Beyond matched pairs and Griliches-type hedonic methods for controlling quality changes in CPI sub-indices

Transcription

1 Beyod mached pars ad Grlches-ype hedoc mehods or corollg qualy chages CPI sub-dces Mahemacal cosderaos ad emprcal eamples o he use o lear ad o-lear hedoc models wh me-depede qualy parameers Paper preseed a he Sh Meeg o he Ieraoal Worg Group o Prce Idces Caberra Ausrala 6 Aprl Tmo Kosmä Yrjö Vara Sascs Flad Uversy o Hels mo.osma@sa. yrjo.vara@hels.

2 . Iroduco 3 The majory o rece research leraure dealg wh qualy adjusme o prce dces s largely cocered o raher echcal ssues. I he realm o usg regresso models or qualy adjusme he requely occurrg opcs are he correc choce o prce deermg acors ucoal orms o he regresso equao ad oher aspecs o he esmao o a correc regresso model. As comes o sascal ageces complg prce dces he doma orm o hg s he radoal mached model - approach. The academc research o hedoc regresso models has so ar bee used he ocal de complao oly a very coservave maer. As a varey orgally chose he CPI sample dsappears rom he mare a ew varey s seleced sead. Hedoc regresso modellg he provdes a mehod o esmae a prevous perod s prce or he qualavely equvale produc. Ths s doe by esmag he prevous perod orecased prce or he ewly seleced produc. The calculao s perormed usg pre-esablshed regresso coeces or cera prce-deermg acors. The basc mechasm however s sll machg he pars. I our opo here are more aural ways o use hedoc mehods de complao where he qualy corol s o based o pre-speced mached pars o observaos see also Hyrö Kue Vara (998). We wll elaborae a raher geeral mahemacal ramewor o hedoc regresso models whch we hope wll mae he uure dscusso o he opc more srucured. The ramewor eplcaes he mahemacal relaoshps bewee varous ypes o hedoc qualy adjusme sraeges. The ormal cosderaos wll be llusraed wh emprcal eamples. The daa we use has bee gahered as a par o sadard CPI daa-colleco. The emprcal resuls preseed here are hus o oly eamples bu also gve dcao o he magude o bas caused by he use o approprae mehods whe aalysg rapdly chagg mares. The srucure o he paper s as ollows: I Chaper we wll mae some remars o currely used pracces o qualy adjusme: () qualy adjusme usg oly mached pars approach () overlap prces ad () he aemp o cure he shorcomgs o radoal approaches by usg regresso esmaes o pach he daa. I () a dsappeared mached par s subsued by a cosruced pached par. The remars we mae are based o he resuls obaed rom our es daa o compuers whe aempg o ollow he above meoed pracces. Tmo Kosmä wroe he rs dra or hs par. I Chaper 3 we epose our ormal ramewor sarg wh he mos geeral vara o hedoc models he case where he coeces o he model are allowed o chage me ad he model cludes secod order erms or ay oher o-lear erms. Lear ad me-vara models cludg he requely used Grlches-ype models are cosdered as specal cases o he geeral model. The relaoshp bewee he varous ypes o hedoc approaches wll be aalysed ad prove mahemacally. Emprcal eamples based o our es daa wll be provded or mos he model ypes we cosder. Yrjö Vara wroe he rs dra or hs par. Chaper 4 cocludes ad suggess opcs or urher research.. The Tradoal Approach. Characerscs o he PC mares - he es daa The daa used hs eperme cosss o 45 prce observaos. I addo o prce each observao also coas 5 smple qualy characerscs o he compuer model: processor ype processor speed sze o he hard ds sze o he memory ad sze o he dsplay. Maeral cosss oly o desop PCs lapops are o cluded he sudy maeral. The daa used was colleced bewee May ad Ocober. For hs sudy he orgally mohly daa ses have bee merged o hree wo-moh ses whch we he ollowg reer o as sprg summer ad all. The daa was colleced as a par o he sadard prce colleco o he Fsh CPI. The oly chage as compared o sadard pracce was ha he ve qualy varables used hs aalyss were coded o he daabase. The ormao sel was already avalable he produc characerscs ha he prce collecors are oblged o collec as a par o he sadard prce colleco procedure. The addoal cos caused by codg o qualy characerscs was eglgble. 3 Ms Mar Suvraa ad Mr Kar Mae rom Sascs Flad have parcpaed he wor relaed o our projec may ways. We epress o hem our mos scere has.

3 Basc characerscs o he daa are gve able below: Table : Characerscs o he es daa Characerscs Number o Mea Mea Mea Mea Share o hgh-ed observaos Prce Processor Memory Hard ds processors speed sze sze (Peum III AMD Ahlo Perod N MHz Mb Gb per ce Sprg Summer Fall Toal Assumg ha our sample gves a represeave pcure o he Fsh PC-mares he ollowg seems o have happeed bewee sprg ad all o year : - Hgh-ed processor ypes (pure Peum ad AMD Ahlo -processors) have creased her mare share - Average processor speed has clearly creased as well as he sze o hard dss - New compuers are equpped wh appromaely same amou o memory hroughou he perod - Uadjused mea prce decreases owards he summer ad he rses aga he all. Machg he pars As saed above he prevale sraegy ocal prce de cosruco s eepg he qualy cosa by ollowg he mached pars sraegy. The ollowg accou ae rom he dra OECD hadboo ha has bee prepared by Jac Trple summarses he pros o he approach a ecelle way: "Prce dees early uversally employ oe udameal mehodologcal prcple: The prce de complg agecy chooses a sample o real oules or sellers ad o produc. I collecs a al perod or base perod prce or each o he producs seleced. I he collecs a some laer dae he prce or eacly he same produc rom he same seller ha was seleced he al perod. The prce de s compued by machg observao by observao he prce a he laer perod wh he al prce. The grea advaages o hs machg mehodology are somemes o eplcly saed ad oher mes o ully apprecaed. The mached model mehodology holds cosa may prce-deermg acors ha are usually o drecly observable. Eamples are characerscs o he realer such as cusomer servce repuao o he mauacurer ec. Machg he prce quoes model by model (ad oule by oule) s o jus a mehodology or holdg qualy chage cosa he ems seleced or prcg. I s also a mehodology or holdg cosa o-observable aspecs o he rasaco ha mgh bas he measure o prce chage. "Trple () pages 3-4). The problem s o course ha machg he pars oe als. Le us ow ry o apply he machg o he pars mehodology o our es daa. Whe machg he sprg daa wh he summer daa we oba 55 machg pars. I he all daa oly 6 compuers ou o he orgally chose oes are le he sample. Machg o he summer daa wh he all daa yelds 7 machg pars. The sample deerorao raes are preseed able. The hgh share o o-maches summer- all comparso s somewha uepeced. From he sample desg po o vew should be appromaely he same as or sprg-summer comparso. Apparely que a umber o ew compuer models were roduced o he mares he all ad reduced he eecve sample sze more ha ormally would be epeced. Table : Share o successul maches Share o Targe Maches successul maches Perod N N per ce Sprg - summer machg Sprg - all machg Summer - all machg I s evde ha relace o he pure mached pars approach does o mae very eecve use o he daa colleced. I our case depedg o he sudy perod hry o eghy perce o he colleced prce observaos ca o be used or de 3

