Consistent aggregation with superlative and other price indices

Various fields of economic analysis (e.g. growth and productivity) and economic policy (e.g. monetary and social policy) rely on accurate measures of price change. Unfortunately, the price index formulae that most price statisticians consider as particularly accurate—the superlative indices of Fisher, Törnqvist, and Walsh—are believed to violate the property of consistency in aggregation. This property, however, is indispensable for economic studies that attempt to disaggregate the overall result into the contributions of individual entities such as sectors of the economy or groups of products. The present paper introduces a thoroughly motivated formal definition of consistency in aggregation and proves that, contrary to general perception, the three superlative price indices can be considered as consistent in aggregation. Furthermore, many other price indices are shown to be consistent in aggregation. The theoretical findings are applied to the Swedish consumer price index.


| INTRODUCTION
In most fields of applied economic analysis, the diversity of individual changes must be aggregated into some single number measuring the average change. Prominent examples are the changes in [Corrections added on 15 December 2020, after first online publication: The alignment of lines, use of italic and font size in definitions and propositions was corrected.] national income, unemployment, money supply or prices. To provide additional insights, the computation of the average change is often conducted in a two-stage procedure, where on the first stage average changes of subgroups are computed and on the second stage these individual results are aggregated into the overall change.
For example, some central banks and many financial analysts decompose the universe of consumer products into the 'core products' (all products except for energy and seasonal food) and the 'non-core products' (energy and seasonal food). The average price change of the core products is called the core inflation, whereas the average price change of the non-core products could be denoted as the non-core inflation.
Core inflation is often considered as a measure of the long-run inflation trend and, therefore, as a key indicator for monetary policy. Averaging the core inflation and the non-core inflation yields the overall inflation rate of the economy. The overall inflation rate is often used for indexing various types of contracts and for transforming nominal values into real values (e.g. national income). The separate compilations of the core inflation and the non-core inflation reveal how strongly the economy's current overall inflation is driven by its long-run inflation trend. 1 Alternatively, the overall inflation rate could be directly computed from the complete universe of products, without decomposing this universe into the core products and the non-core products. This single-stage computation is simpler, but provides fewer insights.
The calculation of an average price change is accomplished by some price index formula. It is considered as a major advantage of a price index formula, when it computes the same overall inflation, regardless of whether it is applied in a single-stage or two-stage calculation. When a price index formula satisfies this postulate, the formula is denoted as consistent in aggregation.
The notion of consistency in aggregation has been alluded to by van Yzeren (1958). Vartia (1976a,b) fleshed out this notion and Blackorby and Primont (1980) formalized and generalized it. Stuvel (1989) and Balk (1995Balk ( , 1996Balk ( , 2008 take the position that in the field of price measurement the general notion proposed by Blackorby and Primont is not adequate. They recommend Vartia's narrow notion. von Auer (2004) contests this position and advocates a definition of consistency in aggregation that is more general than the definition of Balk and Vartia, but less general than that of Blackorby andPrimont. Pursiainen (2005, 2008) provides a rigorous formal treatment of Vartia's original notion.
All of these studies agree that consistency in aggregation of a price index not only requires that the single-stage and the two-stage computations yield the same outcome, but also that three additional conditions (details are spelled out below) must be satisfied. In the following, this common view is referred to as the 'four consensus conditions' of consistency in aggregation. As the above dispute suggests, however, endorsement of the consensus conditions still leaves much space for disagreement.
The present paper's first main contribution is an attempt to settle this dispute. For this purpose, it develops a thoroughly motivated new definition of consistency in aggregation. It is more general than those proposed by von Auer (2004, p. 390), Balk (1995Balk ( , p. 85, 1996, Pursiainen (2005, p. 21;2008, p. 18), Stuvel (1989, p. 36), van Yzeren (1958, and Vartia (1976aVartia ( , pp. 85-89, 1976b. The analysis reveals that several attractive price indices that, hitherto, have been perceived as violating the consensus conditions, turn out to be fully consistent with these conditions. We show that these attractive price indices are consistent in aggregation and, therefore, perfectly appropriate for multi-stage computations in applied empirical analysis. In other words, when using these 1 There is a vivid controversy on the rationale for using the core inflation as a yardstick for monetary policy (e.g. Crone et al.,

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VON AUER ANd WENGENROTH price indices in a two-stage computation, it is possible to recover the same index numbers that would have been obtained if everything had been computed in a single stage. For practical purposes, this is an extremely valuable property.
It would be useful, if among these price indices were a superlative one. The concept of superlative price indices has been introduced by Diewert (1976). These indices are often advocated as generating particularly reliable results and being firmly anchored in economic theory. The most vehemently recommended superlative price indices are the Fisher, Törnqvist, and Walsh indices. However, these indices have been perceived as not being consistent in aggregation (e.g. Balk, 2008, pp. 110-111;Diewert, 2004a, pp. 349-350). The present paper proves that this perception cannot be maintained in the context of our new definition of consistency in aggregation. This is the second contribution of this study. As a third contribution it shows that many other price indices are consistent in aggregation.
This paper is organized as follows. Section 2 provides a more detailed account of the consensus conditions. Section 3 leaves the field of price indices and develops a precise mathematical definition of consistent aggregation rules. In Section 4, we return to the analysis of price indices and provide a rigorous mathematical definition of price indices. In Section 5, we apply this definition to a multi-stage computation of the Swedish consumer price index. The results suggest that superlative price indices may well be consistent in aggregation. Section 6 connects the theoretical concepts developed in Sections 3 and 4. It presents a rigorous mathematical definition of a price index that is consistent in aggregation. In Section 7, we examine whether superlative price indices exist that are consistent in aggregation. In Section 8, we show that many (non-superlative) price indices are consistent in aggregation. An application to the Swedish price data is presented in Section 9. For practical price measurement purposes, additional requirements can be attached to our definition of consistency in aggregation. Section 10 explains these requirements, examines which price index formulae satisfy these requirements and relates the new definition of consistency in aggregation to alternative definitions proposed in the literature. Concluding remarks are contained in Section 11.

