IX Price and Volume Measures: Specific QNA-ANA Issues

Transcription

1 IX Prce and Volume Measures: Secfc QNA-ANA Issues A. Introducton 9.. Ths chater addresses a selected set of ssues for constructng tme seres of rce and volume measures that are of secfc mortance for the uarterly natonal accounts (QNA). In artcular, t dscusses the relatonsh between rce and volume measures n the QNA and n the annual natonal accounts (ANA): namely, () how to aggregate rce and volume measures over tme; (2) how to choose the base erod n the QNA; (3) the freuency of chan-lnkng; and (4) the technues for annual chan-lnkng of uarterly data. In addton, the chater addresses how to deal wth nonaddtvty and resentaton of chan-lnked volume measures n the QNA The 993 SNA does not contan secfc recommendatons for rce and volume measures for the QNA or the relatonsh between rce and volume measures n the QNA and the ANA. The basc rncles for uarterly rce and volume measures n the QNA and the ANA are the same, ncludng the 993 SNA recommendaton of movng away from the tradtonal constant-rce measures to annually chanlnked measures, referably usng suerlatve ndex number formulas such as the Fsher and Tornust formulas. The ssues lsted above rase new roblems, however, many of whch have not satsfactorly been dealt wth to date n the lterature. Conventonal ntertemoral ndex number theory has manly been concerned wth rce and uantty comarsons between searate ars of onts n tme and not wth rce and volume measures n a tme-seres context. In artcular, conventonal ndex number theory has not been concerned wth rce and uantty comarsons between erods of tme of dfferent duraton (e.g., years and uarters) and the relatonsh among Constant rce measures are fxed-base Laseyres-tye volume measures (fxed-rce weghts) and the corresondng rce deflators are Paasche rce ndces. these rce and volume measures for longer tme erods, the corresondng measures for the suberods, and the ont-to-ont measures QNA rce and volume measures should be n the form of tme seres and should be consstent wth corresondng ANA estmates. For QNA rce and volume measures to consttute a tme seres, they must meet the followng four reurements: (a) The data should reflect both the short- and longterm movements n the seres, artcularly the tmng of any turnng onts. (b) The data should allow dfferent erods to be comared n a consstent manner. That s, based on the underlyng tme seres, the data should allow measures of change to be derved between any erod (.e., from the revous erod, the same erod n the revous year, and a artcular erod several years earler). (c) The data should allow erods of dfferent duraton to be comared n a consstent manner. That s, based on the underlyng tme seres, the data should allow measures of change to be derved between any erods of any length (e.g., between the average of the last two uarters and of the revous two uarters or the same two uarters several years earler, from the average of the revous year and of a year several years earler). (d) The data should allow suberods and erods to be comared n a consstent manner (e.g., uarters wth years) Consstency between QNA and ANA rce and volume measures, n rncle, reures ether that the ANA measures are derved from uarterly measures or that consstency s forced on the QNA data usng benchmarkng technues. Ths s true even f the basc reurement that the QNA and ANA measures are based on the same methods of comlaton and resentaton (.e., same ndex formula, base year(s), and reference erod) s met. Strct consstency 47

2 IX PRICE AND VOLUME MEASURES: SPECIFIC QNA-ANA ISSUES between QNA and drect ANA rce and volume measures s generally not ossble because uarterly ndces based on most ndex formulas, ncludng Paasche and Fsher, do not aggregate exactly to ther corresondng drect annual ndces. For fxed-base Laseyres volume ndces, or tradtonal constant rce estmates, consstency reures that the estmates are derved by exlctly or mlctly valung the uanttes at the annual uantty-weghted average of the rces charged n dfferent tme erods of the base year, 2 effectvely mlyng that the annual volume data are derved from the uarterly data 3 (see Secton B) and not drectly. Fnally, for annually chan-lnked Laseyres volume ndces, strct consstency can only be acheved by use of an annual lnkng technue that can result n a break 4 n the estmates between the fourth uarter of one year and the frst uarter of the next year (see Secton D) Consstency between QNA and ANA rce and volume measures also reures that new methods, lke chan-lnkng, are mlemented smultaneously n both the QNA and ANA. Although the 993 SNA recommends movng to chan-lnked volume measures, for countres currently comlng tradtonal constant rce estmates, t would generally be undesrable to comlcate the ntroducton of QNA by also ntroducng new technues for constructng and resentng volume measures at the same tme. It s recommended for these countres to ntroduce channg n a second hase, concurrent wth the ntroducton of chan-lnkng n the ANA. Thus, for countres currently comlng tradtonal constant rce estmates, only the dscusson n Secton B of aggregatng rce and volume measures over tme s of mmedate mortance. B. Aggregatng Prce and Volume Measures Over Tme 9.6. Aggregaton over tme means dervng less freuent data (e.g., annual) from more freuent data (e.g., uarterly). Incorrect aggregaton of rces, or rce ndces, over tme to derve annual deflators can ntroduce errors n ndeendently 2 The corresondng exlct or mlct annual deflators should be derved as current-year uantty-weghted averages of monthly or uarterly fxed-based Paasche rce ndces. 3Ths s artcularly an ssue under hgh nflaton and for hghly volatle tems. 4Ths can occur f there are strong changes n relatve uanttes and relatve rces. comled annual estmates and thus can cause nconsstency between QNA and ANA estmates, even when they are derved from the same underlyng data. When dervng annual constant rce estmates by deflatng annual current rce data, a common ractce s to comute the annual rce deflators as a smle unweghted average of monthly or uarterly rce ndces. Ths ractce may ntroduce substantal errors n the derved annual constant rce estmates, even when nflaton s low. Ths may haen when there are seasonal or other wthn-year varatons n rces or uanttes, and the wthn-year attern of varaton n ether rces or uanttes s unstable Volume measures for aggregated erods of tme should concetually be constructed from erod-total uanttes for each ndvdual homogenous roduct. The corresondng mlct rce measures would be uantty-weghted erod-average rce measures. For examle, annual volume measures for sngle homogenous roducts 5 should be constructed as sums of the uanttes n each suberod. The corresondng mlct annual average rce, derved as the annual current rce value dvded by the annual uantty, would therefore be a uantty-weghted average of the rces n each uarter. As shown n Examle 9., the uanttyweghted average rce wll generally dffer, sometmes sgnfcantly, from the unweghted average rce. Smlarly, for grous of roducts, concetually, annual volume measures can be constructed as a weghted aggregate of the annual uanttes for each ndvdual roduct. The corresondng mlct annual rce deflator for the grou would be a weghted aggregate of the uantty-weghted annual average rces for the ndvdual roducts. Ths annual rce deflator for the grou based on the uantty-weghted annual average rces would generally dffer, sometmes sgnfcantly, from the annual rce deflators derved as a smle unweghted average of monthly or uarterly rce ndces often used n ANA systems deflaton by the latter may ntroduce substantal errors n the derved annual constant rce estmates. 5 Homogenous roducts are dentcal n hyscal and economc terms to other tems n that roduct grou and over tme. In contrast, when there are sgnfcant varatons among tems or over tme n the hyscal or economc characterstc of the roduct grou, each verson should be treated as a searate roduct (e.g., out-of-season frut and vegetables such as old otatoes may be regarded as dfferent roducts than n-season frut and vegetables such as new otatoes). 48