4 cosruco due o he alure o dg a machg varey. Alhough he wo-moh desg o he sudy maeral somewha eaggeraes he sample deerorao he pheomeo s o eglgble a mohly maeral eher. The wasg o colleced daa s o he oly shorcomg o he mached pars approach. I able 3 below we compare he characerscs o machg models wh he o-machg oes. Table 3: Characerscs o machg ad o-machg compuer Number o Mea Mea Mea Mea Share o hgh-ed observaos Prce Processor Memory Hard ds processors speed sze sze (Peum III AMD Ah Perod N MHz Mb Gb per ce Summer sprg maches Summer o-maches Derece per ce Fall sprg maches Fall o maches () Derece per ce Fall Summer maches Fall o maches () Derece per ce The samples o machg pars ge - our es maeral - rapdly based. Aer wo mohs he compuers ha were o he mare he sprg ad have bee ep he sample are clearly o lower qualy ha he replacemes seleced o he summer sample. The same holds also whe we compare summer daa wh all daa ad o course comparg sprg wh all. Ths s que udersadable o he bass o radoal prce-colleco pracces. Prce collecors are advsed o ollow he prce o he oe ad same model as log as s possble. I rapdly chagg mares hs pracce apparely leads o o-represeave samples he course o a couple o mohs. Table 4 shows he perod-o-perod chages ad prce dces based sprg. No eplc qualy adjusmes have bee made. The mached dces show oly very moderae prce chages. The overlap-de where he summer-all machg pars are all ulsed eds up wh 4 perce prce decrease. The "ed-base" de ha oly aes o accou he 6 mached pars or he summer - all comparso shows a he ed o he perod praccally o prce chage a all. I s aurally mpossble o coue he pure ed base de or ay loger perod me as all he models he sample wll vash. I pracce s possble addo o he overlappg prces coue he prce colleco as ohg had happeed.e. cosder he chages he models as rreleva. Ths pracce would he case o compuers lead o rapdly creasg prces. The hrd possbly we eclude he use o hedoc mehods s o apply some orm o judgmeal qualy adjusme. The mpac o hs procedure o observed prce developmes s dcul o oresee. The "pached" de able 4 has bee cosruced by replacg he base perod mssg prces wh esmaed values usg he same ype o lear models whch wll be preseed chaper.3. below. The pached de also shows a very moderae prce decle or he perod. The ma derece compared o ed ad overlap dces s ha he pached de does o dcae prce crease rom summer o all as he oher wo dces do. Table 4: Tradoal prce dces calculaed rom mached (ad pached) samples Fed-base Overlap Pached Sprg Summer Fall Oe could argue ha compuer models should show some orm o "mare le-cycle".e. ha a ew model s roduced o he mare a a hgher prce ad ha he prce o he model he decreases gradually ul dsappears rom he mare see Turvey (999).. The pheomeo does o seem o be very prevale our daa. Fgure. shows he prce developmes our es maeral. Isead o mare le cycle wh couously decreasg prce he more commo case seem o be a "sragh le" dcag ha durg s mare le he prce o a compuer model does o chage. Almos 7 per- 4

5 ce o he compuers dsappearg rom our sample summer showed o prce chage a all durg he wo-moh perod. Alhough here aga our es-sample desg eaggeraes he pheomeo a cosderable share o he prce chages occur combao o he roduco o a ew model. Ths lac o clear mare le cycles or PC models poses real problems o he radoal de cosruco mehods see Vara 976. I he prce chages mos cases occur combao wh he model chages he oly reasoable mehod or descrbg he prce chages s some orm o eplc qualy adjusme. Fgure.: Compuer prces Tes daa se sprg - all Flad 3 5 Prce Euro Oly sample o he daa show Perod.3 Hedoc models o prce-deermg acors A ecessary sep usg hedoc mehods or qualy adjusme s o cosruc a vald hedoc model. I case o compuers here ess que a body o research- ad worg papers o he choce o releva varables ad oher aspecs o model cosruco. As he deals o he modellg approach are o our major cocer here we jus show here summary esmao resuls o he wo models ad he ses o daa whch wll be used as eamples he remag par o hs paper. I boh o he models he log o prce s eplaed by oher log-orm varables. I our lear model he eplaaory varables are he log o processor speed ad he log o he sze o memory. I he o-lear model a secod-order erm squared log o processor speed (acually he square o s devao rom s mea whch allows easy erpreao o he regresso coeces) s corporaed he model as well. Lear model esmaed rom he whole daa se:!" #$%&$%'(#" %!)** - -. %""%/"" -. %(%/( -. 5

6 No - lear model esmaed rom he whole daa se:!" #$%&$%'(#" %!)** - -. %(((( - - %""%/"" -. %(%/( -. Lear model esmaed rom he sprg daa: -!" #$%&$%'(#" %!)** - - %""%/"" -. %(%/( -. No- lear model ed rom he sprg daa: -!" #$%&$%'(#" %!)** - - %($ -- - %""%/"" - -. %(%/( -. Lear model esmaed rom he all daa: - -!" #$%&$%'(#" %!)** %""%/"" - %(%/( --. No-lear model esmaed rom he all daa: - -!" #$%&$%'(#" %!)** - %($ - %""%/"" %(%/( -. 6

7 3. Mahemacal heory o hedoc prce ucos ad o qualy correcos de umbers 3. Cocepual bacgroud ad oao or dere hedoc models Le s sar rom he geeral oao or all dere varaos o hedoc models (HM) we are gog o cosder. We choose o cosder sead o he acual prce P s logarhm y log p as he varable o be eplaed or orecased usg he vecor o releva eplaaory (qualy) varables ad he me. The cocepual bacgroud o all regresso models s he codoal epecao () E( p ) g ( ) The uco () log. g o me ad -vecor deed by () dees he regresso surace o log p. I s a mappg rom R K T o R where K s he umber o eplaaory -varables ad T s he se o me perods cocered. The ucoal orm o g () s deermed by he heorecal (usually hypohecal) jo dsrbuo o he radom vecor ( log p... K ) ( log p ) deed by me. Dere models or hs jo dsrbuo (whch should be careully adjused o he acual problem cosdered o he daa avalable ad o a pror owledge cocerg he depedeces 4.) lead o dere ucoal orms 5. o g () We wll shorly revew some geeral possbles whch lead o dere varaos o HM. We gore hs reame he problems o model buldg ad o esmao as hese are wdely dscussed sadard ecoomerc ad sascal es such as Goldberger (964) or Spaos (986). We deoe he esmaed regresso uco smply by () () es g () es E( log p ). Ths smply gves he sample verso o he sysemac par or he bes orecas o log p erms o eplaaory varables ad me: (3) p log p () () log. Noe ha he varable should o clude ay -speccao (a de relaed o me) or ay observao speccao (or a subde such as or j) because s jus he reely chose symbol or he depede varable he ucoal oao. Ths comme holds or mos orhcomg epressos ad s o repeaed aer hs. The uco () should be applcable o jus ay values o s argume-vecor o jus o her observed values (as esmao) bu also o ay hypohecal -values or o -values rom oher perods as wll be doe aer a whle. 3. The geeral hedoc model These geeral epressos loo very smple because we delberaely leave asde a hs sage all modellg ad esmao problems. For sace all he dere ucoal orms o he HM are hdde our oao (). Dere ucoal orms wll be ae up a a laer sage o our preseao. Ths wll be doe by specalsg he geeral se-up (whch s a easy sep) so we use here rom geeral o specc approach o (ecoomerc) model buldg. Esseal eaures o (3) ca be represeed usg s paral dervaves. Assume or smplcy o preseao ha all - varables are couous 6. Deoe he paral dervave o ay by 4 See Spaos (986) 5 See Rao (965 p. -49) 6 For dscree eger-valued varables paral dervaves are replaced by he eecs o chagg he varable by oe u whle eepg all oher varables cosa he ceers parbus codo. We also assume ha () s couously dereable everywhere a apparely oce assumpo rom he praccal po o vew. 7