| T WO-STAGE COMPUTATION OF PRICE CHANGES
In an economy, a vast number of goods and services are sold. The prices of these 'items' change over time. A price index attempts to measure the items' average price change between a base and a comparison period. It is assumed that during both periods all N items are available and that their prices and quantities are correctly recorded. Let D denote the finite set containing the N items' prices and quantities. A price index formula P (e.g. Laspeyres index) is usually considered as a function that maps the recorded prices and quantities into a single positive number that indicates the N items' average price change.
The single-stage computation of the overall price change applies a given price index formula P to the complete set D. In contrast, a two-stage computation starts by partitioning the set D into several subsets D k . For each of these subsets, a price index P k is computed. In the second stage of this two-stage procedure, the numbers P k are aggregated into the overall price change. Such a two-stage computation provides important additional insights, because it allows to identify the individual forces driving the overall result. Of course, the single-stage and the two-stage computations should be 'consistent'. The precise meaning of 'consistency', however, is difficult to define.
As pointed out before, some consensus conditions exist that spell out the meaning of consistency in more detail. These consensus conditions relate only to price indices P that are continuous in the prices and quantities (continuity axiom) and that are invariant with respect to changes in the units in which the quantities are measured (commensurability axiom). 2 For such price indices, the consensus conditions specify how a two-stage computation should be conducted and how it should relate to the single-stage computation (e.g. von Auer, 2004, p. 385;Balk, 1995, p. 85;Balk, 1996, pp. 358-59;Balk, 2008, pp. 108-109;Vartia, 1976b, p. 124): 1. For all possible partitions of the set D, the two-stage computation of the overall price change of D must yield the same index number as the single-stage computation. 2. On both stages of the two-stage computation, the 'same index formula' must be applied as in the single-stage computation (only the number of variables can be different). 3. In the first stage of the two-stage computation, for each subset D k , a price index number P k and one or more aggregate values are computed. The price index numbers P k and the aggregate value(s) depend only on the prices and quantities of the items in subset D k . 4. In the second stage of the two-stage computation, the index number for the complete set D depends only on the index numbers P k and the aggregate values computed in the first-stage computations.
Unfortunately, these consensus conditions leave much space for ambiguity and disagreement. It is therefore necessary to transform the four consensus conditions into a thoroughly motivated formal definition of consistency in aggregation. As a preliminary step we develop the notion of a consistent aggregation rule.
However, Definition 1 covers also meaningless aggregation rules such as A n (d 1 , …, d n ) = d 1 .
To come closer to the intuitive idea of a meaningful aggregation rule, one needs further properties. In the first place, a meaningful aggregation rule requires that the ordering of the data vector ( d 1 , …, d n ) ∈ I n is irrelevant, that is, A n are symmetric. Second, one would expect that the aggregation of ( d 1 , …, d n ) ∈ I n in one step, A n (d 1 , …, d n ), yields the same result as the following procedure: (1) (d 1 , …, d n ) is partitioned into two arbitrary "groups" (d 1 , …, d m ) ∈ I m and (3) these results are treated as new data and aggregated by A 2 .
These two requirements for a meaningful aggregation rule can be summarized in the following definition: As another example of an aggregation rule for I = ℝ, consider the calculation of a product: with d 1 , …, d n ∈ ℝ. This aggregation rule is symmetric. Since for all m, n ∈ ℕ, the aggregation rule also satisfies condition (1). Therefore, it is a consistent aggregation rule.
Recall that a binary operation ⊕ on I, that is a function I 2 → I, (d 1 , d 2 ) ↦ d 1 ⊕ d 2 , is commutative and associative, if

Proof.
For proving the necessity of Equation (3) assume that A is consistent and define d 1 ⊕ A d 2 = A 2 (d 1 , d 2 ). The symmetry of A 2 precisely means commutativity of ⊕ A . Associativity of ⊕ A follows from The representation for the general case A n is shown by induction on n ∈ ℕ. For n = 1, it follows from A 1 (d j ) = d j . If the necessity of Equation (3) is true for some n ∈ ℕ, we get It is obvious that the existence of a commutative and associative binary operation ⊕ A that satisfies Equation (3) is sufficient for A = (A n ) n ∈ ℕ to be a consistent aggregation rule. ▪ Relationship Equation (3) can be interpreted as a formal specification of the rather vague notion 'same functional form'. For example, multiplication is a commutative and associative binary operation ⊕ of form Equation (3). This confirms that Equation (2) is a consistent aggregation rule.
There is a simple but nevertheless quite general method to produce consistent aggregation rules or, conversely, to prove consistency of some given aggregation rule. It utilizes the concept of a 'quasi-sum'. Proposition 2 In the situation of Definition 3, the aggregation rule is consistent. Proof. In view of Proposition 1, it is to be shown that the binary operation ⊕ defined by Equation (4) is commutative and associative. Commutativity of ⊕ is obvious. For n = 3, one gets which is associativity ▪.

