The CIA (consistency in aggregation) approach A new economic approach to elementary indices

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1 The CIA (consistency in aggregation) aroach A new econoic aroach to eleentary indices Dr Jens ehrhoff*, Head of Section Business Cycle and Structural Econoic Statistics * Jens This ehrhoff, resentation Deutsche reresents Bundesbank the author s ersonal oinions and does not necessarily reflect the views of the Deutsche Bundesbank or its staff. Page

2 Outline. otivation 2. Background 3. Econoic aroach 4. Consistency aroach 5. Eirical results 6. Discussion Eleentary, y dear Watson! (Sherlock Holes) Page 2

3 . otivation Practical consuer rice indices are constructed in two stages:. a first stage at the lowest level of aggregation where rice inforation is available but associated exenditure or quantity inforation is not available and 2. a second stage of aggregation where exenditure inforation is available at a higher level of aggregation. g Although the scoe of the discussion is on the choice of the index forula at the eleentary level, this choice eventually deends on the target rice index at the aggregate level. HICP is not intended to easure the cost of living (COLI), rather, it is a cost of goods index (COGI). Page 3

4 2. Background Bilateral rice indices We secify two accounting eriods, t {, }, for which we have icro rice and quantity data for n coodities (bilateral index context). Denote the rice and quantity of coodity i {,, n} in eriod t by it and q it, resectively. A very sile aroach to the deterination of a rice index over a grou of coodities is the (fixed) basket aroach. Define the Lowe (823) rice index, P Lo, as follows: n i i i q i PLo. n i q i There are two natural choices for the reference basket: the eriod coodity vector q = (q,, q n ) or the eriod coodity vector q = (q,, q n ). Page 4

5 2. Background Laseyres, Paasche and Fisher These two choices lead to the Laseyres (87) rice index P L, if we choose q = q, and the Paasche (874) rice index P P, if we choose q = q : n i i qi i i q i PL, P. n P n q q i i i n i i i According to the CPI anual (ILO et al., 24), the Paasche and Laseyres rice indices are equally lausible. Taking an evenly weighted average of these basket rice indices leads to syetric averages. The geoetric ean, which leads to the Fisher (922) rice index, P F, is defined as: P P P. F L P Page 5

6 2. Background Keynes ure theory of oney In his 93 A Treatise on oney (. 95-2), Keynes deals with the theory of coarisons of urchasing ower. Coarisons of urchasing ower ean coarisons of the coand of oney over two collections of coodities which are in soe sense equivalent to one another, and not over quantities of utility. Alying the ethod of liits establishes that in any case the easure of the change in the value of oney lies between the Laseyres and Paasche rice indices. The crossing of forulae, to which Fisher has devoted uch attention, is, in effect, an attet to carry the ethod of liits soewhat further further (in Keynes oinion) than is legitiate. We can concoct all sorts of algebraic function of P L and P P, and there will not be a enny to choose between the. Page 6

7 2. Background Eleentary indices Suose that there are lowest-level ites or secific coodities in a chosen eleentary category. Denote the eriod t rice of ite by t for t {, } and for ites {,, }. The Dutot (738) eleentary rice index, P D, is equal to the arithetic average of the eriod rices divided by the arithetic average of the eriod rices. The Carli (764) eleentary rice index, P C, is equal to the arithetic average of the ite rice ratios or rice relatives, /. The Jevons (865) eleentary rice index, P J, is equal to the geoetric average of the ite rice ratios or rice relatives, /, or the geoetric average of the eriod rices divided by the geoetric average of the eriod rices. PD, P,. C P J Page 7

