Price and Volume Measures

Transcription

1 8 Price and Volume Measures Price and volume measures in he QNA should be derived from observed price and volume daa and be consisen wih corresponding annual measures. This chaper examines specific aspecs of price and volume measures derived a he quarerly frequency. In paricular, i shows how o aggregae quarerly price and volume measures a he elemenary level using Laspeyres and Fisher index formulas, how o derive quarerly chain volume series using alernaive linking echniques, and how o handle he lack of addiiviy of quarerly chain volume series. Inroducion A primary obecive in compiling quarerly naional accouns (QNA) is o obain an accurae price and volume decomposiion of quarerly ransacions in goods and services. This decomposiion provides he basis for measuring growh and inflaion in macroeconomic aggregaes, such as gross domesic produc (GDP) in volume erms or household consumpion deflaor. To mee his obecive, quarerly changes of ransacions in goods and services a curren prices need o be facored ino wo componens: quarerly price changes and quarerly volume changes. As general principles, QNA price and volume measures should reflec he movemens in quarerly price and volume indicaors and be emporally consisen wih he corresponding price and volume measures derived from he annual naional accouns (ANA). 2 The 2008 SNA (chaper 5) defines basic principles for deriving price and volume measures wihin he sysem of naional accouns in accord wih index number heory and inernaional sandards of price saisics. 2 A key recommendaion in he 2008 SNA, See Chaper 3 for an overview of price and volume indicaors for gross domesic produc by economic aciviies and by expendiure componens. 2 Main references of inernaional sandards of price saisics are he Consumer Price Index Manual: Theory and Pracice (ILO and ohers, 2004a), Producer Price Index Manual: Theory and Pracice also presen in he 993 SNA, is o move away from he radiional naional accouns measures a consan prices 3 oward chain-linked measures. Annual chain indices are superior o fixed-base indices, because weighs are updaed every year o reflec he curren economic condiions. Chaining also avoids he need for re-weighing price and volume series when he base year is updaed every five or en years, which usually generaes large revisions in he hisory of price and volume developmens. 4 The 2008 SNA recommends superlaive index number formulas such as he Fisher and Tornquis formulas; however, a naional accouns sysem based on Laspeyres volume indices (and he associaed implici Paasche price indices) is considered an accepable alernaive for pracical reasons. A summary of he main recommendaions of he 2008 SNA is given in Box The 2008 SNA also conains specific guidance on he compilaion of quarerly price and volume measures. Alhough he same principles apply o boh QNA and ANA, some complicaions derive from he differen frequency of observaion and he overarching requiremen ha quarerly and annual figures (when derived from independen compilaion sysems) should be made consisen wih each oher. The 2008 SNA suggess ha a sound approach o derive quarerly volume esimaes is o calculae annually chained Laspeyres-ype quarerly volume measures from quarerly daa ha are consisen wih annual supply and use ables (SUT) expressed in curren prices and in he prices of he previous year. Using (ILO and ohers, 2004b), and Expor and Impor Price Index Manual: Theory and Pracice (ILO and ohers, 2009). 3 Consan price measures are based on fixed-base Laspeyres volume indices (i.e., weighs aken from a fixed-base year) and corresponding curren period-weighed Paasche price indices. 4 Chain-linked series are sill subec o benchmark revisions (based on comprehensive daa sources available every five or en years) and mehodological revisions (due o changes in naional accouning principles) o curren price daa, which may generae difference in he aggregae price and volume indices.

2 66 Quarerly Naional Accouns Manual 208 Box 8. Main Recommendaions on Price and Volume Measures in he 2008 SNA This box quoes he main recommendaions of he 2008 SNA on expressing naional accouns in volume erms (2008 SNA, paragraph 5.80): (a) Volume esimaes of ransacions in goods and services are bes compiled in a supply and use framework, preferably in conuncion wih, and a he same ime as, he curren value esimaes. This implies working a as deailed a level of producs as resources permi. (b) In general, bu no always, i is bes o derive volume esimaes by deflaing he curren value wih an appropriae price index, raher han consrucing he volume esimaes direcly. I is herefore very imporan o have a comprehensive suie of price indices available. (c) The price indices used as deflaors should mach he values being deflaed as closely as possible in erms of scope, valuaion, and iming. (d) If i is no pracical o derive esimaes of value added in real erms from a supply and use framework and eiher he volume esimaes of oupu and inermediae consumpion are no robus or he laer are no available, hen saisfacory esimaes can ofen be obained using an indicaor of oupu, a leas in he shor erm. For quarerly daa, his is he preferred approach, albei wih he esimaes benchmarked o annual daa. An oupu indicaor derived by deflaion is generally preferred o one derived by quaniy exrapolaion. (e) Esimaes of oupu and value added in volume and real erms should only be derived using inpus as a las resor, since hey do no reflec any produciviy change. (f) The preferred measure of year-o-year movemens of gross domesic produc (GDP) volume is a Fisher volume index; price changes over longer periods being obained by chaining: ha is, by cumulaing he year-o-year movemens. (g) The preferred measure of year-o-year inflaion for GDP and oher aggregaes is, herefore, a Fisher price index; price changes over long periods being obained by chaining he year-o-year price movemens, or implicily by dividing he Fisher chain volume index ino an index of he curren value series. (h) Chain indices ha use Laspeyres volume indices o measure year-o-year movemens in he volume of GDP and he associaed implici Paasche price indices o measure year-o-year inflaion provide accepable alernaives o Fisher indices. (i) () Chain indices for aggregaes canno be addiively consisen wih heir componens whichever formula is used, bu his need no preven ime series of values being compiled by exrapolaing base year values by he appropriae chain indices. A sound approach o deriving quarerly curren value and volume esimaes is o benchmark hem o annual esimaes compiled in a supply and use framework. This approach lends iself o he consrucion of annually chained quarerly volume measures using eiher he Fisher or Laspeyres formula. annual weighs increases consisency wih he annual esimaes and makes he quarerly indices less subec o volailiy due o seasonal effecs and shor-erm irregulariies presen in quarerly daa. 4 The ideal way of producing volume esimaes of QNA aggregaes is o work a a very deailed level. The nex secion discusses some basic principles o derive volume esimaes in he naional accouns a he elemenary aggregaion level, adaped o he quarerly conex. For each individual ransacion, he same esimaion mehod should be used o derive volume esimaes in boh ANA and QNA. As discussed in Chaper 3, for mos marke ransacions, he bes resuls are generally obained by deflaing curren price values using appropriae price indices. Volume exrapolaion should be employed where appropriae price daa are no available or are no observable (e.g., nonmarke oupu), while he applicaion of quaniy revaluaion in he QNA can be considered for hose ransacions where deailed quaniies are available on a quarerly basis. 