumes a price elasticity of 1. This means that if prices of a brand or in a shop double, other prices remaining the same, then the expenditure share of that brand or in that shop will remain constant or, in other words, the share of volume of purchases will halve.

In particular, if one price in an elementary aggregate drops to zero, the GM will also become zero so with this formula, the price index for the aggregate becomes zero. It might be argued that this makes sense; if a brand, or the price in one shop, becomes zero, people will only get this brand or visit this shop. However, in practice this may not be true. For shops, people may be unwilling to travel to a distant shop, especially if they wish to buy other items that are not free in that shop. For brands, people may have needs (clothing may not be available in all sizes) or strong preferences such that they are willing to pay for a brand even if a fairly similar one is free. Thus this can be used as an argument for or against use of the GM.

The assumption of an elasticity of 1 is strong; elasticity may be less than ½ and may even in some circumstances be negative. In this case, GM is unsuitable.

CWSD and BW can also not be formulated as fixed weight indices, nor do they represent constant elasticities. Both of these also become zero if one price in an elementary aggregate drops to zero.

The Axiomatic Approach

It is sometimes argued that the GM is the only valid elementary aggregate formula because it is the only one that satisfies all of a set of supposedly self-evident "axioms". The validity of this argument depends on people accepting all these axioms.

One controversial argument is the "exchangeability axiom". This states that if we have prices a, b, c in say three shops 1, 2, 3 in the base period, and the same prices in a different order, say, b, c, a in a later period, then the elementary aggregate formula must show no price change. Clearly, any formula that is the ratio of some average of prices in the later period divided by the corresponding average in the base period will satisfy this axiom, so RA and GM would be OK. It can be shown that any formula that cannot be expressed as such a ratio will not satisfy this axiom; this includes all the other formulae considered here, AR, CWSD and BW.

It is easy to formulate axioms that other formulae satisfy but GM does not. One is "consistency of aggregation". This says that if the elementary aggregate is split in two, each part aggregated separately and the results combined with weight proportional to the number of price relatives in each aggregate, the result should be the same as if the aggregate had not been split. AR passes this, but the other formulae in general do not.

Which formula is best?

So, which formula is best? That is a tricky question to answer. It partly depends on what we want to measure. For example, a price index that is the best for calculating a Consumers’ Expenditure Deflator for producing GDP may not be the best for measuring a utility-based Cost of Living Index. Since RA and AR assume fixed weights, they are consistent with the Laspeyres method of aggregating the item indices to the overall index. GM is not consistent with Laspeyres.

The rth power mean of the price relatives is a good approximation to a cost of living index if people have an elasticity of substitution between items within the same elementary aggregate of 1-r; on this argument, if this elasticity is 0, the AR is best and if it is 1, the GM (the limit of the rth power mean as r tends to 0) is best. It is often argued that the assumption of fixed weights is implausible, hence GM must be better than AR or RA. However, GM assumes a price elasticity of 1. If the true price elasticity is less than 0.5, or even negative, then GM is worse than the other formulae. In circumstances where price changes are mainly caused by demand rather than supply, low or negative elasticities are common.

The CWSD may be regarded as an unweighted version of the Fisher Ideal Index, the GM of the Laspeyres (base weighted) and Paasche (current weighted) indices. Thus it can be regarded as an "ideal" formula in the same sense as Fisher's. Similarly, the GM may be regarded as an unweighted version of the Törnqvist index.

So in summary, each formula has its advantages and disadvantages. The choice of formula is not easy, and probably different formulae are more suited to different items.

See also

• Averages