8 2. Background Is the Carli index really uward biased? The sole arguent frequently ut forward why the Carli index should be abandoned, is the clai that it has an uward bias with reference to the tie reversal test or circularity test (cf. Diewert, 22): P C (, ) P C (, 2 ) = P C (, ) P C (, ) = P C (, ) for 2 =. But this arguent is useless in the bilateral index context where we can coare the two eriods under consideration directly, i.e. there is no bias at all: P C (, 2 ) = P C (, ) = for 2 =. C(, ) C(, ) In the context of chain indices, the eleentary aggregates only feed into the higher-level indices in which the eleentary yrice indices coaring eriods t- and t (!) are averaged using a set of re-deterined weights (chain indices are non-aggregable); the Dutot, Carli and Jevons indices are, thus, not chain-linked. What is ore, it aarently fell into oblivion that the then chain-linked Laseyres, Paasche, Fisher, Walsh and Törnqvist rice indices are subject to chain drift; i.e. all chain indices are ath deendent, which is the oosite of transitivity. Page 8

9 3. Econoic aroach The CPI anual, aragrahs , has a section in it which describes an econoic aroach to eleentary indices. This section has soeties been used to justify the use of the Jevons index, i.e. the geoetric ean, over the use of the Carli index, i.e. the arithetic ean, or vice versa deending on how uch substitutability exists between ites within an eleentary stratu. This is a isinterretation of the analysis that is resented in this section of the anual. Thus, the econoic aroach cannot be alied at the eleentary level unless rice and quantity inforation are both available. Such inforation is tyically not available, which is exactly the reason eleentary indices are used rather than target indices. (Diewert, 22, Consuer Price Statistics in the UK ) Page 9

10 4. Consistency aroach Consistency in aggregation The consistency in aggregation (CIA) aroach newly develoed (ehrhoff, 2, Jahr. Nationalökon. Statist.) fills the void of guiding the choice of the eleentary index (for which weights are not available) that corresonds to the characteristics of the index at the second stage (where weights are actually available). It contributes to the literature by looking at how nuerical equivalence between an unweighted eleentary index and a weighted aggregate index can be achieved, indeendent d of the axioatic roerties. Consistency in aggregation eans that if an index is calculated stewise by aggregating lower-level indices to obtain indices at rogressively higher levels of aggregation, the sae overall result should be obtained as if the calculation had been ade in one ste. Page

11 4. Consistency aroach Cost of goods index To reiterate, we easure the change in the cost of urchasing a fixed basket of goods and services and not the change in the iniu cost of basket of goods and services, and not the change in the iniu cost of aintaining a given level of utility or welfare. The use of the Dutot and Carli forulae at the eleentary level of aggregation for hoogeneous ites can be erfectly consistent with a Laseyres index concet. The Laseyres rice index can be rewritten in an alternative anner as follows: follows:, L s q q q q P where s is the eriod exenditure share on coodity. l l l q q Page

12 4. Consistency aroach A thought exerient The first case is where the underlying references are Leontief references, i e consuers refer not to ake any substitutions in resonse to changes i.e. consuers refer not to ake any substitutions in resonse to changes in relative rices (zero elasticity): q q = q = q and, hence,. D L P q q P The second case is when the references can be reresented by a Cobb- Douglas function, i.e. consuers vary the quantities in inverse roortion to the changes in relative rices so that exenditure shares reain constant the changes in relative rices so that exenditure shares reain constant (unity elasticity): P P s i = s i = - and, hence,. C L P P Page 2

13 4. Consistency aroach Generalised eans A single corehensive fraework, known as generalised eans, unifies the aggregate and eleentary levels. The generalised ean of order r for the ite rice ratios or rice relatives, /, is defined as follows: P r r if r, r if r. The generalised ean reresents a whole class of unweighted eleentary indices, such as the Carli and Jevons indices for r = and r =, resectively. Page 3

14 4. Consistency aroach Tyical shae Page 4

15 4. Consistency aroach CES aggregator function It is suosed that the unit cost function has the following functional for: c() /( ) if, if, where the are non-negative g consuer reference araeters with. This unit cost function corresonds to a CES aggregator or utility function. The araeter is the elasticity of substitution: When =, the underlying references are Leontief references. When =, the corresonding utility function is a Cobb-Douglas function. Page 5