8 (4) () () ( ) (... K... K ) ( ) ( ) ( ) ( ). K K Noe he symmerc way o deg he chage he h argume whch s he sadard way o sarg o appromae he dervave by he derece quoe umercal aalyss see Comre (966 p. 349) or Kahaer - Moler - Nash (989 p. 3). Ths appromao gves he ordary (ecoomc) erpreao o paral dervaves as he eec o chagg he releva varable by oe u ceers parbus. I HM s () s he (esmaed) eec o he u chage o h varable o log p. Or eve more cocreely () ells how may log perce (also deoed by L% or log-% see Törqvs Vara Vara (985)) he prce creases whe he qualy varable creases by oe u (a sem-elascy). As he oao shows he dervave () depeds usually o boh ad. Corollary: We cao geerally assume or resrc he qualy eecs as depede o me (.e. o be he same or all me perods) or as depede o he values o qualy varables. These are raher resrcve bu esable specal cases o our geeral se-up. For a gve value o he qualy vecor we deoe he eec o dscree chage me smlarly. p p where. (5) () () () log () log () Also hs paral dervave o me (acually he derece quoe c. (4)) s usually a uco o boh ad. Noe ha () esmaes he chage o log-prce or a gve qualy vecor 7. I s he esmaed pure prce (PPC) chage or ha qualy vecor. A useul llusrao o hs udameal dea o a geeral hedoc mehod s gve Fgure. Fgure : Illusrao o PPC he udameal dea o he geeral hedoc mehod log p () ( ) () () () ( ) () 3.3 Sadard qualy pos ad he hedoc prce de Here we cosder wo me perods ad ad or he sae o llusrao he quay vecor s ae as oedmesoal. Fgure shows also he esseal eaures o HM whe moves hgher dmesos. Alhough he gure s resrced o oe-dmesoal oly we verbalse he resuls hgher dmesos. These llusraos ad laer erpreaos were developed by Vara ad Kurjeoja (99) o evaluae wage dscrmao bewee me ad wome. We have 7 Dereces o me depede ucos are aeced by values rom boh perods ad may be ploed a or regarded as properes o he laer perod (as s usually doe) or o he ormer perod (whch would be a ecepoal erpreao). I umercal aalyss hese possbles are dsgushed by reerrg o bacward ad orward dereces respecvely. Acually he derece should be ploed a a compromse value amely a he mea value o he perods ad -.e. a ½. Thereore he proper oao or (5) would be (). 8

9 show also he mea values ad ( ) a hese mea values. The pos o he qualy varables ad predcos (or ed values) o ad log p ( ) ad are reerred as old ad ew sadard qualy pos (SQP) respecvely. The vercal dereces o he old ad ew hedoc ucos () ad () qualy correced (or pure) prce chages (PPC) a hese pos. (6) ( ) ( ) log P ( ) (7) ( ) ( ) log P ( ) PPC a he old SQP PPC a he ew SQP. log P where a hese SQP s measure he Noe ha PPC s gve log-chage orm log p or P s he hedoc prce de (HPI). I (6) he log- log P s calculaed a he old SQP whle (7) s calculaed a he ew SQP. O course chage o he HPI ( ) boh cases argumes ad SQP`s are he same whle he uco chages rom o.ths s he essece o sadard qualy po mehod. We eplcae he erpreao o he compoes o (6) usg (3): (8) log P ( ) log pˆ ( ) log pˆ ( ) ( ) ( ). The hedoc prce ucos (HPF) may be also reerred o as qualy valuao ucos (QVF). Thus () log pˆ () values he dere qualy pos accordg o perod valuaos as () log pˆ () uses perod valuaos. Qualy valuaos are allowed o chage me geeral se-up ad me vara specal valuaos are roduced laer as a specal assumpo. Usg hs suggesve ermology (8) leads o he ollowg aural erpreao. The HPI log ( ) ad ew qualy valuaos a he same old SQP. The erm log p ˆ ( ) shows how he old SQP as log p ˆ ( ) esmaes s valuao usg perod preereces. Ther derece P () log prces or a cosa qualy po as should. Smlarly (9) log P ( ) log pˆ ( ) log p ( ) ( ) ( ) ˆ P compares he old was valued perod log measures he pure chage measures also he eec o chagg valuaos (or log prces) bu or aoher cosa qualy po amely SQP. Thus (8) ad (9) are he aural versos o HPI where boh cases he eecs o qualy chages have bee elmaed or corolled usg sadard qualy pos. These basc deas are urher elaboraed usg OAXACA-ype decomposos ad her geeralsaos o be preseed laer. These mos mpora resuls (8) - (9) ca be easly erred ad remembered usg gure. A. I log P log P he choce o SQP does o maer raher surprsgly. I hs case hedoc modellg does o aec he observed prces as compared o smple comparso because or here s o (or oly lle) qualy chage o he average ( ) ( ) log P ~ ~ log P log P log P () ( ) ( ) alhough chagg quales may have cosderable eec o mcro level. I () ~ values or oher mea prces (preerably geomerc meas whch case log P whe QVF s are lear ucos qualy varables. ~ P ad log ~ P may be deed eher as u p ). Ths resul s easly prove 9

10 B. I here are sysemac qualy chages (.e. ) bu he QVF s (HPF s) are roughly horzoal qualy correcos have oly mmal eecs because () apples appromaely. Ths may a a rs glace seem rval because qualy varables havg oly mmal eecs are o usually cluded hedoc models as hey are o cosdered or called as qualy varables. However s K-dmesoal may coa some qualy varables ha aec he prce cosderably whereas oher qualy varables oly have mor eecs o he prce. I he chage happes o realse oly he dreco o qualy varables havg oly mor eecs he esseally we have he same suao as A ad () apples. Combg cases A ad B shows ha eve here are srog qualy eecs o he mcro level bu eher he average qualy does o chage or he average qualy chage s realsed oly or qualy varables havg mor prce eec he qualy correcos are acually o eeded. O course does' mae ay harm o use HM s eve hese cases. 3.4 The lear me-depede hedoc model We may resrc our reame o he sadard lear case by assumg or reag HPF s as lear ucos o qualy varables. Ths s o as oce as s usually regarded because ow a plae appromaes a regresso surace () whch realy s o-lear. Ths may cause bas o uow magude whch some cases may o be eglgble. The lear regresso models are usually cosdered as easer o esmae hadle ad udersad ha o-lear oes. Ths s more le a geeral aude owards model buldg ha a esablshed ac. O course he regresso surace () happes o be a plae he s aurally o esmae by a correspodg lear ucoal orm () () a K b. Now he paral dervaves have especally smple orms () () b whch are me depede cosas. O he oher had he pure prce chage (PPC) or a gve qualy po amely K (3) () () () ( a a ) ( b b ) whch depeds o boh ad. Ths meas ha pure prce chages or a gve are allowed o deped o. The schemac represeao he prevous Fgure smples he case o lear hedoc model (LHM) as ollows: Fgure : Illusrao o a lear hedoc model log p () () ( ) () For lear LHM we easly derve he ollowg dees:

