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[edit] Introduction

This section deals with formulas for indexes composed of weighted sub-indexes. At the lowest level of aggregation, most sub-indexes are elementary aggregate indexes computed without weights, using only a sample of prices observed for one or more specified product-types, though the weight for each such sub-index is derived from estimates of consumption values for all the products covered by it. These "elementary aggregate indexes" or "micro indexes", are the subject of another chapter.

This section bypasses the issue of whether a Consumer Price Index should be based on consumption viewed as Transactions, as Expenditure or as Use. The term "consumption value" used here should be interpreted to signify consumption in any one of these meanings.

[edit] Basic formulas

[edit] Laspeyres indexes

Consumer Price Indexes are often described as Laspeyres indexes. A Laspeyres index for period t with period 0 as reference-period is simply:

$\frac{{\sum {p_t q_0 } }}{{\sum {p_0 q_0 } }} = \sum {\left( {\frac{{p_0 q_0 }}{{\sum {p_0 q_0 } }} \cdot \frac{{p_t }}{{p_0 }}} \right)}$

that is to say, a weighted average of the price ratios from 0 to t for every single consumption product where the weight for each is its share in reference-period, 0, total consumption value. However, Consumer Price Indexes are not actually compiled using this formula. There are two reasons for this.

The first is that what is compiled is only a sample estimate:

The weights of a Consumer Price Index do not relate to each and every product covered by it but to groupings of products or, with probabilistic product selection, to a sample of products.
The prices used to compute a Consumer Price Index do not relate to to each and every product covered by it, but to a sample of products.
These prices are generally collected only for selected days in a single month

The second reason is that what is estimated are, in practice, not true Laspeyres indexes. They should be described as "fixed-base" or "Laspeyres-type" indexes. There are a number of reasons for this:

The weights in a Consumer Price Index relate to a year, the price reference -period is usually a single month. Thus they relate to periods of different length.
Furthermore, the weight reference-year often antedates the price reference-month. For example, the weights may relate to the year 1999 while the price reference-month is December 2001 for the index from January 2002 onward..
Whereas the Laspeyres concept is defined in terms of an identical set of products in the price reference-period and the current period, Consumer Price Indexes are computed even though some products disappear from the market and new products appear that were not available in the price reference-period.
Some of the products covered in a Consumer Price Index may have no natural quantity units (no q's), for example services for which a percentage fee is charged.

The first two of these last points, the fact that weight and price reference-periods are quite different, mean that theoretical analysis of the relationship between a Laspeyres index and other index concepts, such as a superlative index, cannot be simply applied to the relationship between a monthly Consumer Price Index and such other concepts. The theoretical analysis always assumes that weight and price reference-periods coincide so that the prices of the price reference-period are the prices that help to determine the pattern of weight reference-period consumption. If an index is to be computed with a price reference-period which coincides with its weight reference-period, the price reference-period has to be a whole year since only annual weights are normally available.

[edit] A weighted sum or mean of sub-indexes

Consumer Price Indexes are compiled in practice as a weighted arithmetic mean of sub-indexes:

$\sum {wI_{p:t} }$

where the w are the weights, summing to unity, and the Ip:t are sub-indexes for month t with p as price reference-period. The indexes are almost all what has been called "plutocratic" indexes, as the weights reflect the value shares of its different components in the total value of consumption. A "democratic" index would instead use as weights the average across households of the share of the different components in the consumption value of each household taken separately.

Consumer Price Indexes could also be, but are not, compiled using other formulas, such as a weighted geometric mean of sub-indexes

$\prod {\left( {I_{p:t}^{} } \right)^w }$

The coverage of each sub-index is defined in terms of the type of goods and services covered and also, for many of them, in terms of the location and type of outlet where they are sold.

[edit] Superlative indexes

Superlative indexes compare prices between two periods, each of whose weight reference-period coincides with its price reference-period, using weights which are a symmetric average of weights from both periods. Under strict and unrealistic assumptions they have been shown to provide a very close approximation to a "True cost of living index" as defined by economic theorists. However, they can be accepted as a standard of reference on the commonsense grounds that, for example, to compare 2000 with 1999 prices, both the 1999 and the 2000 pattern of consumption are relevant. The Fisher, Törnqvist and Walsh indexes, are all superlative. Though defined in price and quantity terms, like all national Consumer Price Indexes they can in practice only be estimated using value weights and sub-indexes.

Since the available weighting data relate to calendar years, estimated superlative indexes have to be whole-year to whole-year comparisons. These, of course, can only be made retrospectively once weights for the second year of the comparison become available. Thus they could be used in three ways:

To provide a historical series.
For retrospective evaluation of the Consumer Price Index.
(As recently introduced in Sweden) to provide the index up to the last of the two successive years for which it can be computed, carrying forward the index from that year to the current month as a fixed-base index, using that year as both price reference-year and weight reference-year.

Consider the Edgeworth index which, though not formally a superlative index, provides practically identical results.^[a] This compares annual average prices in two years, say y-1 and y, using a simple average of the annual quantities of those two years, thus relating to a fixed "basket", an obviously meaningful comparison. It is thus defined as: $\frac{{\sum {p_y \left( {q_{y - 1} + q_y } \right)} }}{{\sum {p_{y - 1} \left( {q_{y - 1} + q_y } \right)} }}$

Letting V represent the consumption values of the component sub-aggregates and, as above, the I represent the annual sub-indexes, this can be estimated as:

$\sum {\frac{{V_{y - 1} + \frac{{V_y }}{{I_{y - 1:y} }}}}{{\sum {V_{y - 1} + \frac{{V_y }}{{I_{y - 1:y} }}} }}I_{y - 1:y} }$

Although Fisher, Törnqvist and Walsh indexes yield almost identical results to an Edgeworth index, unlike Fisher and Törnqvist indexes, but like a Laspeyres-type index, the Walsh and Edgeworth indexes have the desirable property that they can be additively decomposed.

As noted above, retrospective comparisons of the Edgeworth or a superlative index with the actual Consumer Price Index can be illuminating. The differences between them are sometimes erroneouslytreated as a measure of the strength of substitution effects between elementary aggregates – shifts in consumption away from those products whose relative prices have risen and towards those whose relative prices have fallen. This would only be legitimate in the extremely unlikely circumstance that all other factors determining the pattern of consumption had remained unchanged over the interval covered by the indexes. These other factors are: changes in disposable income distribution and levels; advertising, fashion, magazine articles, pop stars, TV, weather; the fact that households which bought a durable good or service in the first period usually won't want to repeat the purchase in the second period; and that people and their children are older, some have died and new households have been formed.