It follows from Equation (5) that an aggregation rule A satisfies (4), if and only if
An aggregation rule that satisfies Equation (6) is denoted here as quasi-additive. 4 Proposition 2 says that quasi-additive aggregation rules are consistent. In order to verify that a given aggregation rule A is 4 There is quite some literature on the question which aggregation rules are quasi-additive. A good overview is given in the dissertation of Pursiainen (2005). The minimum of d 1 and d 2 (d 1 , d 2 ∈ I) is a consistent aggregation rule for I = ℝ > 0 which is not quasi-additive. consistent, it is therefore sufficient to find some invertible map Φ : I → M such that Equation (6) holds for all n ∈ ℕ and d i ∈ I.
Of course, the simple summation is additive and therefore also quasi-additive with Φ(d) = d. As another example, consider aggregation rule (2). For I = ℝ > 0 it is consistent, because with Φ(d) = log d Equation (6) is satisfied: We have emphasized the similarity between consistent aggregation rules and the sum or product of numbers or vectors. Aggregating a finite set of data to a kind of average or 'typical value' is an alternative form of aggregation. A simple example is the arithmetic mean for d 1 , …, d n ∈ ℝ. Although symmetric, this aggregation rule fails to be consistent, because it is not associative: except for special cases.
However, the aggregation rule overcomes this problem. By 'inflating' the set I from ℝ to ℝ × ℝ > 0 , the aggregation rule becomes a function of ∑ n i = 1 w i d i and This verifies that aggregation rule (8) satisfies (6). Therefore, it is consistent. The relation between Equations (7) and (8) is that the arithmetic mean (7) is the first component of A n ( (d 1 , 1), …, (d n , 1) ). This shows how consistency is only achieved after augmenting the original aggregation rule. Another simple example is the geometric mean for d 1 , …, d n ∈ ℝ > 0 . Since this aggregation rule is not associative. However, the aggregation rule is a function of This establishes consistency of aggregation rule (10). Then, the geometric mean (9) is the first component of A n ( (d 1 , 1), …, (d n , 1) ) and n is its second component.

| PRICE INDICES AND THEIR ATTRIBUTES
Price index formulae, too, are maps that compute some kind of average, namely the 'overall price change'. Therefore, one can attempt to transform them into a consistent aggregation rule. As a preliminary step, the concept of primary and secondary attributes of a price index must be introduced. The actual transformation of a price index into a consistent aggregation rule is deferred to Section 6. Let S denote the set of integers i = 1, …, N, where each integer represents one of the N items of an economy. All items are available during the base period (t = 0) and the comparison period (t = 1). The period t vector of prices is denoted by p t = (p t 1 , …, p t N ), and the corresponding vector of quantities .
. It is customary to interpret a price index as a mapping of the N-dimensional vectors p 0 , x 0 , p 1 , and x 1 into a single positive number, P ′ (p 0 , x 0 , p 1 , x 1 ).
In practical work, the prices and quantities p 0 i , x 0 i , p 1 i and x 1 i are usually not known. Instead, only the i as well as the price ratios r i = p 1 i ∕ p 0 i are available. 5 However, this does not represent a confinement as long as the applied price index formulae satisfy the commensurability axiom. This axiom postulates that where Λ is some arbitrary N × N diagonal matrix with positive entries i (e.g. von Auer, 2004, pp. 386-387;Eichhorn & Voeller, 1976, p. 24). For a price index that satisfies this axiom, the information in the four vectors p 0 , x 0 , p 1 and x 1 is equivalent to the information contained in the three vectors . This is the commensurability for i = 1∕ p 0 i . As an example, consider the Walsh index: This price index satisfies the commensurability axiom. Therefore, it can be written in the following form: Just like the Laspeyres index, the Walsh index can be interpreted as a weighted arithmetic mean of the price ratios, r i , where the weights represent expenditure weights. However, whereas the Laspeyres weights use only base period expenditures, v 0 i , the weights of the Walsh index are geometric means of the base period expenditures, v 0 i , and the 'deflated' comparison period expenditures, v 1 i ∕ r i . One can show (e.g. von Auer, 2004, p. 393) that all sensible price index formulae satisfy the commensurability axiom. Therefore, any sensible price index can be written in the form P( r, v 0 , v 1 ) and we can take J = ℝ 3 > 0 as the set of possible data, In the price statistics literature, the variables v 0 i and v 1 i are usually denoted as 'values'. However, in the present paper, the term 'value' would have multiple meanings. To avoid confusion, we denote the variables v 0 i and v 1 i as 'expenditures'.
It is customary to interpret a price index formula as a rule that aggregates a finite set of data ( r, v 0 , v 1 ) into a single datum P(r, v 0 , v 1 ) ∈ ℝ > 0 , where it is understood that for each size of the data set, this mapping has the 'same functional form'. As pointed out before, the expression 'same functional form' is not quite appropriate, because these mappings have different domains, and therefore, are totally different objects. As a consequence, the customary definition of a price index formula is not fully satisfactory. To account for different domains, we introduce the following definition of a price index (see also Definition 3.1 in Pursiainen, 2005, p. 21).
> 0 is a sequence P = (P n ) n ∈ ℕ of symmetric and continuous maps The primary purpose of a price index formula is the computation of the overall price change P of the N items in set S, given the data set (r, v 0 , v 1 ) ∈ J N . Restricting the set S to a single item i, the 'overall price change' should be the item's price ratio, r i = p 1 i ∕ p 0 i . Therefore, the price ratio r i is denoted here as the primary attribute of a price index. A price index can be interpreted as a transformation of the primary attribute's values r into some aggregate value P n . However, the value of P n depends not only on r, but also on v 0 and v 1 . Therefore, we denote v 0 i and v 1 i as secondary attributes. More generally, secondary attributes are defined in the following way:

