9. PPI Calculation in Practice

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1 9. PPI Calculaton n Practce A. Introducton 9.1 Ths chapter provdes a general descrpton of the ways PPIs are calculated n practce. The methods used n dfferent countres are not exactly the same, but they have much n common. Both complers and users of PPIs are nterested n knowng how most statstcal offces actually set about calculatng ther PPIs. 9.2 As a result of the greater nsghts nto the propertes and behavor of prce ndces that have been acheved n recent years, t s now recognzed that some tradtonal methods may not necessarly be optmal from a conceptual and theoretcal vewpont. Concerns have also been voced n a number of countres about possble bases that may be affectng PPIs. These ssues and concerns need to be addressed n the Manual. Of course, the methods used to comple PPIs are nevtably constraned by the resources avalable, not merely for collectng and processng prces but also the revenue data needed for weghtng purposes. In some countres, the methods used may be severely constraned by a lack of resources. 9.3 The calculaton of PPIs usually proceeds n two stages. Frst, prce ndces are estmated for the elementary aggregates, and then these elementary prce ndces are averaged to obtan hgher-level ndces usng the relatve values of the revenue weghts for elementary aggregates as weghts. Secton B starts by explanng how the elementary aggregates are constructed and whch economc and statstcal crtera need to be taken nto consderaton n defnng the aggregates. The ndex number formulas most commonly used to calculate the elementary ndces are then presented and ther propertes and behavor llustrated usng numercal examples. The pros and cons of the varous formulas are consdered together wth some alternatve formulas that mght be used. The problems created by dsappearng and new products are also explaned, as are the dfferent ways of mputng for mssng prces. 9.4 Secton C of the chapter s concerned wth the calculaton of hgher-level ndces. The focus s on the ongong producton of a monthly prce ndex n whch the elementary prce ndces are averaged, or aggregated, to obtan hgher-level ndces. Prce updatng of weghts, chan lnkng, and reweghtng are dscussed, wth examples provded. The problems assocated wth ntroducton of new elementary prce ndces and new hgher-level ndces nto the PPI are also covered. The secton explans how t s possble to decompose the change n the overall ndex nto ts component parts. Fnally, the possblty of usng some alternatve and rather more complex ndex formulas s consdered. 9.5 Secton D concludes wth data edtng procedures, snce these are an ntegral part of the process of complng PPIs. It s essental to ensure that the rght data are entered nto the varous formulas. There may be errors resultng from the ncluson of ncorrect data or from enterng correct data napproprately and errors resultng from the excluson of correct data that are mstakenly beleved to be wrong. The secton examnes data edtng procedures that try to mnmze both types of errors. B. Calculaton of Prce Indces for Elementary Aggregates 9.6 PPIs typcally are calculated n two steps. In the frst step, the elementary prce ndces for the elementary aggregates are calculated. In the second step, hgher-level ndces are calculated by averagng the elementary prce ndces. The elementary aggregates and ther prce ndces are the basc buldng blocks of the PPI. B.1 Composton of elementary aggregates 9.7 Elementary aggregates are constructed by groupng ndvdual goods and ndvdual servces nto relatvely homogeneous products and transac- 213

2 Producer Prce Index Manual tons. They may be formed for products n varous regons of the country or for the country as a whole. Lkewse, elementary aggregates may be formed for dfferent types of establshments or for varous subgroups of products. The actual formaton of elementary aggregates thus depends on the crcumstances and the avalablty of nformaton, and they may therefore be defned dfferently n dfferent countres. However, some key ponts should be observed: Elementary aggregates should consst of groups of goods or servces that are as smlar as possble, and preferably farly homogeneous. They should also consst of products that may be expected to have smlar prce movements. The objectve should be to try to mnmze the dsperson of prce movements wthn the aggregate. The elementary aggregates should be approprate to serve as strata for samplng purposes n lght of the samplng regme planned for the data collecton. 9.8 Each elementary aggregate, whether relatng to the whole country, an ndvdual regon, or a group of establshments, wll typcally contan a very large number of ndvdual goods, servces, or products. In practce, only a small number can be selected for prcng. When selectng the products, the followng consderatons need to be taken nto account: () The transactons selected should be ones wth prce movements beleved to be representatve of all the products wthn the elementary aggregate. () The number of transactons wthn each elementary aggregate for whch prces are collected should be large enough for the estmated prce ndex to be statstcally relable. The mnmum number requred wll vary between elementary aggregates, dependng on the nature of the products and ther prce behavor. () The object s to try to track the prce of the same product over tme for as long as possble, or for as long as the product contnues to be representatve. The products selected should therefore be ones that are expected to reman on the market for some tme so that lke can be compared wth lke. B.1.1 Aggregaton structure 9.9 The aggregaton structure for a PPI s dscussed n Chapter 4, Secton C.4, and n Fgure 4.1. Usng a classfcaton of busness products such as PRODCOM, CPC, or CPA, the entre set of produced goods and servces covered by the overall PPI can be dvded nto broad sectons, dvsons, and groups, then further refned nto smaller classes and subclasses. Each elementary aggregate s assgned a product code. Ths enables statstcal offces to aggregate elementary ndces at the lowest level to hgher product classes, groups, dvsons, etc. In addton, each elementary aggregate s assgned an ndustry (actvty) code from a standard ndustral classfcaton such as ISIC or NACE and thus can be aggregated by ndustry from the fourdgt to the three-dgt and hgher levels. The overall PPI should be the same whether aggregated by ndustry or product as long as each elementary aggregate has the same weght n the ndustry and product aggregatons The methods used to calculate the elementary ndces from the ndvdual prce observatons are dscussed below. Workng from the elementary prce ndces, all ndces above the elementary aggregate level are hgher-level ndces that can be calculated from the elementary prce ndces usng the elementary revenue aggregates as weghts. The aggregaton structure s consstent so that the weght at each level above the elementary aggregate s always equal to the sum of ts components. The prce ndex at each hgher level of aggregaton can be calculated on the bass of the weghts and prce ndces for ts components that s, the lower-level or elementary ndces. The ndvdual elementary prce ndces are not necessarly suffcently relable to be publshed separately, but they reman the basc buldng blocks of all hgher-level ndces. B.1.2 Weghts wthn elementary aggregates 9.11 In many cases, the explct revenue weghts are not avalable to calculate the prce ndces for elementary aggregates. Whenever possble, however, weghts should be used that reflect the relatve mportance of the sampled products, even f the weghts are only approxmate. Often, the elementary aggregate s smply the lowest level at whch relable weghtng nformaton s avalable. In ths case, the elementary ndex has to be calculated as 214

