CEP Discussion Paper No 797. (Revised) February Originally published May 2007

Transcription

1 CEP Dscusson Paper No 797 (Revsed) February 2008 Orgnally publshed May 2007 Chan Indces of the Cost of Lvng and the Path- Dependence Problem: An Emprcal Soluton Ncholas Oulton

2 Abstract Ths paper proposes an emprcally feasble method for correctng the path-dependence bas of chan ndces of the cost of lvng. Chan ndces are dscrete approxmatons to Dvsa ndces and t s well known that the latter are path-dependent: the level of a Dvsa ndex s affected not just by the level of prces at the two endponts but also by the path between the endponts. It s also well-known that a Dvsa ndex of the cost of lvng s path-ndependent f and only f all ncome elastctes are equal to one, a restrcton that s decsvely rejected by studes of consumer demand. In theory, the true cost of lvng ndex (or Konüs prce ndex) could be derved by estmatng the expendture functon. But ths seems mpractcal due to data lmtatons: the number of ndependent parameters rses roughly n proporton to the square of the number of commodtes and consumer prce ndces contan hundreds of tems. Ths paper shows how ths problem can n fact be overcome emprcally usng a flexble model of demand lke the Quadratc Almost Ideal Demand System. The proposed method requres data only on prces, aggregate budget shares and aggregate expendture. The method s appled to estmate Konüs prce ndces for 70 products coverng nearly all the UK s Retal Prces Index over , wth each year n turn as the base. The choce of base year for utlty s found to have a sgnfcant effect on the ndex, even n the low nflaton perod snce JEL Classfcaton: C43, D12, E31 Key words: Index number, cost of lvng, Dvsa, Chan, Path-dependence, Almost Ideal Demand System Ths paper was produced as part of the Centre s Productvty and Innovaton Programme. The Centre for Economc Performance s fnanced by the Economc and Socal Research Councl. Acknowledgements I owe thanks to Chuck Hulten for a stmulatng dscusson on the path-dependence problem. I am also grateful to Smon Prce for advce on econometrc ssues, to Andrew Lecester of the Insttute for Fscal Studes for supplyng me wth hs data on retal prces and budget shares and for answerng my queres, and to Bert Balk, John Van Reenen, Kevn Fox, Robert Hll, Hamsh Low, and two anonymous referees for a number of helpful comments. Ths paper also benefted from the comments of semnar partcpants at St Andrews, the London School of Economcs, the Bank of England, and the Centre for Appled Economc Analyss at the Unversty of New South Wales. Ncholas Oulton s a Senor Vstng Research Fellow at the Centre for Economc Performance, London School of Economcs. Emal: n.oulton@lse.ac.uk Publshed by Centre for Economc Performance London School of Economcs and Poltcal Scence Houghton Street London WC2A 2AE All rghts reserved. No part of ths publcaton may be reproduced, stored n a retreval system or transmtted n any form or by any means wthout the pror permsson n wrtng of the publsher nor be ssued to the publc or crculated n any form other than that n whch t s publshed. Requests for permsson to reproduce any artcle or part of the Workng Paper should be sent to the edtor at the above address. N. Oulton, submtted 2007 ISBN

3 1. Introducton The purpose of ths paper s to propose an emprcally feasble method for correctng what I shall call the path-dependence bas of chan ndces of the cost of lvng. Ths bas (see below for a precse defnton) arses because chan ndces are dscrete approxmatons to Dvsa ndces whch, despte ther other desrable propertes, are known to be path-dependent except under mplausble restrctons on consumer preferences. The proposed method employs the Quadratc Almost Ideal Demand System of Banks, Blundell, and Lewbel (1997), a flexble system that s fully consstent wth economc theory. Chan ndces have become ncreasngly popular wth natonal statstcal agences n recent years. In the U.K. the Retal Prces Index (RPI) s an annually chan-lnked Laspeyres ndex (at least approxmately: see Offce for Natonal Statstcs, 1998, for detals). The EU s Harmonsed Index of Consumer Prces, whch under the name Consumer Prces Index (CPI) s used as the Bank of England s nflaton target, s also chaned Laspeyres. In the Unted States, one of the CPI measures publshed snce 2002 by the Bureau of Labor Statstcs s a chaned Törnqvst ndex; ths followed crtcsm by the Boskn Commsson (1996) of the fxed base approach. The 1993 System of Natonal Accounts came down n favour of chanlnkng for GDP (Commsson of the European Communtes Eurostat et al., 1993). Followng that, Eurostat now requres that n all EU countres GDP and ts components on both the output and expendture sdes should be annually chan-lnked Laspeyres. The U.S. adopted chan-lnkng, usng Fsher ndces, n 1997 (Landefeld and Parker, 1997). Canada, Australa and Japan also use chan-lnkng n ther natonal accounts. Chan ndces have the obvous ntutve justfcaton that the weghts are kept up-to-date. A more theoretcal justfcaton s that, f the am s to measure the cost of lvng, then by comparson to fxed base ndces they reduce substtuton bas. Chan ndces can also be justfed as dscrete approxmatons to contnuous Dvsa ndces whch have many desrable propertes. For example, the product of a Dvsa prce ndex and a Dvsa quantty ndex s the value ndex. And Dvsa ndces are consstent n aggregaton: a one-stage Dvsa ndex of food prces computed for (say) apples, oranges, lamb and beef s equal to a two-stage ndex computed frst for frut and meat and then for food as a whole from the sub-ndces for frut and meat. But chan ndces suffer from a drawback that was noted and dscussed by Irvng Fsher (Fsher (1927), who refers to much earler work by Walsh and Westergaard): they fal the 1