Definition 5 A vector valued secondary attribute is a mapping
where Q ≥ 1 and J = ℝ 3 > 0 is the set containing the original data (r, v 0 , v 1 ). This definition implies that a secondary attribute's value corresponding to some item i, z A price index can have alternative vectors of secondary attributes. For example, the Walsh index (12) can be viewed as a function of the primary attribute r and the secondary attribute z = (z 1 , z 2 ) = (v 0 , v 1 ). However, this index can be also interpreted as a function of the primary attribute r and the secondary attribute z = √ v 0 v 1 ∕ r. Before we move on to develop a rigorous definition of consistency in aggregation of price indices, we present an empirical application of a two-stage price index computation.

| APPLICATION TO SWEDISH PRICE DATA: PART A
The underlying data of this empirical application have been acquired from Statistics Sweden. They cover the base year 2010 (t = 0) and the comparison year 2011 (t = 1). The informational set is ( r, v 0 , v 1 ) ∈ J 360 where the elements v 0 i and v 1 i are annual household expenditures on 360 basic headings i and the elements r i are the respective price ratios. As pointed out before, for the purpose of computing a price index, this informational set is as good as the set p 0 , x 0 , p 1 , x 1 .
Both the expenditure and price data are disaggregated at the four-digit level COICOP classification. Table 1 shows an excerpt of the original data set. It lists for each basic heading i the COICOP number, the product group number, the product name, the price ratio r i , and the expenditures v 0 i and v 1 i . The last column can be ignored for the moment.
In 2005, Statistics Sweden implemented the Walsh index (12) for the compilation of its consumer price index (for details see Bäckström & Sammar, 2012, p. 2). Therefore, the same price index is used here. A single-stage computation of the Walsh index (12) yields the index number P Wa = 1.0275, that is, an overall inflation of 2.75%. 6 For the two-stage computation, the 360 items are partitioned into the two subsets S 1 (core inflation) and S 2 (non-core inflation), where the items i = 1, 2, …, 301 are assigned to the subset S 1 while the items i = 302, …, 360 are assigned to the subset S 2 . An item's price ratio, r i , is defined as its primary attribute and the term z i = is chosen as the only secondary attribute of the Walsh index (therefore, at z i no superscript is necessary). For each item i, the value of its primary attribute is listed in Table 1 in the column with the heading r i while the value of its secondary attribute z i = For each subset, the value of its primary attribute (P Wa 1 and P Wa 2 ) is computed by the Walsh index formula 6 Our data and our price index formulae are not completely equivalent to the data and methodology underlying the compilation of the official Swedish consumer price index.  This yields P Wa 1 = 1.0264 and P Wa 2 = 1.0355. In other words, the Swedish core inflation is 2.64%, while the non-core inflation is 3.55%. Recall that the overall inflation rate was 2.75%, that is, slightly larger than the core inflation rate. Note that formula (13) is the same as Equation (12), the formula applied for the single-stage computation. The aggregate values of the secondary attributes are These two numbers are also listed in Table 1.
The second-stage index formula is with K = 2. This is again the same basic formula as in the single-stage computation. Inserting the results of the first-stage computations P Wa 1 , Z 1 and P Wa 2 , Z 2 in the second-stage formula (14) yields the twostage index number P Wa = 1.0275. This is exactly the same index number as in the single-stage computation (13). Furthermore, the Walsh index (13) seems to satisfy all of the four consensus conditions. This suggests that the Walsh index might be consistent in aggregation.
This is a remarkable conjecture, because '…something resembling a consensus has emerged in the index number literature that inflation and growth should be measured using superlative index number formulae … (Hill, 2006, p. 27)'. The Walsh index is one of the three advocated superlative price indices, the others being the Fisher index, and the Törnqvist index, The concept of superlative price indices was introduced by Diewert (1976). A price index receives the title 'superlative', if an aggregator function (utility function or expenditure function) with a 'flexible' functional form exists, such that its corresponding cost of living index yields the same result as the price index. An aggregator function is 'flexible', if it can provide a second-order approximation to an arbitrary twice differentiable linearly homogeneous aggregator function.
It is well known that the superlative indices of Walsh, Fisher and Törnqvist possess a number of desirable properties and that they approximate each other closely (e.g. Diewert, 1978, p. 888;Hill, 2006, p. 27). However, there is a general perception that none of these superlative price indices is consistent in aggregation (e.g. von Auer, 2004, p. 397;Balk, 2008, pp. 107-108;Diewert, 1978, p. 889;Diewert, 2004a, p. 349-350;van Yzeren, 1958, p. 432-433). Even though Diewert (1978, p. 889) shows that these indices are 'approximately consistent in aggregation', empirical studies that conduct a multi-stage analysis usually avoid superlative price indices. Possibly, a superlative index that is merely approximately consistent in aggregation, is not considered as suitable for a multi-stage analysis. The empirical application of the present paper suggests that, contrary to general perception, at least one of the (superlative) indices is consistent in aggregation. In the remaining sections, this conjecture is verified.