3 9. PPI Calculaton n Practce an unweghted average of the prces of whch t conssts. However, even n ths case t should be noted that when the products are selected wth probabltes proportonal to the sze of some relevant varable such as sales, for example, weghts are mplctly ntroduced by the sample selecton procedure. In addton, statstcal offces can work wth establshment respondents to obtan estmated weght data, as dscussed n Chapter For certan elementary aggregates, nformaton about output of partcular products and market shares from trade and ndustry sources may be used as explct weghts wthn an elementary aggregate. Weghts wthn elementary aggregates may be updated ndependently and possbly more often than the elementary aggregates themselves (whch serve as weghts for the hgher-level ndces) For example, assume that the number of supplers of a certan product such as car fuel suppled to garages s lmted. The market shares of the supplers may be known from busness survey statstcs and can be used as weghts n the calculaton of an elementary aggregate prce ndex for car fuel. Alternatvely, prces for water may be collected from a number of local water supply servces where the populaton n each local regon s known. The relatve sze of the populaton n each regon may then be used as a proxy for the relatve revenues to weght the prce n each regon to obtan the elementary aggregate prce ndex for water A specal stuaton occurs n the case of tarff prces. A tarff s a lst of prces for the provson of a partcular knd of good or servce under dfferent terms and condtons. One example s electrcty, for whch one prce s charged durng daytme and a lower prce s charged at nght. Smlarly, a telephone company may charge a lower prce for a call on the weekend than a weekday. Another example may be bus tckets sold at one prce to regular passengers and at lower prces to chldren or senors. In such cases, t s approprate to assgn weghts to the dfferent tarffs or prces to calculate the prce ndex for the elementary aggregate The ncreasng use of electronc recordng for transactons n many countres, n whch both prces and quanttes are mantaned as products are sold, means that valuable new sources of nformaton may become ncreasngly avalable to statstcal offces. Ths could lead to sgnfcant changes n the ways n whch prce data are collected and processed for PPI purposes. The treatment of electronc data transfer s examned n Chapters 6, 7, and 21. B.2 Complaton of elementary prce ndces 9.16 An elementary prce ndex s the prce ndex for an elementary aggregate. Varous methods and formulas may be used to calculate elementary prce ndces. Ths secton provdes a summary of pros and cons that statstcal offces must evaluate when choosng a formula at the elementary level; Chapter 20 provdes a more detaled dscusson The methods statstcal offces most commonly use are llustrated by means of a numercal example n Table 9.1. In the example, assume that prces are collected for four representatve products wthn an elementary aggregate. The qualty of each product remans unchanged over tme so that the month-to-month changes compare lke wth lke. No weghts can be appled. Assume ntally that prces are collected for all four products n every month covered so that there s a complete set of prces. There are no dsappearng products, no mssng prces, and no replacement products. These are qute strong assumptons because many of the problems encountered n practce are attrbutable to breaks n the contnuty of the prce seres for the ndvdual transactons for one reason or another. The treatment of dsappearng and replacement products s taken up later Three wdely used formulas that have been, or stll are, n use by statstcal offces to calculate elementary prce ndces are llustrated n Table 9.1. It should be noted, however, that these are not the only possbltes, and some alternatve formulas are consdered later. The frst s the Carl ndex for = 1,., n products. It s defned as the smple, or unweghted, arthmetc mean of the prce relatves, or prce ratos, for the two perods, 0 and t, to be compared. 215

4 Producer Prce Index Manual Table 9.1. Calculaton of Prce Indces for an Elementary Aggregate 1 January February March Aprl May June July Prces Product A Product B Product C Product D Arthmetc mean prces Geometrc mean prces Month-to-month prce relatves Product A Product B Product C Product D Current to reference month (January) prce relatves Product A Product B Product C Product D Carl ndex Arthmetc mean of prce relatves Month-to-month ndex Chaned month-tomonth ndex Drect ndex on January Dutot ndex Rato of arthmetc mean prces Month-to-month ndex Chaned month-tomonth ndex Drect ndex on January Jevons ndex Geometrc mean of prce relatves or rato of geometrc mean prces Month-to-month ndex Chaned month-tomonth ndex Drect ndex on January All prce ndces have been calculated usng unrounded fgures. (9.1) P 0: t C t 1 p = 0 n p The second s the Dutot ndex, whch s defned as the rato of the unweghted arthmetc mean prces. 216