4 transtvty, or crcularty, test (Dewert, 1987; Balk, 1995). Suppose that we are makng comparsons over four perods and that prces and quanttes return to ther orgnal values n the fourth perod. Then we want the prce ndex (and the quantty ndex) for the fourth perod relatve to the frst to equal one. But smple algebra shows that none of the chan ndces n common use (Laspeyres, Paasche, Törnqvst or Fsher) can satsfy the crcularty test for a comparson over four or more perods. More generally, t can be shown that no chan ndex whch also satsfes other propertes whch we requre of an ndex number can satsfy the crcularty test (see Balk (1995) for a formal proof). Ths falure of chan ndces s wellknown to practtoners as chan ndex drft; t s the man reason why chan-lnkng s not recommended for monthly prce ndces, where due to seasonal factors or promotons prce bounce s qute common. The proof that no chan ndex can satsfy all the desrable tests assumes that prces and quanttes can vary freely and ndependently of each other. But f quanttes are constraned to respond to prces as economc theory requres, then t s possble that the mpossblty result could be avoded. However, t turns out that ths s the case f and only f all ncome elastctes are equal to one (Hulten, 1973; Samuelson and Swamy, 1974). But ths s a very unattractve assumpton to make about consumer demand. After all, one of the oldest emprcal fndngs n ths area s Engel s Law: the share of the budget spent on food declnes as ncome rses. 1 Dvsa ndces suffer from a related problem to that of ther dscrete counterparts: pathdependence. Path-dependence means that the level of a Dvsa ndex at some tme perod T, relatve to ts level n the base perod 0, depends not just on the prce relatves p ( T ) / p (0) of the commodtes n the ndex over ths tme span, but also on the path that prces have followed between the endponts. Dfferent paths, even f they begn and fnsh at the same ponts, produce dfferent values of the Dvsa ndex. So what credence can we gve to comparsons across countres of the average rate of nflaton or the average rate of growth of real output f the results are nfluenced by the partcular paths that the countres have followed wthn the perod studed? Theory gves us a standard by whch to judge any real world consumer prce ndex: the true cost-of-lvng or Konüs ndex (Konüs, 1939). The Konüs prce ndex s defned as the rato of the cost of buyng some reference utlty level at the prces of tme t to the cost of 1 Engel s Law s stll apparently flourshng n the U.K. Blow et al. (2004) fnd that the proporton of the household budget spent on food declned from 25% to 15% between 1975 and

5 buyng the same utlty level n the base perod (perod 0). It has been shown that a Dvsa prce ndex and the Konüs prce ndex are equal f and only f the Konüs ndex s ndependent of the partcular reference level of utlty, whch mples that the utlty functon s homothetc. Another way of puttng ths s that the two ndces are equal f and only f the ncome elastctes of all goods are equal to one, a parallel result to the one for dscrete ndex numbers; see Balk (2005) for a formal proof. So even f chan ndces have allevated one sort of bas, substtuton bas, they have brought n another sort, whch may be called pathdependence bas. 2 The fact that the Konüs prce ndex generally vares wth the reference utlty level can be gven a smple ntutve justfcaton. Consder a household wth a very low standard of lvng spendng 60% of ts budget on food. 3 Suppose the prce of food rses by 20%, wth other prces constant. Then money ncome wll probably have to rse by close to (0.60 x 20% = ) 12%, to leave utlty unchanged, due to the lmted possbltes for substtutng clothng and shelter for food. Compare ths household to a modern day Brtsh one, spendng 15% of ts budget on food (Blow et al., 2004). Now the maxmum rse n ncome requred s only (0.15 x 20 = ) 3% and probably a good bt less as substtuton opportuntes are greater (eg by reducng the order from jumbo fres to regular fres). The soluton to the path-dependence problem seems at frst sght smple: estmate the expendture functon and then compute the Konüs prce ndex. But ths appears to be mpossble n practce. Cost-of-lvng ndces are usually computed from hundreds of components. For example, the U.K. s RPI contans around 650 tems. And usually statstcal agences have only aggregate data on budget shares, prces and expendture. A demand system whch s consstent wth economc theory and s suffcently flexble to be a good ft to the data, such as the Quadratc Almost Ideal Demand System (QAIDS) of Banks 2 et al. (1997), contans ( N + 5N 6) / 2 ndependent parameters, where N s the number of 2 My term path-dependence bas s meant to be exactly analogous to the term substtuton bas. Both refer to the dfference between an ndex number and a specfc Konüs ndex (wth a partcular reference level for utlty). Substtuton bas refers to the dfference between a fxed base ndex lke the Laspeyres and a Konüs ndex. Path dependence bas refers to the dfference between a Dvsa ndex (and so also ts chan ndex approxmatons) and a Konüs ndex. So bas only has meanng wth reference to ths standard and other standards are concevable. 3 Quotng the work of Engel, Marshall (1920, chapter IV) reports that n Saxony n 1857 households headed by a workman wth an ncome of 45 to 60 a year spent 62% of ther ncome on food. 3

6 commodtes. If ths were estmated for the RPI on annual data, 4 wth all the cross-equaton restrctons mposed, at a mnmum over 328 years of data would be requred for each product! Clearly ths s out of the queston. It s true that household data are also avalable, eg n the U.K. the Famly Expendture Survey, and these sorts of data are used n practce to estmate systems lke the QAIDS. Even so, such systems are usually estmated for only half a dozen or so commodty groups. One reason for ths s that the expendture shares derved from these surveys are at a relatvely hgh level of aggregaton, and are really for commodty groups, not ndvdual commodtes. 5 The correspondng prces are therefore themselves ndex numbers. Also, these surveys yeld less data than at frst appears. The households n the sample keep expendture dares for only two weeks. So for many products a household s expendture s recorded as zero (eg expendture on summer holdays f the household s dary s kept for two weeks n wnter). The upshot s that any attempt to correct the pathdependence bas n conventonal consumer prce ndces usng the theory of demand, at the level of detal at whch these ndces are constructed, seems mpossble n practce. The purpose of ths paper s to show that ths last concluson s wrong. We can n fact estmate a true cost-of-lvng ndex usng only the aggregate data commonly avalable to natonal statstcal agences the same data as these agences use to estmate conventonal ndex numbers. Testng the economc theory of demand requres a huge amount of data. But ths s not the pont of the present exercse. Here we accept that some demand system lke the QAIDS s a good approxmaton to consumer behavour. Then subject to ths assumpton we can compute a correcton to a conventonal chan ndex of the cost-of-lvng. It turns out that ths requres not very much data at all. The reason s that the correcton nvolves estmatng only the parameters relatng to ncome elastctes and ths can be done qute parsmonously provded one does not attempt to recover all the other parameters (e those relatng to prce elastctes). The plan of the paper s as follows. In secton 2 I consder the path-dependence problem for Dvsa ndces n more detal and characterse the dfference between the growth rates of a Konüs ndex and of a Dvsa ndex. Secton 3 shows how ths dfference between the two ndces can be estmated n practce, usng the QAIDS as a mantaned hypothess. Secton 4 4 The prces n the RPI are collected monthly but the budget shares are only avalable annually. 5 The Famly Expendture dvdes household expendture nto 14 major categores and 77 sub-categores. 4