5 When deailed quarerly daa on oupu and inermediae consumpion are available, volume esimaes of value added should be derived using a double indicaor mehod. Volume esimaes of oupu and inermediae consumpion should be derived

3 Price and Volume Measures 67 independenly using appropriae price or volume indices. However, quarerly daa on deailed inermediae inpus may no be available or may be so wih a long ime lag. In hese cases, he calculaion of quarerly value added in volume should be based on single indicaor mehods. Typically, a fixed relaionship beween oupu and value added in volume erms is assumed. The nex secion elaboraes furher on using alernaive single indicaor mehods o bes approximae he double indicaor approach. 6 Sric consisency beween QNA and direc ANA price and volume measures is only guaraneed when annual and quarerly changes are aggregaed using he same sysem of weighs. Coherenly, wih he 2008 SNA, he preferred soluion o achieve fully consisen QNA and ANA price and volume measures is o calculae Laspeyres-ype volume indices wih annual weighs from he previous year. When he annual overlap (AO) echnique is used for chain-linking quarerly indices, 5 annually chained Laspeyres-ype quarerly volume measures are also consisen wih he corresponding annual Laspeyres volume measures. Quarerly indices based on oher index formulas, including Paasche and Fisher, or linked wih oher echniques (e.g., he one-quarer overlap [QO] echnique) do no aggregae exacly o heir corresponding direc annual indices. In such cases, consisency beween QNA and ANA price and volume measures requires eiher ha he ANA measures are derived as he annual sum of QNA measures or ha consisency is forced on he QNA daa using benchmarking echniques. 7 Nowihsanding he pracical advanages of Laspeyres-ype volume indices, a price and volume decomposiion based on superlaive indices (like Fisher) remains a heoreically superior soluion for boh ANA and QNA. The Fisher formula is a symmeric one, one in which price and quaniy relaives are aggregaed using weighs from boh he base period and he curren period, and provides a beer aggregaion of elemenary price and quaniy relaives beween he wo periods han he Laspeyres formula (which uses he base period) and he Paasche formula (which uses he curren period). This chaper 5 As menioned in his chaper, he annual overlap echnique may inroduce a break in he chain volume series beween one year and he nex. However, his happens only if here are srong changes in quaniy weighs wihin he year (see Annex 8.). illusraes a soluion o develop a Fisher-based price and volume esimaion sysem in he QNA based on (rue) quarerly and annual Fisher indices. 8 Price and volume series should guaranee imeseries characerisics: ha is, daa from differen periods should be comparable in a consisen manner. Sequence of price and volume indices having differen weigh periods (e.g., volume series a previous year s prices) are no comparable over ime and should no be presened in he form of ime series. Chain-linking is a necessary operaion o ransform annual and quarerly links from he previous year (or from he previous quarer, in he case of quarerly Fisher indices) ino consisen ime series. This chaper provides guidance on how o calculae quarerly chain volume series using alernaive linking echniques. Furhermore, i discusses how o resolve some pracical issues arising from he lack of addiiviy of chain-linked measures, including he calculaion of addiive conribuions o percen changes from nonaddiive chain volume series based on he Laspeyres and Fisher formulas. 9 Sricly adhering o he 2008 SNA principles, his chaper emphasizes he advanages of compiling chain-linked measures. Many counries, however, are sill compiling radiional consan price esimaes in boh he ANA and QNA and are far from implemening chain-linked measures. These counries will find useful o examine he specific QNA mehodological issues presened in he firs hree secions (basic principles, emporal consisency of price and volume measures, and choice of index formula for QNA volume measures), because hese issues apply equally o consan price esimaes. On he oher hand, he discussion on chainlinking presened in he remainder of he chaper is more relevan for hose counries ha have already implemened chain-linking in he QNA or ha plan o implemen i soon. Basic Principles for Deriving Volume Measures a he Elemenary Aggregaion Level 0 Volume measuremen relaes o decomposiion of ransacion values a curren prices ino heir price and volume componens. The aim of his decomposiion is o analyze how much of he change is due o

4 68 Quarerly Naional Accouns Manual 208 price movemens and how much o volume changes. 6 This decomposiion is admissible for ransacions in goods and services for which i is possible o assume ha he curren value is composed of a price and a quaniy componen. In addiion o pure ransacions in goods and services, volume measures can be compiled for ransacions such as axes and subsidies on producs, rade margins, consumpion of fixed capial, and socks of invenories and produced fixed asses. The accouning framework makes i possible o define and consruc volume measures for value added, alhough value added does no represen any observable flow of goods and services ha can be facored ino a price and volume componen direcly. This secion discusses some basic principles for deriving volume measures a he elemenary aggregaion level in he naional accouns and how hey should be implemened in he QNA conex. Volume esimaes of naional accouns should sar from a very deailed level. 