Comparing Class Level Chain Drift for Different Elementary Aggregate Formulae Using Locally Collected CPI Data


 Poppy Bennett
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1 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng Gareth Clews 1, Anselma DobsonMcKttrck 2 and Joseph Wnton Summary The Consumer Prces Index (CPI) s a measure of consumer prce nflaton; t s a fxed basket prce ndex constructed from a combnaton of weghted and unweghted arthmetc and geometrc means of prce relatves. The content of the basket s updated every year and so are the weghts; ths s to ensure that the selecton of products n the basket and ther weghts reman representatve of the overall pattern of expendture of households covered by the CPI. These changes lead to a dscontnuty n the ndex seres. To remove ths effect, the ndex seres before and after the change are lnked to produce a contnuous seres and ad comparson of the prce ndex level across years. A prce ndex can be used to measure nflaton n a number of ways. The most common s to look at how the ndex has changed over a year; ths s calculated by comparng the prce ndex for the latest month wth the same month a year ago. Makng ths comparson nvolves crossng a lnk. Comparng ndex numbers across a tme nterval contanng one or more lnks can gve dfferent values to the same comparson wthout a lnk. The dfference between an ndex number calculated wth a lnk and an ndex number calculated over the same perod drectly (wthout a lnk), s known as chan drft 3. If we make a comparson across a lnk where the basket and the weghts are unchanged then whether we see chan drft depends on the type of ndex formula we use. If the ndex s bult from purely geometrc formulae at both elementary aggregate and hgher levels then there s no drft a comparson of ndex number dfferences from a lnked and drect ndex seres wll gve the same answer. However, f some arthmetc formulae are used ether at the elementary aggregate level, or the upper levels, or both, then the drft wll not be zero 4. Measurng chan drft n the CPI s complcated by effects such as the changng of the basket of tems each year and the weghts. What we are nterested n s the dfferences between chaned and drect ndces due to the choce of formula Placement student from the Unversty of the West of England Channg s repeated lnkng usually at regular tme ntervals; f the comparson of ndex numbers s made across a tme perod contanng several lnks then the drft s cumulatve. It s possble for the drft to be zero wth arthmetc formulae, but ths s only n very specal crcumstances Offce for Natonal Statstcs 1
2 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng Ths paper nvestgates the magntude of the drft for three choces of elementary aggregate formula; the Carl, Dutot and Jevons, where the lnk occurs at the class level as s the case for the CPI. In all cases, the upper level ndex formula s the Lowe as s used n the CPI. The calculatons are made usng locally collected prce data and magntudes of drft are calculated for each class. It s wdely quoted that the use of the (arthmetc) Carl formula at the elementary aggregate level wll lead to a larger drft than the (geometrc) Jevons. The Dutot formula s expected to behave somewhat smlarly to the Jevons formula despte ts use of an arthmetc mean. These comparsons are complcated by the fact that the lnk s carred out at the class level, so elementary aggregate ndces are combned usng the Lowe ndex formula (a weghted arthmetc average). The emprcal results show that the drft from usng the Carl formula does exceed the drft from the Jevons n most, but not all cases. The average (absolute) drft across all classes n 2009 for the Carl s 0.67, for the Jevons s 0.11 and for the Dutot s The correspondng Fgures for 2010 are 0.47, 0.06 and There are a few classes where the drft of the Jevons exceeds the drft for the Carl 5 out of 65 classes n 2009 and 10 out of 64 n The drft when usng Jevons s often smlar to when usng Dutot and can be hgher or lower. An understandng of the factors that contrbute to the magntude of the drft wll be the subect of further work. Acknowledgements The Authors would lke to thank Natale Weaden, Robert O Nell, Ra Sanderson and Jeff Ralph for ther contrbutons to ths research. 