[edit] Indexes and their sub-indexes

Usually, Consumer Price Indexes are compiled in several stages by aggregation, that is to say as weighted averages of sub-indexes, most of which are in turn weighted averages of sub-sub-indexes, some of which in turn may be weighted averages of sub-sub-sub-indexes. These weights are shares in consumption values, and for a given set of overall consumption values, V, it makes no difference how this aggregation is done. Thus, for example, if, for month t with p as price reference-month, there are four sub-sub-indexes A₁, A₂, B₁ and B₂, the Consumer Price Index computed from all four of them is identically equal to the index computed from the two sub-indexes for A and B:

$\begin{array}{c} {\rm{CPI}} = \frac{{{\rm{V}}^{{\rm{A}}_{\rm{1}} } }}{{\sum V }}I_{p:t}^{A_1 } + \frac{{{\rm{V}}^{{\rm{A}}_{\rm{2}} } }}{{\sum V }}I_{p:t}^{A_2 } + \frac{{{\rm{V}}^{{\rm{B}}_{\rm{1}} } }}{{\sum V }}I_{p:t}^{B_1 } + \frac{{{\rm{V}}^{{\rm{B}}_{\rm{2}} } }}{{\sum V }}I_{p:t}^{B_2 } \\ \equiv \frac{{{\rm{V}}^{\rm{A}} }}{{\sum V }}I_{p:t}^A + \frac{{{\rm{V}}^{\rm{B}} }}{{\sum V }}I_{p:t}^B \\ \end{array}$

where $I_{p:t}^A = \frac{{{\rm{V}}^{{\rm{A}}_{\rm{1}} } }}{{{\rm{V}}^{{\rm{A}}_{\rm{1}} } + {\rm{V}}^{{\rm{A}}_{\rm{2}} } }}I_{p:t}^{A_1 } + \frac{{{\rm{V}}^{{\rm{A}}_{\rm{2}} } }}{{{\rm{V}}^{{\rm{A}}_{\rm{1}} } + {\rm{V}}^{{\rm{A}}_{\rm{2}} } }}I_{p:t}^{A_2 }$ and similarly for B.

The consumption values, V, relate to the weight reference-period, though they may be price-updated as discussed below. As already noted, the weight reference-period is usually a full year antedating the price reference-period which is usually a single month. However the index can be expressed as if the price reference-period were a whole year. Thus if December 1999 is the actual price reference-month, an index for May 2000 can be transformed to express it with 1997 as price reference-year as:

$I_{\overline {97} :May2000} = I_{\overline {97} :Dec.99} \cdot I_{Dec.99:May2000}$

where the bar over 1997 signifies an annual average for that year.

It is rarely possible to compute annual average prices directly. In consequence, price re-referencing has to be done by using an average of monthly sub-indexes. Using indexes based on the price reference-period previously used, say December 1996, the average required for $I_{\overline {97} :Dec.99}$ is:

$I_{\overline {97} /Dec.99}^{} = \frac{{I_{Dec.96:Dec.99}^{} }}{{\frac{1}{{12}}\left( {I_{Dec.96:Jan.97}^{} + I_{Dec.96:Feb.97}^{} \cdots + I_{Dec.96:Dec.97}^{} } \right)}}$

$I_{\overline {97} :Dec.99}^{} = \frac{{I_{Dec.96:Dec.99}^{} }}{{\sqrt[{12}]{{I_{Dec.96:Jan.97}^{} \times I_{Dec.96:Feb.97}^{} \cdots \times I_{Dec.96:Dec.97}^{} }}}}$

As monthly consumption values are known to differ from month to month, a weighted average would be preferable to these unweighted, i.e. equally weighted, averages. If purchase quantities and prices are positively or negatively correlated within a year, a weighted average of twelve monthly indexes will exceed or fall short of their simple average. Such correlations certainly exist for certain index components:

The availability of fresh products varies seasonally, their prices moving inversely. Simple averages of monthly fresh product indexes will exceed their weighted averages. Hence weight updating from the weight reference-year to the price reference-month using the simple average will yield lower price-updated fresh product weights than if the weighted averages are used.
Clothing purchases are particularly large in months when there are Sales.
December purchases exceed their monthly average for many products, so that in a year with marked inflation there is a positive correlation.

Price-quantity correlations are likely to be small for highly aggregate subindexes, however, except for years with rapid inflation.

Examples for such subindexes are provided by some UK data.

Year	Food	Household goods	Clothing & footwear
Year	Weighted minus simple average % points
1992	0.0	0.1	0.2
1993	0.0	0.1	0.3
1994	0.0	0.1	0.3
1995	0.1	0.3	0.4
1996	0.0	0.1	0.5

These relate to seasonally unadjusted monthly retail sales volumes by three groups of outlets: Predominantly food stores, Household goods stores and Textile and clothing stores. Matching them with three approximately corresponding monthly price sub-indexes, allows calculation of the differences between weighted and simple average annual indexes shown in the table below. They are very small, but they do exist.

In the case of monthly fresh product indexes, it is clear that simple averages will exceed their weighted averages because of the marked negative price-quantity correlations for such items. (I have found no data to illustrate this, though users of the Rothwell method obviously have such data.) Weight updating from the weight reference-year to the price reference-month using the simple average will therefore yield lower price-updated fresh product weights than if the weighted averages are used.

Since monthly quantity data are unavailable for the majority of elementary aggregate indexes and of subindexes, monthly quantity weights have to be monthly deflated expenditures. Denoting the monthly indexes as I_m and the monthly expenditures as V_m, the weighted averages will thus have to be computed as:

$\sum\limits_{m = 1}^{m = 12} {\frac{{\frac{{E_m }}{{I_m }}}}{{\sum\limits_{m = 1}^{m = 12} {\frac{{E_m }}{{I_m }}} }}I_m } = \frac{{\sum\limits_{m = 1}^{m = 12} {E_m } }}{{\sum\limits_{m = 1}^{m = 12} {\frac{{E_m }}{{I_m }}} }}$

[edit] Successive indexes

Whenever weights are revised and/or the sample of prices is revised, a new index is introduced and must be chained onto the old index. The following imaginary example shows this, supposing that it happens in January 2000, with prices collected both for the old sample and for the new sample in December 1999. It also shows how any index can be rescaled to make the Index Reference Period different from the price reference-period by dividing it by the value of the index for the selected Index Reference Period. Such rescaling obviously leaves rates of price change unaltered, but affects the absolute differences between successive index values.