11 Theorem. For LHM gve () or ay se o observaos (o ecessarly a sample) (... ) we have or all : (4) ( ) ( ) or () (). K... Proo. () a a a K a b K b b K b K (). Smply a arhmec average (deoed by a log bar) o a lear uco s he correspodg value o he uco a he arhmec mea value (deoed by a shor bar argume). Ths s easly remembered by movg he log bar above he uco above s argume ad mag shor. Ths resul holds ecessarly or lear ucos bu usually als or olear ucos. Oly by luc (or very specal crcumsaces) 4 hold or o-lear ucos as wll be demosraed laer. The case o o-lear ucos s closely relaed o Jese s equaly (see Rao (968 p.46) or Chug (968 p. 45 8)) ad Iô s ormula see Björ (998 pp ). Jese s equaly hods or ay cove uco () o ay real radom varable havg a arbrary dsrbuo ad saes ha (4b) E () ( E) provded ha he epecaos es. For ay e se o -values hs mples (4c) ( ) ( ).e. () (). For cocave ucos he equaly s reversed. Theorem. I LHM s ed by OLS (or by ay oher esmao mehod ha orces he sum o resdual o zero) o a sample log p... rom he perod. The addo o (4) also ( ) (5) Thereore boh log p log p log p logp log p ad ( ) ( ). p p e log log e s zero. Proo. (5) s jus he codo ha he sum o resduals ( ) Theorem 3. Decomposg he chage ( ) ( ) ca move rom po ( ( ) ) o ( ( ) ) o HPI ad he eec o chagg quales ECQ. I Fgure 3 we rs va A gvg (6) ad he va B gvg 7.

12 Fgure 3: log p A () B () (6) ( ) ( ) [ ( ) ( )] [ ( ) ( )] (7) ( ) ( ) [ ( ) ( )] [ ( ) ( )] log P ( ) LQCF ( ) where LQCF s he Logarhmc Qualy Correco Facor. Noe ha (6) ad (7) are acually algebrac dees because he cross erms where upper dees he HPF ad s argume are dere ca be cacelled ou. The erpreao o he par (HPI ECQ) ( log P ( ) LQCF ( ) eeds rs some eor o be ully udersood ad remembered. (8) ( ) ( ) [ ( ) ( )] [ ( ) ( )] log P ( ) LQCF ( ) wh s obvous erpreao o aoher par o (HPI ECQ) amely ( log ( ) LQCF ( ) P. Noe ha here s a smlar asymmery or varao he superscrps as he well-ow decomposos o value raos o Laspeyres ad Paasche dces where Laspeyres may be appled eher as a prce or a volume de ad Paasche appears always as he oher par. Mehodologcal comme. Recogsg he valdy o (7) ad (8) s very dere rom realsg he relevace o hem. Acually he algebrac rvaly o (8) s revealed by he ollowg dervao o. Tae ay real umber z ad orce zero he orm z z he epresso (9) ( ) ( ) [ ( ) z] [ z ( )] ad group erms as dcaed. Ths s a dey or ay z. Choose z ( ) o ge (8) whle ( ) z gves (7). Hece somehg mpora arses rom a mahemacal rvaly. Noe ha he erpreao does o hold or a arbrary z (9) bu requre he careully adjused choce o z gvg (7) ad (8). Ths s a easy ormal proo o (7) ad (8) whch has very lle o do wh he acual meag ad relevace o he re- erms o he wo dere (HPI ECQ) pars suls 8. Ther relevace sems rom he decomposo o ( ) ( ) ( ( ) ( log P LQCF ) ad ( log ( ) LQCF ( ) P cluded he llusraos. Ths smple dey s a geeralsao o OAXACA-decomposo ha s radoally gve or lear HPF's. Theorem 4. Geeralsao o heorem 3 o o-lear hedoc prce uco HPF. Decomposos (7) ad (8) ad her proos are vald as such also or o-lear HPF s as llusraed Fgure. Noe however ha Theorems ad 5 do o hold or o-lear HPF s see Lemma 3 ad Theorem 6. 8 C. Wgese (967 p.9e): Each proo proves o merely he ruh o he proposo proved bu also ha ca be proved hs way.

13 We have decded o pospoe he dervao o more amlar ad pracce mpora lear HPF s so ha he reader may beer apprecae he relevace ad meag o he decomposos above. Theorem 5. I lear HPF:s ()... ad () are ed separaely o -specc samples ( p ) log ad usg OLS (or ay oher esmao mehod whch orces he sum o resduals o zero) he he ollowg (HPI ECP) decomposos hold as dees log p log p ( ) ( ) () log P ( ) LQCF ( ) log p log p ( ) ( ) () log P ( ) LQCF ( ) Proo. Combe heorems ad 3. Beore gog ay urher we prese here emprcal resuls based o he above cosderaos: Emprcal eample : Sadard Qualy po approach lear me depede model Esmaed average log prces HPI s ad LQCF s HPF or QVF based o: Hedoc Prce Ide HPI Sadard Pure chage prces Qualy po SQP Sprg valuaos Fall valuaos log-us log-perces Sprg (old SQP) log-% Fall (ew SQP) log-% LQCF (ECQ) log-us 5 87 log-perces 5 log-% 87 log-% Acual observed mea values (c. Fgure ) are gve bold. Esmaed geomerc mea prces HPI s ad QCF s I orgal us ( HPF or QVF based o: Hedoc Prce Ide HPI Sadard Pure chage prces Qualy po Sprg valuaos Fall valuaos per ce Sprg (old SQP) % Fall (ew SQP) % QCF (ECQ) 4 8 per ce % 9 % Acual observed mea values (c. Fgure ) are gve bold. 3

14 We prese also he resuls showg eac correspodece wh our decomposos ad oaos: Logarhmc Logarhmc Logarhmc Sadard Qualy Po Pure Prce Ide Qualy Correco Facor Sum o hese Lear me depede HPF s or QVF s SQP PPI LQCF Sum o hese Name Noao Noao Value Noao Value Value Equaos Old SQP New SQP log ( ) log ( ) Weghed average o decomposos (7) ad (8): P Overall SQP () P -77 ( ) P -94 ( ) log -86 / ( ) LQCF 87 (7) & (53) LQCF 5 (8) & (55) LQCF 96 (6) 3.5 The o-lear me-depede hedoc model Theorem 6. I o-lear HPF s () ad () are ed o -specc samples usg ay esmao mehod whch orces he sum o resduals o zero he () ad () are o usually dees bu hold appromaely. The appromao error s relaed o he rs equaly (whle he laer holds sll as a dey) ad depeds o he degree o curvaure (he secod order paral dervaves o () ad () ) a ad ad o he varaces ad covaraces o he -varables. I ac he appromao error depeds o he dereces (or chages) o he secod paral dervaves ad secod ceral momes (varaces ad covaraces) bewee he perods cosdered. Ths echque maes use o some raher sraghorward properes o o-lear appromao heory ha are applcable much wder coes. As hey seem o be o well ow we wll demosrae her useuless a elemeary way. Lemma. Le us sar rom he smple case o wo-dmesoal -vecor ( ) ad cosder a uco () ( ) ha s quadrac hese wo argumes. Tae ay ed po ( ) he varables a hs sage ad cosder he ollowg specal represeao o () () a b ( ) b ( ) c ( ) c ( ) d ( )( ) : R R o ecessarly he mea o Ths represeao 9 o () s uque ad has uque coeces whch deped o course o he po ( ) or... ad suppose ha ( ) y chose. Cosder wo -specc samples ( ) s ed o hem separaely usg ay esmao mehod ha orces he sum o resduals o zero so ha assumpos o he heorem 6 apply. I we ow specy he po above ( ) as he -specc mea ( ) ( ) we derve hs smple specal case quadrac uco o wo varables a mos mpora resul whch geeralses o arbrary wce couously dereable ucos o ay umber o argumes: (3) ( ) ( ) y ( ) [ c Var( ) c Var( ) d Cov( )] 9 Noe ha ay vald represeao o ay uco descrbes ad dees he uco compleely ad hereore ay o s (ely may) represeaos reveals all he properes o. The uco as a mappg should o be med wh ay o s represeaos because s somehg hey all descrbe a specc way. Much couso arses whe hs s o udersood. 4