1. Introducton The degree to whch dfferent elementary aggregate (EA) formulae contrbute to chan drft s a topc whch enters the dscusson of whch s the most approprate formula to use at the lowest level of aggregaton for an nflaton measure. One of the crtcsms of the Carl formula s that ts arthmetc nature wll result n a hgher chan drft than the geometrc Jevons formula. Whle ths may seem to be somewhat natural, the actual stuaton s complcated by the fact that chan lnkng doesn t occur at the elementary aggregate level, but at the class level. To reflect the formula used n the CPI, a comparson has to be made wth the dfferent elementary aggregates combned wth the Lowe formula up to class level. The topc of chan drft and the choce of elementary aggregate formula was rased durng the RPI Consultaton; however, there was no resource avalable to nvestgate t at the tme. There appears to have been lttle nvestgaton of ths topc n the lterature. The most elementary scenaro n whch chan drft occurs s a prce bounce stuaton. Ths s where the prce starts at a specfc level before ncreasng (or decreasng) n the next perod and then returnng to ts ntal value n the followng perod. Ths can also be extended across any number of perods wth only the requrement that the start and end prce are equal. We consder the choce of elementary aggregate formula on the chan drft at hgher levels of aggregaton. Offce for Natonal Statstcs 2
3 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng The structure of ths paper s as follows. Frst we provde a bref overvew of the terms used n ths paper and a descrpton of the constructon of the CPI. We then descrbe the data used followed by a theoretcal analyss of the problem. We nclude our results before concludng. 1.1 Defnng drft To defne chan drft, frst consder three tme perods, t=0, 1 and 2 and an arbtrary ndex formula f( ab, ). The drect ndex, Λ, between t = 0 and t = 2 s defned as applcaton of the ndex formula to the start and end data, ( 0, 2) f ( 0, 2) Λ = (1) A lnked ndex, Λ, across the tme perod [0,2], wth a lnk at tme perod t = 1 s gven by: ( 0, 2) f ( 0,1) f ( 1, 2) * Λ = (2) In ths paper, chan drft s defned as the dfference Λ (0,2) Λ(0,2). The lmted lterature on drft ncludes the rato of a lnked and drect ndex and defnes ths as a drft factor; however, the rato and dfference are related so there s no ssue n consderng a dfference[1]. 1.2 Structure of the CPI The CPI measures the change n prces for goods and servces purchased by households over tme. It s a measure of nflaton based on average household expendture on tems n a fxed basket. In an deal stuaton the CPI would be calculated usng the prces for all transactons occurrng throughout the UK. However, ths s not avalable so there are two approxmatons made. The frst s to collect prces for tems on shelves and consder ths as a proxy for the prces pad by consumers for tems. Secondly only a sample of all avalable tems from a sample of locatons s taken. Ths sample forms the basket. The tems n the basket are classfed based on the nternatonal classfcaton system for household consumpton expendtures, known as Classfcaton of Indvdual Consumpton by Purpose (COICOP). Ths s a herarchcal structure. Each level s an aggregaton of somewhat homogeneous subgroups. For example, the alcohol class s made up of separate subclasses for beer, wne and sprts, each of whch s also an aggregate of more detaled types of alcohol. Fgure 1, below, provdes a dagrammatc example of ths construct. The basket remans fxed, leadng to use of the term fxed basket, for one calendar year and s updated to reflect changes n consumer spendng habts every February. Offce for Natonal Statstcs 3