Year	Old Index.	New Index	Chain New to Old	Divide by 1.250 to rescale to	Divide by 1.476 to rescale to
Year	Index Reference Dec. 1995	Index Reference Dec. 1999	Index Reference Dec. 1995	Index Reference average 1997	Index Reference June 2000
1997 average	1.250		1.250	1.000	0.84688
…	…	…	…	…	…
…	…	…	…	…	…
December 1999	1.440	1.000	1.440	1.152	0.97561
…	…	…	…	…	…
June 2000		1.025	1.4760	1.1808	1.000
…	…	…	…	…
December 2000		1.066	1.53504	1.22803	1.040

These indexes are expressed with an Index Reference Value of 1. In practice, they are usually expressed with an Index Reference Value of 100. In this case the December 2000 index rescaled to a 1997-average index Reference Value would be $\frac{{153.504}}{{125}} \times 100 = 122.803$ . Similarly, when two indexes with Index Reference Values of 100 are multiplied together, their product has to be divided by 100, for example $\frac{{106.6 \times 144}}{{100}} = 153.504$ .

[edit] Price-updating of weights

[edit] The method

The weights used in a fixed-base index Consumer Price Index are usually consumption values which have been price-updated from the weight reference-year to the price reference-month. Thus if V₉₇ is the value of consumption in the weight-reference-year of 1997 while the price reference-month is December 1999, the weights used from January 2000 onwards will be:

$\frac{{\left( {{\rm{V}}_{{\rm{97}}}^{} \times I_{\overline {97} :Dec.99}^{} } \right)}}{{\sum\limits_{} {\left( {{\rm{V}}_{{\rm{97}}}^{} \times I_{\overline {97} :Dec.99}^{} } \right)} }}$

where the bar over 97 signifies that these indexes relate December 1999 prices to annual average prices of 1997. (Preferably each would be a monthly-weighted mean of the twelve indexes Jan 97:Dec.99, Feb.97:Dec.99….. ….Dec.97:Dec.99.)

The index for May 2000 will then be computed by chaining the index computed with these new weights on to its predecessor Consumer Price Index, base b, with December 99 as the link month: $\begin{array}{c} CPI_{b:May00} = CPI_{b:Dec.99} \times \sum {\frac{{\left( {{\rm{V}}_{{\rm{97}}}^{} \times I_{\overline {97} :Dec.99}^{} } \right)}}{{\sum\limits_{} {\left( {{\rm{V}}_{{\rm{97}}}^{} \times I_{\overline {97} :Dec.99}^{} } \right)} }}I_{Dec.99:May00} } \\ = CPI_{b:Dec.99} \times \frac{{\sum {\left( {{\rm{V}}_{{\rm{97}}}^{} \times I_{\overline {97} :May00}^{} } \right)} }}{{\sum\limits_{} {\left( {{\rm{V}}_{{\rm{97}}}^{} \times I_{\overline {97} :Dec.99}^{} } \right)} }} \\ \end{array}$

The index can be interpreted in terms of ratios of revalued weight reference-period consumption values. Thus consider the ratio of the May 2000 index to the February 2000 index, the left-hand expression below. The central expression shows this calculated with price-updated weights, but it reduces to the right-hand expression which is the ratio of the sum of consumption values revalued to May to their sum revalued to February.

$\frac{{CPI_{Dec.99:May2000} }}{{CPI_{Dec.99:Feb2000} }} = \frac{{\sum {\frac{{V_{\overline {97} } I_{\overline {97} :Dec,99} }}{{\sum {V_{\overline {97} } I_{\overline {97} :Dec,99} } }}} I_{Dec.99:May2000} }}{{\sum {\frac{{V_{\overline {97} } I_{\overline {97} :Dec,99} }}{{\sum {V_{\overline {97} } I_{\overline {97} :Dec,99} } }}} I_{Dec.99:Feb2000} }} = \frac{{\sum {V_{\overline {97} } I_{\overline {97} :May2000} } }}{{\sum {V_{\overline {97} } I_{\overline {97} :Feb2000} } }}$

The ratio of the May 2000 index to the May 99 index is $\frac{{CPI_{b/Dec.99} \times \frac{{\sum {{\rm{V}}_{{\rm{97}}}^{} \times I_{\overline {97} :May00}^{} } }}{{\sum\limits_{} {\left( {{\rm{V}}_{{\rm{97}}}^{} \times I_{\overline {97} :Dec.99}^{} } \right)} }}}}{{CPI_{b:May.99} }}$

[edit] International indexes

For consistent aggregation of national indexes into an index for a group of countries, the weights used both within countries and for countries must be expressed at prices for the same point of time. For the European Harmonised Index of Consumer Prices, price-updating from each December to the following December is required, and any real change in the weights may be introduced only in December.

[edit] Deficient information

The fact that price-updating involves separate price-updating of each component weight by its own sub-index creates difficulties when a new set of weights includes new components. Thus if the new set of weights based on 1997 consumption values included a new component $V_{97}^{B_3 }$ , the index $I_{\overline {97} :Dec.99}^{B_3 }$ would be needed, and this requires that B₃ prices will have had to be collected ever since December 1996. Yet the need to collect these prices may not have been discovered until well after the end of 1997 when the consumption value data were compiled. No estimate of $I_{\overline {97} :Dec.99}^{B_3 }$ will then be available and it may be necessary to assume that B3 prices moved in parallel with those of some other sub-aggregate for which a sub-index has been compiled.

[edit] The significance of price-updating

For Consumer Price Indexes, whose weights necessarily relate to the past, it turns out that the choice between :

Price-updating weights to the price reference-period, implying constant quantity ratios;
Not updating them, implying constant value shares;

is related to the choice of formula to be used in computing that index.

Indexes for May 2000 calculated using weights price-updated from 1997 to a price reference-period of December 1999, for example, provide estimates of: $\frac{{\mbox{Value of 1997 annual consumption at May 2000 prices}}}{{\mbox{Value of 1997 annual consumption at December 1999 prices}}}$

Price-updating thus preserves the 1997 consumption volume pattern. This result would provide an estimate of $\frac{{\mbox{Value of December 1999 consumption at May 2000 prices}}}{{\mbox{Value of December 1999 consumption at December 1999 prices}}}$

only if December 1999 relative volumes of the different components of consumption were the same as in 1997.

Without any price-updating, the indexes for May 2000 would be estimates of the same thing under a different assumption, namely that December 1999 value shares, wⁱ, were the same as the value shares, wⁱ, in 1997 annual consumption.