3 Aggregatng Prce and Volume Measures Over Tme Examle 9.. Weghted and Unweghted Annual Averages of Prces (or Prce Indces) When Sales and Prce Patterns Through the Year are Uneven Constant Prce Value Current Unt Value At Unweghted At Weghted Prce Unweghted Weghted Average Average 999 Quantty Prce Value Average Prce Average Prce 999 Prces Prces () (2) (3) (4) (5) (3)/() (6) (4) () (7) (5) () ,5 7,5 6, ,5 2,5 2, , 5 45, 9, , 9, 8, , ,6 4 48, 9, % Change from 999 to 2.% 6.7% 2.% 6.7%.%.% Drect Deflaton of Annual Current Prce Data 2 at 999 rces 96/(4/5) 96/.8 2, % change from 999 (2/9-) 33.3% Ths examle hghlghts the case of an unweghted annual average of rces (or rce ndces) beng msleadng when sales and rce atterns through the year are uneven for a sngle homogenous roduct.the roducts sold n the dfferent uarters are assumed to be dentcal n all economc asects. In the examle, the annual uanttes and the uarterly rces n uarters wth nonzero sales are the same n both years, but the attern of sales shfts toward the second uarter n 998.As a result, the total annual current rce value ncreases by 6.7 ercent. If the annual deflator s based on a smle average of uarterly rces then the deflator aears to have droed by 2 ercent.as a result, the annual constant rce estmates wll wrongly show an ncrease n volume of 33.3 ercent. Consstent wth the uantty data, the annual sum of the uarterly constant rce estmates for 999 and 2, derved by valung the uanttes usng ther uantty-weghted average 999 rce, shows no ncrease n volumes (column 7).The change n annual current rce value shows u as an ncrease n the mlct annual deflator, whch would be mlctly weghted by each uarter s roorton of annual sales at constant rces. Prce ndces tycally use unweghted averages as the rce base, whch corresonds to valung the uanttes usng ther unweghted average rce.as shown n column 6, ths results n an annual sum of the uarterly constant rce estmates n the base year (999) that dffers from the current rce data, whch t should not. Ths dfference, however, can easly be removed by a multlcatve adjustment of the comlete constant rce tme seres, leavng the erod-toerod rate of change unchanged.the adjustment factor s the rato between the annual current rce data and sum of the uarterly constant rce data n the base year (9/) Conseuently, to obtan correct volume measures for aggregated erods of tme, deflators should take nto account varatons n uanttes as well as rces wthn the erod. For examle, annual deflators could be derved mlctly from annual volume measures derved from the sum of uarterly volume estmates obtaned usng the followng two-ste rocedure: (a) Benchmark the uarterly current rce data/ ndcator(s) to the corresondng annual current rce data. (b) Construct uarterly constant rce data by deflatng the benchmarked uarterly current rce data. Euvalently, the annual volume measure could be obtaned by deflatng usng an annual deflator that weghts the uarterly rce ndces by the constant rce values of that tem for each uarter. Ether way of calculaton acheves annual deflators that are uantty-weghted average annual rce measures A more dffcult case occurs when the annual estmates are based on more detaled rce and value nformaton than s avalable uarterly. In those cases, f seasonal volatlty s sgnfcant, t would be ossble to aroxmate the correct rocedure usng weghts derved from more aggregated, but closely related, uarterly data. 9.. The ssue of rce and uantty varatons also aly wthn uarters. Accordngly, when monthly data are avalable, uarterly data wll better take nto 6 The corresondng formulas are rovded n Annex