7 The mos disaggregaed level in he naional accouns defines he level a which ransacions in curren values are deflaed or exrapolaed using available price or volume indices. To obain accurae resuls, i is desirable for he price and volume indices o be as homogeneous as possible. The more deailed are he indices, he more homogeneous are he produc groups measured by he indices. In he naional accouns, hese indices are considered elemenary price indices, even hough hey are already aggregaions of more deailed price indices. When he ype of producs of he index is homogeneous, he differen underlying weighing mehodologies can be assumed o be irrelevan and he price and volume changes from he indices can be used as price deflaor or volume exrapolaor for an elemenary ransacion of QNA. 2 In he QNA, he elemenary level of aggregaion should be decided on he basis of he ANA deail and 6 The expression volume change in he naional accouns includes boh quaniy changes and qualiy changes. Changes in qualiy over ime should be recorded as changes in volume and no as changes in price. Composiional changes should also be recorded as changes in volume, such as hose resuling from a shif from or o higher qualiy producs. 7 Working a a deailed level means ha, for example, volume esimaes of gross domesic produc (GDP) by indusry should be derived from volume esimaes of deailed economic aciviies, or ha volume esimaes of GDP by expendiure be derived from volume esimaes of deailed caegories of demand aggregaes. he scope of price and volume indicaors available on a quarerly basis. The ANA classificaion (by produc, by indusry, by expendiure funcion, ec.) generally defines he fines level of disaggregaion possible for he QNA. Ideally, QNA price and volume measures should be derived a he same deail level used in he ANA. More ofen, he QNA deail is more aggregaed han he ANA deail due o he reduced se of informaion available a he quarerly level. I is unnecessary and inefficien o keep he same ANA disaggregaion in he QNA when he quarerly informaion se does no permi o disinguish nominal price and volume measures a ha deail. 3 Prices and volumes are inrinsic componens of nominal values. Denoe wih c (, sy) he value a curren prices of an elemenary QNA ransacion for quarer s of year y, wih s =234,,, and y =2,,... 8 A he micro level, his ransacion can be hough of as he sum of a (finie) number of individual price volume ransacions: where, () (, sy) ( sy, ) (, sy) ( sy, ) c = c = p q is an index for ransacions included in he aggregae c (, sy), (, sy p ) is he price of ransacion in quarer s of year y, and (, sy q ) is he volume (quaniy plus qualiy effecs) of ransacion in quarer s of year y. (, sy The enire se of individual ransacions c ), including heir price and quaniy deails, are rarely direcly observable. In he QNA, he quarerly value c (, sy) is derived using some quarerly value indicaor (direcly in nominal erms or derived as he combinaion of price quaniy indices). For any given year, he quarerly figures (equaion ()) are made consisen wih he corresponding (generally more comprehensive) annual observaion C y hrough benchmarking. 4 As noed in his chaper, he mos frequen soluion adoped by counries for calculaing consisen 8 Differenly from previous chapers, noaion in his chaper shows he ime dimension in superscrip and he iem dimension in subscrip. This noaion is used in many price index heory exbooks and adoped by he 2008 SNA (chaper 5). Lower-case leers denoe quarerly observaions, wih quarer and year indicaed in brackes. Upper-case leers denoe annual observaions.

5 Price and Volume Measures 69 price and volume measures in boh ANA and QNA is o use annual weighs. 9 This approach should be followed for boh annual and quarerly daa. For chainlinked measures, he weighs should be updaed every year. The volume measure associaed wih equaion (), denoed wih k y ( s, y), is expressed as he quaniies of quarer s of year y valued a he prices of he previous year y : where y k (, s y) y s y sy = k (, ) = P (, ) q, (2) P y is a weighed average price of ransacion in year y (for a discussion on how bes o calculae weighed averages of quarerly price indices, see Main Principles of Seasonal Adusmen of his chaper). Equaion (2) provides he quarerly volume measure a he (weighed average) prices of he previous year (or a previous year s prices) of he elemenary ransacion for quarer s of year y. 5 By conras, a consan price measure is expressed as follows: b k (, s y ) b s y sy = k (, ) = Pq (, ), (3) b where he quarerly quaniies of quarer s of year y are valued a he average prices of a base year b. The advanage of using he volume esimae a previous year s prices in equaion (2) insead of he consan price measure in equaion (3) is ha he weighs are updaed every year and are no aken from a fixed (and ofen disan) base year. 6 When deailed quaniies in he curren quarer and prices of he previous year are available, he volume measure k y ( s, y) can be obained by quaniy revaluaion. This mehod may provide accurae price and volume decomposiion, as long as qualiy changes are incorporaed in he quaniies observed. This approach lends iself very well for homogeneous producs, where qualiy changes are less likely o occur. Quaniy revaluaion finds some applicaions for agriculural producs, whose quarerly quaniies may be derived from work-in-progress models based on deailed crop forecass, or for highly concenraed indusries, such as oil-producing indusries which ofen 9 The following discussion can easily be adaped o calculaing indices from he previous quarer, as required o derive quarerly Fisher indices. y provide deailed daa on heir quarerly producion hrough oil-relaed business associaions. 7 More commonly, volume measures k y ( s, y) are calculaed using one of wo alernaive mehods: price deflaion and volume exrapolaion. 0 Price Deflaion 8 The volume esimae k y ( s, y) is derived by dividing he curren price value c (, sy) by an appropriae price index. Ideally, he volume esimae k y ( s, y) should be derived using a quarerly Paasche-ype price index : PP y (, s y) = p (, sy) ( sy, ) q y P In effec, i is easily shown ha PP c (, sy) y (, s y) = p p P (, sy) q. (4) (, sy) ( sy, ) q (, sy) ( sy, ) q y ( sy, ) q y (, sy) y ( s, y) = P q = k. (5) Paasche-ype price indices are rarely available for naional accouns purposes. 2 They require weighs from every period and are difficul o calculae in pracice. Price indices are usually calculaed using he Laspeyres formula wih a fixed-base year, wih weighs aken from a survey conduced in ha year. 3 Denoing wih LP b ( s, y) a Laspeyres-ype price index wih a fixed-base year b, i is possible o calculae a price relaive of quarer s of year y from he previous year y as follows: LP y (, s y) LP = LP b (, s y) b y, (6) 0 Chaper 3 idenifies wheher price deflaion or volume exrapolaion are mos suiable for gross domesic produc componens by economic aciviy and by expendiure caegories. A quarerly Paasche-ype price index is a weighed harmonic average of price relaives wih weighs from he curren quarer. 2 A noable excepion of Paasche-ype aggregaion is uni value indices in merchandise rade saisics. 3 In pracice, saisical offices do no calculae Laspeyres-ype indices bu Lowe indices, where he weigh period precedes he base period. On he relaionship beween Lowe, Laspeyres, and Paasche price indices, see he Consumer Price Index Manual: Theory and Pracice (ILO and ohers, 2004a).