7 then apples ths method to actual data, the U.K. s Retal Prces Index (RPI), at the level of 70 commodtes over the perod Fnally, secton 5 concludes. 2. The Path-Dependence Problem for Cost-of-Lvng Indces The Konüs prce ndex s defned as the rato of the cost of buyng the reference utlty level at the prces of tme t to the cost of buyng the same utlty level at the prces prevalng n the base perod. Wthout loss of generalty we can number tme perods so that the base perod s perod 0. Then the Konüs prce ndex, or true cost of lvng ndex, s defned as: K P ( t,0) E[ p( t), u(0)] = K P (0,0) E[ p(0), u(0)] (1) where E[, ] s the expendture functon and p ( t) = ( p1, p2,..., p N ) s the prce vector at tme t. K (Usually, we would normalse so that P (0,0) = 1). At least for economsts, the Konüs ndex s the theoretcal deal, to whch real world prce ndces aspre. The connecton between Konüs and Dvsa prce ndces can be seen by dfferentatng equaton (1) logarthmcally wth respect to tme and applyng Shephard s Lemma: 6 K d ln P ( t,0) d ln E[ p( t), u(0)] ln E[ p( t), u(0)] d ln p ( t) = = dt dt ln p ( t) dt = d ln p ( t) s ( t) dt 0 (2) where 0 ln E[ p( t), u(0)] p t q s ( t) = =, = 1,..., N ln p p ( t) q ( t) ( ) 0 ( t) 0 6 Shephard s Lemma states that E / p = q, where q s the quantty demanded of the th commodty at prces p and utlty level u: q = q ( p, u), the Hcksan demand functon. 5

8 and q 0 ( t ) s the quantty of the th commodty that would be demanded at prces p(t) f utlty were held constant at the base perod level. The 0 s are the hypothetcal shares n total expendture, f utlty were held constant at the base perod level but prces were at ther actual, observed levels. These shares could also be called compensated shares, by analogy wth compensated (or Hcksan) elastctes. By contrast, a Dvsa prce ndex 7 s defned by: D d ln P ( t) d ln p ( t) = s ( t) dt dt (3) where p ( t) q ( t) ln E[ p( t), u( t)] s ( t) = =, = 1,..., N, p ( t) q ( t) ln p ( t) applyng agan Shephard s Lemma. The weghts n the Dvsa ndex are observed, actual shares, as opposed to the unobserved, compensated shares of the Konüs ndex (see Balk (2005) for ths way of charactersng the two ndces). 8 So, ntutvely, for the two prce ndces to be equal, the two sets of shares have to be equal: s 0 ( t) = s ( t). Ths means that the value shares have to be ndependent of the utlty level, whch mples that the utlty functon s homothetc, e all ncome elastctes are equal to one. 9 7 For a general dscusson of Dvsa ndces, ncludng the path-dependence problem, see Hulten (1973). 8 Prevous results on the relatonshp between chan, Dvsa and Konüs ndces nclude Dewert (1981), Feenstra and Rensdorf (2000) and Balk (2004). Suppose a utlty functon exsts whch ratonalses the data but may be non-homothetc. Dewert (1981) showed that there exsts a utlty level whch s ntermedate between the levels at the endponts of the nterval under study such that a Konüs prce ndex over ths nterval, wth utlty fxed at the ntermedate level, s bounded below by the Paasche and above by the Laspeyres. Balk (2004) showed that when the growth of prces s pecewse log lnear a chaned Fsher prce ndex approxmates a Konüs prce ndex over an nterval when the reference utlty level s fxed at that of some ntermedate pont n the nterval. A somewhat more precse result for the Almost Ideal demand system s due to Feenstra and Rensdorf (2000). If prces are growng at constant rates, they show that the Dvsa ndex between two tme perods equals the Konüs prce ndex when the reference utlty level s a weghted average of utlty levels along the path. 9 Here I gnore the trval case where all prces grow at the same rate, n whch case any pattern of weghts whch sum to one wll produce the same value for the prce ndex. Balk (2005) provdes a formal proof that the two ndces are equal f and only f the utlty functon s homothetc (hs Theorem 1). 6

9 If ncome elastctes are not all equal to one, then n general compensated and actual shares dffer. But there s more to t than that, for n ths case the Dvsa ndex s path dependent. Ths means that ts level at some tme perod T, relatve to ts level n the base perod 0, depends not just on the prce relatves p ( T ) / p (0), but also on the path that prces have followed between the endponts. So dfferent paths, even f they begn and fnsh at the same ponts, produce dfferent values of the Dvsa ndex. In comparng the cost of lvng n perod T wth the cost n perod 0, only prces at 0 and T would seem to be relevant. Any prces strctly wthn the nterval [0,T] would seem rrelevant. And the path between the endponts should not nfluence the comparson between the stuatons at the two endponts. Suppose that prces n perods 0 and T are dentcal, and also that quanttes are dentcal. D D Then we should certanly want the Dvsa prce ndex P ( T ) / P (0) (and also the Dvsa quantty ndex) to equal one for ths perod. But ths s not guaranteed unless all ncome elastctes equal one. Despte ts apparent smlarty to the Dvsa ndex, the Konüs ndex s not pathdependent. That s to say, we can recover the level of the prce ndex by ntegraton: T K d ln P ( t,0) T ln E[ p( t), u(0)] d ln p dt = dt dt ln p dt 0 0 Td ln E[ p( t), u(0)] = dt 0 dt = ln E[ p( T ), u(0)] ln E[ p(0), u(0)] K K = ln P ( T,0) ln P (0,0) The reason ths works s that the compensated shares depend only on prces, snce by defnton utlty s beng held constant. The actual shares on the other hand depend not only on prces but also on the level of utlty whch vares over the path, unless utlty s homothetc, n whch case shares depend only on prces. Path-dependence or -ndependence s a mathematcal concept, a property of lne ntegrals lke equatons (2) and (3) above. The man mathematcal result s that a lne ntegral lke (3) s path-ndependent f and only f there exsts a potental functon φ( p ) such that 7