5 9. PPI Calculaton n Practce (9.2) P 0: t D 1 = n 1 n p p t 0 The thrd s the Jevons ndex, whch s defned as the unweghted geometrc mean of the prce relatve or relatves, whch s dentcal to the rato of the unweghted geometrc mean prces. (9.3) P 0: t J 1 n t p = = 0 p t ( p ) 0 ( p ) 1 n 1 n The propertes of the three ndces are examned and explaned n some detal n Chapter 20. Here, the purpose s to llustrate how they perform n practce, to compare the results obtaned by usng the dfferent formulas, and to summarze ther strengths and weaknesses Each month-to-month ndex shows the change n the ndex from one month to the next. The chaned month-to-month ndex lnks together these month-to-month changes by successve multplcaton. The drect ndex compares the prces n each successve month drectly wth those of the reference month, January. By smple nspecton of the varous ndces, t s clear that the choce of formula and method can make a substantal dfference n the results obtaned. Some results are strkng n partcular, the large dfference between the chaned Carl ndex for July and each of the drect ndces for July, ncludng the drect Carl The propertes and behavor of the dfferent ndces are summarzed n the followng paragraphs and explaned n more detal n Chapter 20. Frst, the dfferences between the results obtaned by usng the dfferent formulas tend to ncrease as the varance of the prce relatves, or ratos, ncreases. The greater the dsperson of the prce movements, the more crtcal the choce of ndex formula and method becomes. If the elementary aggregates are defned so that the prce movements wthn the aggregate are mnmzed, the results obtaned become less senstve to the choce of formula and method Certan features dsplayed by the data n Table 9.1 are systematc and predctable and follow from the mathematcal propertes of the ndces. For example, t s well known that an arthmetc mean s always greater than, or equal to, the correspondng geometrc mean the equalty holdng only n the trval case n whch the numbers beng averaged are all the same. The drect Carl ndces are therefore all greater than the Jevons ndces, except n May and July when the four prce relatves based on January are all equal. In general, the Dutot ndex may be greater or less than the Jevons ndex but tends to be less than the Carl ndex One general property of geometrc means should be noted when usng the Jevons formula. If any one observaton out of a set of observatons s zero, ts geometrc mean s zero, whatever the values of the other observatons. The Jevons ndex s senstve to extreme falls n prces, and t may be necessary to mpose upper and lower bounds on the ndvdual prce relatves of, say, 10 and 0.1, respectvely, when usng the Jevons. Of course, extreme observatons are often the results of errors of one knd or another, and so extreme prce movements should be carefully checked n any case Another mportant property of the ndces llustrated n Table 9.1 s that the Dutot and the Jevons ndces are transtve, whereas the Carl ndex s not. Transtvty means that the chaned monthly ndces are dentcal to the correspondng drect ndces. Ths property s mportant n practce, because many elementary prce ndces are n fact calculated as chan ndces that lnk together the month-to-month-ndces. The ntranstvty of the Carl ndex s llustrated dramatcally n Table 9.1, n whch each of the four ndvdual prces n May returns to the same level as t was n January, but the chaned Carl ndex regsters an ncrease of almost 14 percent over January. Smlarly, n July, although each ndvdual prce s exactly 10 percent hgher than n January, the chaned Carl ndex regsters an ncrease of 29 percent. These results would be regarded as perverse and unacceptable n the case of a drect ndex, but even n the case of the chaned ndex, the results seems so ntutvely unreasonable as to undermne the credblty of the chaned Carl ndex. The prce changes between March and Aprl llustrate the effects of prce bouncng, n whch the same four prces are observed n both perods, but they are swtched between the dfferent products. The monthly Carl ndex from March to Aprl ncreases, whereas both the Dutot and the Jevons ndces are unchanged. 217

6 Producer Prce Index Manual 9.24 The message emergng from ths bref llustraton of the behavor of just three possble formulas s that dfferent ndex numbers and methods can delver very dfferent results. Index complers have to famlarze themselves wth the nterrelatonshps between the varous formulas at ther dsposal for the calculaton of the elementary prce ndces so that they are aware of the mplcatons of choosng one formula rather than another. However, knowledge of these nterrelatonshps s not suffcent to determne whch formula should be used, even though t makes t possble to make a more nformed and reasoned choce. It s necessary to appeal to addtonal crtera to settle the choce of formula. Two man approaches may be used, the axomatc and the economc approaches. B.2.1 Axomatc approach to elementary prce ndces 9.25 As explaned n Chapters 16 and 20, one way to decde on an approprate ndex formula s to requre t to satsfy certan specfed axoms or tests. The tests throw lght on the propertes possessed by dfferent knds of ndces, some of whch may not be obvous. Four basc tests llustrate the axomatc approach. Proportonalty Test: If all prces are λ tmes the prces n the prce reference perod (January n the example), the ndex should equal λ. The data for July, when every prce s 10 percent hgher than n January, show that all three drect ndces satsfy ths test. A specal case of ths test s the dentty test, whch requres that f the prce of every product s the same as n the reference perod, the ndex should be equal to unty (as n May n the example). Changes n the Unts of Measurement Test (or Commensurablty Test): The prce ndex should not change f the quantty unts n whch the products are measured are changed for example, f the prces are expressed per lter rather than per pnt. The Dutot ndex fals ths test, as explaned below, but the Carl and Jevons ndces satsfy the test. Tme Reversal Test: If all the data for the two perods are nterchanged, then the resultng prce ndex should equal the recprocal of the orgnal prce ndex. The Carl ndex fals ths test, but the Dutot and the Jevons both satsfy the test. The falure of the Carl ndex to satsfy the test s not mmedately obvous from the example but can easly be verfed by nterchangng the prces n January and Aprl, for example, n whch case the backward Carl ndex for January based on Aprl s equal to 91.3, whereas the recprocal of the forward Carl ndex s 1/132.5, or Transtvty Test: The chaned ndex between two perods should equal the drect ndex between the same two perods. The example shows that the Jevons and the Dutot ndces both satsfy ths test, whereas the Carl ndex does not. For example, although the prces n May have returned to the same levels as n January, the chaned Carl ndex regsters Ths llustrates the fact that the Carl ndex may have a sgnfcant bult-n upward bas Many other axoms or tests can be devsed, as presented n Chapter 16, but the above (summarzed n Table 9.2) are suffcent to llustrate the approach and also to throw lght on some mportant features of the elementary ndces under consderaton here The sets of products covered by elementary aggregates are meant to be as homogeneous as possble. If they are not farly homogeneous, the falure of the Dutot ndex to satsfy the unts of measurement, or commensurablty, test can be a serous dsadvantage. Although defned as the rato of the unweghted arthmetc average prces, the Dutot ndex may also be nterpreted as a weghted arthmetc average of the prce relatves n whch each rato s weghted by ts prce n the base perod. 1 However, f the products are not homogeneous, the relatve prces of the dfferent products may depend qute arbtrarly on the quantty unts n whch they are measured Consder, for example, salt and pepper, whch are found wthn the same CPC subclass. Suppose the unt of measurement for pepper s changed from grams to ounces, whle leavng the unts n whch salt s measured (say, klos) unchanged. Because an ounce of pepper s equal to grams, the prce of pepper ncreases by more than 28 tmes, whch effectvely ncreases the 1 Ths can be seen by rewrtng equaton (9.1) as 1 0 t 0 p ( p p ) 0:t PD = n. 1 0 p n 218