6 70 Quarerly Naional Accouns Manual 208 ha is, he raio beween he fixed-base index for quarer s of year y and he fixed-base index for year y. Replacing PP y ( s, y) wih he (fixed-base) Laspeyres-ype price index LP y ( s, y) in equaion (6) will provide an approximae volume measure k y ( s, y). 4 Working a a deailed elemenary level is crucial for assuming ha a fixed-base Laspeyres price index is close o he ideal curren period-weighed Paasche price index. Volume Exrapolaion 9 This mehod requires an annually weighed Laspeyres-ype quarerly volume index, which is defined as follows: LQ y (, s y) = 4 y (, sy) P q P y y Q, (7) where Q y is he annual quaniy of ransacion in year y. The volume measure k y ( s, y) can be derived ideally by exrapolaing he (rescaled) curren price value of he previous year using he index LQ y ( s, y) : ha is, y y (, sy) y y C LQ = P Q 4 P P y (, sy) q y y Q = y (, P sy ) y ( s, y q = k ). (8) Likewise prices, he available volume indices are normally fixed-base Laspeyres-ype indices. Similar o he price relaive calculaed in equaion (6), a (fixed-base) quaniy relaive from he previous year can be calculaed as follows: LQ y (, s y) LQ = LQ b (, s y) b y (9) and used in equaion (8) o exrapolae he volume change from he previous year. A consan prices, he 4 Consan price daa can be derived direcly by dividing he curren price daa c (, sy) wih a fixed-base Laspeyres price index LP b ( s, y) or by exrapolaing he curren price daa in he base year 4 Cb wih a fixed-based Laspeyres volume index LQ b ( s, y). volume index LQ b ( s, y) can be used direcly o exrapolae he curren price daa in he base year. 20 In mos counries, quarerly GDP is derived from he producion approach. This fac resuls from a greaer availabiliy of quarerly daa by economic aciviy compared wih expendiure and income ransacions. Therefore, i assumes paricular relevance how volume esimaes of quarerly value added are calculaed. As discussed in Chaper 3, he bes mehod o derive volume measures of value added is o use double indicaor mehods a volume measure of value added as he difference beween a direc esimae of oupu in volume and a direc esimae of inermediae consumpion in volume (each of which can be derived eiher by direc revaluaion, deflaion, or volume exrapolaion). However, in pracice, he informaion needed for obaining independen and reliable volume esimaes of oupu and inermediae consumpion may no be available or may no be of sufficien qualiy. In paricular, o derive a proper deflaor for inermediae consumpion for each aciviy, deailed daa on inermediae consumpion by produc in he curren quarer is needed. 2 In he QNA, simplified approximaion mehods someimes need o be used. 5 One such simplified mehod is o use volume indicaors o exrapolae value added. This is called he single exrapolaion echnique. The single exrapolaion echnique, using a volume esimae of oupu 6 o exrapolae value added, is based on an underlying assumpion of a consan relaionship beween oupu, inermediae consumpion, and value added in volume erms. This assumpion usually holds rue in he shor run for many indusries in periods of economic sabiliy, while i is a highly quesionable assumpion in he long run and for counries wih rapid srucural changes. The fixedraio assumpion in volume erms should be checked coninuously looking a he annual benchmarks of naional accouns, making sure ha here are no sudden changes in he oupu-o-inermediae consumpion raio beween one year and he nex. 5 For an empirical assessmen of he differences beween double deflaion and single indicaor mehods, see Alexander and ohers (207). 6 As noed in Chaper 3, inpu-relaed volume indicaors (such as deflaed wages or employmen daa) may be considered o exrapolae value added when informaion on oupu is absen or less reliable (one example is nonmarke oupu).