4 IX PRICE AND VOLUME MEASURES: SPECIFIC QNA-ANA ISSUES account varatons wthn the erod f they are bult u from the monthly data. 9.. In many cases, varaton n rces and uanttes wthn years and uarters wll be so nsgnfcant that t wll not substantally affect the estmates. Prmary roducts and hgh-nflaton countres are cases where the varaton can be artcularly sgnfcant. Of course, there are many cases n whch there are no data to measure varatons wthn the erod A related roblem that can be observed n uarterly data s the annual sum of the uarterly constant rce estmates n the base year dfferng from the annual sum of the current rce data, whch should not be the case. Ths dfference can be caused by the use of unweghted annual average rces as the rce base when constructng monthly and uarterly rce ndces. As shown n Annex 9., deflatng uarterly data wth deflators constructed wth unweghted average rces as the rce base corresonds to valung the uanttes usng ther unweghted annual average rce rather than ther weghted annual average rce. Ths dfference n the base year between the annual sum of the uarterly constant rce estmates and the annual sum of the current rce data can easly be removed by a multlcatve adjustment of the comlete constant rce tme seres, leavng the erod-to-erod rate of change unchanged. The adjustment factor s the rato between the annual current rce data and the sum of the ntal uarterly constant rce data based on the unweghted annual average rces n the base year, whch, for a sngle roduct, s dentcal to the rato of the weghted and unweghted average rce Two dfferent concets and measures of annual change n rces are llustrated n Examle 9., whch both are vald measures of economc nterest. The frst showng a declne n rces of 2 ercent based on unweghted annual average rces corresonds to a measure of the average change n rces. The second showng an ncrease n rces of 6.7 ercent based on weghted annual average rces corresonds to change n average rces. As shown n Examle 9., only the latter fts n a value/volume/rce measurement framework for tme erods, as reured by the natonal accounts, n contrast to the measurement framework for onts n tme addressed n conventonal ndex number theory. In Examle 9., the annual value change s 6.7 ercent, and the correct annual volume change s an undsutable. ercent, because the annual sum of the uanttes s unchanged and the uanttes refer to a sngle homogenous roduct An aarent dffculty s that the changes shown by the weghted annual average rce measure fal the fundamental ndex number axom that the measures should reflect only changes n rces and not changes n uanttes. Thus, the weghted annual average rce measure aears to be nvald as a measure of rce change. The 6.7 ncrease n average rces from 997 to 998 results from changes n the uanttes transacted at each rce and not from ncreases n the rces, and therefore does not satsfy basc ndex number tests such as the dentty and roortonalty tests. For that reason, t can be argued that Examle 9. shows that, n rncle, t s not ossble to factor changes n values for tme erods nto measures of rce and uantty changes that are each accetable as ndex numbers n ther own rght. The basc ndex number tests and conventonal ndex number theory, however, are concerned wth rce and uantty comarsons between searate ars of onts n tme rather than wth rce and uantty comarsons between tme erods and, conseuently, not wth measures of the change n average rces from one erod to another. To measure the change n average rces, for a sngle homogenous roduct, each erod s average rce should be defned as the total value dvded by the corresondng uanttes wthn that erod; that s, they should be unt values. From Examle 9., t s clear that annual average rces for natonal accountng uroses cannot be realstcally defned wthout reference to the corresondng uanttes and therefore should be calculated usng a weghted average wth uarterly/subannual uanttes as weghts. C. Choce of Prce Weghts for QNA Volume Measures. Laseyres-Tye Volume Measures 9.5. The tme-seres reurements and the QNA- ANA consstency reurement mly that the uanttyweghted average rces of a whole year should be used as rce weghts for ANA and QNA Laseyrestye volume measures. 7 Use of the rces of one 7 The term Laseyres-tye s used to cover the tradtonal constant rce measures, fxed-base Laseyres volume ndces, and chanlnked Laseyres volume ndces. 5

5 Choce of Prce Weghts for QNA Volume Measures artcular uarter, the rces of the corresondng uarter of the revous year, the rces of the corresondng uarter of a fxed base year, or the rces of the revous uarter are not arorate for tme seres of Laseyres-tye volume measures n the natonal accounts for the followng reasons: Consstency between drectly derved ANA and QNA Laseyres-tye volume measures reures that the same rce weghts are used n the ANA and the QNA, and that the same rce weghts are used for all uarters of the year. The rces of one artcular uarter are not sutable as rce weghts for volume measures n the ANA, and thus n the QNA, because of seasonal fluctuatons and other short-term volatltes n relatve rces. Use of weghted annual average rces reduces these effects. Therefore, weghted annual average rces are more reresentatve for the other uarters of the year as well as for the year as a whole. The rces of the corresondng uarter of the revous year or the corresondng uarter of a fxed base year are not sutable as rce weghts for volume measures n the QNA because the derved volume measures only allow the current uarter to be comared wth the same uarter of the revous year or years. Seres of year-to-year changes do not consttute tme seres that allow dfferent erods to be comared and cannot be lnked together to form such tme seres. In artcular, because they nvolve usng dfferent rces for each uarter of the year, they do not allow dfferent uarters wthn the same year to be comared. For the same reason, they do not allow the uarters wthn the same year to be aggregated and comared wth ther corresondng drect annual estmates. Furthermore, as shown n Annex., changes from the same erod n the revous year can ntroduce sgnfcant lags n dentfyng the current trend n economc actvty. The rces of the revous uarter are not sutable as rce weghts for Laseyres-tye volume measures for two reasons: (a) The use of dfferent rce weghts for each uarter of the year does not allow the uarters wthn the same year to be aggregated and comared wth ther corresondng drect annual estmates. (b) If the uarter-to-uarter changes are lnked together to form a tme seres, short-term volatlty n relatve rces may cause the uarterly chan-lnked measures to show substantal drft comared to corresondng drect measures. Ths s llustrated n Examle Quarterly Laseyres-tye volume measures wth two dfferent base-erod 8 rce weghts may be used: (a) The annual average of a fxed-base year, resultng n the tradtonal constant rce measures, whch s euvalent to a fxed-based Laseyres volume ndex. (b) The annual average of the revous year, resultng n the annually chan-lnked uarterly Laseyres volume ndex The tradtonal volume measures at the constant rce of a fxed base year, the fxed-based uarterly Laseyres volume ndex, and the short-term lnk n the annually chan-lnked uarterly Laseyres volume ndex can be exressed n mathematcal terms as the followng: At the constant average rces of a fxed base year: The fxed-based uarterly Laseyres: LQ ( y), (9..a) (9..b) Short-term lnk n the annually chan-lnked uarterly Laseyres: y LQ ( y ) (, y) where CP,y LQ (,y) LQ (y ) (,y) CP y,, y,,,,, y,,, y,, y, y, (9..c) s the total value n uarter of year y measured at the annual average rces of year. reresents a Laseyres volume ndex measurng the volume change from the average of year to uarter n year y wth average of year as base and reference erod; 9 reresents a Laseyres volume ndex measurng the volume change from the average of year y to uarter n year 8 The term base erod s defned n aragrah 9.22 as meanng () the base of the rce or uantty ratos beng weghted together (e.g., erod s the base for the uantty rato), and (2) the rcng year (the base year) for constant rce data. 9 The term Reference erod s defned n aragrah 9.22 as meanng the erod for whch the ndex seres s exressed as eual to. 5