This alternative assumption would mean that the ratio of the December 1999 values of each component to its 1997 value was the same for all of them, say a ratio of R. Hence the ratio of the December 1999 total consumption value to its 1997 value would also be R. Expressing the argument in terms of the universe of prices, p, and quantities, q, where the i are the component products; if, for all i: ${\rm{ }}w^i = \frac{{p_{Dec.99}^i q_{Dec.99}^i }}{{\sum\limits_i {p_{Dec.99}^i q_{Dec.99}^i } }} = \frac{{p_{\overline {97} }^i q_{97}^i }}{{\sum\limits_i {p_{\overline {97} }^i q_{97}^i } }}\;{\rm{then}}\;\frac{{p_{Dec.99}^i q_{Dec.99}^i }}{{p_{\overline {97} }^i q_{97}^i }} = \frac{{\sum\limits_i {p_{Dec.99}^i q_{Dec.99}^i } }}{{\sum\limits_i {p_{\overline {97} }^i q_{97}^i } }} = R$

However, since the weights sum to unity, R equals $\prod\limits_i {R^{w^i } }$ , and so: $\prod\limits_i {R^{w_i } } = \prod\limits_i {\left( {\frac{{p_{Dec.99}^i q_{Dec.99}^i }}{{p^i _{\overline {97} } q^i _{97} }}} \right)^{w^i } = \quad } \prod\limits_i {\left( {\frac{{p_{Dec.99}^i }}{{p^i _{\overline {97} } }}} \right)^{w^i } } \times \quad \prod\limits_i {\left( {\frac{{q_{Dec.99}^i }}{{q^i _{97} }}} \right)^{w^i } }$

(I am indebted to Jörgen Dalén for this formulation).

This decomposition into price and quantity components means that, under this assumption of constant value shares, the estimator of the price index would be $\prod\limits_i {\left( {I_{\overline {97} :Dec.99}^i } \right)^{w^i } }$ , not $\sum\limits_i {w^i I^i _{\overline {97} :Dec.99} }$ . In this case, the estimator of the price index for May 2000, with December 1999 as price reference-period, should be $\prod\limits_i {\left( {I_{Dec.99:May.2000}^{} } \right)} ^{w^i }$ where the weights wⁱ are the 1997 weights without any price updating.

[edit] When to price-update

This shows that whether to price-update, and whether to use the weighted arithmetic or geometric mean of sub-indexes, are linked questions which ought to be answered consistently.

Price-updating and the weighted arithmetic mean of sub-indexes should be chosen when it is expected that the price reference-period pattern of consumption will be closer to the weight reference-period volume pattern of consumption than to its value-share pattern.
When, alternatively, it is expected to be closer to the weight reference-period value-share pattern, the weights should not be price-updated and the weighted geometric mean of sub-indexes should be chosen.

The first alternative is appropriate when relative price changes and relative quantity changes are uncorrelated. The second is appropriate when they move inversely. This is more likely when the components are close substitutes in consumption. The textbook example of this is chicken and beef, a fall in chicken prices relative to beef prices inducing consumer substitution of chicken for beef.

The choice between these alternatives is not necessarily an either/or choice, because constant relative quantities and the weighted arithmetic mean may be more appropriate for some aggregates than for others. Substitution effects are more likely to be important within lower-level aggregates such as Meat, than between Meat and other food lower-level aggregates, that is to say within the higher-level food aggregate. Substitution effects between, for example, Food and Clothing, are even less likely, so that there is a presumption that constant relative quantities and the weighted arithmetic mean are appropriate when the overall Consumer Price Index is computed as a weighted mean of the highest-level aggregates such as Food, Clothing, Transport etc. Substitution effects are also unlikely in response to differential price changes in different regions of the country. Thus the aggregates for which constant value shares and the weighted geometric mean may be appropriate are regional low-level aggregates.

The decision of when to use arithmetic means and when to use geometric means should be made by investigating, for different aggregates, which would best approximate the most recently estimated weights of their sub-aggregates:

the preceding set of weights of their sub-aggregates with each weight price-updated to the year of the most recently estimated weights and divided by their new sum, implying constant relative volumes

the preceding set of weights of their sub-aggregates without any price-updating, implying constant value shares.

[edit] An alternative to price-updating

There is a good case for using a full year as link period, with chaining year upon year instead of using a single month.

For example, instead of price-updating the 1997 annual weights to the link month of December 1999, the subindexes would be price-backdated to their average for 1997 which would be used as the link year. Thus the index for May 2000 becomes: $\begin{array}{c} CPI_{b:May00} = CPI_{b:\overline {97} } \times \sum {\frac{{V_{97} }}{{\sum {V_{97} } }}I_{\overline {97} :May00} } \\ \\ \end{array}$

with $I_{\bar 9\bar 7:May.00} = I_{Dec.99:May00} \times \frac{{I_{b:Dec.99} }}{{\sqrt[{12}]{{I_{b:Jan.97} \times I_{b:Feb.97} \cdots I_{b:Dec.97} }}}}$ or its arithmetic equivalent.

The ratio of this May 2000 index to the old May 99 index is: $\frac{{CPI_{b:\overline {97} } \times \sum {\frac{{V_{97} }}{{\sum {V_{97} } }}I_{\overline {97} :May00} } }}{{CPI_{b:May99} }}$

To understand the difference between using a whole year rather than a single month, compare:

the index for May 00 with monthly linking in December 1999

with

what it would be with yearly linking in 1997 by dividing the former by the latter:

$\frac{{CPI_{b:Dec.99} \times \sum {\frac{{{\rm{V}}_{{\rm{97}}}^{} \times I_{\overline {97} :Dec.99}^{} }}{{\sum\limits_{} {\left( {{\rm{V}}_{{\rm{97}}}^{} \times I_{\overline {97} :Dec.99}^{} } \right)} }}I_{Dec.99:May00} } }}{{CPI_{b:\overline {97} } \times \sum {\frac{{V_{97} }}{{\sum {V_{97} } }}I_{\overline {97} :May00} } }} = \frac{{CPI_{b:Dec.99} }}{{CPI_{b:\overline {97} } }} \div \sum {\frac{{V_{97} }}{{\sum {V_{97} } }}} I_{\overline {97} :Dec.99}^{}$

This ratio is seen to equal the year 1997 to December 1999 change calculated with the old b weights, divided by the same change calculated retrospectively using the new 1997 weights. Comparing the May 99 to May 00 twelve-month change in the monthly linked index with that in the yearly linked index yields exactly the same expression. If substitution effects dominate, it may come out slightly below unity. (It can be said that the old index is upward biased on account of substitution effects only if b was both its price reference-period and its weight reference-period, so that b quantities were optimal with respect to b prices.)

The advantages of yearly links in the Consumer Price Index are several:

The weights, not being price-updated, can be simply described as weight reference-year value proportions.
The index can be simply described as a Laspeyres index if no geometric mean formulas are used, as it compares the current month's value of weight reference-year consumption with its weight reference-year value.
The index can accommodate Rothwell-type sub-indexes for seasonal products of constant quality.
Retrospective comparison can be made with a superlative index.