4 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng Fgure 1: Dagrammatc explanaton of the structure of the COICOP classfcaton system Prces are collected for tems n the basket and are used, alongsde weghts  from a varety of sources  whch descrbe expendture on those tems, to calculate prce ndces at each level of the COICOP structure. These weghts are calculated from expendture shares for the classfcaton and are avalable at all levels other than the stratum level (see Fgure 2 below). At the most detaled level of descrpton weghts are not avalable. Hence, some unweghted averages are used n the CPI. These are referred to as elementary, or unweghted, aggregates. Wthn the CPI ths form of aggregate s calculated only where weghtng nformaton s not avalable. In practce the formula used where weghts are not avalable s the Jevons prce ndex or n some cases the Dutot prce ndex. The Jevons prce ndex s an unweghted geometrc mean of a group of prce relatves, where a prce relatve for a gven product s the rato of the prces n the current perod to the prces n the base perod. The Dutot prce ndex s the rato of the arthmetc mean of prces n the current perod to the arthmetc mean of prces n the base perod. At hgher levels, where we have weghts for each component, a weghted arthmetc mean s used. Ths s often descrbed as beng of Laspeyres type and s known as a Lowe ndex, where the weghts are calculated from expendture nformaton n a perod before the prce reference (base) perod. Fgure 2 provdes a dagrammatc nterpretaton of the structure of the CPI. Ths Fgure s a smplfed verson of the structure descrbed n the CPI Techncal Manual [2] whch provdes more nformaton on the structure and weghts used at each level. Dfferent weghts wthn the CPI are updated at dfferent tmes n the year. In January, the class weghts are updated. Followng ths, n February, the basket tself s updated as descrbed above, along wth the assocated tem weghts. As a result of ths the CPI s chan lnked twce every year. Ths s done by calculatng a January based ndex based on the prces for the December n the prevous year. We then multply ths by the ndex for the precedng years from the base perod (2005) and the newly calculated ndex n the current month wth prces based n January. Ths means that January ndces contan only one chan lnk whlst those for the rest of the year have two. The formula used for ths calculaton s ncluded n the Annex. Offce for Natonal Statstcs 4
5 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng Fgure 2. The structure of the CPI, ndcatng where weghts are appled 2. Data The data used n ths analyss are the locally collected prce quotes used n the calculaton of the CPI 5. To obtan these data, prce collectors n the feld vst shops and outlets wthn sampled locatons. These data do not comprse the whole coverage of prces used and account for around 57% of the total weght. In the CPI the prce quotes collected from stores s supplemented by data collected centrally for products that are not bought from shops; for such thngs as TV lcences whch are unformly prced across the country; and for products for whch local collecton s dffcult. Fgure 3 shows a comparson between the month on same month n prevous year growths (commonly referred to as the twelve month growth) n the publshed CPI and the CPI calculated usng only the locally collected data. All locally collected elementary aggregates are constructed usng the Jevons prce ndex. 5 Avalable from the ONS webste: Offce for Natonal Statstcs 5
6 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng Fgure 3: Twelve month growth rates of the CPI when calculated usng only locally collected data and the full CPI 3. Methodology Measurng chan drft n the CPI s complcated by effects such as the changng of the basket of tems each year and the weghts. The dfferences between chaned and drect ndces caused by updatng weghts and the basket are a necessary result of those changes whch are made to ensure that the CPI remans relevant. The element of chan drft that we are most nterested n s the dfferences between chaned and drect ndces due to the choce of formula. Ideally, f lnkng occurs wthout a change to weghts or basket composton then there wll be no chan drft. To measure only the effect of the choce of formula on chan lnkng the ndces, wthout other contrbutng factors t s necessary to mantan the same basket of tems and assocated weghts. We have chosen to use an artfcal lnk durng the calendar year, whch mantans the fxed basket and weghts. Our three tme perods are May, June and July usng data for both 2009 and Offce for Natonal Statstcs 6