10 ln φ( p( t)) s ( t) =, = 1,..., N ln p ( t) e shares depend only on prces. 10 But ths s equvalent to requrng the utlty functon to be homothetc. If ths condton holds, the expendture functon can be wrtten as E[ p( t), u( t)] = E[ p ( t),1] u( t) and then the potental functon φ( p ) s n fact the expendture functon, snce now s ( t) = ln E( p ( t),1) / ln p ( t). The sze of the path-dependence bas n a Dvsa ndex s a functon of the gap between actual and compensated budget shares, s t s t 0 ( ) ( ). But t s also and prmarly an emprcal queston. After all, f all prces rse at the same rate, then the bas would be zero, snce the Konüs and Dvsa prce ndces would be equal, whatever the dfference between compensated and actual budget shares. To estmate the bas, we need a theory of consumer demand that fts the facts emprcally, the topc of the next secton The Quadratc Almost Ideal Demand System A good startng pont s the almost deal (AI) demand system of Deaton and Muellbauer (1980a) and (1980b, chapter 3). Ths s fully consstent wth economc theory and also possesses the property of exact aggregaton. Most mportantly n the present context, ncome elastctes can dffer from one. Applyng Shephard s Lemma to the AI expendture functon leads to a set of share equatons: s ( t) = α + γ j ln p j + β ln z( t), = 1,..., N (4) j Here z s deflated expendture: z ( t) = x( t) / P( t), where x( t ) s total expendture, = p ( t) q x ( t) ( t), and P s a prce ndex defned by: 10 See Hulten (1973), wth references to the mathematcal lterature, eg Apostol (1957), chapter An alternatve approach, complementary to the present one, has been proposed by Hll (1999) and (2004). Roughly speakng, he suggests usng a chan ndex (for ntertemporal, cross-country or panel comparsons) but choosng the lnks n the chan so that on average the growth of prces between any two lnks s as close to proportonal as possble. 8

11 + α ln p ( t) + 1 ln P ( t) = α 0 2 γ j ln p ( t)ln p ( t) (5) j j We normalse by settng p ( 0) = 1, all. Consstency wth economc theory requres that the parameters of the system satsfy the followng addng-up and symmetry restrctons: α = 1; γ = 0, ; β = 0; γ = γ, j j j j j The Quadratc Almost Ideal Demand System of Banks, Blundell and Lewbel (1997), or QAIDS, s a generalsaton of the AI system. It too s consstent wth economc theory and exactly aggregable. Emprcally, ts strength s that t allows Engel curves to be quadratc whch provdes a much better ft to the data; t turns out that for many commodtes Engel curves are not lnear n the log of deflated expendture as would be requred by the AI system (Banks et al., 1997; Blow et al., 2004). The share equatons correspondng to the QAIDS are smlar to those of the AI system, except that there s an addtonal term n the square of deflated expendture: λ s ( ) ln ( ) ln ( ) [ ln ( )] 2 t = α + γ j p j t + β z t + z t, = 1,..., N j β p ( t) (6) In addton to the prevous parameter restrctons, consstency wth economc theory also requres that λ = 0. The ndrect utlty functon for the QAIDS s gven by Banks et al. (1997) as: ln u( t) = ln z( t) β p ( t) + ln z( t) λ ln p ( t) (7) 9

12 (applyng my notaton to ther equaton (3)). 12 Let us pck a partcular year as the base year for utlty, say year R. Now, by approprate choce of currency and quantty unts, we can normalse so that ln z( R ) = 0. Then ln u( R ) = 0 also. Equvalently, we can choose utlty unts so that ln u( R ) = 0 ; then ln z( R ) = 0. So by settng the left hand sde of equaton (7) equal to 0, we can solve for the level of deflated expendture requred to keep utlty at the R base year level (zero), when the prces of year t preval; ths s denoted by z ( t ) : R ln z ( t) R 0 = ln z ( t) = 0 β R p ( t) + ln z ( t) λ ln p ( t) (8) For the QAIDS, the compensated shares (the budget shares that would preval f utlty were held constant at ts base year level) are obtaned from equaton (6): R R R λ R s ( t) = α + γ j ln p j ( t) + β ln z ( t) + ln z ( t) j β p ( t) R = α + γ ln p ( t), = 1,..., N j j j 2 (9) usng (8). I now wrte R α rather than smply α snce the values of the α depend on the normalsaton adopted for ln z( R ) : see the next sub-secton below. Pluggng (9) nto (6) and solvng, R λ s ( t) = s ( t) β ln z( t) ln z( t), = 1,..., N p β [ ] 2 (10) Ths s the relatonshp between compensated and actual shares that we are seekng and on whch we shall rely for the emprcal analyss. Also, by settng t = R n (10), we obtan: R s ( R) = s ( R), = 1,..., N (11) 12 By settng all the λ equal to zero, we obtan the ndrect utlty functon and the expendture functon of the AI system. 10