16 4. Consistency aroach Laseyres and Paasche rice indices A generalised ean of order r equal to the elasticity of substitution () yields aroxiately the sae result as the Laseyres rice index. Hence, if the elasticity of substitution is one (Cobb-Douglas references), for exale, r ust equal one and the Carli index at the eleentary level will corresond to the Laseyres rice index as target index. However, if the Paasche rice index should be relicated, the order of the generalised ean ust equal inus the elasticity of substitution, in the above exale inus one. Thus, the haronic index gives the sae result and therefore, in this case it should be used at the eleentary level. Only if the elasticity of substitution is zero (Leontief references), the Jevons (Dutot) index corresonds to both the Laseyres and Paasche rice indices which in this case coincide. Page 6

17 5. Eirical results Alcoholic beverages in the UK As an eirical alication, detailed exenditure data fro Kantar Worldanel for eleentary aggregates within the COICOP grou of alcoholic beverages in the UK are analysed. The data cover the eriod fro January 23 to Deceber 2; the data set consists of transaction level data, which records inter alia urchase rice and quantity, and includes 92,948 observations after outlier identification. The elasticity of substitution is estiated in the fraework of a log-linear linear odel by eans of ordinary least squares. (Note that the consuer reference araeters are reoved via differencing roducts coon to adjacent onths and, thus, there is no need for alication of seeingly unrelated regression.) As a robustness check to the CES odel based results, the generalised ean which iniises relative bias and root ean squared relative error to the desired aggregate index is found directly by nuerical otiisation techniques. (Rather than at the aggregate transaction ti level, l like the econoetric ethod, this analysis, however, is erfored one level above at the eleentary level.) Page 7

18 5. Eirical results COICOP structure for alcoholic beverages Page 8

19 5. Eirical results Findings on substitution behaviour The edian elasticity of substitution is.5, ranging fro.7 to 3.8. All estiates are statistically significantly greater than zero; for 8 out of 9 sub-classes the difference to iso-elasticity is insignificant, while for the reaining sub-classes substitution is found to even exceed unity elasticity. In sirits, consuers are ore willing to substitute between different tyes of whiskey (S 4) than is the case for brandy or vodka (S and S 2). For both red and white wines, substitution is ore ronounced for the New World (W 5 and W 7) than for Euroean wines (W 4 and W 6). Also, the elasticity of substitution tends to be higher for 2 cans and 2 bottles of lager (B 3 and B 5), resectively, than for 4 acks (B 2 and B 4). These results are consistent with the findings of Elliott and O Neill (22). Furtherore, coaring the CES regression results with the direct calculation of the generalised eans, the outcoes do not change qualitatively. In articular, the Carli index erfors rearkably well at the eleentary level of a Laseyres rice index, questioning the arguent of its uward bias in fact, it is the Jevons index that has a downward bias. Page 9

20 5. Eirical results CES estiation results Page 2

21 5. Eirical results Laseyres rice index: robustness Page 2

22 5. Eirical results Laseyres rice index: bias Page 22

23 5. Eirical results Laseyres rice index: tie series Page 23

24 6. Discussion The roble of aggregational consistency deonstrates the need for a weighting at the lowest ossible level. This would ean that, in the trade-off between estiated weights/weights fro secondary sources on the one hand and the eleentary bias of unweighted indices on the other, the balance would often ti in favour of weighting. The biases at the eleentary level can, in soe cases, reach such large diensions that they becoe relevant for the aggregate index. There is a rice to be aid at the uer level for subotial index forula selection at the lower level; thus, the need for two-staged rice indices to be accurately constructed becoes obvious. Disaggregation is a anacea! Insofar as no inforation on weights is available, studies on substitution can hel in guiding the choice of the otial eleentary index for a given easureent target. Often, even an exert judgeent on substitutability outerfors the test aroach. Page 24