7 9. PPI Calculaton n Practce Table 9.2. Propertes of Man Elementary Aggregate Index Formulas Formula Formula propertes Carl Arthmetc mean of prce relatves Dutot Relatve of arthmetc mean prces Jevons Geometrc mean of prce relatves Proportonalty yes yes yes Change-of-unts of measurement yes no yes Tme reversal no yes yes Transtvty no yes yes Allows for substtuton no no yes weght gven to pepper n the Dutot ndex by more than 28 tmes. The prce of pepper relatve to salt s nherently arbtrary, dependng entrely on the choce of unts n whch to measure the two goods. In general, when there are dfferent knds of products wthn the elementary aggregate, the Dutot ndex s unacceptable conceptually The Dutot ndex s acceptable only when the set of products covered s homogeneous, or at least nearly homogeneous. For example, the Dutot ndex may be acceptable for a set of apple prces, even though the apples may be of dfferent varetes, but not for the prces of dfferent knds of fruts, such as apples, pneapples, and bananas, some of whch may be much more expensve per tem or per klo than others. Even when the products are farly homogeneous and measured n the same unts, the Dutot ndex s mplct weghts may stll not be satsfactory. More weght s gven to the prce changes for the more expensve products, but they may well account for only small shares of the total revenue wthn the aggregate, n practce. Purchasers are unlkely to buy products at hgh prces f the same products are avalable at lower prces It may be concluded that from an axomatc vewpont, both the Carl and the Dutot ndces, although they have been and stll are wdely used by statstcal offces, have serous dsadvantages. The Carl ndex fals the tme reversal and transtvty tests. In prncple, t should not matter whether we choose to measure prce changes forward or backward n tme. We would expect the same answer, but ths s not the case for the Carl ndex. Chaned Carl ndces may be subject to a sgnfcant upward bas. The Dutot ndex s meanngful for a set of homogeneous products but becomes ncreasngly arbtrary as the set of products becomes more dverse. On the other hand, the Jevons ndex satsfes all the tests lsted above and also emerges as the preferred ndex when the set of test s enlarged, as shown n Chapter 20. From an axomatc pont of vew, the Jevons ndex s clearly the ndex wth the best propertes, even though t may not have been used much untl recently. The Jevons ndex also allows for some substtuton effects consstent wth a untary elastcty of substtuton. There seems to be an ncreasng tendency for statstcal offces to swtch from usng Carl or Dutot ndces to Jevons. B.2.2 Economc approach to elementary prce ndces 9.31 The objectve of the economc approach s to estmate for the elementary aggregates an deal (or true ) economc ndex that s, one consstent wth the economc theory of revenue-maxmzng producers explaned n Secton F of Chapter 20. The products for whch respondents provde prces are treated as a basket of goods and servces produced by establshments to provde revenue, and producers are assumed to arrve at ther decson about the quanttes of outputs to produce on the bass of revenue-maxmzng behavor. As explaned n Chapters 1, 15, and 17, an deal theoretcal economc ndex measures the rato of revenues between two perods that an establshment can attan when faced wth fxed technologes and nputs. Changes n the ndex arse only from changes n prces. The technology s assumed to be held fxed, although the revenue-maxmzng producer can make substtutons between the products produced 219