7 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng For the CPI structure we compared equaton (1) wth equaton (2) usng the formulae ( 0, 2) w ( 0,1) w ( 1, 2) * Λ = Ε Ε (3) And ( 0, 2) w ( 0, 2) Λ = Ε (4) where w s the weght for product up to the class level 6, Ε ( 0, 2) s the elementary aggregate for product n perod 2 based on perod 0, that s, ths s the stratum ndex for stratum. We are comparng three elementary aggregate formulae, the Jevons, Dutot and the Carl: 1 n p ( b) J Ε ( ab, ) = [Jevons] p ( a) D ( ab, ) Ε = p ( b) p ( a) [Dutot] Ε 1 p ( b) C ( ab, ) = n p a [Carl] Pror to calculatng the ndces, a cleanng process was appled to the locally collected prce data. The process ncluded removng mssng or zero prce relatves and also those prce relatves larger than a threshold value of 10. Ths follows the CPI methodology used by ONS. The process of calculatng prce ndces and aggregatng these to hgher levels followed that used n the CPI. The approach usng the Carl or Dutot elementary aggregates s dentcal to the constructon of the CPI wth only the formula used for the stratum level ndces changed. The stratum ndces ndces were then aggregated usng the assocated weghts and the Lowe formula 7 gven n (3) and (4). Ths resulted n drect and chaned class level ndces, and thus the value of the chan drft, for each of the elementary aggregate formulae. The comparson s made at the class level as ths s the frst level where ndces are chaned n the CPI. The chan drft was calculated for each class for both of the elementary aggregate formulae for each of the years 2009 and Ths s n effect a compound weght; we have flattened the structure of the CPI so that nstead of aggregaton across the homogenous groups we consder the sum across all stratum ndces. Ths greatly reduces the complexty of the formulae we present wthout changng the result. A Laspeyrestype (arthmetc) ndex wth weghtng nformaton taken from expendture data dated pror to the prce reference perod. Offce for Natonal Statstcs 7
8 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng 4. Analyss In ths secton, we use equatons (3) and (4) to make theoretcal conclusons about the relatve sze and drecton of the chan drft that can be expected from each elementary aggregate formula. Λ Λ as defned by equatons (3) and (4). We have the followng * Consder ( 0, 2) ( 0, 2) 0, 2 0, 2 w 0,1 w 1, 2 w (0, 2) * Λ Λ = Ε Ε Ε (, ) = wwε 0,1 Ε 1, 2 wε (0, 2) 2 = w Ε 0,1 Ε 1, 2 + wwε 0,1 Ε 1, 2 wε (0, 2) (, ) (5) Ths formula does not, at ths pont, have any dependence on the choce of elementary aggregate formula. By consderng the elementary aggregate beng one of Jevons, Dutot or Carl, n turn, we can somewhat smplfy equaton (5) for each stuaton. We should note here that ths formula also apples to any level of aggregaton. The dfference between each level s the weght appled to each elementary aggregate. In any numercal applcaton the weghts should be handled carefully. 4.1 Chan drft when usng a Jevons elementary aggregate J ab, ab, Ε J 0,1 Ε J 1, 2 =Ε J 0, 2 If Ε s a geometrc mean of prce relatves, Ε then equaton (5) becomes * 2 J J J (, ) J J J = ww Ε ( 0,1) Ε ( 1, 2) w( 1 w) Ε (0, 2) (, ) J J J = ww Ε ( 0,1) Ε ( 1, 2) ww Ε ( 0, 2) (, ) (, ) (, ) Λ 0, 2 Λ 0, 2 = w Ε 0, 2 + wwε 0,1 Ε 1, 2 wε (0, 2) J J J = ww Ε 0,1 Ε 1, 2 Ε 0, 2. Ths equaton s nether postve nor negatve defnte (or ndeed semdefnte). I.e. a pror the chan drft can be ether postve or negatve. 8 and 4.2 Chan drft when usng a Dutot elementary aggregate Ε D 0,1 Ε D 1, 2 =Ε D 0, 2 9. Hence, the equaton for chan drft when usng It s easly seen that the Dutot formula at the elementary aggregate level s gven equaton (6) wth the Jevons J D ab, Ε ab,. It s also elementary aggregate Ε replaced by the Dutot elementary aggregate (6) 8 9 A short proof of ths equalty can be found n Annex B A short proof of ths equalty can be found n Annex B Offce for Natonal Statstcs 8