6 IX PRICE AND VOLUME MEASURES: SPECIFIC QNA-ANA ISSUES y wth the average of year y as base and reference erod;,,y s the rce of tem n uarter of year y;,y s the uantty-weghted arthmetc average of the rce of tem n the uarters of year y ;, s the uantty-weghted arthmetc average of the rce of tem n the uarters of year,,y s the uantty of tem n uarter of year y;,y s the smle arthmetc average of the uanttes of tem n the uarters of y ; and, s the smle arthmetc average of the uanttes of tem n the uarters of year. 2. Fsher-Tye Volume Indces 9.8. The Fsher volume ndex, beng the geometrc average of a Laseyres and a Paasche volume ndex, uses rce weghts from two erods the base erod and the current erod. Quarterly Fsher ndces wth three dfferent base-erod weghts may be used: (a) The annual average of a fxed-base year, resultng n the fxed-based uarterly Fsher ndex. (b) The annual average of the revous year, resultng n the annually chan-lnked uarterly Fsher ndex. (c) The average of the revous uarter, resultng n the uarterly chan-lnked uarterly Fsher ndex The fxed-based uarterly Fsher volume ndex and the short-term lnks n the annually and uarterly chan-lnked uarterly Fsher volume ndex can be exressed n mathematcal terms as the followng: Fxed-based uarterly Fsher: FQ LQ PQ ( y) ( y),, ( y, ),,,, y,,,,,, ; y,, y,,,, y,,, (9.2.a) Short-term lnk n the annually chan-lnked uarterly Fsher: FQ LQ PQ ( y ) (, y) ( y ) (, y) ( y ) (, y) (9.2.b) Short-term lnk n the uarterly chan-lnked uarterly Fsher: FQ LQ PQ ( t ) ( t ) ( t) ( t ) ( t) where t FQ A (,y) LQ A (,y) PQ A (,y) P P t, t, t, t, t, t, t, t, (9.2.c) s a generc symbol for tme, whch s more convenent to use for erod-to erod measures than the uarter n year y notaton used for most formulas n ths chater; reresents a Fsher volume ndex measurng the volume change from erod A to uarter n year y wth erod A as base and reference erod; reresents a Laseyres volume ndex measurng the volume change from erod A to uarter n year y wth erod A as base and reference erod; reresents a Paasche volume ndex measurng the volume change from erod A to uarter n year y wth erod A as base and reference erod; and,a s the rce of tem n erod A. Perod A s eual to the average of year for the fxed-based Fsher, to the average of the revous year for the annually chan-lnked Fsher, and to the revous uarter for the uarterly chan-lnked Fsher For the same reasons as for Laseyres-tye volume measures, the followng alternatve erods are not sutable as base erods for tme seres of Fsher-tye volume ndces: One artcular fxed uarter. The corresondng uarter of the revous year. The corresondng uarter of a fxed base year. y, y,, y,, y,, y, y, y,, y, 52

7 Choce of Prce Weghts for QNA Volume Measures D. Chan-Lnkng n the QNA. General 9.2. The 993 SNA recommends movng away from the tradtonal fxed-base year constant rce estmates to chan-lnked volume measures. Constant rce estmates use the average rces of a artcular erod, the base erod, to weght together the corresondng uanttes. Constant rce data have the advantage for the users of the comonent seres beng addtve, unlke alternatve volume measures. The attern of relatve rces n the base year, however, s less reresentatve of the economc condtons for erods farther away from the base year. Therefore, from tme to tme t s necessary to udate the base erod to adot weghts that better reflect the current condtons (.e., wth resect to roducton technology and user references). Dfferent base erods, and thus dfferent sets of rce weghts, gve dfferent ersectves. When the base erod s changed, data for the dstant ast should not be recalculated (rebased). Instead, to form a consstent tme seres, data on the old base should be lnked to data on the new base. Change of base erod and chan-lnkng can be done wth dfferent freuences; every years, every 5 years, every year, or every uarter/month. The 993 SNA recommends changng the base erod, and thus conductng the chan-lnkng, annually The concets of base, weght, and reference erod should be clearly dstngushed. Index number termnology s not well establshed nternatonally, whch can lead to confuson. In artcular, the term base erod s sometmes used for dfferent concets. Smlarly, the terms base erod, weght erod, and reference erod are sometmes used nterchangeably. In ths manual, followng 993 SNA and the current domnant natonal accounts ractce, the followng termnology s used: Base erod for () the base of the rce or uantty ratos beng weghted together (e.g., erod s the base for the uantty rato,t /, ), and (2) the rcng year (the base year) for the constant rce data. The erod length should be a year, as recommended n the revous secton. Ths should be done for each seres, aggregates as well as subcomonents of the aggregates, ndeendently of any aggregaton or accountng relatonsh between the seres. As a conseuence, the chan-lnked comonents wll not aggregate to the corresondng aggregates. No attemts should be made to remove ths chan dscreancy, because any such attemt mles dstortng the movements n one or several of the seres. Weght erod for the erod(s) from whch the weghts are taken. The weght erod s eual to the base erod for a fxed-base Laseyres ndex and to the current erod for a fxed-base Paasche ndex. Symmetrc fxed-base ndex formulas lke Fsher and Tornust have two weght erods the base and the current erod. Reference erod for the erod for whch the ndex seres s exressed as eual to. The reference erod can be changed by smly dvdng the ndex seres wth ts level n any erod chosen as the new reference erod Chan-lnkng means constructng long-run rce or volume measures by cumulatng movements n short-term ndces wth dfferent base erods. For examle, a erod-to-erod chan-lnked ndex measurng the changes from erod to t (.e., CI t ) can be constructed by multlyng a seres of short-term ndces measurng the change from one erod to the next as follows: CI I I I I... I t ( t ) t t I( τ ) τ τ (9.3) where I (t ) τ reresents a rce or volume ndex measurng the change from erod t to t, wth erod t as base and reference erod The corresondng run, or tme seres, of chan-lnked ndex numbers where the lnks are chaned together so as to exress the full tme seres on a fxed reference erod s gven by CI CI I CI 2 I I 2 CI I I I... t CI t I( τ ) τ τ (9.4.a) Chan-lnked ndces do not have a artcular base or weght erod. Each lnk (I (t ) t ) of the chan-lnked ndex n euaton (9.4.a) has a base erod and one or two weght erods, and the base and weght erod(s) are changng from lnk to lnk. By the same token, the full run of ndex numbers n 53