13 R snce ln z( R) = ln z ( R) = 0 by normalsaton. That s, compensated and actual shares are equal n the base year. Actually, ths result s ndependent of any normalsatons as follows from consderaton of the expendture functon: R s ( R) = ln E[ p ( R), u( R)]/ ln p = s ( R) In the emprcal work reported below t proves convenent to make the base year for utlty the same as the reference year for the prce ndex, e we set R = So (10) and (11) then become 0 λ s ( ) ( ) ln ( ) [ ln ( )] 2 t = s t β z t z t, β p s (0) = s (0), = 1,..., N 0 (12) snce ln z (0) = 0 by normalsaton. 3.1 A new nterpretaton of the AI and QAIDS prce ndex P The AI prce ndex P defned by equaton (5) has never tll now been gven a clear nterpretaton. The fact that t depends only on prces and not on utlty or deflated expendture suggests that t mght have some connecton wth the Konüs prce ndex. The followng argument shows that the connecton s a very close one: the prce ndex P s n fact the Konüs prce ndex for the AI system and the QAIDS. R As before choose year R as the base year. Then from (8) we have that ln z ( t ) = 0. Correspondng to deflated expendture z R (t) there s also a level of nomnal expendture requred to keep utlty at ts level n year R whch we can wrte as x R (t). So from the defnton of z( t ) we have: 13 A further natural normalsaton s to choose currency and quantty unts so that x (0) = 1. Together wth the other normalsatons ths then mples that P (0) = 1, whch fxes α 0 : α 0 = 0. However, ths s not requred for any of the results n the present paper. 11

14 R R R 0 = ln z ( t) = ln[ x ( t) / P( t, R)], e x ( t) / P( t, R) = 1 (13) I now wrte the AI prce ndex P as P( t, R ) snce as I am about to show t depends on the R base year R. Now x ( t ) s the cost of purchasng the perod R level of utlty at the prces of R R perod t, so x ( t)/ x (0) s an ndex of the cost of purchasng the perod R level of utlty, at the prces prevalng n year t, relatve to the expendture requred at perod 0 prces. In other words: x R ( t)/ x R (0) = E[ p( t), u( R)]/ E[ p (0), u( R)] = P K ( t, R)/ P K (0, R) Snce equaton (13) holds for t = 0 too, we have by dvson that R R x ( t) / x (0) = P( t, R) / P(0, R) So the last result shows that the prce ndex P of the AI system and the QAIDS, gven by equaton (5), s not just any old prce ndex but s dentcal to the Konüs prce ndex wth base perod R: K K P ( t, R) / P (0, R) = P( t, R) / P(0, R) (14) Surprsngly, as far as I know ths result has not been ponted out before. The result may appear puzzlng at frst sght snce we know that there s a dfferent Konüs prce ndex for each choce of base year for utlty (unless tastes are homothetc), whle there seems to be only one AI system prce ndex, gven by equaton (5). The paradox s resolved by notng that the α parameters whch partly determne the growth rate of P depend on the choce of base year for utlty. By settng t = R n (9) and usng (11): α = s ( R) γ ln p ( R), = 1,..., N (15) R j j j (When R = α = s ). However the values of the other parameters n the defnton of the 0 0, (0) prce ndex n equaton (5), the γ j, are nvarant to the choce of base year snce these are 12

15 sem-elastctes of budget shares wth respect to prces, wth utlty held constant. Thus, as asserted, the AI system (and QAIDS) prce ndex P s dentcal to the Konüs prce ndex f demand s correctly descrbed by ths system. And the growth rate of ths prce ndex vares wth the choce of base year, through the α parameters. 3.2 Estmatng the compensated shares emprcally An emprcal counterpart to the share equatons can be wrtten by transformng equatons (6) to dscrete tme and addng an error term: s = + ln( p / p ) + ln z + ln y +, = 1,..., N; t = 0,..., T 1 (16) 0 N t α γ 2 j jt 1t β j t λ t ε = t Here ε t s the error term, we have put N 2 ln yt [ln zt ] / p β t =, and we have used the fact that γ = 0,, to express the share equatons n terms of relatve prces (takng the frst j= 1 j good as the numerare). 2 In the QAIDS, there are ( N N) / 2 ndependent γ j parameters and the α, β, and λ 2 number a further 3( N 1) ndependent parameters, for a total of ( N + 5N 6) / 2. So f the number of commodtes s at all large, t s qute mpractcal to estmate such a system, because the number of γ j parameters explodes. But for the purpose of estmatng a cost-oflvng (Konüs) ndex, we don t need to! The trck s to fnd a parsmonous way of estmatng N γ ln( / 2 j p jt p1 t ) as a lnear combnaton of parameters and varables, wthout tryng to j= recover the ndvdual parameters. There are two ways to do ths. The frst way comes from notng that accordng to equaton (10) we only need estmates of the parameters relatng to ncome elastctes (the β and λ ) n order to derve compensated shares from actual shares; we do not need estmates of the parameters relatng to prce elastctes. 14 If we had access to household survey data for one or more perods, we could estmate these ncome elastcty parameters econometrcally, assumng that all households face the same prces n a gven 14 I gnore here the fact that the AI system prce ndex P used to estmate deflated expendture s a functon of these parameters. The teratve procedure explaned below gets round ths dffculty. 13

16 perod. Here however we follow a second approach whch only requres aggregate data on budget shares and prces. Frst we can note a specal case n whch the effects of relatve prces can be exactly captured by just one varable. Suppose that all relatve prces are growng at constant but possbly dfferent rates: ln( p / p ) = µ t, j = 2,..., N jt 1t j where the µ j are the growth rates of the relatve prces and the frst product s taken as the 2 N N numerare. Then γ ln( / 2 j p jt p1 t ) = t γ j= j jµ j = δt, say. In ths case the effect of = relatve prces s captured entrely by a tme trend, wth a dfferent coeffcent n each share equaton (subject to the cross-equaton restrcton that δ = 0 ). A more general case can be treated by usng prncpal components as a data reducton technque. 15 We can collapse the relatve prce data nto M prncpal components, where M < N 1 s to be chosen emprcally. Then the share equatons (16) can be wrtten as: s = + PC + ln z + ln y +, = 1,..., N ; t = 0,..., T 1 (17) 0 M t α θ 1 k kt β k t λ t ε = t where PC kt s the kth prncpal component of the N 1 relatve prces and the θ k are coeffcents subject to the cross-equaton restrctons θ = 0, k. 16 The success of ths strategy wll depend on whether the varaton n relatve prces can be captured by a farly small number of prncpal components small that s n relaton to the number of tme seres observatons, T. We have now reduced the problem to estmatng a system of equatons, each of whch contans only M + 3 coeffcents ( the θ, α, β, and λ ). We must also take account of the cross-equaton restrctons: k k α = 1; β = 0; λ = 0; θ = 0, k 0 k 15 For a textbook exposton of prncpal components, see Johnson and Wchern (2003, chapter 8). 16 The specal case just dscussed s where the whole varaton n relatve prces can be captured by one prncpal component, a tme trend. 14