| CONSISTENT AGGREGATION OF PRICE INDICES
A price index P in the sense of Definition 4 is not an aggregation rule in the sense of Definition 1, because the maps P n have values in ℝ > 0 instead of J = ℝ 3 > 0 . However, a price index formula P can be transformed to become an aggregation rule in the sense of Definition 1. The first step is to transform the original data set: In a second step, an aggregation rule must be specified in the sense of Definition 1, A n d 1 , …, d n .
For example, the Walsh index (12) can be transformed into an aggregation rule A Wa This aggregation rule has two components. The first one is the price index formula (12). It maps the data set I n into ℝ > 0 , that is, into some aggregated value of the primary attribute. This aggregate value depends on the individual values of the primary and secondary attribute. The aggregation rule's second component is a mapping that transforms the individual values of the secondary attribute z i into some aggregate value ∑ n i = 1 z i . This aggregate value exclusively depends on the individual values of the secondary attribute. Finally, the two components are combined to form an aggregation rule A for I = ℝ 2 > 0 in the sense of Definition 1. The superscript "Wa" emphasizes that this aggregation rule corresponds to the Walsh Index.
Formula (11) suggests that an alternative form for an aggregation rule relating to the Walsh index exists: This alternative aggregation rule has three instead of two components. The first one is the price index formula (11). In contrast to the first component of A Wa n , the first component of A Wa ′ n does not depend on the values of the primary attribute, but only on the values of the secondary attribute z i .
As a second example, consider the Fisher index P Fi defined by formula (15). It can be transformed into the aggregation rule This aggregation rule has five components. The first component is the price index formula (15). As was true for the aggregation rule A Wa ′ n , this first component does not depend on the values of the primary attribute. Similarly, the Törnqvist index can be written in the form We owe this insight to Bjørn Kjos-Hanssen.
The corresponding aggregation rule is Again, the first component does not depend on the primary attribute.
In Definitions 4 and 5, a price index P and its secondary attribute z were defined. Consistency of an aggregation rule was defined in Definition 2. Building on these three definitions, we define consistency of aggregation of a price index formula P with an explicit reference to its secondary attribute: Definition 6 A price index P = (P n ) n ∈ ℕ is consistent in aggregation with respect to a secondary attribute z : This definition emphasizes the continuity of the consistent aggregation rule A. In the Appendix (Proposition 9), it is shown that a neglect of continuity has absurd consequences.
In Definition 6, the phrase 'with respect to a secondary attribute z' is important, because the definition allows for a wide range of possible secondary attributes and not all of them may appear appealing. Different views on what constitutes an admissible secondary attribute have given rise to a wide variety of definitions of consistency in aggregation. However, a discussion of these definitions and of what constitutes an admissible secondary attribute, is deferred to Section 10. For the time being, we examine whether price indices exist that are consistent in aggregation with respect to some secondary attribute z in the broad sense of Definition 6. We begin with the three superlative price indices (Fisher, Törnqvist, Walsh).

| CONSISTENCY OF SUPERLATIVE PRICE INDICES
To prove that some price index is consistent in aggregation with respect to some z, one has to transform the price index in an aggregation rule that is consistent.
Proposition 3 Let P = (P n ) n ∈ ℕ be a price index and z = z 1 , …, z Q : J → M ⊆ ℝ Q > 0 a secondary attribute such that M is stable under addition and for some continuous function h : M → ℝ > 0 we have

VON AUER ANd WENGENROTH
Then P is consistent in aggregation with respect to z with the aggregation rule for I = ℝ > 0 × M defined by Proof. We only have to check consistency of the stated aggregation rule (with the obvious additional definition A 1 (r, z) = (r, z)). This follows from Proposition 1 and the representation since (r, z) ⊕ (s, w) = (h(z + w), z + w) defines a commutative and associative binary operation on I. ▪ Proposition 3 is extremely useful in the context of superlative price indices. Diewert (1976, pp. 131-135) derives two related price index families that are superlative. One family is denoted as the quadratic-mean-of-order-s price indices: For s = 2, formula (22) yields the Fisher index (15). The second family is denoted as the implicit quadratic-mean-of-order-s price indices. The label indicates that this family is implicitly derived from the family of quadratic-mean-of-order-s quantity indices and the expenditure ratio (23) yields the Walsh index (11). (20) is consistent in aggregation with respect to the secondary attribute z = ln(r)v 0 , v 0 , ln (r)v 1 , v 1 . Proof. As the first component of the aggregation rule A s n corresponding to the family of price indices (22), we can invoke Proposition 3 and define the function A n ( (r 1 , z 1 ), …, (r n , z n ) ) = h

Proposition 4 The Fisher index (15) is consistent in aggregation with respect to the secondary attri-
A n ( (r 1 , z 1 ), …, (r n , z n ) ) = (r 1 , z 1 ) ⊕ ⋯ ⊕ (r n , z n ), For s = 2, function (24) simplifies to the Fisher index (15). For the first component of the aggregation rule A Tö n defined in Equation (21), we can choose with z = ln(r )v 0 , v 0 , ln (r)v 1 , v 1 . ▪ Proposition 5 The Walsh index (11) is consistent in aggregation with respect to the secondary attribute z = and, for s = 1, with respect to the secondary attribute Proof. The first component of the aggregation rule A Wa ′ n defined in Equation (18) is , v 1 , v 0 . For s = 1, this function simplifies to the Walsh index defined in Equation (11). ▪ Economic theory and practical considerations imply that not all secondary attributes are equally appealing. In Section 10, we will demonstrate that this is the reason why different notions of consistency in aggregation have been proposed in the literature. We will introduce four potential requirements that secondary attributes should possibly satisfy. For example, the secondary attributes z = ( corresponding to the aggregation rule A Wa ′ n satisfy only two of these requirements. However, we know that the aggregation rule A Wa n is another candidate for the Walsh index. In contrast to the aggregation rule A Wa ′ n , the aggregation rule A Wa n has a single secondary attribute: This attribute satisfies three of the four potential requirements. But what can be said about the consistency in aggregation of A Wa n with respect to z = √ v 0 v 1 ∕r? Since the secondary attribute of the aggregation rule A Wa n does not have the form specified in Proposition 3, we need an alternative route to prove consistency. It turns out that this alternative route is also useful to prove the consistency in aggregation of many other price indices with respect to some secondary attribute. Consider again the weighted arithmetic mean defined by Equation (8). Utilizing Proposition 2, we could show that this aggregation rule is consistent. The aggregation rule A Wa n defined in Equation (17) is a special case of Equation (8) Therefore, the Walsh index is consistent in aggregation with respect to the secondary attribute z = √ v 0 v 1 ∕ r. The example reveals that Proposition 2 provides an elegant route to prove an aggregation rule's consistency and that a similar route exists for proving that a price index is consistent in aggregation with respect to some secondary attribute z.