7 Price and Volume Measures 7 22 An alernaive, less saisfacory, approximaion is o use a price indicaor (e.g., he price deflaor for oupu, inermediae consumpion, or a wage index) o deflae value added direcly. This is known as he single deflaion echnique. The single deflaion echnique, using he price deflaor for oupu as he deflaor for value added, is based on an underlying assumpion of a consan relaionship beween he price deflaors for oupu, inermediae consumpion, and value added. While here is reason o expec he relaionship beween oupu, inermediae consumpion, and value added in volume erms o change only gradually, here is no reason o expec a sable relaionship beween he price deflaors for oupu, inermediae consumpion, and value added. This is a highly quesionable assumpion o rely on because price relaives may change abruply, even in he shor erm. For his reason, he single deflaion echnique should be avoided. 23 When simplified mehods such as he single exrapolaion echnique are used, i is srongly recommended o esimae all he componens of he producion accoun in volume erms, and no only value added. Furhermore, i is recommended o derive esimaes based on more han one esimaion echnique and o assess he esimaes and he validiy of he underlying assumpions by inspecing and comparing he implici deflaors for oupu, inermediae consumpion, and value added, or by assessing he inermediae consumpion shares a he quarerly frequency. Aggregaing Price and Volume Measures Over Time 24 Aggregaion over ime means deriving less frequen daa (e.g., annual) from more frequen daa (e.g., quarerly). Incorrec aggregaion of prices, or price indices, over ime o derive annual deflaors can inroduce errors in independenly compiled annual esimaes and hus can cause inconsisency beween QNA and ANA esimaes, even when hey are derived from he same underlying daa. When deriving annual volume esimaes by deflaing annual curren price daa, a common pracice is o compue he annual price deflaors as a simple unweighed average of monhly or quarerly price indices. This pracice may inroduce subsanial errors in he derived annual volume esimaes, even when inflaion is low. This may happen when a. here are seasonal or oher wihin-year variaions in prices or quaniies and b. he wihin-year paern of variaion in eiher prices or quaniies is unsable. 25 Volume measures for aggregaed periods of ime should concepually be consruced from period-oal quaniies for each individual homogenous produc. The corresponding implici price measures would be quaniy-weighed period-average price measures. For example, annual volume measures for single homogenous producs 7 should be consruced as sums of he quaniies in each subperiod. The corresponding implici annual average price, derived as he annual curren price value divided by he annual quaniy, would herefore be a quaniy-weighed average of he prices in each quarer. As shown in Example 8., he quaniy-weighed average price will generally differ, someimes significanly, from he unweighed average price. Similarly, for groups of producs, concepually, annual volume measures can be consruced as a weighed aggregae of he annual quaniies for each individual produc. The corresponding implici annual price deflaor for he group would be a weighed aggregae of he quaniy-weighed annual average prices for he individual producs. This annual price deflaor for he group based on he quaniy-weighed annual average prices would generally differ, someimes significanly, from he annual price deflaors derived as a simple unweighed average of monhly or quarerly price indices ofen used in ANA sysems deflaion by he laer may inroduce subsanial errors in he derived annual volume esimaes. 26 Consequenly, o obain correc volume measures for aggregaed periods of ime, deflaors should ake ino accoun variaions in quaniies as well as prices wihin he period. For example, annual deflaors could be derived implicily from annual volume measures derived from he sum of quarerly volume 7 Homogenous producs are idenical in physical and economic erms o oher iems in ha produc group and over ime. In conras, when here are significan variaions among iems or over ime in he physical or economic characerisic of he produc group, each version should be reaed as a separae produc (e.g., ou-of-season frui and vegeables such as old poaoes may be regarded as differen producs han in-season frui and vegeables such as new poaoes).

8 72 Quarerly Naional Accouns Manual 208 Example 8. Weighed and Unweighed Annual Averages of Prices (or Price Indices) When Sales and Price Paerns Through he Year Are Uneven Volume esimaes Quaniy Price Curren Price Value Unweighed Average Price Uni Value Weighed Average Price A Unweighed Average 200 Prices A Weighed Average 200 Prices Quarer () (2) (3) (4) (5) = (3)/() (6) = (4)*() (7) = (5)*() q q ,500 7,500 6,750 q ,500 2,500 2,250 q , ,000 9,000 q q ,000 9,000 8,00 q , q , ,000 9,000 Change from 200 o 20 (%) Direc Deflaion of Annual Curren Price Daa a 200 prices 9,600/(40/50) = 9,600/0.8 = 2,000 Change from 200 (2,000/9,000 ) 00 = 33.3% This example highlighs he case of an unweighed annual average of prices (or price indices) being misleading when sales and price paerns hrough he year are uneven for a single homogenous produc. The producs sold in he differen quarers are assumed o be idenical in all economic aspecs. In he example, he annual quaniies and he quarerly prices in quarers wih nonzero sales are he same in boh years, bu he paern of sales shifs oward he second quarer of 20. As a resul, he oal annual curren price value increases by 6.67 percen. If he annual deflaor is based on a simple average of quarerly prices, hen he deflaor appears o have dropped by 20 percen. As a resul, he annual consan price esimaes will wrongly show an increase in volume of 33.3 percen. Consisen wih he quaniy daa, he annual sum of he quarerly volume esimaes for 200 and 20, derived by valuing he quaniies using heir quaniy-weighed average 200 price, shows no increase in volumes (column 7). The change in annual curren price value shows up as an increase in he implici annual deflaor, which would be implicily weighed by each quarer s proporion of annual sales in volume erms. Price indices ypically use unweighed averages as he price base, which corresponds o valuing he quaniies using heir unweighed average price. As shown in column 6, his resuls in an