17 These addng-up restrctons are automatcally satsfed f the system s estmated equatonby-equaton usng OLS, though f the regressors are endogenous or the errors are non-normal ths mght not be the best method. There s one loss from usng prncpal components: we can no longer mpose the symmetry restrctons. 17 Fnally, we must recognse that z t and y t are measured wth error, snce both nvolve ntally unknown parameters. We can proceed here n the same way as do emprcal workers who are seekng to estmate all the parameters of the QAIDS (eg Banks et al., 1997): that s, use teraton. Start wth an ntal estmate of z t and y t, and then estmate the system of (17). Use the regresson estmates of the unknown coeffcents to obtan updated estmates of z t and y t. Then re-estmate the system (17) usng these updated estmates, and contnue to terate tll convergence s reached. In more detal, we can generate an ntal estmate of the prce ndex P usng a chaned Törnqvst ndex (a chaned Laspeyres or chaned Fsher ndex would serve as well): (1) 1 ln P = ( s + s ) ln p, P t 2 t t 1 t (1) 0 = 1 where a superscrpt number n parentheses denotes the number of the estmaton round; n ths case (1) denotes the frst round. Intally, we can set our estmates of the β to zero. Then ln z = ln[ x / P ] and ln y = {ln[ x (1) t t (1) t (1) t t / P (1) t ]} 2 In the next round, we can update the prce ndex by (2) 1 0 (1) 0(1) ln P = ( s + s ) ln p, P t 2 t t 1 t (2) 0 = 1 Here the 0(1) s t are the frst round predctons of the compensated shares from (17), derved as: 17 For example, suppose that N = 3 and that the specal case of all relatve prces changng at constant rates apples. Then, droppng the thrd equaton, takng the frst product as the numerare, and mposng all the constrants, the relatonshp between the δ and the γ j s as follows: δ1 = γ 12µ 2 ( γ11 + γ 12) µ 3, δ2 = γ 22µ 2 ( γ12 + γ 22) µ 3. These relatonshps mply no further restrctons on δ 1 and δ 2. 15

18 s = ˆ + ˆ PC, = 1,..., N; t = 0,..., T 1 0(1) (1) M (1) t α θ k = 1 k kt where a hat (^) denotes a regresson estmate. Then we can update z t and y t by (2) (2) (2) (2) 2 ˆ(1) ln zt ln[ xt / Pt ] and ln yt {ln[ xt / Pt ]} p β t = = Assumng convergence, the fnal estmate of P wll be (an approxmaton to) the Konüs prce ndex, when perod 0 s the base year for utlty. It wll be a chan ndex number, but t wll be path-ndependent. Ths method could be appled by natonal statstcal agences, usng exactly the same data as they employ to construct real world consumer prce ndces. Though I do not pursue ths pont n the present paper, the same method could also be appled to the constructon of cross-country prce ndces (nternatonal comparsons of purchasng power) Changng the base year Once we have estmated a Konüs prce ndex wth zero as the base, we can use the followng relatonshps to estmate a Konüs prce ndex wth some other perod, say R, as the base. Takng frst dfferences n equaton (9), we obtan: 0 s R ( t) = γ j ln p j ( t) = s ( t), = 1,..., N (18) j e the changes n compensated share are ndependent of the choce of base year. Also, R accordng to (11), s ( R) = s ( R), = 1,..., N, e the compensated and actual shares are equal n the base year. Hence from (18) R t 0 ( ) = ( ) + ( ), 1 ; 1,..., τ R 1 τ > = = + s t s R s T t R N R 0 ( ) ( ), 0 ; 1,..., τ t 1 τ = + = s R s t < R = N (19) 18 Neary (2004) uses the QAIDS to estmate Konüs prce ndces and real ncome for 60 countres n 1980, based on 11 commodty groups. 16

19 Once we have estmated the compensated share changes wth 0 as the base year (from the estmated 0 s t, see above), we can then recover the levels of the compensated shares when R s the base year from (19). These compensated shares can then be used to estmate a Konüs prce ndex wth perod R rather than perod 0 as the base. So we only need to estmate the parameters of nterest once, for one base year. Then we can calculate a Konüs prce ndex wth any other year as the base; there s no need for any further econometrc estmaton. 4. The Method n Practce The results reported here are based on a dataset of prces and budget shares for 70 products n the U.K. s Retal Prces Index (RPI) over the perod The dataset was orgnally put together by the Insttute for Fscal Studes: see the Data Appendx for more detal and for descrptve statstcs. 19 These 70 products account for vrtually 100% of the tems n the RPI n the earler years though the coverage gradually falls after 1992 to reach 91% n Total expendture (x) s measured on a per capta bass. It s estmated as total fnal consumers expendture by U.K. households n the U.K. and abroad n current prces (ONS code: ABJQ), multpled by the proporton of total expendture on the All tems RPI (e the overall ndex of retal prces) that s covered by the prces ncluded n the present study, dvded by the populaton. The mean nflaton rate as measured by a number of conventonal prce ndces s shown n Table 1. All the chaned measures are smlar to each other; ndeed, the chaned Fsher and Törnqvst ndces are dentcal to two decmal places. However the fxed weght ndces, whch use ether the frst year (1974) or the last (2004), dffer more markedly. Interestngly, the Paasche ndex grew more rapdly than the Laspeyres, contrary to the normal expectaton. The frst step was to estmate the prncpal components of the 69 log relatve prces. 20 The proporton of the varaton explaned by successve components s gven n Table 2. The frst 19 The data on U.K. retal prces and budget shares were kndly suppled by Andrew Lecester of the Insttute for Fscal Studes (IFS). A very smlar dataset underles Blow et al. (2004). 20 Snce the varables to be summarsed are log relatve prces and so have the same unts, the prncpal components were based on the covarance matrx, not the correlaton matrx, e 17