9 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng the case, a pror, that the chan drft resultng n the use of Dutot at the elementary aggregate level may be postve or negatve and s not necessarly larger or smaller than that of the Jevons or Carl formulae. 4.3 Chan drft when usng a Carl elementary aggregate The use of an unweghted arthmetc mean of prce relatves, the Carl formula, at the elementary aggregate level provdes a more complcated problem than both the Jevons and Dutot elementary aggregate formulae. Here C C 1 p 1 1 p 2 E ( 0,1) E ( 0,1) = n p 0 n p 1 1 p 1 p 2 = 2 n p 0 p 1 1 p 1 pk 2 = 2 n k p 0 pk 1 (7) Ths formula does not easly smplfy to the drect ndex plus or mnus some other term as t does for the other formulae and so no smple argument for the consderaton of the chan drft n the hgher level s possble. Let ε be the chan drft n the elementary aggregates ε : = E 0,1 E 1, 2 E 0, 2 C C C There s no a pror condton whch means that ths chan drft must be postve, negatve, greater than that usng any other elementary aggregate formula n the report or less than t 10. We can, C C C however, wrte E ( 0,1) E ( 1, 2) = E ( 0, 2) + ε and then C 2 C C ( w w wε ) ww (8) Λ 0, 2 Λ 0, 2 = 1 E 0, E 0,1 E 1, 2 Equaton (8) can then be used to provde a condton on the sgn of the chan drft n the hgher level ndex n terms of weghts, elementary aggregates and the chan drft at the lowest level. Ths formula s a good deal more complcated and features more terms than the other formulae. It s plausble to compare the chan drft theoretcally usng ths formula and those for the other aggregates; however, ths only provdes theoretcal condtons and can only represent what happens n practce once we use the collected prces. We shall then provde emprcal evdence to make statements on whch aggregates cause greater or lesser chan drft gven the dstrbutons of prce relatves. (, ) 10 Ths s a typcal argument aganst the Carl elementary aggregate, whch theoretcally s not true. It s, however, backed up by emprcal evdence where the dstrbuton of prces s known. Offce for Natonal Statstcs 9
10 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng 5. Results 5.1 Chan drft Before we provde the results of the comparson of the values for chan drft when usng the dfferent formulae t s worth notng the assumptons and condtons for whch these results hold. The weghts for all perods n the chan are the same because they are chosen wthn the year; n the producton of the CPI these wll change across the lnks. Ths means that for the actual ndex our equatons do not apply exactly, but refer to a smplfed stuaton. A closed form formula for the chan drft, as n equatons (6), (7) and (8) can be obtaned for varable weghts. As noted n secton 2, the data used are locally collected prce data for 2009 and 2010 only; ths does not cover the entrety of the CPI collecton. The mplementaton of the channg used s also artfcal n order to remove the effect of updatng weghts and the basket. Ths means that the results do not relate drectly to any chan drft present n the CPI but rather provde a comparson of the relatve sze of the chan drft from usng dfferent formulae on CPI prce data. In practce the chan drft s also compounded year on year as the ndces are chan lnked twce a year from the base year of Ths has not been consdered as t makes the formulae more complcated and ntroduces other effects due to problems such as changes to the basket and weghts. We do not suggest that a drect ndex should be the target measure for nflaton as chan lnkng s necessary to ensure that the ndex remans relevant through basket and weght updates. Ths report only descrbes the effect that the chan lnkng procedure has on the value compared to the case where chan lnkng was not performed and focuses on dfferences n chan drft arsng from the choce of formula used to calculate elementary aggregates. These assumptons do not nvaldate the research; however, any numbers gven here are not drectly equal to any chan drft wthn the CPI. The results we present n ths paper should not be nterpreted as any knd of bas n the CPI, but only taken to show the effect of the chan lnkng procedure on the hgher level Laspeyres type ndex formula usng CPI data. Fgure 4 shows comparson of the chaned vs. drect ndces for each elementary aggregate n 2009 and There s a postve correlaton between drect and chaned ndces usng all three elementary aggregates, whch s to be expected. The dotted lne on the graph corresponds to those values for whch the chaned ndex concdes exactly wth the drect. We