8 IX PRICE AND VOLUME MEASURES: SPECIFIC QNA-ANA ISSUES euaton (9.4.a) derved by channg each lnk together does not have a artcular base erod t has a fxed reference erod The reference erod can be chosen freely wthout alterng the rates of change n the seres. For the chan-lnked ndex tme seres n euaton (9.4.a), erod s referred to as the ndex s reference erod and s conventonally exressed as eual to. The reference erod can be changed smly by dvdng the ndex seres wth ts level n any erod chosen as a new reference erod. For nstance, the reference erod for the run of ndex numbers n euaton (9.4.a) can be changed from erod to erod 2 by dvdng all elements of the run by the constant CI 2 as follows: CI CI CI I I CI2 CI CI 2 I 2 CI2 2 CI 2 CI 2 CI2 3 CI 3 CI 2 I 2... t CI2 t CI t CI 2 I τ τ (9.4.b) The chan-lnked ndex seres n euaton (9.3) and euatons (9.4.a) and (9.4.b) wll consttute a erod-to-erod chan-lnked Laseyres volume ndex seres f, for each lnk, the short-term ndces (I (t ) t ) are constructed as Laseyres volume ndces wth the revous erod as base and reference erod. That s, f I ( ) τ t, LQ w t, t t,, t V ( t ) t ( t ) t t, t, t,, t, t (9.5.) where LQ (t ) t reresents a Laseyres volume ndex measurng the volume change from erod t to t, wth erod t as base and reference erod;,t s the rce of tem n erod t (the rce weghts );,t s the uantty of tem n erod t; w,t s the base erod share weght, that s, the tem s share n the total value of erod t ; and V t s the total value at current rces n erod t Smlarly, the chan-lnked ndex seres n euaton (9.3) and euatons (9.4.a) and (9.4.b) wll consttute a erod-to-erod chan-lnked Fsher volume ndex seres f, for each lnk, the short-term ndces (I (t ) t ) are constructed as Fsher volume ndces wth the revous erod as base and reference erod as n euaton (9.2.c) Any two ndex seres wth dfferent base and reference erods can be lnked to measure the change from the frst to the last year 2 as follows: CI t I (t h) I (t h) t (9.6) That s, each lnk may cover any number of erods For nstance, f n euaton (9.6) t and h 5, the resultng lnked ndex (CI ) consttutes a 5-year chan-lnked annual ndex measurng the change from year to year. Examle 9.2 rovdes an llustraton of the basc chan-lnkng technue for annual data wth t 5 and h Growth rates and ndex numbers comuted for seres that contan negatves or zeroes such as changes n nventores and cro harvest data generally are msleadng and meanngless. For nstance, consder a seres for changes n nventores at constant rces that s n erod one and +2 n erod two. The corresondng growth rate between these two erods s 3 ercent ( ((2/ ) ) ), whch obvously s both msleadng and meanngless. Smlarly, for a seres that s n erod one and n erod two, the corresondng growth rate from erod one to two would be 9 ercent. Conseuently, for such seres, only measures of contrbuton to ercentage change n the aggregates they belong to can be made (see Secton D.7. for a dscusson of measure of contrbuton to ercentage change n ndex numbers). 2. Freuency of Chan-Lnkng n the QNA The 993 SNA recommends that chanlnkng should not be done more freuently than annually. Ths s manly because short-term volatlty n relatve rces (e.g., caused by samlng errors and seasonal effects) can cause volume measures that are chan-lnked more freuently than annually to show substantal drft artcularly so 2 As long as they have one erod n common, that s, there s at least one overlang erod. For nstance, n euaton (9.6) wth t and h 5, year 5 reresents the overla. Smlarly, n Examle 9.2, year reresents the overla. 54

9 Chan-Lnkng n the QNA Examle 9.2. Basc Chan-Lnkng of Annual Data The 993 SNA Examle The examle s an elaborated verson of the llustraton rovded n the 993 SNA. (993 SNA Table 6., ages ) Basc Data Year Year Year 5 v v 5 5 v 5 Item A Item B Total Constant rce Data Base Year Base Year Year Year Year 5 Year Year Year Item A Item B Total Laseyres Volume Indces for the Total Fxed-Based Year Year Year 5 Year as base and reference Perod-to-erod rate of change 87.% 5.5% Year as base and reference Perod-to-erod rate of change 74.4% 2.4% Re-referenced to year (year as base) Chan-Lnked Index Year Perod-to-erod rate of change 87.% 2.4% Year / Perod-to-erod rate of change 87.% 2.4% The Laseyres fxed-base volume ndex for the total wth year as base and reference erod was derved as 62/62, 6/62 87., 34/ Smlarly, the Laseyres fxed-base volume ndex for the total year wth as base and reference erod was derved as 25/ , 28/28, 245/ And the Laseyres fxed-base volume ndex for the total wth year as base and year as reference erod was derved as 57.3/57.3, / , 2.4/ for nonsuerlatve ndex formulas lke Laseyres and Paasche as llustrated n Examle 9.3. Smlarly, short-term volatlty n relatve uanttes can cause rce measures that are chan-lnked more freuently than annually to show substantal drft. The urose of chan-lnkng s to take nto account long-term trends n changes n relatve rces, not temorary short-term varatons Suerlatve ndex formulas, such as the Fsher ndex formula, are more robust aganst the drft roblem than the other ndex formulas as llustrated n Examle 9.3. For ths reason, a uarterly chan-lnked Fsher ndex may be a feasble alternatve to annually chan-lnked Fsher or Laseyres ndces for uarterly data that show lttle or no shortterm volatlty. The uarterly chan-lnked Fsher ndex does not aggregate exactly to the corresondng drect annual Fsher ndex. 3 For chan-lnked Fsher ndces, consstency between QNA and ANA rce and volume measures can only be acheved by dervng the ANA measures from the uarterly measures or by forcng consstency on the data wth the hel of benchmarkng technues. There s no reason to beleve that for nonvolatle seres the average of an annually chan-lnked Fsher wll be closer to a drect annual Fsher ndex than the average of a uarterly chan-lnked Fsher. 3 Nether does the annually-lnked, nor the fxed-based, Fsher ndex. 55