20 sx prncpal components account for 97.8% of the varaton. Wth 13 components the cumulatve proporton rses to 99.7%. On ths bass, t was decded to employ sx prncpal components (see below for the effect of ncludng more or fewer prncpal components). However a potental problem for the emprcal analyss s the hgh degree of multcollnearty between the prncpal components, deflated expendture (ln(z)) and deflated expendture squared (ln(y)). The multple correlaton coeffcent between the sx prncpal components and ln(z) s and that between the sx prncpal components and ln(y) s 0.994; the smple correlaton coeffcents between the frst prncpal component and ln(z) and ln(y) are respectvely and Next, equaton (17) was estmated for each of the 70 products over the perod by OLS. 21 Convergence of the estmates of equaton (17) was rapd. There was very lttle change n the estmated growth rate of the Konüs prce ndex (wth 1974 as the base) after three teratons; nevertheless a further fve teratons were carred out by whch tme the mean growth rate was stable up to the 7 th decmal place. Once convergence was reached, Konüs prce ndces were then constructed wth each of the years 1974 to 2004 n turn servng as the base, usng equatons (19) to generate the compensated shares for the reference years The results of the eghth and fnal round of estmates appear n Table On the whole the model fts qute well, as measured by R 2 (last column). The Breusch-Godfrey LM test for seral correlaton suggests that at the 5% level frst-order seral correlaton s present n only 15 of the 70 equatons. Stll, as a precauton, the t ratos are based on Newey-West standard errors whch are robust to seral correlaton. On the bass of a Wald test, the sx prncpal components of log relatve prces are jontly sgnfcant (non-zero) at the 5% level n 64 out of 70 equatons. A Wald test shows that the coeffcents on ln(z) and ln(y), the β and λ, are jontly nonzero at the 5% level for 37 products; that s, for each of these 37 products the ncome elastcty dffers sgnfcantly from one, whle for the other 33 products t does not (Table 3). the varables were not standardsed. Estmaton was done by Stata s pca command, wth the covarance opton. 21 Snce the regressors are the same n each share equaton, estmaton by SUR would lead to dentcal results. 22 These Konüs prce ndces were estmated usng the Törnqvst formula. 23 In a small number of cases, affectng nne products, the estmated compensated shares were negatve n a few years. In these cases the estmates of the compensated shares were set to zero and the sum of the shares was constraned to equal one. 18

21 (On the bass of t tests, ln(z) s ndvdually sgnfcant at the 5% level for 25 products, ln(y) for 36 products). So despte the hgh multcollnearty we have already noted between the prncpal components, ln(z) and ln(y), ncome elastctes are stll found to dffer sgnfcantly from one n the majorty of cases. The mportance of the λ parameters s apparent n the Engel curves. These show budget shares as a functon of log real ncome per capta wth prces held constant. Only n 14 out of 70 cases are the Engel curves approxmately lnear even when attenton s confned to the range of real ncome observed over the study perod As explaned n the prevous secton, these regresson results can be used to generate 31 dfferent Konüs prce ndces, one for each possble base year for utlty over The average growth rates of these 31 Konüs prce ndces over three ntervals, , and the whole perod , appear n the left hand panel of Table 4. The mean of the average growth rates of these 31 ndces s close to the conventonal chaned Laspeyres ndex of the 70 component prces. In fact, over the whole perod the mean of the 31 average growth rates, 6.20 per cent per annum, s almost exactly the same as that of the chaned Laspeyres, 6.21 per cent per annum (Table 1); at 6.15 per cent per annum, the values for the chaned Fsher and chaned Törnqvst ndces are also close. Ths s n accordance wth the theoretcal predctons of Balk (2004) and Feenstra and Rensdorf (2000), even though these were establshed under more restrctve condtons than apply here. Nevertheless there s qute a lot of varaton between ndces wth dfferent base years. Thus over the whole 31 year perod the mnmum average growth rate s 5.91 per cent per annum (usng 2004 as the base) 24 whle the maxmum s 6.39 per cent per annum (wth 1978 as the base), a dfference of 0.48 percentage ponts. Surprsngly, there s almost as much varaton between ndces wth dfferent base years n the more recent, low-nflaton perod as there s n the hgh-nflaton perod In the low-nflaton perod, the maxmum average growth rate (2.97 per cent per annum) s found when 1993, 1995 or 1996 are used as the base year, the mnmum (2.34 per cent per annum) when 2004 s the base; the dfference between mnmum and maxmum s 0.63 percentage ponts. By contrast the dfference between mnmum and maxmum average growth rates n the hgh-nflaton perod s 0.69 percentage ponts (comparng 2002 wth 1978 or 1979 as the base). 24 Recall that 5.91 per cent per annum s the answer to the followng queston: gven the money ncome of the average household n 2004, what s the change n ts money ncome between 1974 and 2004 (expressed as an annual percentage rate) whch would have allowed the household to enjoy ts 2004 utlty level n 1974? 19

22 Table 4 n conjuncton wth Table 1 also reveals the sze of the path-dependence bas of a conventonal ndex number lke the chaned Laspeyres. Dependng on the base year for utlty and the tme perod, ths can be as large as or per cent per annum (Table 5). The relatonshp between the average growth rates of the Konüs ndces and the base year can be seen more clearly n Fgures 1-3. Over the frst sub-perod, , and over the whole perod, , the relatonshp between the average rate of nflaton and the base year s roughly lnear and negatve. Over the second sub-perod, , there s stll a negatve relatonshp but only for base years later than Snce on average real ncome rose steadly over , a negatve relatonshp means that the less well off were more adversely affected by nflaton than were the more prosperous. Ths s an nterestng emprcal fndng though not one that would necessarly generalse to other perods and other countres. 4.1 IV estmates As mentoned earler, estmaton by OLS mght be questoned snce the real expendture varables (z and y) are measured wth error (gven that the QAIDS prce ndex P s tself an estmate) and also may be endogenous: a rse n the prce of some good may lead households to draw down ther lqud reserves. The share equatons were therefore also estmated by IV (2SLS), usng as nstruments one lag of ln(z) and of ln(y) and the chaned Laspeyres measure of the overall nflaton rate. 25 The ratonale for usng the latter as an nstrument s that households are more lkely to suffer unwelcome surprses and so are more lkely to draw down ther savngs when nflaton s hgh. In fact, the partal R 2 statstcs of Shea (1997) and Bound et al. (1995) are both qute hgh when all three nstruments are ncluded. But when only the lags of ln(z) and ln(y) are ncluded the same statstcs suggest that these two nstruments by themselves are weak. Wth all three nstruments ncluded, Hansen s J statstc suggests that we can reject the null of no correlaton between the nstruments and the errors at the 5% level n only 9 out of 70 cases. The resultng IV estmates of the 31 Konüs prce ndces were remarkably smlar to the OLS ones as the rght hand panel of Table 4 shows. 26 The correlaton coeffcent between the 25 The use of GMM s lkely not justfed due to the small number of tme seres observatons on each share (Baum et al., 2003). 26 The IV estmates were produced wthn Stata by the vreg2 command wrtten by Baum et al. (2003). 20