Since yearly and monthly linking both require year-97 to December-00 indexes for each and every component, V97, the admitted difficulty of estimating them retrospectively for new components introduced into the 1997 weights does not affect the choice between them.

[edit] Chaining and aggregation

[edit] Decomposition

Index users often want to ask questions of the kind illustrated by the following examples:

How much of the change in the index was due to food price changes?
What is the division of the index between its durable and non-durable components?
What would the index be if shelter costs were omitted?

All of these require some decomposition of the index, but the possibilities are limited.

[edit] Transitivity and Intransitivity

For indexes that are transitive it makes no difference whether the index for May 2000 is computed by directly relating May 2000 prices to December 1999 prices or whether it is computed by chaining from month to month. Transitivity means that, for example:

${\rm{I}}_{{\rm{Dec}}{\rm{.99:May 2000}}} = {\rm{I}}_{{\rm{Dec}}{\rm{.99:Jan 2000}}} \times I_{Jan.2000:Feb.2000} \times {\rm{I}}_{{\rm{Feb}}{\rm{.2000:Mar}}{\rm{.2000}}} \times {\rm{I}}_{{\rm{Mar}}{\rm{.2000:Apr}}{\rm{.2000}}} \times {\rm{I}}_{{\rm{Apr}}{\rm{.2000:May}}{\rm{.2000}}}$

"Elementary aggregate" or "micro" indexes calculated as unweighted ratios of arithmetic or geometric mean prices of the same unchanged set of sampled products are transitive. They are discussed in another chapter. But the indexes under consideration here, which are calculated as weighted sums of lower-level indexes, are not transitive. Chaining January to February with February to March, and so on, using the same value share weights throughout, would amount to altering the implicit quantities from month to month. The indexes therefore have to be calculated as fixed-base indexes by directly relating each month's prices to December 1999 prices.

The result of this is that, for these non-transitive indexes, the price change from, say, February 2000 to May 2000 has to be calculated as:

$\frac{{\sum\limits_i {w^i I_{Dec.99:May2000}^i } }}{{\sum\limits_i {w^i I_{Dec.99:Feb2000}^i } }}$

which could perhaps be described as chain linking May with February indirectly via December 1999.

[edit] Additivity of sub-indexes with given weighting

Such I_Feb:May sub-indexes cannot be weighted and summed to obtain the overall ratio of the May CPI to the February CPI. Index ratios are not additive, as can simply be demonstrated for two components, A and B :

${\rm{CPI}}_{{\rm{Feb:May}}} = \frac{{w^A I_{Dec.99:May.2000}^A + w^B I_{Dec.99:May.2000}^B }}{{w^A I_{Dec.99:{\rm{Feb}}.2000}^A + w^B I_{Dec.99:{\rm{Feb}}.2000}^B }} \ne w^A \frac{{I_{Dec.99:May.2000}^A }}{{I_{Dec.99:{\rm{Feb}}.2000}^A }} + w^B \frac{{I_{Dec.99:May.2000}^B }}{{I_{Dec.99:{\rm{Feb}}.2000}^B }}$

The proportional increase (or decrease) in the overall Consumer Price Index between February and May 2000 is:

$\begin{array}{l} {\rm{CPI}}_{{\rm{Feb:May}}} - 1 = \frac{{w^A I_{Dec.99:May.2000}^A + w^B I_{Dec.99:May.2000}^B - w^A I_{Dec.99:{\rm{Feb}}.2000}^A - w^B I_{Dec.99:{\rm{Feb}}.2000}^B }}{{w^A I_{Dec.99:{\rm{Feb}}.2000}^A + w^B I_{Dec.99:{\rm{Feb}}.2000}^B }} \\ \quad \quad \quad \quad \quad = \frac{{w^A \left( {I_{Dec.99:May.2000}^A - I_{Dec.99:{\rm{Feb}}.2000}^A } \right) + w^B \left( {I_{Dec.99:May.2000}^B - I_{Dec.99:{\rm{Feb}}.2000}^B } \right)}}{{{\rm{CPI}}_{Dec.99:{\rm{Feb}}.2000} }} \\ \end{array}$

Separating the absolute increase (or decrease) in the overall index into additive A and B components is simpler :

$\begin{array}{l} \quad {\rm{CPI}}_{{\rm{Dec}}{\rm{.99:May}}{\rm{.2000}}} \quad minus\quad {\rm{CPI}}_{{\rm{Dec}}{\rm{.99:Feb}}{\rm{.2000}}} \\ = \left( {w^A I_{Dec.99:May.2000}^A + w^B I_{Dec.99:May.2000}^B } \right) - \left( {w^A I_{Dec.99:{\rm{Feb}}.2000}^A + w^B I_{Dec.99:{\rm{Feb}}.2000}^B } \right) \\ = w^A \left( {I_{Dec.99:May.2000}^A - I_{Dec.99:{\rm{Feb}}.2000}^A } \right) + w^B \left( {I_{Dec.99:May.2000}^B - I_{Dec.99:{\rm{Feb}}.2000}^B } \right) \\ \end{array}$

These expressions, and those that follow, apply equally to decomposition of a sub-index into the sub-sub-indexes for its components.

[edit] Additivity with reweighting

If a new set of weights based on more up-to-date consumption value data is introduced, the index has to be chained. An aggregation problem then arises for all comparisons that overlap the weight change. Consider, for example, a comparison of May 2000 with May 1999, supposing that:

Old weights, w₀ were used up to December 1999, with December 1997 as price reference-month. The indexes from May 1999 to December 1999, $I M a y .99: D e c .99$ , thus equal the ratios $\frac{{I_{Dec.97:Dec.99} }}{{I_{Dec.97:May.99} }}$ ;
New weights, wn, were used thereafter for the index from January 1999 onwards and December 1999 replaced December 1997 as the price reference-month;

The overall Consumer Price Index for the twelve-month interval from May 1999 to May 2000 must be computed aggregatively by chaining together the overall Consumer Price Index from May 1999 to December 1999 and the overall Consumer Price Index from December 1999 to May 2000: $\left( {w_o^A I_{May.99:Dec.99}^A + w_o^B I_{May.99:Dec.99}^B } \right)\quad \times \quad \left( {w_n^A I_{Dec.99:May.2000}^A + w_n^B I_{Dec.99:May.2000}^B } \right)$

It cannot be obtained as a weighted sum of the two chained sub-indexes $I_{May.99:Dec.99}^A \times I_{Dec.99:May.2000}^A \quad {\rm{,}}\quad I_{May.99:Dec.99}^B \times I_{Dec.99:May.2000}^B$

unless the two sets of weights are identical.