8 Producer Prce Index Manual n response to changes n ther relatve prces. In the absence of nformaton about quanttes or revenues wthn an elementary aggregate, an deal ndex can be estmated only when certan specal condtons are assumed to preval There are two specal cases of some nterest. The frst case s when producers contnue to produce the same relatve quanttes whatever the relatve prces. Producers prefer not to make any substtutons n response to changes n relatve prces. The cross-elastctes of supply are zero. The technology by whch nputs are translated nto outputs n economc theory s descrbed by a producton functon, and a producton functon wth such a restrctve reacton to relatve prce changes s descrbed n the economcs lterature as Leontef. Wth such a producton functon, a Laspeyres ndex would provde an exact measure of the deal economc ndex. In ths case, the Carl ndex calculated for a random sample of products would provde an estmate of the deal economc ndex that the products are selected wth probabltes proportonal to the populaton revenue shares The second case occurs when producers are assumed to vary the quanttes they produce n nverse proporton to the changes n relatve prces. The cross-elastctes of supply between the dfferent products they produce are all unty, the revenue shares remanng the same n both perods. Such an underlyng producton functon s descrbed as Cobb-Douglas. Wth ths producton functon, the geometrc Laspeyres 3 ndex would provde an exact measure of the deal ndex. In ths case, the Jevons ndex calculated for a random sample of products would provde an unbased estmate of the deal economc ndex provded that the products are selected wth probabltes proportonal to the populaton expendture shares. 2 It mght appear that f the products were selected wth probabltes proportonal to the populaton quantty shares, the sample Dutot would provde an estmate of the populaton Laspeyres. However, f the basket for the Laspeyres ndex contans dfferent knds of products whose quanttes are not addtve, the quantty shares, and hence the probabltes, are undefned. 3 The geometrc Laspeyres s a weghted geometrc average of the prce relatves, usng the revenue shares n the earler perod as weghts. (The revenue shares n the second perod would be the same n the partcular case under consderaton.) 9.34 From the economc approach, the choce between the sample Jevons ndex and the sample Carl ndex rests on whch s lkely to approxmate more closely the underlyng deal economc ndex n other words, whether the (unknown) crosselastctes are lkely to be closer to unty or zero, on average. In practce, the cross-elastctes could take on any value rangng up to + for an elementary aggregate consstng of a set of strctly homogeneous products that s, perfect substtutes. 4 It may be conjectured that for demand-led ndustres where producers produce less of a commodty whose relatve prce has ncreased to meet the reduced quantty demanded, the average crosselastcty s lkely to be closer to unty. Thus, the Jevons ndex s lkely to provde a closer approxmaton to the deal economc ndex than the Carl ndex. In ths case, the Carl ndex must be vewed as havng an upward bas. However, there are some establshments n ndustres, ncludng utltes, n whch supply s relatvely unresponsve to demand changes, and the Carl ndex would be more approprate, gven that samplng s wth probablty proportonal to base-perod revenue shares. And, yet agan, there would be establshments n ndustres n whch quanttes produced ncrease as prces ncrease, and, wth probablty samplng proportonal to base-perod revenues, nether the Carl nor the Jevons ndex would be approprate from the economc approach The nsght provded by the economc approach s that the Jevons and Carl ndces can be justfed from the economc approach dependng on whether a sgnfcant amount of substtuton s more lkely than no substtuton, especally as elementary aggregates should be delberately constructed to group together smlar products that are close substtutes for each other The Jevons ndex does not mply, or assume, that revenue shares reman constant. Obvously, the Jevons can be calculated whether changes do or do not occur n the revenue shares n practce. What the economc approach shows s that f the revenue shares reman constant (or roughly constant), then the Jevons ndex can be expected to provde a good estmate of the underlyng deal 4 It should be noted that n the lmt when the products really are homogeneous, there s no ndex number problem, and the prce ndex s gven by the rato of the unt values n the two perods, as explaned below. 220

9 9. PPI Calculaton n Practce economc ndex. Smlarly, f the relatve quanttes reman constant, then the Carl ndex can be expected to provde a good estmate, but the Carl ndex does not actually mply that quanttes reman fxed. Reference should be made to Secton F of Chapter 20 for a more rgorous statement of the economc approach It may be concluded that on the economc approach, as well as the axomatc approach, the Jevons emerges as the preferred ndex n general, although there may be cases n whch lttle or no substtuton takes place wthn the elementary aggregate, and the Carl mght be preferred. The ndex compler must make a judgment on the bass of the nature of the products actually ncluded n the elementary aggregate Before leavng ths topc, t should be noted that t has thrown lght on some of the samplng propertes of the elementary ndces. If the products n the sample are selected wth probabltes proportonal to expendtures n the prce reference perod, The sample (unweghted) Carl ndex provdes an unbased estmate of the populaton Laspeyres, and The sample (unweghted) Jevons ndex provdes an unbased estmate of the populaton geometrc Laspeyres These results hold, regardless of what the underlyng economc ndex may be. B.3 Chaned versus drect ndces for elementary aggregates 9.40 In a drect elementary ndex, the prces of the current perod are compared drectly wth those of the prce reference perod n a chaned ndex, prces n each perod are compared wth those n the prevous perod, the resultng short-term ndces beng chaned together to obtan the long-term ndex, as llustrated n Table Provded that prces are recorded for the same set of products n every perod, as n Table 9.1, any ndex formula defned as the rato of the average prces wll be transtve that s, the same result s obtaned whether the ndex s calculated as a drect ndex or as a chaned ndex. In a chaned ndex, successve numerators and denomnators wll cancel out, leavng only the average prce n the last perod dvded by the average prce n the prce reference perod, whch s the same as the drect ndex. Both the Dutot and the Jevons ndces are therefore transtve. As already noted, however, a chaned Carl ndex s not transtve and should not be used because of ts upward bas. Nevertheless, the drect Carl remans an opton Although the chaned and drect versons of the Dutot and Jevons ndces are dentcal when there are no breaks n the seres for the ndvdual products, they offer dfferent ways of dealng wth new and dsappearng products, mssng prces, and qualty adjustments. In practce, products contnually have to be dropped from the ndex and new ones ncluded, n whch case the drect and the chaned ndces may dffer f the mputatons for mssng prces are made dfferently When a replacement product has to be ncluded n a drect ndex, t often wll be necessary to estmate the prce of the new product n the prce reference perod, whch may be some tme n the past. The same happens f, as a result of an update of the sample, new products have to be lnked nto the ndex. Assumng that no nformaton exsts on the prce of the replacement product n the prce reference perod, t wll be necessary to estmate t usng prce relatves calculated for the products that reman n the elementary aggregate, a subset of these products, or some other ndcator. However, the drect approach should be used only for a lmted perod. Otherwse, most of the reference prces would end up beng mputed, whch would be an undesrable outcome. Ths effectvely rules out the use of the Carl ndex over a long perod, because the Carl ndex can be used only n ts drect form anyway, beng unacceptable when chaned. Ths mples that, n practce, the drect Carl ndex may be used only f the overall ndex s chaned annually, or at ntervals of two or three years In a chaned ndex, f a product becomes permanently mssng, a replacement product can be lnked nto the ndex as part of the ongong ndex calculaton by ncludng the product n the monthly ndex as soon as prces for two successve months are obtaned. Smlarly, f the sample s updated and new products have to be lnked nto the ndex, ths wll requre successve old and new prces for the present and the precedng month. However, for a chaned ndex, the mssng observaton wll affect the ndex for two months, snce the mssng observaton s part of two lnks n the chan. Ths s not 221