15 5 Ths ucoal equao s he basc mahemacal resul leadg o boh uvarae ad muldmesoal Iô s lemma whch s he basc ool couous me ace models see Björ (997). I shows how he mea o values o a uco o several (here wo) varables s epressed erms o value o he uco calculaed a mea value o s argumes. These values are o he same (uless he uco s lear or he argumes have o varao) alhough our uo somemes als o oce hs. The derece (3) o hese wo depeds o or o he secod order dervaves or he curvaures o he uco (a he mea po as wll be see laer) ad o he correspodg varaces ad covaraces o he sample o argume pos. We repea (3) a more uve orm or a easy reerece (3b) ( ) ( ) ( ) ( ) ( ) [ ] Cov d Var c Var c where Ths s easy o vsualse usg he coceps o he log upper bar (whch reers o he mea o he ed values) ad he shor upper bar (whch reers o he mea o he argumes o he uco). I he log upper bar s subsued by a shor upper bar or s argumes he covarace - curvaure-erm (o he Iô-ype) mus be added as a correco. Proo. Cosder ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( )( ) [ ] ( ) ( ) ( ) [ ] cov d Var c Var c a d c c b b a e y y where ( ) ( ) Var. Dvdg by ad og ha ( ) a gves (3). A sraghorward cosequece o Lemma s Lemma. Assume he same as Lemma. (4) ( ) ( ) ( ) [ ] ( ) [ ] ( ) ( ) [ ] [ ] ( ) ( ) where he derece o he covarace-ype erms s (4b) [ ] ( ) ( ) ( ) [ ] ( ) ( ) ( ) [ ]. cov cov d Var c Var c d Var c c Var Noe rom he rs le o (4) how move he shor upper bar versos rom her log upper bar averages. However whe erms are grouped as he secod le dereces bewee log ad shor bar versos wll be good appromaos or each oher because he derece (4b) s usually early zero. Ths appromao mus be good c c d d ad ( ) ( ) Var Var ( ) ( ) cov cov ( ).e. he curvaures ad he sample covaraces are roughly me vara. Ths usually holds pracce. Equao (4) s eve closer o Iô s lemma ha Lemma. Ne we geeralse he Lemmas ad or arbrary quadrac ucos K-dmesoal spaces. Le s epla rs why he represeao () was chose or R : R. Calculae he rs wo paral dervaves o : Here we have ad o as a dvsor here ad he sample covarace. Frs degree erms vash because ( ).

16 (5) ( ) ( ) ( ) b d ( ad l are dere) l l (6) ( ) ( ) ( ) ( ) c We see e.g. ha parameers ( b b c c d ) he chose po ( ) are eacly he rs ad secod paral dervaves o ( ) calculaed a o ecessarly he mea po. Ths eplas why also he mulpler ½ appears (). Now reur o he geeral case where (... ) ad (... ). A sraghorward geeralsao (whch s ormally smpler ha lemma ) s: Lemma 3. Le ( y... ) K K K : R be ay couously wce dereable uco ed such a way o he samples R K ha he sum o resduals equals zero. The we have y (7) ( ) ( ) ( ) where he Iô-ype covarace-curvaure erm equals (7b) K K K l K ( ) Var( ) l ( ) cov( l ) l l< ( ) cov( ). l Noe ha (7) s eve smpler s oao ha s wo-dmesoal specal verso. Now we see a very compac orm how a average o ed values ( ) s appromaed by he value o he uco ( ) a he average po o s argume vecor. Noe ha or lear ucos hese mus be equal. For o-lear ucos appears ad by Jese s equaly mus be posve (egave) or cove (cocave) ucos. Equao becomes a dey or all quadrac ucos : R ad a secod degree Taylor-appromaos or ay uco (havg couous secod paral dervaves he eghbourhood o see Aposol (957 p. 4)). Noe ha covaraces cov ( ) are calculaed oly oce he mddle epresso (where l > ad mulpler ½ appears asymmercally) bu appears wo mes he las epresso. l The las represeao s he mos symmerc oe ad correspods o he Taylor-epresso o several varables. Lemma 3 coas all prevous cases ad oher eresg resuls as s specal cases. We revew hese shorly. I () s (a mos) quadrac bu secod order dervaves l () vash we are bac o he lear case. Theorems 5 are derved as a specal case because all Σ-epressos vash ad ( ) ( ) ( ) R K y. Also (8) y log p log p. We rewre (7) or ay () well appromaed by a quadrac Taylor-appromao These problems sem rom he dey ( ) ( ) where s calculaed oly oce he mddle epresso bu appears wce he las epresso. These hree dere represeaos dee he same quadrac uco ( ). We mus always decde whch o hem (or perhaps somehg else) we are usg ad o ge coused. 6

17 (9) log p ( ) ( ) ( ) Here he rs equaly holds srcly because sum o resduals was orced o zero esmao bu secod holds oly as a appromao. Subracg equaos or dere me perods rom each oher ad arragg erms we ge (3) log p log p log p ( ) ( ) log p These are src equales because sums o resduals vashed. Ths raher surprsg resul holdg or arbrary () s whch seems o be uow hedoc leraure. Because p logg( p ) we also have a eresg resul G( p ) (3) log G( p ) ( ) ( ) log he logarhm o geomerc mea o prces whch holds whou error or ay o-lear HPF s ed separaely o respecve samples (whe resduals sum o zero). Usg he appromao par o (9) we ge ally (3) ( ) ( ) [ ( ) ] [ ( ) ] [ ( ) ( )] [ ] [ ( ) ( )] K K K K (33) δ l ( ) cov( l ) l ( ) cov( l ) l l Loo a gure 4 or a llusrao o (3). The vercal les a old ad ew SQP s mee he ucos a blac crossg pos amely a he shor bar pos ( ) ad ( ).The hollow pos below hem have dred away because o (9) bu ayhow he dereces bewee log ad shor bar pos rema appromaely equal. Ths s a raher volved deduco alhough all s pars are almos sel-evde. I pracce epresso δ [ ] volvg covarace erms wll cacel away (because hey are dereces o smlar erms o perod mus perod ). Thereore a smple bu powerul heorem resuls. Theorem 6. For ay o-lear HPF s ed separaely o respecve samples we have (3). Usually eve δ may be egleced whch case (34) G log G where (... ) K ( p ) ( p ) ( ) ( ) ( ) ( ) ( ) ( ) s a arbrary vecor o qualy varables. Furhermore he derece ( ) ( ) s decomposed he aural way e.g. as ollows (35) ( ) ( ) log P ( ) LQCF ( ) [ ( ) ( )] [ ( ) ( )] Equaos (3) ad (3) ogeher epla why eacly he uweghed geomerc mea should be used o aggregae prces (or equvalely prce raos) o a mcro de level as s geerally suggesed 7