annual sum of he quarerly volume esimaes in he base year (200) ha differs from he curren price daa, which i should no. As explained above and in his chaper, quarerly weighed prices should be used o derive annual prices. The difference beween unweighed and weighed annual prices in he base year, however, can easily be removed by a muliplicaive adusmen of he complee consan price ime series, leaving he period-o-period rae of change unchanged. The adusmen facor is he raio beween he annual curren price daa and sum of he quarerly volume daa in he base year (9,000/0,000). esimaes obained using he following hree-sep procedure: a. benchmark he quarerly curren price daa/ indicaor(s) o he corresponding annual curren price daa, b. consruc quarerly volume daa by dividing he benchmarked quarerly curren price daa by he quarerly price index, and c. derive he annual volume daa as sum of he quarerly volume daa. Equivalenly, he annual volume measure could be obained by deflaing, using an annual deflaor ha weighs he quarerly price indices by he volume values of ha ransacion for each quarer. Eiher way of calculaion achieves annual deflaors ha are quaniyweighed average annual price measures. 27 The procedure described above guaranees he bes resuls of deflaion if i is possible o obain a reliable measuremen of he quarerly paern a curren prices. If he curren price indicaor used o

9 Price and Volume Measures 73 decompose he annual value is deemed o provide an inaccurae quarerly decomposiion of he year (e.g., seasonal effecs which are no fully represenaive of he ransacion), he annual volume daa could be affeced by a disored allocaion of weighs o quarerly prices. When i is no possible o derive accurae quarerly decomposiion of curren price daa, unweighed averages of sub-annual indices represen a feasible choice for he ANA. 28 A more difficul case occurs when he annual esimaes are based on more deailed price and value informaion han is available quarerly. In hose cases, if seasonal volailiy is significan, i would be possible o approximae he correc procedure using weighs derived from more aggregaed, bu closely relaed, quarerly daa. 29 The issue of price and quaniy variaions also applies wihin quarers. Accordingly, when monhly daa are available, quarerly daa will beer ake ino accoun variaions wihin he period if hey are buil up from he monhly daa. 30 In many cases, variaion in prices and quaniies wihin years and quarers will be so insignifican ha i will no subsanially affec he esimaes. Comparing weighed and unweighed averages can help idenify he producs for which he disincion is mos relevan. Primary producs and high-inflaion counries are cases where he variaion can be paricularly significan. Of course, here are many cases in which here are no daa o measure variaions wihin he period. 3 A relaed problem ha can be observed in quarerly daa a consan prices of a fixed-base year is he annual sum of he quarerly volume esimaes in he base year differing from he annual sum of he curren price daa, which should no be he case. This difference can be caused by he use of unweighed annual average prices as he price base when consrucing monhly and quarerly price indices. Deflaing quarerly daa wih deflaors consruced wih unweighed average prices as he price base corresponds o valuing he quaniies using heir unweighed annual average price raher han heir weighed annual average price. This difference in he base year beween he annual sum of he quarerly volume esimaes and he annual sum of he curren price daa can easily be removed by a muliplicaive adusmen of he complee volume series, leaving he period-o-period rae of change unchanged. The adusmen facor is he raio beween he annual curren price daa and he sum of he iniial quarerly volume daa based on he unweighed annual average prices in he base year, which, for a single produc, is idenical o he raio of he weighed and unweighed average price. Index Formula for QNA Volume Measures 32 Using he same noaion inroduced earlier, he applicaion of revaluaion, deflaion, or volume exrapolaion mehods a he mos deailed level in he QNA generaes a se of elemenary volume indices: q where y (, s y) k y (, s y) = y C 4 denoes a generic QNA ransacion, y s y q, (0) (, ) is a volume index from year y o quarer s of year y for he -h ransacion, y s y k (, ) is he volume esimae of quarer s of year y a previous year s prices, and 4 is he (rescaled) annual value a curren prices in he previous year. C y Because numeraor and denominaor are valued using he same se of prices, he raio measures a volume movemen from year y o quarer s of year y. The formula is addiive wihin he year and coincides wih he annual volume index. I is also addiive across QNA ransacions: he same formula can be used o exrapolae higher-level aggregaes. Equaion () provides he links o form chain-linked volume series, which is discussed in secion Chain-Linking in he QNA. 33 In a consan price sysem, equaion () is modified as follows: q where b (, s y) k b (, s y) = b y K 4, () b ( s, y q ) is a fixed-base volume index of quarer s of year y for ransacion,

10 74 Quarerly Naional Accouns Manual 208 b ( s, y) k is he esimae of quarer s of year y a consan prices of a (fixed) base year b, and b y K 4 is he (rescaled) consan price daa in he previous year. Because equaion () derives fixed-base indices (i.e., indices expressed wih a common base year), here is no need for using linking echniques beween differen years. Linking, however, is sill necessary when he base year changes and he rebased series need o be linked o he series in he old base year. The echniques inroduced in secion Chain-Linking in he QNA are also relevan for linking consan prices series wih differen base years. 