10 IX PRICE AND VOLUME MEASURES: SPECIFIC QNA-ANA ISSUES Examle 9.3. Freuency of Chan-Lnkng and the Problem of Drft n the Case of Prce and Quantty Oscllaton Observaton/Quarter Quarter Quarter 2 Quarter 3 Quarter 4 Prce tem A (A) Prce tem B (B) Quanttes tem A (A,t ) Quanttes tem B (B,t ) Total value (Vt ) Volume Indces Fxed-based Laseyres (-based) Fxed-based Paasche (-based) Fxed-based Fsher (-based) Quarterly chan-lnked Laseyres Quarterly chan-lnked Paasche Quarterly chan-lnked Fsher Fxed-Based Laseyres Index: I,, t,, t t,, V Fxed-Based Paasche Index: I t, t, Vt t t,, t,, Quarterly Chan-Lnked Laseyres Index: t t CL t I, τ, τ, τ ( τ ) τ τ, τ, τ Quarterly Chan-Lnked Paasche Index: t t CL t I, τ, τ, τ ( τ ) τ τ, τ, τ I t 2 [ ]/4] 7.5 I t 2 [ ]/4] 67.5 I t 4 [ ]/4]. I t 2 [4/( ] 2.6 I t 3 [3/( ] 93.8 I t 4 [4/( ]. I t 3 I t 2 [ ]/4] 8.6 I t 4 I t 3 [ ]/4] 86. I t 3 I t 2 [3/( ] 2.6 I t 2 I t 3 [4/( ] 5.9 In ths examle, the rces and uanttes n uarter 4 are the same as those n uarter, that s, the rces and uanttes oscllate rather than move as a trend.the fxed-base ndces corresondngly show dentcal values for and 4, but the chan-lnked ndces show comletely dfferent values.ths roblem can also occur n annual data f rces and uanttes oscllate and may make annual channg narorate n some cases. It s more lkely to occur n data for shorter erods, however, because seasonal and rregular effects cause those data to be more volatle. Furthermore, observe that the dfferences between the and 4 data for the uarterly chan-lnked Laseyres and the uarterly chanlnked Paasche ndces are n ooste drectons; and, corresondngly, that the uarterly chan-lnked Fsher ndex drfts less.ths s a unversal result. The examle s based on Szultc (983) For Laseyres-tye volume measures, consstency between QNA and ANA rovdes an addtonal reason for not chan-lnkng more freuently than annually. Consstency between uarterly data and corresondng drect annual ndces reures that the same rce weghts are used n the ANA and the QNA, and conseuently that the QNA should follow the same change of base year/chan-lnkng ractce as n the ANA. Under those crcumstances, the annual overla lnkng technue resented n the next secton wll ensure that the uarterly data aggregate exactly to the corresondng drect ndex. Moreover, under the same crcumstances, any dfference between the average of the uarterly data and the drect annual ndex caused by the referred one-uarter overla technue wll be mnmzed Thus, n the QNA, chan-lnked Laseyrestye volume measures should be derved by comlng uarterly estmates at the average rces of the revous year. These uarterly volume measures for each year should then be lnked to form long, consstent tme seres the result consttutes an annually chan-lnked uarterly Laseyres ndex. Alternatve 56

11 Chan-Lnkng n the QNA lnkng technues for such seres are dscussed n the next secton. 3. Choce of Index Number Formulas for Annually Chan-Lnked QNA Data The 993 SNA recommends comlng annually chan-lnked rce and volume measures, referably usng suerlatve ndex number formulas such as the Fsher and Tornust formulas. The ratonale for ths recommendaton s that ndex number theory shows that annually chan-lnked Fsher and Tornust ndces wll most closely aroxmate the theoretcally deal ndex. Fsher and Tornust ndces wll, n ractce, yeld almost the same results, and Fsher, beng the geometrc average of a Laseyres and a Paasche ndex, wll be wthn the uer and lower bounds rovded by those two ndex formulas. Most countres 4 that have mlemented chan-lnkng n ther natonal accounts, however, have adoted the annually chan-lnked Laseyres formula for volume measures wth the corresondng annually chan-lnked Paasche formula for rce measures, 5 and the Euroean Unon s statstcal offce (Eurostat) s reurng member states to rovde annually chan-lnked volume measures usng the Laseyres formula Annual chan-lnkng of uarterly data mles that each lnk n the chan s constructed usng the chosen ndex number formula wth the average of the revous year (y ) as base and reference erod. The resultng short-term uarterly ndces must subseuently be lnked to form long, consstent tme seres exressed on a fxed reference erod. Alternatve annual lnkng technues for such seres wll be dscussed n Secton D.3. Whle the dscusson n Secton D.3 focuses on Laseyres ndces, the technues llustrated and 4 The use of chan-lnked measures for offcal natonal accounts data was oneered by the Netherlands (985) and Norway (99). Subseuently, a large number of countres have adoted, or are n the rocess of adotng, chan-lnkng for ther offcal measures. Currently, only the Unted States has oted for a chan-lnked Fsher ndex formula nstead of the chan-lnked Laseyres formula. The Unted States adoted n 996 an annually chan-lnked uarterly Fsher-lke formula usng annual weghts n both the Laseyres and the Paasche art of the ndex but changed to a standard uarterly chan-lnked Fsher ndex n Laseyres volume measures reure that the corresondng rce measures are based on the Paasche formula so that the roduct of the volume and rce ndces s eual to the corresondng value ndex. 6 Euroean Commsson Decson of November 3, 998, clarfyng the Euroean System of Accounts 995 rncles for rce and volume measures, and Eurostat (999) aragrah the ssues dscussed are alcable to all annually chan-lnked ndex formulas. The Laseyres, Paasche, and Fsher annually chan-lnked uarterly volume ndex formulas for each short-term lnk n the chan are gven as Short-term lnk n annually chan-lnked Laseyres: y LQ ( y ) (, y) y y,, w (9.7.a) Short-term lnk n annually chan-lnked Paasche: y PQ ( y ) (, y) (9.7.b) Short-term lnk n annually chan-lnked Fsher: FQ LQ PQ ( y ) (, y) ( y ) (, y) ( y ) (, y) where w y, y y,, y, y,, y,,, y, y,,, y,, y,, y, y, y,, y, y,, y,, y,, y,, (9.7.c) s the base erod share weght, that s, the tem s share n the total value n year y ; and,,y s the rce of tem n uarter of year y Countres have oted for the annually chanlnked Laseyres formula nstead of the annually chan-lnked Fsher formula for volume measures manly for the followng reasons: Exerence and theoretcal studes ndcate that annual chan-lnkng tends to reduce the ndex number sread to the degree that the exact choce of ndex number formula assumes less sgnfcance (see, for examle, 993 SNA, aragrah 6.5). The annually chan-lnked uarterly Fsher ndex does not aggregate to the corresondng drect y,, y,, y, y, y,, y, 57