18 (36) ( ) ( ) log P ( ) LQCF ( ) [ ( ) ( )] [ ( ) ( )] These decomposos ocus a parcular pars o HPF s amely a he old ad ew SQP s (or beer a old ad ew observaos) ad orge almos everyhg else. I a more symmerc reame a weghed mea o hese decomposos (weghed by umbers o observaos) s ae. Ths leads o a smlar decomposo where he erms he resulg (HPI ECQ)-par are weghed meas o correspodg erms (34) ad (35). I may be show by drec calculaos or by mpuaos boh drecos o be cosdered laer ha hs leads o (HPI ECQ) ( P log () LQCF ( )) where s he overall mea o qualy varables or he weghed mea o old ad ew SQP s. Now we have derved decomposos or hree aural choces o SQP s. O course hese are oly hree possbles. I hedoc mpuao all observed qualy pos are used as SQP s o produce PPC s or all prces whch acually meas comparg dsaces o old ad ew HPF s or all daa pos ad ag he average o hese as he HPI. We reur o mpuao echques Chaper 3.7. Fgure 4: Illusrao o (34)- (36) log p ( ) () Equaos (35) ad (36 )are dees ad her erpreaos as qualy corolled prce dces log P ( ) ad log P ( ) ad as Logarhmc Qualy Correco Fuco LQCF ( ) ad ( ) LQCF are evde rom he gure. For arbrary lear HPF s wh me depede coeces appromao (34) urs ou o a equaly ad all equaos (34)-(35) are dees. Noe ha all hese equaos qualy varables are acually K- dmesoal vecors ad gure 4 s eded o reveal oly her esseal eaures. To calculae SQP-esmaes based erms le ( ) ad ( ) oe has o be careul o able o cocerae o s correc erpreao whch ders rom commo uve erpreaos. We have based our reame o a raher absrac applcao o geeral mahemacal coceps alhough he epressos may o commucae hs ad may appear a he rs glace eve rval. To gve a cocree eample we cosder our smple quadrac uco o our secod emprcal eample where ( ) s wo-dmesoal ad hereore e.g. ( ) s he old SQP. Ths () ( ) was esmaed usg he sadard OLS-mehod he orm a b b c ( ) (36b) () ( ) where lspeed ad s squared devao rom s ew mea (see above) was reaed as he hrd (pseudo)varable as should OLS esmao. However () ( ) mus be reaed as a uco o wo depede argumes. I should o be cosdered as a uco o hree depede varables as appears (36b) especally s wre usg a ew (pseudo)varable z ( we "specy" our regresso equao he ollowg way: ). (36c) ( z) ( z) a b b c z Ths s he usual way o represeg o-lear regresso ucos whe hey are esmaed bu some serous problems may arse rom hs coveo. Here z cao be reaed "as a depede varable" because s values are deermed eacly by. I hs s o ae o accou (whch may happe by accde some sadard roues sascal programs le SAS are used) serous errors occur whou us ocg ayhg. 8

19 Smply we have (36d) ( ) ( ) ( ) a b b c a b b where we have a eresg represeao o zero he las compoe o he mddle epresso! O he oher had we ge a perhaps oddly loog ormula or he "cross erm" where upper dces o he uco ad he SQP der: a b b c ( ). (36e) ( ) ( ) Noe ha coeces are rom perod uco ad he quadrac erm cludes o course he ew mea o he secod qualy varable bu he mea po or whch he value o he uco s calculaed s he old SQP. Thereore (36e) cludes hs quadrac erm whch s o cluded (36d). We wa o emphasse ha a parcular represeao o a uco (whch s a cocep oally depede o he way s epressed.e. depede o he parcular represeao chose e.g. o esmae regresso aalyss) should o be allowed o couse he reame. Ordarly ecoomss ad sascas seem o oally uaware o he eed o hs dsco as hey mapulae log ad complcaed mahemacal epressos a mechacal way. Thereore a uco ad (s ely may possble) represeaos or mahemacal epressos mus be careully dsgushed rom each oher wheever possble by scg rmly o he geeral mahemacal oao o a uco whou eve meog he complcaed ad parly arbrary epressos used o specy. Ths s a eresg case o he decao problem: a uco ca ever dey s represeao because here are ely may represeaos eve or he smples ucos say or a uco () 3 6 deed or all real umbers. We have e.g. () 3 6a 6( a) 6( ½) (6 /()) ec. whch may be especally suable or parcular purposes revealg dere properes o hs (). Especally o-lear rasormaos o varables (say o-lear co-ordae rasormaos) easly couse careless researchers. Emprcal eample : Sadard Qualy po approach o-lear me depede model Esmaed average log prces HPI s ad LQCF s HPF or QVF based o: Hedoc Prce Ide HPI Sadard Pure chage prces Qualy po SQP Sprg valuaos Fall valuaos log-us log-perces Sprg (old SQP) log-% Fall (ew SQP) log-% LQCF (ECQ) log-us 8 73 log-perces 8 log-% 73 log-% Esmaed geomerc mea prces HPI s ad QCF s I orgal us ( HPF or QVF based o: Hedoc Prce Ide HPI Sadard Pure chage prces Qualy po Sprg valuaos Fall valuaos per ce Sprg (old SQP) % Fall (ew SQP) % QCF (ECQ) per ce 5 % 75 % 9

20 These resuls ad her average decomposo are show below a more accurae oao. Logarhmc Logarhmc Logarhmc Sadard Qualy Po Pure Prce Ide Qualy Correco Facor Sum o hese No-lear me depede HPF s or QVF s SQP PPI LQCF Sum o hese Name Noao Noao Value Noao Value Value Equaos Old SQP New SQP log ( ) log ( ) Weghed average o decomposos (35) ad (36): P Overall SQP () P -69 ( ) P -5 ( ) log -9 / ( ) LQCF 73 4 (35) LQCF 8 3 (36) LQCF 96 4 (6) 3.6 Decomposos or me-vara qualy valuaos We are ready o pass o o more popular ad resrced HM s where qualy valuaos are me vara. I hs case HPF s (or QVF s) are separable me ad qualy dmesos whch s represeed by he ollowg propery o () : (37) () a () where a does o deped o ad () does o deped o. Now he paral dervaves o () sasy (38) () () (39) () () () depede o a a depede o boh ad. The pure prce chace PPC or a gve s ow depede o because HPF s or dere perods are jus shed versos o a me vara QVF () because o (37). Ths s a very uve easy ad beauul case bu may o adequaely represe acual qualy valuaos. Thereore s valdy cao be decded a pror bu should be esed agas he prevous more geeral HM s where qualy valuaos were allowed o chage me. Such ess are sadard procedures regresso aalyss. Fgure 5: Illusrao o o-lear Grlches-ype hedoc models (o-lear GTHM) log p ( ) () () () I we specy urher he model (37) by resrcg () o a lear uco o he qualy varables we ge he Grlches ype HM (GTHM): (4) () a K b.