10 Producer Prce Index Manual the case for a drect ndex, where a sngle, nonestmated mssng observaton wll affect only the ndex n the current perod. For example, when comparng perods 0 and 3, a mssng prce of a product n perod 2 means that the chaned ndex excludes the product for the last lnk of the ndex n perods 2 and 3, whle the drect ndex ncludes t n perod 3 (snce a drect ndex wll be based on products wth prces avalable n perods 0 and 3). However, n general, the use of a chaned ndex can make the estmaton of mssng prces and the ntroducton of replacements easer from a computatonal pont of vew, whereas t may be nferred that a drect ndex wll lmt the usefulness of overlap methods for dealng wth mssng observatons. Ths s dscussed further n Secton B The drect and the chaned approaches also produce dfferent by-products that may be used for montorng prce data. For each elementary aggregate, a chaned ndex approach gves the latest monthly prce change, whch can be useful for both edtng data and mputng mssng prces. By the same token, however, a drect ndex derves average prce levels for each elementary aggregate n each perod, and ths nformaton may be a useful byproduct. However, the avalablty of cheap computng power and spreadsheets allows such byproducts to be calculated whether a drect or a chaned approach s appled, so that the choce of formula should not be dctated by consderatons regardng by-products. B.4 Consstency n aggregaton 9.46 Consstency n aggregaton means that f an ndex s calculated stepwse by aggregatng lower-level ndces to obtan ndces at progressvely hgher levels of aggregaton, the same overall result should be obtaned as f the calculaton had been made n one step. For presentatonal purposes, ths s an advantage. If the elementary aggregates are calculated usng one formula, and the elementary aggregates are averaged to obtan the hgherlevel ndces usng another formula, the resultng PPI s not consstent n aggregaton. However, t may be argued that consstency n aggregaton s not necessarly an mportant or even approprate crteron. Also t may be unachevable, partcularly when the amount of nformaton avalable on quanttes and revenues s not the same at the dfferent levels of aggregaton. In addton, there may be dfferent degrees of substtuton wthn elementary aggregates compared wth the degree of substtuton between products n dfferent elementary aggregates As noted n Secton B.2.2 above, the Carl ndex would be consstent n aggregaton wth the Laspeyres ndex f the products were to be selected wth probabltes proportonal to revenues n the prce reference perod. However, ths s typcally not the case. The Dutot and the Jevons ndces are also not consstent n aggregaton wth a hgherlevel Laspeyres. However, as explaned below, the PPIs actually calculated by statstcal offces are usually not true Laspeyres ndces anyway, even though they may be based on fxed baskets of goods and servces. As also noted earler, f the hgherlevel ndex were to be defned as a geometrc Laspeyres ndex, consstency n aggregaton could be acheved by usng the Jevons ndex for the elementary ndces at the lower level, provded that the ndvdual products are sampled wth probabltes proportonal to revenues. Although unfamlar, a geometrc Laspeyres ndex has desrable propertes from an economc pont of vew and s consdered agan later. B.5 Mssng prce observatons 9.48 The prce of a product may not be collected n a partcular perod, ether because the product s mssng temporarly or because t has permanently dsappeared. The two classes of mssng prces requre dfferent treatments. Temporary unavalablty may occur for seasonal products (partcularly for frut, vegetables, and clothng) because of supply shortages or possbly because of some collecton dffculty (for example, an establshment was closed or a respondent was on vacaton). The treatment of seasonal products rases a number of partcular problems. These are dealt wth n Chapter 22 and wll not be dscussed here. B.5.1 Treatment of temporarly mssng prces 9.49 In the case of temporarly mssng observatons for products, one of four actons may be taken: Omt the product for whch the prce s mssng so that a matched sample s mantaned (lke s compared wth lke), even though the sample s depleted. 222