23 OLS and IV estmates of the mean growth rates of the 31 ndces over s However, the problem noted earler the tendency for some of the estmated compensated shares to be negatve after 1990 s a bt more serous now. After the negatve shares are set to zero (but pror to the sum of the shares beng constraned to sum to one), the sum of the IV estmates of the compensated shares rses steadly to reach n 2004; the same sum s only for the OLS estmates. So n ths case nothng much seems to be ganed by movng from OLS to IV. But at any rate t s reassurng that the OLS and IV estmates of the Konüs prce ndces are so smlar. 4.2 Varyng the number of prncpal components The decson to nclude just sx prncpal components n the estmaton of equaton (17) mght be crtcsed as arbtrary. So I also estmated ths equaton and the resultng Konüs ndces usng alternately one through nne prncpal components (e settng M equal to successvely 1,2,...,9 n equaton (17)). The effect on the mean nflaton rate over (usng ether 1974 or 2004 as the base year for utlty) s llustrated n Table 6. Here a dfferent knd of convergence s apparent. For example, wth just one prncpal component ncluded, the mean nflaton rate wth 2004 as the base year s 6.20 % p.a., whle wth sx ncluded t s 5.91%. In ths case the nflaton rate falls as the number of prncpal components s ncreased, but wth lttle further effect once four are ncluded. A smlar comment apples when 1974 s the base: there s agan lttle further effect on the estmated nflaton rate once four prncpal components have been ncluded (though now the nflaton rate rses as the number of prncpal components s ncreased). Hence the use of sx prncpal components as n Table 3 can be defended as a reasonable compromse between the desre to account for as much of the varaton of relatve prces as possble and the need to conserve degrees of freedom. 5. Conclusons I have argued that chan ndces typcally suffer from path-dependence bas by comparson wth true cost-of-lvng ndces. I have proposed a method of removng ths bas, usng a flexble model of consumer demand that s known to ft the data well (at least at a hgh level 21

24 of aggregaton), the QAIDS. Although testng the QAIDS requres a large amount of data, usng t to remove the bas requres much less data. I have appled ths method to estmate Konüs prce ndces for 70 products coverng most of the U.K. s Retal Prces Index over , wth each year n turn as the base. In ths case t turns out that annual data suffcent to estmate nne coeffcents for each commodty are all that s requred. The choce of base year s found to have a sgnfcant effect on the ndex, even n the low nflaton perod snce For example, wth 1993 as the base year for utlty, the average growth rate of the Konüs prce ndex over s 2.97 per cent per annum; wth 2004 as the base year t s 2.34 per cent per annum, a dfference of 0.63 percentage ponts. The path-dependence bas of a conventonal ndex number lke the chaned Laspeyres ndex of the RPI can be as large as or per cent per annum (dependng on the base year for utlty and the tme perod). Judgng by these results, the method proposed here could be mplemented n practce on the consumer prce ndex at a detaled level by any statstcal agency possessng 30 years or more of annual data on prces and budget shares. It could also be used to mprove the measurement of cross-country dfferences n lvng standards, usng the knd of data generated by the World Bank s Internatonal Comparson Program. These results do however pont to an ssue that as far as I am aware s unresolved: what s the best base year for utlty? Fxed base ndces, where t s the weghts (prces or quanttes) that are fxed, are no longer popular wth statstcal agences. But now the problem of a fxed base for the weghts seems to reappear n the guse of a fxed base for utlty. A closely related ssue has been dscussed extensvely n the lterature on nternatonal comparsons of purchasng power and real ncome (see eg Caves et al., 1982; Dewert, 1987; Hll, 1999 and 2004; Neary, 2004). In the cross-country context, the base country plays the same role as does the base year n the tme seres context. It s wdely held that the ndex number for real ncome should be nvarant to the choce of base country. One could take the purchasng pattern of a sngle country, eg the Unted States, as the base but t s far from clear that ths s approprate f we want to compare the real ncomes of Albana and Zamba. So the ndex numbers commonly employed for nternatonal comparsons (Geary, EKS, or the measure proposed by Caves et al. (1982)) represent some sort of average of the blateral ndces based on each country n turn as the base. However, t s not clear that the arguments for base-country nvarance carry over to the tme seres context, the concern of the present paper. Here the choce of base would seem to depend on the purpose at hand. The choce for 22

25 a central bank targetng nflaton mght be dfferent from the choce of a statstcal agency or an economc hstoran measurng real GDP. 23

26 Table 1 Conventonal prce ndces: average annual growth rates, % p.a Chaned Törnqvst Chaned Fsher Chaned Laspeyres Chaned Paasche Laspeyres (base 1974) Paasche (base 2004) Source Offce for Natonal Statstcs, Insttute for Fscal Studes and own calculatons. All ndces are for 70 products coverng most of the tems n the RPI. Table 2 Prncpal component analyss of 69 log relatve prces: U.K. Retal Prces Index, Egenvalue Dfference Proporton Cumulatve Component Component Component Component Component Component Component Component Component Component Component Component Component Note The prncpal components were estmated from the logs of the prces for the 70 products wthn the U.K. RPI lsted n Table A.1, wth the frst product ( Bread ) taken as the numerare; each relatve prce takes the value 1 n 1974 (0 n logs). 24