21 Here he dervaves are very smple (4) () b (depede o ad me) Ths meas ha he valuao o qualy compoes s cosa cross-seco ad s me-vara. I. e: For all possble qualy pos he pure prce chage s he same cosa (he assumpo o "cosa prce chage"). (4) () a a (depede o ) Fgure 6: Illusrao o he lear Grlches-ype hedoc model (lear GHTM) log p () () I GTHM HPF s are o-lear ucos o or plaes. They are urhermore shed versos o oe me vara QVF () b whch s a lear uco o all qualy varables. O course all our prevous resuls hold or GTHM or s o-lear geeralsaos (37) reerred o as o-lear GTHM. We prese he ma resuls or GTHM s as Theorem 7. Ma heorem or me vara qualy valuaos. Cosder ay o-lear HPF s havg me vara qualy valuaos sasyg (37). Whe hese have bee ed o respecve me specc samples usg ay esmao mehod orcg he sum o resduals o zero resuls o heorem 6 apply. Especally have here (43) G( p ) ( ) ( ) ( ) G p [ a ( )] [ a ( )] [ a a ] ( ) ( ) log [ ] The log bars ca be rasormed o shor bars usg (36) ad (37) gvg [ ] ( ) ( ) [ a a] ( ) ( ) (44) [ a a ] [ ( ) ( ) d ] where d s a usually eglgble derece o he covarace erms (or he Iô-erm) sasyg (45) d K K l K K l l l K K ( ) cov( l ) ( ) cov( ) l l K K ( ) cov( l ) ( ) cov( ). l l l l Here l () () are he secod dervaves o he me vara QVF (). I () s quadrac (44) l holds as a dey ad geerally s a secod degree Taylor-appromao or a arbrary me vara o-lear QVF ().

22 Neglecg he usually mor derece o he covarace erms d ( ) ( ) [ a a ] [ ( ) ( )] (46) log P LQCF( ) where he qualy correced prce de log P [ a a ] log P ( ) P ( ) [ a a ] we have s ow depede o he sadard qualy po (e.g. log ) ad he logarhmc qualy correco acor (47) ( ) ( ) LQCF( ) s me vara (e.g. LQCF ( ) LQCF ( ) ( ) ( ) ). Emprcal eample 3: Esmao resuls o-lear Grlches-ype model. - -!" #$%&$%'(#" %!)** %(( - - %""%/"" -. %(%/( - -. %% -. Ths leads o he ollowg summary Logarhmc Logarhmc Logarhmc Sadard Qualy Po Pure Prce Ide Qualy Correco Facor Sum o hese No-lear Grlches-ype hedoc model GTHM SQP PPI LQCF Sum o hese Name Noao Noao Value Noao Value Value Equaos ay SQP lacg log P -89 ( ) LQCF 79 (46) For easy reerece we sae he very specc resuls or lear GTHM () a he covarace erm d vashes decally ad eve (46) becomes a dey. b a separae heorem 8. Here Theorem 8. Ma heorem or lear me vara qualy valuaos or lear GTHM s. Cosder ay lear HPF s havg me vara qualy valuaos or sasyg (4). These have bee ed o respecve samples usg ay esmao mehod orcg he sum o resduals o zero. Now heorem 7 apples wh d ad (46) as a dey. We have [ ] G( p ) (48) log ( ) ( ) [ a a ] ( ) ( ) G( p ) ( ) ( ) ( ) ( ) (49) LQVC ( )

23 Thereore he log-chage o geomerc mea prces s decomposed a uque way coag o me or qualy po depede choces o wo acors a qualy correced prce de ad a qualy correco as ollows G( p ) (5) log G( p ) All equaos (48) - (5) hold as dees. [ a a ] [ ( ) ( )] log P LQCF( ). The specal codos leadg o hese smple ad beauul resuls cao be assumed a pror because hey are usually reused by ess based o emprcal observaos. Bu heorem 8 gves a deally smple ad beauul case agas whch more geeral (ad more realsc) models should be corased. These cases are cosdered our precedg heorems. Emprcal eample 4: esmao resuls lear Grlches-ype model --!"(#"(!" #$%&$%'(#" %!)** %""%/"" -. %(%/(. %% Ths leads o he ollowg summary Logarhmc Logarhmc Logarhmc Sadard Qualy Po Pure Prce Ide Qualy Correco Facor Sum o hese SQP PPI LQCF Sum o hese Name Noao Noao Value Noao Value Value Equaos Lear Grlches-ype hedoc model GTHM ay SQP lacg log P -83 ( ) LQCF 93 (5) 3.7 Hedoc mpuao (or erpolao) ad s coeco o sadard qualy po mehods I he so called Hedoc Impuao or Ierpolao (HI or shor) we may esmae ( mpue ) or all old log-prce-qualy pos (log p ) ew log-prces log p ( ) correspodg o eacly he same old characerscs usg he ew verso o our HPF amely (5) log p ( ) ( ) or all. Le us cosder rs he smpler case o lear HPF s whch are me depede. Usg (5) we are ready o calculae he pure (or qualy corolled) log-prce chages as ollows log p ) (5) log p ( ) - ( - log p whch geomerc erms s a comparso bewee observed prce pos ad her mpued prces lyg o he HPF below (or above). Ths s a vercal moveme he gure uderled by he erm mpuao. Ierpolao reers o a compua- 3

24 oally dere bu equvale horzoal moveme whe we correc or qualy chages he case o mached pars o prces. Averagg over all old observaos gves (53) (log p ( ) log ) p ( ) logg( p ) ( ) ( ) by (5) ( ) ( ) by leary log P ( ) by (7). Noe ha he asymmery o he rs wo epressos vashes we sar rom esmaed old prces log p ( ) sead o he observed oes. The res o he equao holds also or hese because he sum o he resduals s zero. Also hs procedure s cluded Hedoc Impuao. We have derved a very mpora ad uvely sgca coeco bewee HI ad SQP-mehod saed as Theorem 9: Cosder ay me depede lear HPF s. I old log-prces or her esmaes are mpued usg he ew HPF ad he resulg qualy corolled log-prce chages are averaged over all observaos he HPI calculaed a he old qualy po log P ( ) arses as he resul. I hs sese hedoc mpuao o old prces ad SQP-mehod a he old SQP are equvale. We derve a smlar resul ew log-prce-qualy pos (log p j j ) are mpued bacwards ad he resulg pure logprce chages (oe he chage he order he derece) (54) log p j - log ( ) p log p - ( ) or all j j j j are averaged: (55) (log p log ( )) j p j log G( p ) ( ) ( ) ( ) ( ) ( ) by leary log P ( ) by (8). As above more symmerc epressos arse sead o acual ew log-prces her esmaes are ae as a sarg po (55). We have prove Theorem : Cosder ay me depede lear HPF s. I ew log-prces or her esmaes are mpued usg he old HPF ad he resulg qualy corolled log-prce chages are averaged over all observaos he HPI calculaed a he ew qualy po log P ( ) arses as he resul. I hs sese hedoc mpuao o ew prces ad SQP-mehod a he ew SQP are equvale. Bu we are ready o aac more specc problems o HM because he cocepual ad mahemacal seup has bee clared whou ay assumpos how we have arrved a our me-specc HPF's or QVF's () (... K ). I we pool or combe mpuaos orward (53) ad bacwards (54).e. use boh old ad ew observaos hedoc mpuao we ac calculae he weghed average o he resulg pure prce dces wh umber o observaos ad as weghs. A sraghorward calculao shows ha hs weghed average o pure prce dces cocdes wh a pure prce de calculaed a a smlarly weghed qualy po reerred as overall mea or SQP w w (56) where /( ) We have w 4