11 9. PPI Calculaton n Practce Carry forward the last observed prce. Impute the mssng prce by the average prce change for the prces that are avalable n the elementary aggregate. Impute the mssng prce by the prce change for a partcular comparable product from a smlar establshment. Omttng an observaton from the calculaton of an elementary ndex s equvalent to assumng that the prce would have moved n the same way as the average of the prces of the products that reman ncluded n the ndex. Omttng an observaton changes the mplct weghts attached to the other prces n the elementary aggregate Carryng forward the last observed prce should be avoded wherever possble and s acceptable only for a very lmted number of perods. Specal care needs to be taken n perods of hgh nflaton or when markets are changng rapdly as a result of a hgh rate of nnovaton and product turnover. Whle smple to apply, carryng forward the last observed prce bases the resultng ndex toward zero change. In addton, there s lkely to be a compensatng step-change n the ndex when the prce of the mssng product s recorded agan, whch wll be wrongly mssed by a chaned ndex but wll be ncluded n a drect ndex to return the ndex to ts proper value. The adverse effect on the ndex wll be ncreasngly severe f the product remans unprced for some length of tme. In general, carryforward s not an acceptable procedure or soluton to the problem unless t s certan the prce has not changed Imputaton of the mssng prce by the average change of the avalable prces may be appled for elementary aggregates when the prces can be expected to move n the same drecton. The mputaton can be made usng all the remanng prces n the elementary aggregate. As already noted, ths s numercally equvalent to omttng the product for the mmedate perod, but t s useful to make the mputaton so that f the prce becomes avalable agan n a later perod, the sample sze s not reduced n that perod. In some cases, dependng on the homogenety of the elementary aggregate, t may be preferable to use only a subset of products from the elementary aggregate to estmate the mssng prce. In some nstances, ths may even be a sngle comparable product from a smlar type of establshment whose prce change can be expected to be smlar to the mssng one Table 9.3 llustrates the calculaton of the prce ndex for an elementary aggregate consstng of three products, where one of the prces s mssng n March. The upper part of Table 9.3 shows the ndces where the mssng prce has been omtted from the calculaton. The drect ndces are therefore calculated on the bass of A, B, and C for all months except March, where t s calculated on bass of B and C only. The chaned ndces are calculated on the bass of all three prces from January to February and from Aprl to May. From February to March and from March to Aprl, the monthly ndces are calculated on the bass of B and C only For both the Dutot and the Jevons, the drect and chan ndces now dffer from March onward. The frst lnk n the chaned ndex (January to February) s the same as the drect ndex, so that the two ndces are dentcal numercally. The drect ndex for March gnores the prce decrease of product A between January and February, whle ths s taken nto account n the chaned ndex. As a result, the drect ndex s hgher than the chaned ndex for March. On the other hand, n Aprl and May, where all prces agan are avalable, the drect ndex catches the prce development, whereas the chaned ndex fals to track the development n the prces In the lower half of Table 9.3, the mssng prce for product A n March s mputed by the average prce change of the remanng products from February to March. Whle the ndex may be calculated as a drect ndex comparng the prces of the present perod wth the reference perod prces, the mputaton of mssng prces should be made on bass of the average prce change from the precedng to the present perod, as shown n the table. Imputaton on the bass of the average prce change from the prce reference perod to the present perod should not be used, snce t gnores the nformaton about the prce change of the mssng product that has already been ncluded n the ndex. The treatment of mputatons s dscussed n more detal n Chapter

12 Producer Prce Index Manual Table 9.3. Imputaton of Temporarly Mssng Prces January February March Aprl May Prces Product A Product B Product C Omt mssng product from the ndex calculaton Carl ndex Arthmetc mean of prce relatves Drect ndex Dutot ndex Rato of arthmetc mean prces Month-to-month ndex Chaned month-to-month ndex Drect ndex Jevons ndex Rato of geometrc mean prces or geometrc mean of prce relatves Month-to-month ndex Chaned month-to-month ndex Drect ndex Imputaton Carl ndex Arthmetc mean of prce relatves Impute prce for A n March as 5 (9/8 + 4/3)/2 = 6.15 Drect ndex Dutot ndex Rato of arthmetc mean prces Impute prce for A n March as 5 [(9 + 4)/(8 + 3)] = 5.91 Month-to-month ndex Chaned month-to-month ndex Drect ndex Jevons ndex Rato of geometrc mean prces or geometrc mean of prce relatves Impute prce for A n March as 5 (9/8 4/3) 0.5 = 6.12 Month-to-month ndex Chaned month-to-month ndex Drect ndex B.5.2 Treatment of products that have permanently dsappeared and ther replacements 9.55 Products may dsappear permanently for varous reasons. The product may dsappear from the market because new products have been ntroduced or the establshments from whch the prce has been collected have stopped sellng the product. When products dsappear permanently, a replacement product has to be sampled and ncluded n the ndex. The replacement product should deally be one that accounts for a sgnfcant proporton of sales, s lkely to contnue to be sold for some tme, and s lkely to be representatve of the sampled prce changes of the market that the old product covered. 224

13 9. PPI Calculaton n Practce 9.56 The tmng of the ntroducton of replacement products s mportant. Many new products are ntally sold at hgh prces that then gradually drop over tme, especally as the volume of sales ncreases. Alternatvely, some products may be ntroduced at artfcally low prces to stmulate demand. In such cases, delayng the ntroducton of a new or replacement product untl a large volume of sales s acheved may mss some systematc prce changes that ought to be captured by PPIs. It may be desrable to try to avod forced replacements caused when products dsappear completely from the market and to try to ntroduce replacements when sales of the products they replace are decreasng and before they cease altogether Table 9.4 shows an example where product A dsappears after March and product D s ncluded as a replacement from Aprl onward. Products A and D are not avalable on the market at the same tme, and ther prce seres do not overlap. To n- Table 9.4. Dsappearng Products and Ther Replacements wth No Overlap January February March Aprl May Prces Product A Product B Product C Product D Carl ndex Arthmetc mean of prce relatves Impute prce for D n January as 9 /[(5/3 + 10/7) 0.5] = 5.82 Drect ndex Dutot ndex Rato of arthmetc mean prces Impute prce for D n March as 9 /[(5 + 10)/(4 + 9)] = 7.80 Month-to-month ndex Chaned month-to-month ndex Impute prce for D n January as 9 /[(5 + 10)/(3 + 7)] = 6.00 Drect ndex Jevons ndex Rato of geometrc mean prces or geometrc mean of prce relatves Impute prce for D n March as 9/[(5/4 10/9) 0.5 ] = 7.64 Month-to-month ndex Chaned month-to-month ndex Impute prce for D n January as 9/[(5/3 10/7) 0.5 ] = 5.83 Drect ndex Omt the prce Dutot ndex Rato of arthmetc mean prces Month-to-month ndex Chaned month-to-month ndex Jevons ndex Rato of geometrc mean prces or geometrc mean of prce relatves Monthly ndex Chan month-to-month ndex