12 IX PRICE AND VOLUME MEASURES: SPECIFIC QNA-ANA ISSUES annual ndex; 7 the annually chan-lnked Laseyres ndex lnked, usng the annually overla technue resented n Examle 9.4.a, does. 8 Chan volume measures n monetary terms 9 based on the annually chan-lnked Laseyres formula wll be addtve n the reference year and the subseuent year, 2 whle volume measures based on the Fsher ndex wll not. The Laseyres formula s smler to work wth and to exlan to users than the Fsher ndex. For nstance, tme seres of annually chan-lnked Laseyres ndces easly can be converted nto seres of data valued at the constant average rces of the revous year that are addtve f corresondng current rce data are made avalable. Ths feature makes t easy for users to construct ther own aggregates from ublshed data. The formulas for comutng contrbuton to ercentage change are easer for data based on the chan-lnked Laseyres formula than for data based on the Fsher ndex. The Fsher formula s not consstent n aggregaton wthn each lnk; t s only aroxmately consstent n aggregaton. The Laseyres formula, n contrast, s addtve wthn each lnk. Ths makes t easer to combne chan-lnkng wth comlaton analytcal tools lke suly and use (SU) tables and nut-outut tables that reure addtvty of comonents Technues for Annual Chan-Lnkng of Quarterly Data Two alternatve technues for annual chanlnkng of uarterly data are usually aled: annual overlas and one-uarter overlas. In addton to these two conventonal chan-lnkng technues, a thrd technue sometmes s used based on changes 7 Nether does the uarterly-chan lnked, nor the fxed-based, uarterly Fsher ndex. 8 However, ths may not be a decsve argument for two reasons. Frst, smulatons ndcate that, n ractce, the dfference between a drect annual Fsher and the average of a uarterly Fsher may often not be sgnfcant and may easly be removed usng benchmarkng technues. Second, the referred uarterly overla technue resented n Secton D.3., even when used for Laseyres ndces, also ntroduces dfferences between drect annual ndces and the average of uarterly ndces. 9 See Secton D.7. and artcularly aragrah 9.48 for a dscusson of chan volume measures n monetary terms. 2 See Examle 9.5.a for an llustraton of ths and Secton D.5. for a dscusson of the nonaddtvty roerty of most ndex number formulas besdes the fxed-based Laseyres formula. 2 The frst two countres to adot chan-lnkng for ther offcal natonal accounts rce and volume measures both dd t wthn an SU comlaton framework. from the same erod n the revous year (the overthe-year technue ). Whle, n many cases, all three technues gve smlar results, n stuatons wth strong changes n relatve uanttes and relatve rces, the over-the-year technue can result n dstorted seasonal atterns n the lnked seres. Whle standard rce statstcs comlaton exclusvely uses the one-uarter overla technue, the annual overla technue may be more ractcal for Laseyrestye volume measures n the natonal accounts because t results n data that aggregate exactly to the corresondng drect annual ndex. In contrast, the one-uarter overla technue and the over-the-year technue do not result n data that aggregate exactly to the corresondng drect annual ndex. The oneuarter overla rovdes the smoothest transton between each lnk, however, n contrast to the annual overla technue that may ntroduce a ste between each lnk. Examles 9.4.a, 9.4.b, 9.4.c, and Chart 9. rovde an llustraton of these three chan-lnkng technues. (A formal resentaton of the two frst methods s gven n Annex 9.2.) 9.4. The technue of usng annual overlas mles comlng estmates for each uarter at the weghted annual average rces of the revous year, wth subseuent lnkng usng the corresondng annual data to rovde lnkng factors to scale the uarterly data uward or downward. The technue of one-uarter overlas reures comlng estmates for the overla uarter at the weghted annual average rces of the current year n addton to estmates at the average rces of the revous year. The rato between the estmates for the lnkng uarter at the average rces of the current year and at the average rces of the revous year then rovdes the lnkng factor to scale the uarterly data u or down. The over-the-year technue reures comlng estmates for each uarter at the weghted annual average rces of the current year n addton to estmates at the average rces of the revous year. The year-on-year changes n these constant rce data are then used to extraolate the uarterly constant rce data of the chosen reference erod To conclude, there are no establshed standards wth resect to technues for annually chan-lnkng of QNA data, but chan-lnkng usng the one-uarter overla technue, combned wth benchmarkng to remove any resultng dscreances between the uarterly and annual data, gves the best result. In many crcumstances, however, the annual overla technue may gve smlar results. The over-the-year technue should be avoded. 58

13 Chan-Lnkng n the QNA Examle 9.4.a. Quarterly Data and Annual Chan-Lnkng Annual Overla Laseyres Volume Index Annual sums and averages n bold. Chan- Lnked At Constant Prces of: Index Quant- Quant- Total at Index Index Index - tes tes Prce Prce current Rate of Basc data A B A B rces Level Level Level Level Change ,73. 3, % % % % ,594. 3, , % % % % ,779. 3, , % % % % , , % Indeendently chan-lnked annuals 997 3, , , , , , Ste : Comle estmates for each uarter at the annual average rces of the revous year; the annual data beng the sum of the four uarters. e.g.: Ste 2: Convert the constant rce estmates for each uarter nto a volume ndex wth the average of last year. e.g.: 998 [87./(373./4)] [85.7/(373./4)] / Ste 3: Lnk the uarterly volume ndces wth shftng base and reference year usng the annual ndces as lnkng factors (usng 997 as the reference erod for the chan-lnked ndex). e.g.: Observe that the unweghted annual average of the derved chan-lnked uarterly ndex seres s eual to the ndeendently derved chan-lnked annual data. e.g.: 2 [ ]/4.5 Fnally, observe that the change from, e.g., to 2, n the chan-lnked seres based on annual overla dffers from the corresondng change n the chan-lnked ndex based on a one-uarter overla n the next examle. e.g.: 2/ based on annual overla.3% 999/ based on one uarter overla (and 999 rces).5% Ths s the ste n the seres ntroduced by the annual overla technue. 59