This is unfortunate. It means, for example, that for comparisons overlapping a weight change, such as from May 1999 to May 2000, it is impossible to divide the overall Consumer Price Index ratio or its proportional change into an A component and a B component.

It is, however, possible to decompose the absolute change in the overall Consumer Price Index into additive A and B components.

Write: CPI_{Dec 97:May 2000} minus CPI_{Dec 97:May 97} as: $CPI_{Dec97:Dec99} \times \;\left( {w_n^A I_{Dec.99:May.2000}^A + w_n^B I_{Dec.99:May.2000}^B } \right)\; - \,\;\left( {w_o^A I^A _{Dec.97:May.99} + w_o^B I^B _{Dec.97:May.99} } \right)$

and then rearrange as:

$\begin{array}{l} \quad \\ \left( {CPI_{Dec97:Dec99} \quad \times \quad w_n^A I_{Dec.99:May.2000}^A } \right) - w_o^A I^A _{Dec.97:May.99} \\ + \\ \left( {CPI_{Dec97:Dec99} \quad \times \quad w_n^B I_{Dec.99:May.2000}^B } \right) - w_o^B I^B _{Dec.97:May.99} \quad \\ \end{array}$

A weight-change-overlapping sub-index for the A or B component regarded separately would require a separate computation, using the corresponding two sets of weights for its sub-sub-indexes and chaining through December 1999. There is no single set of higher level weights which will combine such chained lower-level sub-indexes overlapping a reweighting into a higher level index.

An imaginary numeric example, with the figures unrounded, follows. The price reference-period for the New index is December 1999. The indexes excluding Food are calculated using only Clothing, Housing and Everything-else weights. The May to May indexes in the bottom two lines are calculated as the product of an index for May 1999 to December 1999 (derived by dividing the December index by the May index) and an index for December 1999 to May 2000.

	Old index					New Index
	Weight	May 1999 index	December 1999 index	May index * Weight	December index * Weight	Weight	May 2000 index	May index * Weight
Food	0.2	1.06	1.07	0.212	0.214	0.15	1.04	0.156
Clothing	0.1	1.05	1.08	0.105	0.108	0.15	1.02	0.153
Housing	0.4	1.01	1.02	0.404	0.408	0.3	1.01	0.303
Everything-else	0.3	1.02	1.04	0.306	0.312	0.4	1.02	0.408
Overall index	1			1.027	1.042	1		1.0200
Excluding Food	0.8			1.01875	1.035	0.85		1.0146706
Overall index May 1999 to May 2000				= 1.042 / 1.027 * 1.0200				1.034898
Index excluding Food May 1999-May 2000				=1.035 /1.01875 * 1.0164706				1.032684

This all assumes an unchanged set of sub-indexes. But sometimes the set is changed. For example, reverting to the case where there are subindexes for A and B, reweighting might entail the introduction of a new and additional sub-index for a component of consumption, C. This may have been too small to have been included in the old weights, w_o, or may have been deliberately excluded from coverage when they were estimated, but is now to be included in the index computed with new weights, w_n, from December 1999 onward. In this case, the overall Consumer Price Index from May 1999 to May 2000 will be: $\left( {w_o^A I_{May.99:Dec.99}^A + w_o^B I_{May.99:Dec.99}^B } \right) \times \left( {w_n^A I_{Dec.99:May.2000}^A + w_n^B I_{Dec.99:May.2000}^B + w_n^C I_{Dec.99:May.2000}^C } \right)$

[edit] Prices that were zero

A special case of the introduction of new weights arises when government policy changes so that prices have to be paid for a group of goods or services, C, which were previously provided free. A new sub-index has to be created in order to include them in the index. This, for once, requires information about quantities, which should, however, be available from the branch of government that provided the free goods or services.

Suppose that the new sub-index is introduced from the beginning of 1999 and consider an index for, say, May 2000 calculated using 1997 weights price-updated to December 1999. It provides an estimate of:

$\frac{{\mbox{Market value of 1997 annual consumption at May 2000 prices}}}{{\mbox{Market value of 1997 annual consumption at December 1999 prices}}}$

Writing p^c and q^c for the prices and quantities of the C goods or services this becomes: $\frac{{V_{97}^A I_{\overline {97} :Dec.99}^A I_{Dec.99:May2000}^A + V_{97}^B I_{\overline {97} :Dec.99}^B I_{Dec.99:May2000}^B + \sum {p_{May2000}^c q_{1997}^c } }}{{V_{97}^A I_{\overline {97} :Dec.99}^A + V_{97}^B I_{\overline {97} :Dec.99}^B }}$

so that $\frac{{\sum {p_{May2000}^c q_{1997}^c } }}{{V_{97}^A I_{\overline {97} :Dec.99}^A + V_{97}^B I_{\overline {97} :Dec.99}^B }}$ has to be added to the index computed only for A and B.

The opposite case arises when a product which had its own weight disappears from the market as, for example, when leaded petrol ceases to be sold. This should not be allowed to affect the index. If its weight were simply added to the weight for a similar product or product group, for example unleaded petrol, the index might change even if the prices of everything remaining in the index had not changed. Chaining is therefore necessary, for example, the last petrol price index which included leaded petrol should be moved forward by the index for unleaded petrol.

[edit] Reweighting without price-updating

Having examined the introduction of new weights which are price-updated to the price reference-month, consider the introduction of new weights without any price updating.

Suppose that:

Old weights, w_o were used up to December 1999, with December 1997 as price reference-month.
New weights, wn, were used thereafter, but December 1997 continues to be used as the price reference-month;

and consider the overall Consumer Price Index for May 2000. This can be computed by chaining the overall Consumer Price Index from December 1997 to December 1999, computed with the old weights, with the overall Consumer Price Index from December 1999 to May 2000, computed with the new weights: $\left( {w_o^A I_{Dec.97:Dec.99}^A + w_o^B I_{Dec.97:Dec.99}^B } \right)\quad \times \quad \frac{{w_n^A I_{Dec.97:May.2000}^A + w_n^B I_{Dec.97:May.2000}^B }}{{w_n^A I_{Dec.97:Dec.99}^A + w_n^B I_{Dec.97:Dec.99}^B }}$

In this case too, no decomposition into the separate contributions of A prices and B prices to the overall increase is possible for periods overlapping December 1999.