Proposition 6
Let P = (P n ) n ∈ ℕ be a price index with a secondary attribute z : , where M is stable under addition. Set I = ℝ > 0 × M and assume that there is a continuous function f : I → L (where L is either ℝ > 0 or ℝ) such that, for each m ∈ M, the partial function r ↦ f(r,m) is invertible on ℝ > 0 and that , m) � v 0 j v 1 j ∕ r j , the Walsh index, P Wa n , can be expressed as in Equation (25) with and The function f satisfies the assumptions stated in Proposition 6. ▪ Definition 6 leaves a lot of, perhaps too much, scope of discretion in our choice of secondary attributes. As pointed out before, in Section 10 we will return to this issue. There we discuss additional requirements that restrict the set of admissible secondary attributes and the way these attributes are aggregated.
Before we move on to show that many other price index formulae are consistent in aggregation with respect to some secondary attribute z, we emphasize a general insight from our discussion of the Walsh index. We learnt that for this price index formula different secondary attributes are available, including . Of course, in the context of a single-stage computation, the choice of the secondary attribute does not affect the index number. All options are equivalent. However, this equivalence does not carry over to a two-stage computation. When z = √ v 0 v 1 ∕ r is chosen, the index number coincides with the index number obtained from the single-stage computation. However, when z = v 0 , v 1 is chosen, a different index number is produced. Therefore, in a two-stage computation, the choice of the secondary attribute matters. This important difference between single-stage and two-stage computation applies not only to the Walsh index, but to all price indices P in the sense of Definition 4.
In Propositions 4, 5 and 7, we identified secondary attributes that ensure for superlative price indices that the single-stage and two-stage computation yield the same index number. Next, we identify such secondary attributes for price indices that are not superlative.

| CONSISTENCY OF OTHER PRICE INDICES
Besides the superlative price indices of Fisher, Törnqvist, and Walsh, numerous other price indices exist that are consistent in aggregation with respect to some secondary attribute z. Tables 2 and 3 are consistent in aggregation with respect to the secondary attributes specified in the last column of these tables.

Proposition 8 The price index formulae P listed in
Proof. Tables 2 and 3 list for each price index the corresponding function f( r i , z i ) and the secondary attributes z q i . Via Proposition 6, the function f( r i , z i ) yields an explicit construction for the aggregation rule A required in the definition of consistency. ▪ Table 2 lists a number of traditional price indices, whereas many indices that von Auer (2014) categorizes as generalized unit value indices are listed in Table 3.

| APPLICATION TO SWEDISH PRICE DATA: PART B
The three superlative price indices and all price indices listed in Tables 2 and 3 have been applied to the Swedish data set described in Section 5. Table 4 reports the Swedish overall, core and non-core inflation rates as measured by the various price indices. In the last four columns, the table shows the aggregate values of the respective secondary attributes. The various index formulae produce very similar results for the overall inflation (second column). The same is true for the core inflation as well as for the non-core inflation (fourth column). As expected, the largest values are produced by the Laspeyres index, whereas the Paasche index generates the smallest values.

RELATED LITERATURE
The studies of Vartia (1976a,b) are the first formal treatments of consistency in aggregation. A more general definition of consistency in aggregation is proposed by Blackorby and Primont (1980, p. 96) T A B L E 2 Traditional price indices and their secondary attributes