14 IX PRICE AND VOLUME MEASURES: SPECIFIC QNA-ANA ISSUES Examle 9.4.b. Quarterly Data and Annual Chan-Lnkng One-uarter overla Annual sums and averages n bold. At Constant Prces of: Chan-lnked ndex 997 Total at Index Index Index - current Rate of Basc data 2 2 rces Level LeveI Level Level Change ,73. 3, % % % ,594. 3, , % % % % ,779. 3,7. 3, % % % % , , Ste : Comle estmates for each uarter at the annual average rces of the revous year; the annual data beng the sum of the four uarters. Ste 2: Comle estmates for the fourth uarter of each year at the annual average rces of the same year. e.g.: Ste 3: Convert the constant rce estmates for the uarters of the frst year after the chosen reference year (997) nto a volume ndex wth the average of the reference year e.g.: 998 [87.4/(373./4)] [85.7/(373./4)] 7.24 Ste 4: Convert the constant rce estmates for each of the other uarters nto a volume ndex wth the fourth uarter of last year e.g.: 999 [96.6/97.55] [936.45/97.55] 3.5 Ste 5: Lnk together the uarterly volume ndces wth shftng base usng the fourth uarter of each year as lnk. e.g.: The resultng lnked seres s referenced to average 997. Fnally, observe that the unweghted annual average of the derved chan-lnked uarterly ndex seres dffers from the ndeendently derved chan-lnked annuals n examle 9.4.a. e.g.: 2 [ ]/ Chan-Lnked Measures and Nonaddtvty In contrast to constant rce data, chan-lnked volume measures are nonaddtve. To reserve the correct volume changes, related seres should be lnked ndeendently of any aggregaton or accountng relatonshs that exst between them; as a result, addtvty s lost. Addtvty s a secfc verson of the consstency n aggregaton roerty for ndex numbers. Consstency n aggregaton means that an aggregate can be constructed both drectly by aggregatng the detaled tems and ndrectly by aggregatng subaggregates usng the same aggregaton formula. Addtvty, n artcular, mles that at each level of aggregaton the volume ndex for an aggregate takes the form of a weghted arthmetc average of the volume ndces for ts comonents wth the base-erod values as weghts (993 SNA, aragrah 6.55). That s the same as reurng that the aggregate be eual to the sum of ts comonents when the current rce value of the aggregate and the comonents n some reference erod are multled, or extraolated, wth the aggregate ndex and the comonent ndces, resectvely, resultng n chan volume measures exressed n monetary terms. It follows that, at the most detaled level, addtvty s the same as reurng that the value obtaned by extraolatng the aggregate s eual to the sum of the comonents valued at the reference erod s rces. Thus, the addtvty reurement effectvely defnes the fxed-base Laseyres ndex and standard constant rce data. 6

15 Chan-Lnkng n the QNA Examle 9.4.c. Quarterly Data and Annual Chan-Lnkng The Over-the-Year Technue Laseyres Volume Index () Par of years at the same rces. () Chan-lnkng usng changes from the same uarter n the revous year. Annual sums and averages n bold. Chan-Lnked At Constant Prces of : Index Quant- Quant- Total at Index -tes tes Prce Prce Current Rate of Basc Data A B A B Prces Level Level Level Level Change ,73. 3, % % % % ,594. 3, , % % % % ,779. 3,7. 3, % % % % , , Ste : Ste 2: Ste 3: Ste 4: Ste 5: Comle estmates for each uarter at the annual average rces of the revous year. e.g.: Comle estmates for each uarter at the annual average rces of the same year. e.g.: Convert the constant rce estmates for each uarter of the frst year after the chosen reference year (997) nto a volume ndex wth the average of the revous year e.g.: 998 [87.4/(373./4)] [85.7/(373./4)] 7.24 For the other years, based on the constant rce estmates derved n stes and 2, calculate the volume change from the same uarter of the roceedng year as the followng: e.g.: 999/ / / / Lnk the uarterly volume ndces wth shftng base and reference year usng the changes from the same erod of the revous year as lnkng factors (extraolators). e.g.: Observe that the unweghted annual average of the derved chan-lnked uarterly ndex seres s only aroxmately eual to the ndeendently derved chanlnked annuals. e.g.: 2 [ ]/ Fnally, observe that the rate of change from 4 n one year to n the next year n the chan-lnked seres based on the over-the-year technue dffers substantally from the corresondng changes n chan-lnked ndex based on a one-uarter overla n the revous examle. e.g.: 999/ based on the over-the-year technue (6.23/7.24 ).9% 999/ based on one-uarter overla (and 998 rces): (8.3/7.24 ).% 2/ based on the over-the-year technue (7.67/. ) 3.% 2/ based on one-uarter overla (and 999 rces): (.6/. ).5% Observe also that the rate of change from 4 n one year to n the next year n the chan-lnked seres based on the over-the-year technue dffers substantally from the corresondng changes n the constant rce measures based on the average rces of the current year. That s, 999/ based on average 999 rces (953.8/936.5 ).5% These dfferences between the 4-to- rate of change n the chan-lnked seres based on the over-the-year technue and the corresondng rate of change based on drect measurements are the stes n the seres ntroduced by the technue. Notce also that, n ths examle, the break aears to ncrease over tme, that s, the breaks are cumulatve.the breaks wll be cumulatve f there s a trend-wse change n relatve rces and relatve uanttes, as n ths examle. 6