[edit] Price-updating without real change

Note that these problems attached to reweighting arise only when there are real changes in the weights. Merely price-updating them to a new reference-period, n, does not prevent the combination of sub-indexes to produce a higher-level index.. Price-updating multiplies each old weight, w_o, by its sub-index and then divides by the sum of all the weights each multiplied by its sub-index, thus preserving the sum of the price-updated weights, w_n, as unity. In this case the index from May 1999 to May 2000, the product of the 1999 May to December index and the December 1999 to May 2000 index $\left( {w_o^A I_{May.99:Dec.99}^A + w_o^B I_{May.99:Dec.99}^B } \right)\quad \times \quad \left( {w_n^A I_{Dec.99:May.2000}^A + w_n^B I_{Dec.99:May.2000}^B } \right)$

can be expressed as:

$\begin{array}{l} \quad \quad \left( {w_o^A I_{May.99:Dec.99}^A + w_o^B I_{May.99:Dec.99}^B } \right)\quad \\ \times \quad \left( \begin{array}{l} \quad w_o^A \frac{{w_o^A I_{May.99:Dec.99}^A }}{{\left( {w_o^A I_{May.99:Dec.99}^A + w_o^B I_{May.99:Dec.99}^B } \right)}}I_{Dec.99:May.2000}^A \\ + w_o^B \frac{{I_{May.99:Dec.99}^B }}{{\left( {w_o^A I_{May.99:Dec.99}^A + w_o^B I_{May.99:Dec.99}^B } \right)}}I_{Dec.99:May.2000}^B \\ \end{array} \right) \\ \end{array}$

which simplifies to:

${\rm{CPI}}_{{\rm{May99:May 2000}}} = \left[ {w_o^A \left( {I_{May.99:Dec.99}^A I_{Dec.99:May.2000}^A } \right) + w_o^B \left( {I_{May.99:Dec.99}^B I_{Dec.99:May.2000}^B } \right)} \right]$

The A and B components are obviously additive.

[edit] Seasonal products

[edit] The problem

The computation of a Consumer Price Index involves a comparison between:

The current month' s cost to consumers of (a sample of) the products that constituted the weight reference-year' s consumption.
The price reference-period cost to consumers of (the sample of) the products that constituted the weight reference-year's consumption.

This creates an obvious problem in months when some products are wholly unavailable, or too largely unavailable to allow their prices to be collected. The most important such "seasonal" products are fresh fruit, vegetables and fish, cut flowers, package holidays and, in many cases, sporting goods and clothing..

[edit] Practical alternatives

There are three ways out of this dilemma. All three are unsatisfactory, but one of them has unavoidably to be chosen. They are:

Completely omit seasonal products from the index, thus limiting its coverage to products that are available throughout the year.
Pretend that unavailable seasonal products are available and invent prices for them. There two ways of doing this:
1. Carry forward the last collected prices unchanged,
2. Extrapolate the last collected prices by the month-to-month change in the prices of available products.
In both cases annual weights will be used. Prices can either be compared with prices (including any fictitious prices) for a weight reference-month, or they can be compared with average prices for a weight reference-year
Use monthly weights and a price reference-year. The computation for each month will compare the current price for each product available in that month with the corresponding price reference-year weighted average price of that product. For each group of seasonal products a sub-index is computed using monthly weights equalling quantities from that month in the weight reference-year valued at weighted average prices from the price reference-year. Thus the August sub-index for such such a group will compare the current August value of August weight reference-year consumption with its value at price reference-year weighted average prices.

Monthly quantities are required both to compute the weights of the available fruits for each month and to compute the weighted annual average price of each fruit in the price reference-year. It may be better to derive the weights from data for several years, since the seasonal pattern can vary between years.

In the absence of monthly data, weights will simply have to be set at zero for months of non-availability and at one for months of availability.

Which of these is chosen can have a considerable effect upon the index.

Whether the index views consumption as Transactions, as Expenditure or as Use will make a difference for products where the timing of transactions, payments and use can differ. Thus transactions in which consumers book a July package holiday may be made as early as January, payment of part of the price may follow in a later month, and the holiday can only be taken in July.

[edit] Pros and Cons

Method (2.ii) assumes that, if the products had remained available, their prices would have moved in the same way as the prices of the products within the broader consumption category which did remain available. This is equivalent to giving a greater weight to these other products. But though this way of imputing fictitious prices is algebraically equivalent to the use of monthly weights, its implicit monthly weights are unreasonable. To see why, consider the simple case where annual expenditures on oranges and on apples are each 40, spread evenly over the twelve months, while annual expenditure on cherries is 20, spread evenly over only the two months of July and August. Then, if cherry prices are extrapolated forward according to the movement of apple and orange prices, starting each September and continuing monthly until the following June, apples and oranges each acquire effective weights of 50 per cent during these ten months. This is reasonable. In July and August, apples and oranges have weights of 40 per cent and cherries have a weight of 20 per cent. But if cherries account for 20 per cent of total annual expenditure on the three types of fruit, they must account for vastly more than 20 per cent of expenditure on fruit in July and August! The implicit July and August weights are, therefore, not reasonable for measuring month-to-month changes.

Both methods (2.i) and (2.ii) have the advantage as compared with (3) that they can cope with differences between years in the timing of availability. If cherries become available a month earlier than usual, their actual current price can be used instead of a fictitious price. But method (3) requires that a current cherry price be collected only for each month for which cherries have a weight. If, on the other hand, cherries cease to be available a month earlier than normal, a fictitious price will have to be used under all of these methods.

A drawback with method (3) is that the index may change between two months because of a weight change, even though no price has changed and availabilities are unaltered. Furthermore, it may seem unclear what an index comparison between two successive months in different seasons signifies. Is it meaningful, for example, to say that the June 2001 cost of buying the June 2000 basket of fruit compared with its cost at average 2000 prices exceeds the May 2001 cost of buying the May 2000 basket compared with its cost at average 2000 prices?

$\frac{{\sum {{\rm{P}}_{June01} Q_{June00} } }}{{\sum {{\rm{P}}_{\overline {00} } Q_{June00} } }} > \frac{{\sum {{\rm{P}}_{May01} Q_{May00} } }}{{\sum {{\rm{P}}_{\overline {00} } Q_{May00} } }}$

If method (3) is employed, as many statisticians recommend, the use of a whole year as price reference-period for seasonal products may require the use of a whole year as weight and price reference-period for all other sub-indexes as well. This can be avoided if the sub-indexes for seasonal products are rescaled from their price reference-year to a price reference-period of a single month.

Leaving aside any such rescaling, in method (3), a sub-index for May 2002

with 2000 as both weight reference-year and price reference-year would be an estimate of:

$\frac{{\sum {p_{2002,May} q_{2000,May} } }}{{\sum {\bar p_{2000} q_{2000,May} } }} = \sum {\frac{{p_{2002,May} }}{{\bar p_{2000} }}\left( {\frac{{\bar p_{2000} q_{2000,May} }}{{\sum {\bar p_{2000,May} q_{2000,May} } }}} \right)}$

where the summation is over all products and p with a bar signifies the quantity-weighted annual average price. The weight for each product (the bracketed term) is not its share in weight reference-period May value; it is its share calculated using weighted average 2000 prices to value the May quantities of products.