see that n most nstances the class ndces are above ths lne, meanng that the chaned ndex s larger than the drect. Ths s an artefact of usng an arthmetc mean at the hgher level and occurs for all choces of elementary aggregate. The effect s notceably larger n the ndces whch use the Carl formula at the lower level. It s worth notng that there are occasons where the chan drft n the class ndex usng a Jevons elementary aggregate s larger than that of the ndex number makng use of the Carl formula. Ths s rare, occurrng n only 15 nstances out of 129 across the dataset; where ths s the case, the drft s also small: the average absolute drft usng the Jevons n these cases s In the 114 cases where the absolute drft s greater n the Carl than the Jevons, the average absolute drft usng the Carl s Offce for Natonal Statstcs 10
11 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng Fgure 4: A comparson of the chaned vs. drect class level ndces for each of the elementary aggregate formulae for 2009 and 2010 Offce for Natonal Statstcs 11
12 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng The chan drft values are compared n Fgure 5. Fgure 5 shows that for all three elementary aggregates, the chan drft s postve n the maorty of cases (96%, 73% and 72% postve for Carl, Jevons and Dutot respectvely). Ths s most strkng for Carl whch often has large values of drft; Fgure 5 shows that the values of drft are clustered around zero for Dutot and Jevons but are often large for Carl. It s also notceable that the use of a Jevons and Dutot elementary aggregates n the CPI structure causes a suffcent proporton of classes to exhbt negatve chan drft, where the drect ndex s n fact larger than that when chan lnked. The use of a Carl elementary aggregate can be seen to create negatve chan drft for some classes but ths effect s seen far less frequently. Fgure 5 ndcates that there s no drect relatonshp between the ndex level tself and the amount of chan drft. We note here that one value for whch the chaned ndex exceeded 120 and the drect ndex was 105 was removed from the 2009 data; ths ndex used the Carl formula at the elementary aggregate level. The datum was removed to provde clarty of the behavour of the rest of the ndces n the Fgure and has been omtted throughout. Fgure 5. Chan drft vs. chaned ndex number for all classes Offce for Natonal Statstcs 12
13 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng Fgure 6 shows that the use of the Carl formula at the lowest level does not always cause larger absolute chan drft than the use of ether of the other two formulae, though when ths s the case the drft from the Carl s often much larger than the drft from ether Jevons or Dutot. Fgure 6 also shows that the chan drft from usng the Jevons formula at the elementary aggregate level s often very smlar to the chan drft from the Dutot. Fgure 6: Chan drft by class for each choce of elementary aggregate n 2009 and 2010 It s expected n the lterature that a Carl elementary aggregate wll nduce larger chan drft than the use of ether of the other choces of formula n the CPI at the class level. Fgure 6 supports ths clam. The average of the absolute values of the chan drft s gven n the table below. Offce for Natonal Statstcs 13
14 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng Table 1. Average of the absolute values of the chan drft across all classes n each year for each choce of elementary aggregate formula. Average absolute chan drft values Year Carl Jevons Dutot At ths pont t s worth consderng how the class ndces usng Jevons and Dutot elementary s the absolute value of aggregates compare. Fgure 7 shows exp{ d }, where the chan drft when usng the Jevons formula at the stratum level and the use of Dutot. Fgure 7. d s that correspondng to Exponentaton of the dfference n absolute chan drft at the class level between the Jevons and Dutot elementary aggregates. Offce for Natonal Statstcs 14