34 Elemenary volume indices (equaion () or ()) need o be aggregaed o derive QNA volume esimaes. This secion discusses how o aggregae elemenary indices using he Laspeyres and Fisher formulas. Laspeyres-Type Formula 35 A Laspeyres-ype index aggregaes elemenary indices using weighs from he base period. The base period for he QNA elemenary volume indices shown in equaion () is he previous year y. 8 An annually weighed Laspeyres-ype quarerly volume index LQ y ( s, y) can be calculaed as he weighed average of elemenary volume indices of quarer s of year y wih weighs from year y : where n y (, s y) y s y y (, ) = LQ = q W n y C (2) y (, s y) = q = n = C y is he index for ransacions in he aggregae, n is he number of ransacions in he aggregae, y ( s, y q ) is he elemenary volume index of ransacion from year y o quarer s of year y as shown in equaion (), 8 For he sake of clariy, he following noaion is based on volume indices a previous year s prices. However, any index aggregaion formulas presened in his secion apply equally o fixed-base indices. C y is he annual value a curren prices of ransacion for year y, y C is he sum of all he annual values in he aggregae a curren prices for year y, and W y is he share of C y in he aggregae for year y. Calculaion of annually weighed Laspeyres-ype volume measures from elemenary volume indices is shown in Example Combining equaion () and equaions (2) (9), equaion (2) can be rewrien as follows: where LQ y (, s y) = 4 n = n y (, sy) q P = P y y Q, (3) (, sy q ) is he quaniy of ransacion in quarer s of year y, P y is he price of ransacion in year y, and Q y is he quaniy of ransacion in year y. Equaion (3) shows ha a Laspeyres-ype index is he raio beween he quaniies of he curren quarer valued a he (average) prices of he previous year and he rescaled annual value of he previous year a curren prices. This noaion is commonly found in he presenaion of index numbers; however, i is difficul o apply in pracice because, as noed before, price and quaniies of QNA ransacions are no available in mos siuaions. For his reason, equaion (2) is used in pracice and is applied in he examples hroughou his chaper. 37 As discussed earlier, annual weighs for Laspeyresype volume indices are generally preferable over quarerly weighs. Use of he prices of one paricular quarer, he prices of he corresponding quarer of he previous year, he prices of he corresponding quarer of a fixed-base year, or he prices of he previous quarer are no appropriae for ime series of Laspeyresype volume measures in he naional accouns for he following reasons: Consisency beween direcly derived ANA and QNA Laspeyres-ype volume measures requires ha he same price weighs are used in he ANA

11 Price and Volume Measures 75 Example 8.2 Deriving Annual and Quarerly Volume Measures Using Laspeyres-Type Formula Curren Prices Elemenary Price Indices (Previous Year = 00) Elemenary Volume Measures (in Moneary Terms) Elemenary Volume Indices (Previous Year = 00) Laspeyres Volume Index (Previous Year = 00) Laspeyres Volume Measure (in Moneary Terms) () (2) (3) = ()/(2) 00 (4) (5) (6) A B Toal A B A B Sum A B Toal Toal , , , , , , , , , , , ,564. q q q q Sum , , ,5.0 q q q q Sum , , ,528.7 q q q q Sum , , ,564. (Rounding errors in he able may occur.) Deflaion a he Elemenary Level This example explains how o derive volume esimaes of wo ransacions a he mos deailed level (A and B) and how o derive a volume index using an annually weighed Laspeyres-ype formula. Annual and quarerly daa a curren prices of he wo ransacions from 200 o 203 are presened in column, wih he quarerly spli available from q 20. On average, ransacion A shows a 6.5 percen increase a year, while ransacion B declines a a.9 percen annual rae: oal increase is.4 percen a year. The relaive size of ransacions A and B is revered afer hree years. Column 2 conains he elemenary price indices for A and B of each quarer compared wih he previous year, as explained in equaions () (9). Volume esimaes for A and B are obained by price deflaion in column 3. For insance, volume esimaes of A for he quarers of 20 are calculaed as follows: q 20: (59.7/02.0) 00 = 56.6 q2 20: (63.2/02.5) 00 = 59.2 q3 20: (67.4/03.0) 00 = 62.5 q4 20: (69.7/03.0) 00 = The same operaions are done using he annual daa. As explained in his chaper, annual price changes are derived as weighed average of he quarerly indices wih weighs given by he quarerly volume esimaes in column 3. Noe ha because annual indices are weighed average of quarerly indices, he sum of he quarerly volume esimaes corresponds o he independenly calculaed annual volume figure. This condiion is also me for he oal aggregae. Elemenary Volume Indices Elemenary volume indices are shown in column 4. For he annual daa, hey are derived implicily by dividing he annual volume measures in column 3 by he curren price value in he previous year. For insance, he annual index for 20 for ransacion A is 643./600 = For he quarerly daa, he elemenary volume indices are derived by dividing he quarerly volume measures in column 3 by he rescaled curren price value in he previous year (see equaion (9)). The quarerly index for q 20 for ransacion A is 56.6/(600/4) = Laspeyres-Type Volume Indices and Laspeyres-Type Volume Measures in Moneary Terms The annually weighed Laspeyres-ype volume indices in column 5 are calculaed as a weighed average of he elemenary volume indices in columns 4. The weighs are he share a curren prices from he previous year. The annual indices are calculaed as follows: 20: 07.8 (600/,500) (900/,500) = : 3.05 (660/,54.9) (854.9/,54.9) = : (759/,528.6) (769.5/,528.6) =

12 76 Quarerly Naional Accouns Manual 208 Similar o he annual indices, he quarerly indices are calculaed using weighs from he previous year. For he quarers of 20, q 20: (600/,500) (900/,500) = 00.7 q2 20: 06.5 (600/,500) (900/,500) = 00.6 q3 20: (600/,500) (900/,500) = q4 20: (600/,500) (900/,500) = For he quarers of 202, q 202: (660/,54.9) (854.9/,54.9) = q2 202: 07.9 (660/,54.9) (854.9/,54.9) = q3 202: 3.35 (660/,54.9) (854.9/,54.9) = q4 202: (660/,54.9) (854.9/,54.9) = Volume esimaes in moneary erms are derived by muliplying he Laspeyres volume indices by he oal curren price value in he previous year. For 20 and 202, 20: 00.73,500 =, : 00.9,54.9 =,528.7 q 20: 00.7 (,500/4) = q 202: (,54.9/4) = q2 20: 00.6 (,500/4) = q2 202: (,54.9/4) = q3 20: (,500/4) = q3 202: (,54.9/4) = q4 20: (,500/4) = q4 202: 0.85 (,54.9/4) = I is easily shown ha he sum of he quarerly volume measures in moneary erms corresponds o he corresponding annual volume measure. This condiion is verified wihin each link using he Laspeyres-ype formula. In addiion, noe ha he quarerly volume measures in moneary erms are equal o he sum of he deflaed elemenary ransacions shown in column 3 a boh annual and quarerly levels. and he QNA, and ha he same price weighs are used for all quarers of he year. The prices of one paricular quarer are no suiable as price weighs for volume measures in he ANA, and hus in he QNA, because of seasonal