[edit] No weights available

(The graph can be uploaded as image: Fruit price indexes; Turvey's imaginary data)

[edit] Appendix: An in-between formula

A variant of the $\sum {wI_{p:t} }$ formula suggested by Brent Moulton can provide a result which falls somewhere between the two extremes so far examined, providing an approximation to a superlative index – if substitution effects are the only factor causing weights to alter. If adopted, consistency requires that this variant be applied both to the price-updating of weights and the computation of the index using those weights.

This variant is (omitting the i superscripts) $\left( {\sum {wI_{p:t}^k } } \right)^{\frac{1}{k}}$ where k is a constant which will be lower the greater is substitutability between the sub-aggregates and hence the greater are demand elasticities.. In the case of no substitutability, implying zero price elasticities, k = 1 and the formula reduces to $\sum {wI_{p:t} }$ . The corresponding formula for weights price-updated from the weight reference-period, w, to the price reference-period, t, is: $\frac{{w_p = \left( {w_w I_{w:p} } \right)^k }}{{\sum {w_p = \left( {w_w I_{w:p} } \right)^k } }}$

With k = 1 this formula gives the same weights as the price-updated weights normalised to sum to unity, while with k almost equal to 0 the formula gives almost equal weights.

When past weights are available for two years, application of the formula to the earlier weights will allow those values of k to be determined which best predict the actual later weights for each aggregate and its sub-aggregates. There is then a presumption that:

The formula should apply these values of k to update the latest available weights to the price reference-period currently being used
Each index and its sub-indexes should then be computed as $\left( {\sum {wI_{p:t}^k } } \right)^{\frac{1}{k}}$ , using the same values of k.

Where high-level aggregate indexes are combined to update weights, k = 1 will be likely to give the least bad answer. Since such aggregates are not substitutes for one-another, straightforward price-updating is appropriate But k < 1 may well be appropriate for lower-level aggregates where substitutability may exist.

Turning now from the price-updating of weights to the computation of the index, the problem is to find, for any aggregate and its sub-aggregates, that value of k which brings the index for that aggregate computed as, for example, $\left( {\sum {w_{87} I_{87:94}^k } } \right)^{\frac{1}{k}}$ closest to a corresponding superlative index that compares the later year with the earlier year. A "superlative" index uses some kind of symmetric average of weights reflecting the consumption patterns of the two terminal years. Such an index can, of course, only be computed retrospectively, so can only be used as a standard against which an actual monthly Consumer Price Index can be checked after the event. If k turns out to have been stable this might justify using its past value in current computations.

Theoretical discussions of ideal indexes and superlativeness assume that the weight and price reference-periods coincide and are of the same length. The pure Laspeyres and Paasche indexes and the (superlative) Fisher index to which this theory relates can be calculated only for whole years and not for months. Moulton proved that the above formula yields an exact true cost of living index for a representative consumer with unchanging homothetic preferences with a constant elasticity of substitution equal to 1-k. A representative consumer is presumably an adult hermaphrodite with a non-integer number of children.

Shapiro, M. and Wilcox, D., in their paper "Alternative strategies for aggregating prices in the CPI"^[1] have applied this idea in a one-stage computation of the overall U.S. Consumer Price Index from 9,108 area/product sub-indexes and in a two-stage computation, in which they first applied $\left( {\sum {w_{} I_{}^k } } \right)^{\frac{1}{k}}$ to the 207 product indexes, then used $\sum {wI_{} }$ to aggregate over the 44 areas. The justification of this two-stage approach was the eminently reasonable assumption that substitutability between areas is extremely low, so that k ≈ 1.

As they computed their indexes from December to December instead of from whole-year to whole-year, they had to use a Fisher- or Törnqvist-type approximation to a pure Fisher or Törnqvist index.. Their Laspeyres-type index weights elementary aggregate indexes based on December in the preceding year by expenditures for the whole of that preceding year; their Paasche-type weights them by expenditures for the whole of the current year. Their Törnqvist-type index uses the simple mean of the weights for the two years. They note that this method could be refined in two ways. One would centre the price reference-period within the period over which the expenditures are calculated, e.g. basing the elementary aggregate indexes on mid-year rather than December prices. The other would price-update or down-date expenditures to the price reference-month (and perhaps the current month).

Percentage points difference between annual change of $\left( {\sum {w_{} I_{}^k } } \right)^{\frac{1}{k}}$ index and Törnqvist index annual change
	k = 0.4	k = 0.3	k = 0.2
1987-8	.06	.04	.02
1988-9	.02	-.0	-.02
1989-90	-.0	-.05	-.09
1990-1	-.08	-.09	-.09
1991-2	.03	0	-.20
1992-3	.07	.04	.02
1993-4	.06	.04	.02
1994-5	.04	.02	.01
Mean difference	.02	0	-.02
Standard Deviation	.05	.04	.05
Mean annual index change	3.35	3.32	3.30

Their result, formulated in terms of 1-k, indicated that the value of k which gave the closest approximation to the superlative index was 0.3. This followed from the following comparison of December to December changes in the chained index computed for different values of k, with the changes in an annually chained Törnqvist index:

The difference from the previous calculation is due to the fact that this one relates to a large number of low-level aggregates instead of a small number of high-level aggregates, so that much more substitutability between aggregates would be likely.

The greater complexity of $\left( {\sum {wI_{p:t}^k } } \right)^{\frac{1}{k}}$ as compared with $\sum {wI_{p:t} }$ and $\prod {I_{p:t}^w }$ has two consequences. The first is that it is more difficult to explain to users. The second is that the change in it between a pair of months cannot be decomposed to show the contribution of the various components to the aggregate change. Perhaps it should only be employed to construct sub-indexes for low-level aggregates which would then be combined to compute higher-level indexes and the overall index employing one of the two simpler formulas.

[edit] Notes

^[a] An equally meaningful concept, hitherto applied in Sweden, relates to twelve monthly price changes from the beginning to the end of a year, from pby to pey, using the annual quantities of the whole of that year:

$\frac{{\sum {p_{ey} q_y } }}{{\sum {p_{by} q_y } }}$

Using monthly subindexes from December of year y-1 to December of year y, this can be estimated as:

$\sum {\frac{{V_y }}{{\sum {V_y } }}} I_{y - 1,12:y,12}$

[edit] References

^ Shapiro, M. and Wilcox, D. "Alternative strategies for aggregating prices in the CPI" in Federal Reserve Bank of St Louis Review, May/June 1997 Volume 79, no.3