14 Producer Prce Index Manual clude the new product n the ndex from Aprl onward, an mputed prce needs to be calculated ether for the base perod (January) f a drect ndex s beng calculated, or for the precedng perod (March) f a chaned ndex s calculated. In both cases, the mputaton method ensures that the ncluson of the new product does not, n tself, affect the ndex In the case of a chaned ndex, mputng the mssng prce by the average change of the avalable prces gves the same result as f the product s smply omtted from the ndex calculaton untl t has been prced n two successve perods. Ths allows the chaned ndex to be compled by smply channg the month-to-month ndex between perods t 1 and t, based on the matched set of prces n those two perods, on to the value of the chaned ndex for perod t 1. In the example, no further mputaton s requred after Aprl, and the subsequent movement of the ndex s unaffected by the mputed prce change between March and Aprl In the case of a drect ndex, however, an mputed prce s always requred for the reference perod to nclude a new product. In the example, the prce of the new product n each month after Aprl stll has to be compared wth the mputed prce for January. As already noted, to prevent a stuaton n Table 9.5. Dsappearng and Replacement Products wth Overlappng Prces January February March Aprl May Prces Product A Product B Product C Product D Carl ndex Arthmetc mean of prce relatves Impute prce for D n January as 6 /(5/10) = Drect ndex Dutot ndex Rato of arthmetc mean prces Chan the monthly ndces based on matched prces Month-to-month ndex Chaned month-to-month ndex Dvde D s prce n Aprl and May wth 10/5 = 2 and use A s prce n January as base prce Drect ndex Impute prce for D n January as 6 /(5/10) = Drect ndex Jevons ndex Rato of geometrc mean prces or geometrc mean of prce relatves Chan the monthly ndces based on matched prces Month-to-month ndex Chaned month-to-month ndex Dvde D s prce n Aprl and May wth 10/5 = 2 and use A s prce n January as base prce Drect ndex Impute prce for D n January as 6 /(5/10) = Drect ndex

15 9. PPI Calculaton n Practce whch most of the reference perod prces end up beng mputed, the drect approach should be used only for a lmted perod of tme The stuaton s somewhat smpler when there s an overlap month n whch prces are collected for both the dsappearng and the replacement product. In ths case, t s possble to lnk the prce seres for the new product to the prce seres for the old product that t replaces. Lnkng wth overlappng prces nvolves makng an mplct adjustment for the dfference n qualty between the two products, snce t assumes that the relatve prces of the new and old product reflect ther relatve qualtes. For perfect or nearly perfect markets, ths may be a vald assumpton, but for certan markets and products t may not be so reasonable. The queston of when to use overlappng prces s dealt wth n detal n Chapter 7. The overlap method s llustrated n Table In the example, overlappng prces are obtaned for products A and D n March. Ther relatve prces suggest that one unt of product A s worth two unts of product D. If the ndex s calculated as a drect Carl ndex, the January base-perod prce for product D can be mputed by dvdng the prce of product A n January by the prce rato of A and D n March A monthly chaned ndex of arthmetc mean prces wll be based on the prces of products A, B, and C untl March, and from Aprl onward by B, C, and D. The replacement product s not ncluded untl prces for two successve perods are obtaned. Thus, the monthly chaned ndex has the advantage that t s not necessary to carry out any explct mputaton of a reference prce for the new product If a drect ndex s calculated as the rato of the arthmetc mean prces, the prce of the new product needs to be adjusted by the prce rato of A and D n March n every subsequent month, whch complcates computaton. Alternatvely, a reference perod prce of product D for January may be mputed. However, ths results n a dfferent ndex because the prce relatves are mplctly weghted by the relatve reference perod prces n the Dutot ndex, whch s not the case for the Carl or the Jevons ndex. For the Jevons ndex, all three methods gve the same result, whch s an addtonal advantage of ths approach. B.6 Other formulas for elementary prce ndces 9.64 A number of other formulas have been suggested for the prce ndces for elementary aggregates. The most mportant are presented below and dscussed further n Chapter 20. B.6.1 Laspeyres and geometrc Laspeyres ndces 9.65 The Carl, Dutot, and Jevons ndces are all calculated wthout the use of explct weghts. However, as already mentoned, n certan cases there may be weghtng nformaton that could be exploted or developed n the calculaton of the elementary prce ndces. If the reference perod revenues for all the ndvdual products wthn an elementary aggregate, or estmates thereof, were avalable, the elementary prce ndex could tself be calculated as a Laspeyres prce ndex, or as a geometrc Laspeyres. The Laspeyres prce ndex s defned as (9.4) t 0: t 0 p 0 PL = w, w 1 0 = p, where the weghts, w 0, are the revenue shares for the ndvdual products n the reference perod. If all the weghts were equal, equaton (9.4) would reduce to the Carl ndex. If the weghts were proportonal to the prces n the reference perod, equaton (9.4) would reduce to the Dutot ndex The geometrc Laspeyres ndex s defned as (9.5) P 0: t JW W0 t ( p ) 0 ( p ) 0 W t p 0 =, w 1 0 = 0 W =, p where the weghts, w 0, are agan the revenue shares n the reference perod. When the weghts are all equal, equaton (9.5) reduces to the Jevons ndex. If the revenue shares do not change much between the weght reference perod and the current perod, then the geometrc Laspeyres ndex approxmates a Törnqvst ndex. 227

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