Secondary attributes z q i
Banerjee (GUV-3) a who also introduce the notion of primary and secondary attributes. We take their definition as a starting point for the following discussion. As in our Definition 6, Blackorby and Primont (1980, p. 96) postulate that a secondary attribute of some item i must exclusively use information that specifically relates to this item. In contrast to Definition 6, however, they allow only for quasi-additive aggregation of secondary attributes and they T A B L E 4 Numerical illustration a -results of one-and two-stage aggregation Name One-stage aggregation do not preclude 'external information', that is, information other than prices and quantities (e.g. quality of an item). Blackorby and Primont (1980, p. 96) are well aware of the (too) general nature of their approach. They conclude: 'Thus, unless there is some a priori notion of how the attributes are defined, this generalized consistency in aggregation notion does not seem helpful'. The present paper has argued that in the specific context of price measurement such an a priori notion exists. In a price index computation, the only available pieces of information are those in the informational set I. The secondary attributes must exclusively use information from set I, that is prices and quantities, or equivalently, price ratios, r i , and expenditures, v 0 i and v 1 i . Accordingly, Definition 6 precludes any "external information".
Nevertheless, some practitioners may still regard Definition 6 as too general, because it neglects some additional requirements that one possibly wants to attach to the secondary attributes and their aggregation. We discuss four increasingly restrictive requirements (Requirements A-D).
In Propositions 4, 7 and 8, we listed 15 price indices that are consistent in aggregation with respect to some secondary attribute z in the sense of Definition 6 (Proposition 5 is redundant here, because of Proposition 7). Adding the new requirements reduces the number of price indices that are considered as consistent in aggregation. Only 4 out of the 15 price indices satisfy all four requirements. Unfortunately, there is no agreement as to which requirements are sensible and necessary and which are not. So far, the different positions are not thoroughly related to each other and it would be overoptimistic to expect a complete agreement on the issue. However, our formalized exposition adds more structure to the dispute and, as a result, may create a greater consensus. Definition 6 in conjunction with the list of additional requirements enables us to pinpoint the differences in past attempts of defining consistency in aggregation. Therefore, the following discussion also provides a comprehensive review of the price index literature on consistency in aggregation.
From an economic perspective, the secondary attributes z q i (q = 1, …, Q and i = 1, …, N) must reflect the importance of item i. The item's importance is best measured by a function z q i (r, v 0 i , v 1 i ) that aggregates the item's expenditures v 0 i and v 1 i in a meaningful way. For example, the Walsh index uses the secondary attribute z i = √ v 0 i v 1 i ∕ r i , that is the geometric average of the base period expenditures and the deflated comparison period expenditures. Obviously, the relative importance of items i and j should be invariant with respect to currency changes. The following requirement formalizes this postulate.
Requirement A The ratios of the secondary attributes are invariant with respect to proportional changes of all expenditures: for q = 1, …, Q and i, j = 1, …, n.
The secondary attributes of all price indices listed in Propositions 4, 7, and 8 fulfil Requirement A. This list includes the superlative indices of Fisher, Walsh, and Törnqvist. A price index that is consistent in aggregation with respect to a secondary attribute z, is a consistent aggregation rule, A n , that determines how the individual values of each secondary attribute, z q i ( q = 1, …, Q and i = 1, …, n), are transformed into the respective aggregated value, Z q . It seems reasonable to postulate that in this transformation an aggregated value, Z q , depends only on the individual values of the secondary attribute q: z q 1 , …, z q n . Which types of transformation are acceptable? Since the secondary attributes, z q i , represent the weighting system of the index compilation, one , may demand that the aggregated values of the secondary attributes, Z q k , must be expressed in the same units as the individual values, z q i . Simple summation of the individual z q i -values preserves the units of measurement. Furthermore, simple summation ensures that the sum Z q = ∑ K 1 Z q k does not change as we remove some item i from subset k and assign it to some other subset l. The preceding demands can be combined in the following requirement: Requirement B The secondary attributes are aggregated additively: This precludes other quasi-additive aggregator functions such as multiplication. All secondary attributes of the price indices listed in Propositions 4, 7, and 8 satisfy also Requirement B.
Consensus condition (ii) stated that on both stages of a two-stage computation the 'same functional form' must be applied. As an extensive interpretation of this condition, one may postulate that any functional relationship between the secondary attributes of the individual items must carry over to the aggregated secondary attributes. For example, consider the four secondary attributes of the Fisher index (15). They are linked by Let Z q = ∑ i ∈ S z q i (q = 1, …, 4) denote the aggregate values of the secondary attributes. Since the relationships between the secondary attributes of the Fisher index (15) do not carry over to their aggregated counterparts. The same applies to the Törnqvist index (20). More formally, the postulate can be stated in the following way: Requirement C If a map g exists, such that z i is the vector of all secondary attributes except for attribute q, then Z q = g(P, Z − q ), where Z q is the aggregate value of all z q i with i ∈ S, P is the price index with respect to set S, and Z − q are the aggregated values of all secondary attributes except for attribute q.
The Walsh index (12) has only one secondary attribute: Therefore, no violation of Requirement C can arise. A price index with two secondary attributes may or may not satisfy Requirement C. For example, the Walsh-Vartia index, can be interpreted as a price index with the two secondary attributes v 0 i and v 1 i . Between these two attributes no functional relationship exists. Accordingly, Requirement C is fulfilled. However, when a price index has more than two secondary attributes (e.g. all price indices listed in Table 3 and the Fisher index), Requirement C is usually violated. As a function of three variables (r, v 0 , v 1 ), the transformed A n (r 1 , z 1 1 , …, z Q 1 ), …, (r n , z 1 n , …, z Q n ) = P n , information ( r, z 1 , …, z Q ) lies in a low-dimensional 'variety' of the set I = ℝ > 0 × M ⊆ ℝ 1 + Q so that one expects that many functional relations should hold in that variety. In a sense, Requirement C would only allow 'linear' relations (respecting the aggregation rule ⊕ A on I) but no non-linear relations.
The secondary attributes of the Walsh index (12) and of all price indices listed in Table 2 fulfil Requirements A, B and C, while the secondary attributes of the Fisher index (15) and the Törnqvist index (20) violate Requirement C. It should be noted that this result doesn't rule out that the Fisher index and the Törnqvist index might be consistent in aggregation with respect to some alternative secondary attributes z that happen to satisfy Requirements A, B and C.
From a theoretical perspective, it would be interesting to derive conditions under which such alternative secondary attributes may exist. This task is left for future research. Furthermore, for practitioners, such an analysis would be of less relevance. They are primarily interested in findings that identify reasonable secondary attributes of a price index, such that this index is consistent in aggregation with respect to these secondary attributes. Our Propositions 4, 5, 7, and 8 represent such findings.
Since the secondary attributes of the Walsh, Marshall-Edgeworth, Walsh-2 and Theil indices violate Requirement D, the only remaining price indices are the Laspeyres, Paasche, Walsh-Vartia and Vartia indices.
Is it possible to provide some justification for Requirement D? In an extremely extensive interpretation of consensus condition (ii), one may postulate that the relationship between a secondary attribute z q i and the three basic variables r i , v 0 i and v 1 i from which this attribute is computed, must carry over to the aggregated values. For example, the Marshall-Edgeworth index, P ME , has the secondary attribute z i = v 0 i + v 1 i ∕ r i . However, for the aggregate value Z = Therefore, the relationship for the individual items does not carry over to their aggregated counterparts. In fact, formal correspondence between the computation of a z q i -value and the computation of its aggregated counterpart Z q requires that z 1 i = v 0 i and/ or z 2 i = v 1 i , that is Requirement D. von Auer (2004, pp. 386-391) criticises Requirement D as being too restrictive and proposes a milder version. Besides v 0 i and v 1 i , he allows also for the 'hybrid' secondary attributes v 0 With these four admissible secondary attributes, the Marshall-Edgeworth index would re-enter the list of price indices that are consistent in aggregation.
Definition 6 does not award the label 'consistent in aggregation'. Instead, it awards the label 'consistent in aggregation with respect to a secondary attribute z'. We propose to reserve the label 'consistent in aggregation' for those price indices that satisfy Definition 6 and from Requirements A-D those that are deemed as indispensable.
Unfortunately, there is no consensus on the list of indispensable requirements. Some index users may consider Requirements A and B as indispensable, but not the other two requirements. Then, all price indices listed in Table 4 are consistent in aggregation. If Requirements A-C are deemed as compulsory, the Fisher index, the Törnqvist index and all price indices listed in Table 3 drop out. If index users regard all four requirements as indispensable, then the Laspeyres, the Paasche, the Walsh-Vartia and the Vartia index remain in the list, while the Walsh, the Marshall-Edgeworth, the Walsh-2 and the Theil index no longer qualify for the label 'consistent in aggregation'. 8 The latter four indices share with the Laspeyres and the Paasche index another property that is often appreciated in applied work. The indices can be written in the following form: with Z k = ∑ i ∈ S k z i . For example, in Sections 5 and 9, we applied the Walsh index to the Swedish consumer price index data and obtained P Wa = 1.0275, that is an overall inflation of 2.75%. In applied work, one may want to decompose these 2.75% into the contribution of the core inflation and the contribution of the non-core inflation. We know from our calculations that the core inflation was 2.64% (P Wa 1 = 1.0264) and the non-core inflation 3.55% (P Wa 2 = 1.0355). To compute the individual contributions, though, these two numbers are not sufficient. We also need weights that reflect the importance of the items assigned to core inflation relative to the items assigned to non-core inflation. These weights can be obtained from the secondary attribute. In Equation (26) the weight of each subset k is quantified by Z k ∕ ∑ K l = 1 Z l . In our Swedish example, Equation (26) becomes The aggregated values of the secondary attributes were Z 1 = 1 257 744.8 and Z 2 = 178 317.6. Inserting all numbers in Equation (27) yields Even though the core inflation is much smaller than the non-core inflation, the contribution of the core inflation to the overall inflation of 2.75% is 2.31%, whereas the contribution of the non-core inflation is merely 0.44%. A detailed exposition of the decomposition properties of price indices can be found in Balk (2008, pp. 140-151).