25 (57) w log P ( ) w log P ( ) log P ( ) because log P ( ) s a lear uco as a derece o wo lear ucos o he same argumes. Ths resul was show already Vara ad Kurjeoja (99) he coe o wage dscrmao bewee me ad wome. We sae he resul show above as Theorem : Impuao boh drecos or a lear me depede HPI boh old ad ew observaos (or esmaed values o log-prces) are mpued ad he pure log-prce chages are averaged over all observaos he log o he HPI (57) log P ( ) calculaed a he overall mea (56) o he qualy varables arses as he resul. I hs sese hedoc mpuao o boh old ad ew prces ad he SQP-mehod a he overall mea are equvale. Theorems 9 ad may be geeralsed or o-lear HPF s. Proos ollow rom he Theorem 6 ad are omed here. Thereore we have Theorem : Cosder ay o-lear me depede HPF s whch are esmaed such a way ha resduals sum o zero. The Theorems 9 ad hold appromaely. The appromao error s esmaed by d gve (33); hs d s eglgble mos applcaos. For sace Theorem 9 geeralses o (58) G( p ) log ( ) ( ) G( p ) [ ( ) ( )] [ ( ) ( )] log P ( ) LQCF ( ) Ths s clearly a dey where he (HPI ECQ)-par s calculaed by aggregag mcro-level PPC s ad ECQ s accordg o orward mpuao sead o SQP-mehod. By heorem HPI s ad ECQ s calculaed by mpuao ad SQPmehods sasy (59) log P ( ) log P ( ) (6) LQCF ( ) LQCF ( ) Noe ha (59) ad (6) meas are calculaed or he ed values he le had sde whch s ypcal hedoc mpuao whle hey appear as mea values o he argumes (HPI ECQ)-pars whe SQP-mehods or erpreaos are used. I he case o lear me depede HPF s ad QVF s hese wo erpreaos cocde I he lear case does o maer wheher we sar rom he (HPI ECQ)-par (log P ( ) LQCF ( )) o he SQP-ype or o he par (log P ( ) LQCF ( )) o he hedoc mpuao ype. They gve eacly he same resuls or lear HM s by Theorem 9 ad oly he praccal compuaos are dere. Bu he case o o-lear me depede HM`s hey gve dere resuls ad provde hereore dere geeralsaos. The rs o hem geeralses he SQP-mehod or o-lear models whle he laer leads o hedoc mpuao. They allocae he d erms (plus oher possble hgher order erms caused by o-leary) dere ways o (HPI ECQ)-pars. Furher research s eeded o evaluae hese problems. Resuls o our emprcal eamples are colleced he ollowg summary able where hey are easly compared. 5

26 Summary Table o emprcal eamples Logarhmc Logarhmc Logarhmc Sadard Qualy Po Pure Prce Ide Qualy Correco Facor Sum o hese SQP PPI LQCF Sum o hese Name Noao Noao Value Noao Value Value Equaos Lear me depede HPF s or QVF s Old SQP New SQP log ( ) log ( ) Weghed average o decomposos (7) ad (8): P Overall SQP () P -77 ( ) P -94 ( ) log -86 / ( ) LQCF 87 (7) & (53) LQCF 5 (8) & (55) LQCF 96 (6) No-lear me depede HPF s or QVF s Old SQP New SQP log ( ) log ( ) Weghed average o decomposos (35) ad (36): P Overall SQP () No-lear Grlches-ype hedoc model GTHM ay SQP lacg P -69 ( ) P -5 ( ) log -9 / ( ) log P -89 ( ) LQCF 73 4 (35) LQCF 8 3 (36) LQCF 96 4 (6) LQCF.79. (46) Lear Grlches-ype hedoc model GTHM ay SQP lacg log P -83 ( ) LQCF 93 (5) 6

27 4 Coclusos We have reerred o he dual aure o hedoc models by gvg wo dere erpreaos o amely s HPFerpreao (as comparg he me specc Hedoc Prce Fucos or a gve he qualy po. I hs erpreao we cosder pure prces chages whe he qualy po s ed). The oher s s QVF-erpreao (as a Qualy Valuao Fuco or a gve me perod. I hs erpreao we are eresed how chages quales aec he prce whe me perod s ed). We have coceraed he paper mosly o HPF-erpreao o he hedoc models bu he umercal eamples llusrae boh hese vew. We have derved geeralsaos o he popular (bu resrced) OAXACA-ype decomposos o he log-chage geomerc mea prces o he log-chage o pure prces (or he log o he pure prce de log P ) ad logarhmc qualy correco acor (or he eec o chagg quales o log-prces LQCF). The mos symmerc orm o hese geeralsed OAXACA-decomposos s a dey or lear me depede hedoc models (ad a appromao or o-lear oes) G( p ) (6) log log P ( ) LQCF ( ) G( p ) where he LQCF s a weghed mea o LQCF s based o old ad ew QVF s gve (7) ad (8): (6) LQCF( ) w LQCF ( ) w LQCF ( ). I he case o lear me-depede QVF s (6) reduces o a sgle average QVF whose coeces are weghed averages o me specc QVF s whch eplas our upper de ½ hese epressos. (Proos o hese saemes based o Theorem are omed here or brevy.) Thereore we may duplcae our resuls by cocerag o QVF s ad LQCF s sead o HPF s ad log P s elmag eecs o qualy chages o he le sde epresso o (58). All hese mpora resuls are eher dees (or her appromaos).e. ucoal equaos see Echhor (978) arsg rom he basc ucoal seup where log-prces are descrbed usg a HPF or QVF o ype () where s a arbrary vecor o qualy varables. Very lle has bee sad o he choce o hese qualy varables he umber o hem or o he oher aspecs o hedoc model buldg such as he speccao o he ucoal orm o () (... ) or s esmao ad esg. Ayhg reasoable ca be doe cocerg hese aspecs o he hedoc modellg ad ohg more has bee assumed ha ha he resduals sum o uy. Ths a mpora po udersadg our resuls. We have bee able o separae geeral mahemacal aspecs o he hedoc modellg ad mag qualy correcos rom more specc problems relaed o model buldg ad esmao o hese models. We regard hs as our major corbuo he eld. Aer hs we are ready o hadle hese more specc ad echcal problems whch vary rom oe suao o aoher ad are esseally problem specc ad daa depede. The mehodologcal commes epla also why we have gve very ew reereces o a wde leraure o hedoc modellg. I HM eher oo may o s problems are cosdered a he same me or some specc aspecs o hedoc modellg (perhaps srogly coeced o a specal suao applcao or o some "assumed" ucoal orm such as lear GTHM ec) has bee aaced. Thereore mos o he leraure s eher oo geeral (or coused) o lead o ay useul resuls or oo specc o be very eresg. I some saces raher modes progress has emerged despe o he cosderable resources vesed he projecs o qualy adjusme. I our vew ad erpreao geeral ad specc problems may have bee med up hese eors such a way ha progress has slowed dow ad less ha epeced commo udersadg has emerged. The heerogeey o he problems ad he problem solvers (accompaed by varyg commeaors ad pracoers o he eld o prce dces) dees a oo comple socal evrome o produce a geeral agreeme o wha should be doe. To be more specc here seems o preval a subcoscous commme o raher uecessary bu wdely ulsed coveos (see Leamer (983)) o scg o Laspeyres ype prce dces ad o "mached models" approach. Ths has eecvely hdered he aalyss o wder problems ad more geeral echques e.g. he realm o hedoc modellg. I seems uecessary o sc o lear or o-lear versos o Grlches ype o hedoc models GTHM ad o avod me depede lear or o-lear HPF s where eher SQP-mehod or mpuao has o be appled. Evdely ryg o ae o accou me eecs he valuao o qualy varables has bee (msaely) regarded as oo dcul. Our resuls show ha hs s o problem he geeral seup. Commo wsdom has also sressed esmao problems o HPF's bu 7