15 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng The ponts to the left of the the vertcal lne at 1 correspond to the nstances where the chan drft usng a Dutot elementary aggregate exceeds that when usng Jevons. Those ponts to the rght of 1 are the classes for whch chan drft when usng a Jevons formula exceeds that when usng Dutot. We see that there are more nstances where the chan drft usng a Jevons elementary aggregate formula s larger than that when usng a Dutot, and consstently so over the two years, but the dfference s usually of smaller magntude than those ponts to the left of the lne. 5.2 Prce Bounce Ths work explored the prevalence of prce bouncng across the classes. In the context of ths research a prce bounce corresponds to prces beng at a set level n May, before changng value n June and then returnng to ther ntal level n July. Ths s the smplest case of the phenomenon and the prce bounce occurs over the chan lnk. Cases where the prce vares for more months before returnng to ther ntal level, asymmetrc comparsons around the chan lnk perod or prces returnng to nearby ther ntal level are out of scope of ths work and further complcate the matter. There s some evdence that there s greater chan drft n classes wth a hgher proporton of prce bounce, however ths s not the only reason for chan drft. Further nvestgaton of prce bounce and the relaton of prce bounce to chan drft s needed to understand ths more. Fgure 8. Proporton of Prce Bounce aganst absolute Chan Drft from the Carl formula at the elementary aggregate per Class, 2009 and 2010 Offce for Natonal Statstcs 15
16 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng 6. Conclusons Ths work has used the locally collected prce data and an artfcal, mdyear lnk to explore the relatve magntudes of chan draft n class level ndces when usng varous elementary aggregate formulae. It shows that the use of a Carl formula at the elementary aggregate level does lead to greater chan drft than usng a Jevons or Dutot formula for most classes. There are a number of areas where addtonal work could be carred out. Frstly, tryng to dentfy what factors contrbute to the dfferent szes of the drft for dfferent classes would be nterestng and why there are nstances where the Jevons drft exceeds the Carl. The actual stuaton of the real CPI s, of course, more complcated wth a double lnk; t would be nterestng to see the effect of a second lnk. It would also be nterestng to see the compound effect of multple lnks and lnks over longer perods. References [1] Peter von der Lppe, 2005, Index Theory and Prce Statstcs, Peter Lang, ISBN [2] ONS, 2014, Consumer Prces Techncal Manual, [3] Bert Balk, 2008, Prce and Quantty Index Numbers, Cambrdge Unversty Press, ISBN Offce for Natonal Statstcs 16
17 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng Annex A. Chan lnkng n the CPI The Consumer Prce Index ncorporates changes to weghts and a change to the basket every year n January (hgher level weghts) and February (basket and tem weghts). Ths forces the requrement of a chan lnk for each change so that the ndex s chan lnked twce for each ndex n y n year y wth a base perod of January n year c Λ ( bx, ) the double chan lnked ndex b = ( basemonth,baseyear ) s. Usng the same notaton as n the man text, x = ( month,year ) for wth base perod n c Λ (( Jan, y n),( t, y) ) = Λ( ( Jan, y v),( Dec, y v) ) Λ( ( Dec, y v),( Jan, y ( v 1 ))). v= 1 Λ (( Jan, y),( t, y) ) The square bracket denotes the compoundng of the chan lnks over tme so as to create a contnuous seres from the base perod to the current month n the current year. In essence the chan lnk uses the last value from the prevous year along wth a specal ndex calculated usng January prces aganst a prevous December base perod to translate the current year s seres so that there are no level shfts. Ths process s a pure translaton of the seres and does not affect the behavour. Offce for Natonal Statstcs 17
18 Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng Annex B. Relatonshp between Drect and Chaned elementary Aggregates B1. Jevons Prce Index 1 1 n n p p (1) (2) J J Ε ( 0,1) Ε ( 1, 2) = p( 0) p( 1) B2. Dutot Prce Index 1 1 n n (1) (2) (1) (2) p p p p = = p( 0) p( 1) p( 0) p( 1) 1 n p (2) = = Ε p ( 0) J ( 0, 2) p (1) p (2) D D Ε ( 0,1) Ε ( 1, 2) = p(0) p(1) p(1) p (2) p(2) = p(0) p(1) = = Ε p(0) D ( 0, 2) Offce for Natonal Statstcs 18
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