| CONCLUDING REMARK S
The computation of the overall price change is often conducted in a two-stage (or multi-stage) procedure, where on the first stage price changes of subgroups are computed and on the second stage price changes of the subgroups are aggregated into the overall price change. In the literature, it has been postulated that the price index formula applied in such a multi-stage analysis should be consistent in aggregation. The present paper has argued that consistency in aggregation is a complex concept that requires a careful definition. Blackorby and Primont (1980) distinguished between the primary attribute of a price index and its secondary attributes. 8 In the proofs of Propositions 4 and 5, secondary attributes were identified such that the quadratic-mean-of-order-s price indices defined by Equation (22) as well as the implicit quadratic-mean-of-order-s price indices defined by (23) are consistent in aggregation with respect to these secondary attributes. These attributes satisfy Requirements A and B, but not Requirements C and D. (26) 2.75 = 2.31 + 0.44.
Combining this distinction with the general concept of a consistent aggregation rule, yields a thoroughly motivated basic definition of consistency in aggregation, specifically designed for the context of price measurement. Surprisingly many price index formulae satisfy this basic definition of consistency in aggregation. Among these are the price indices of Fisher, Törnqvist, and Walsh. This is a remarkable finding because these indices are known as superlative price indices. In the literature, there has been a general perception that superlative price indices are particularly reliable for single-stage computations, but that they are not consistent in aggregation and therefore unsuitable for multi-stage computations. Our findings show that this perception cannot be sustained in the context of our new definition of consistency in aggregation.
It was argued that further requirements can be added to the basic definition of consistency in aggregation. Such additional requirements shrink the list of price indices that are consistent in aggregation. Four such requirements were discussed, with Requirements A and B being the most obvious ones, and Requirement D being the most contentious one. From an applied perspective, Requirements A-C appear particularly relevant. The Törnqvist index and the Fisher index only satisfy Requirements A and B, whereas the Walsh index satisfies all three requirements. Furthermore, the Walsh index allows for a simple additive decomposition of the overall price change into the contributions of individual subgroups (e.g. core inflation and non-core inflation).
In the national statistical offices of the 20th century, these appealing properties of the Walsh index would have received little attention, because all superlative price indices require information on the transacted quantities of the comparison period, and such information was simply not available. As a result, superlative indices, though serving as a theoretical benchmark, have played hardly any role in the routine procedures of official price statistics. With the advent of scanner price data in official price measurement, however, the situation has completely changed. In many countries (e.g. Belgium, Denmark, Iceland, the Netherlands, Norway, Sweden and Switzerland), scanner data have replaced traditional price collectors. Today, scanner data cover important segments of the consumption basket and they provide information about the comparison period's expenditures, prices, and therefore, quantities. With such data at hand, superlative price indices can be applied. This paper has shown that these indices are even suitable for the compilation of some overall price change by a multi-stage procedure and that the Walsh index is the primary candidate for this purpose.