flucuaions and oher shor-erm volailiies in relaive prices. Use of weighed annual average prices reduces hese effecs. Therefore, weighed annual average prices are more represenaive for he oher quarers of he year as well as for he year as a whole. The prices of he corresponding quarer of he previous year or he corresponding quarer of a fixed-base year are no suiable as price weighs for volume measures in he QNA because he derived volume measures only allow he curren quarer o be compared wih he same quarer of he previous year or years. Series of year-o-year changes do no consiue ime series ha allow differen periods o be compared and canno be linked ogeher o form such ime series. In paricular, because hey involve using differen prices for each quarer of he year, hey do no allow differen quarers wihin he same year o be compared. For he same reason, hey do no allow he quarers wihin he same year o be aggregaed and compared wih heir corresponding direc annual esimaes. Furhermore, as shown in Chaper, changes from he same period in he previous year can inroduce significan lags in idenifying he curren rend in economic aciviy. The prices of he previous quarer are no suiable as price weighs for Laspeyres-ype volume measures for wo reasons: a. The use of differen price weighs for each quarer of he year does no allow he quarers wihin he same year o be aggregaed and compared wih heir corresponding direc annual esimaes. b. If he quarer-o-quarer changes are linked ogeher o form a ime series, shor-erm volailiy in relaive prices may cause he quarerly chain-linked measures o show subsanial drif compared o corresponding direc measures. 38 In sum, he Laspeyres formula offers a very convenien soluion o achieve consisency beween ANA and QNA volume measures. As shown in Example 8.2, he sum of annually weighed Laspeyres-ype quarerly volume measures (i.e., he quarerly volume esimaes a previous year s prices) maches he independenly derived Laspeyres-ype annual volume measures (i.e., he annual volume esimae a previous year s prices). Moreover, he quarerly volume esimaes a previous year s prices are addiive wihin each link (quarer or year). Laspeyres-ype indices have hese properies because annual and quarerly indices use he same se of weighs. Fisher indices, as explained in paragraph 8.76, do no have hese

13 Price and Volume Measures 77 properies and need o be reconciled when hey are calculaed a differen frequencies. 39 Because Laspeyres-ype volume esimaes in moneary erms are addiive in each period, volume esimaes of aggregaes can simply be derived as he sum of he elemenary volume componens (see Example 8.2). As noed a he beginning of his subsecion, equaion (2) can be used o calculae Laspeyres-ype volume indices from boh elemenary iems and aggregaes. They can be derived by dividing he sum of elemenary volume componens for a paricular quarer by he (rescaled) aggregae esimae a curren prices of he previous year (i.e., by applying equaion () on he aggregae esimaes). Fisher-Type Formula 40 A Fisher index is he geomeric mean of he Laspeyres and Paasche indices. A Fisher index is a symmeric index, one ha makes equal use of he prices and quaniies in boh he periods compared and rea hem symmerically. Symmeric indices saisfy a se of desirable properies in index number heory (like he ime reversal es) and are o be preferred for economic reasons because hey assign equal weigh o he wo siuaions being compared. 9 4 Calculaion of annually weighed quarerly Fisher-ype indices is complicaed. They should be derived as symmeric annually weighed Laspeyres-ype and Paasche-ype quarerly volume indices. However, he (implici) Paasche-ype quarerly index corresponding o he annually weighed Laspeyres-ype quarerly index shown in equaion (2) has weighs from he curren quarer (i.e., he curren period). This would make he geomeric average of Laspeyres and Paasche indices (i.e., he Fisher index) emporally asymmeric, because he weigh srucure would be aken from he previous year and he curren quarer. 42 The 2008 SNA illusraes a soluion o calculae symmeric annually weighed quarerly Fisherype indices (paragraphs ). For each pair of consecuive years, Laspeyres-ype and Paasche-ype quarerly indices are consruced for he las wo quarers of he firs year and he firs wo quarers of 9 Oher symmeric (and superlaive) indices are he Walsh and Törnqvis indices. Deails on he heory of symmeric and superlaive indices can be found in he Consumer Price Index Manual: Theory and Pracice (ILO and ohers, 2004a). he second year. The annual value shares are aken from he wo years o consruc Laspeyres-ype and Paasche-ype quarerly indices. The annually chained Fisher-ype indices are derived as he geomeric mean of hese wo indices. The resuling quarerly Fisher indices need o be benchmarked o annual chain Fisher indices. A he end of he series (when Paasche indices using annual weighs from he curren year are impossible o calculae), rue quarerly Fisher indices can be used o exrapolae he annually chained Fisher-ype indices. 43 True quarerly Fisher indices provide resuls ha are no exacly consisen wih corresponding annual Fisher indices; neverheless, hey are usually close enough when quaniy and price weighs are relaively sable wihin he year. When he Fisher formula is chosen in he ANA, he preferred soluion for he QNA is o calculae rue quarerly Fisher indices (wih quarerly weighs) and benchmark hem o he corresponding annual Fisher indices. 20 The benchmarking process forces he quarerly volume measures o be consisen wih he annual ones. Before benchmarking, he difference beween he annual and quarerly indices should be invesigaed carefully o deec possible drifs in he chain quarerly series (see he drif problem in he secion Frequency of Chain-Linking ). 44 To calculae quarerly Fisher volume indices, quarerly Laspeyres volume indices and quarerly Paasche volume indices 2 are necessary. They can be calculaed as follows: LQ PQ n c = q n = n = ( q ) = = c c n = c (4), (5) 20 The Unied Saes adops his soluion o calculae consisen annual and quarerly Fisher price and volume indices in he naional accouns (see Parker and Seskin, 997). 2 Quarerly Paasche volume indices adop as weighs he curren price daa for he mos recen quarer. Because daa for he las quarer may be subec o large revisions, Paasche indices could be more volaile over ime han he corresponding Laspeyres indices.