ons methodology working paper series no 3 a note on
TRANSCRIPT
ONSMethodologyWorkingPaperSeriesNo3
AnoteondistributionsusedwhencalculatingestimatesofconsumptionoffixedcapitalCraigMcLarenandChrisStapenhurstMarch2015
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0 Summary
Thisisatechnicalreferencepaperwhichdescribesdistributionsusedwhencalculatingestimatesofcapitalconsumption.WefocusondistributionsusedthatcanbeusedwithinthePerpetualInventoryMethod.Inpractice,thechoiceofretirementdistributionsforcapitalconsumptioncalculationscanbesomewhatarbitrary,sothispaperdescribespropertiesofdistributionstohelpunderstandtherelativemeritsofdifferentapproachesusedinthatcontext.
1 Introduction
ThePerpetualInventoryMethod(PIM)isawellestablishedmethodtoestimatecapitalstocksandconsumptionwithinNationalAccounts.WithintheOfficeforNationalStatistics(ONS),outputfromthePIMispublishedwithintheCapitalStocksandConsumptionofFixedCapitalrelease(ONS,2014).ForadetailedoverviewofthePIM,andrelatedconceptswithinthewidercontextoftheNationalAccountsframework,seeOECD(2009).
Describedbriefly,thePIMsumsflowsofinvestmentsoverpreviousperiods,subtractingretiredcapitalandcorrectingforreductionsinproductivity(value)togivethestock(value)ofproductivecapital.Incorporatingappropriatepriceprofilegivesthenetcapitalstock.Finally,therateofreturnandholdinggainsandlossesareusedtoderivethevalueofcapitalservices.
Thispaperisconcernedwiththeestimationofcapitalstocksfromhistoricalinvestmentsdataandassumptionsaboutassetretirementanddegradation.Forthisthefollowingarerequired:
Aclassificationofcapitalinputsintoreasonablyhomogenoustypes,i.e.exhibitingsimilarcharacteristicsintermsofdegradationandretirement
andthenforeachclassofassets:
Atimeseriesofinvestmentsineachclassofcapital(forourpurposesweassumepriceshavebeendeflatedappropriately)
Anage‐priceorage‐efficiencyprofile(collectively,age‐profiles)forasingleassetineachclassofcapital
Aretirementdistributionforeachclassofcapital
Oncethesearegiven,foreachassetclasstheage‐profileforasingleassetiscombinedwitharetirementdistributiontogiveanage‐profileforacohortofassets.Thisisthenappliedtotheseriesofinvestmentsofthatclass;finallywesumacrossallclassestogivethecapitalstock.
Theaimofthispaperistocatalogueanddescribesomeofthemoreprevalentfunctionalformsusedforage‐profilesandretirementdistributionsintoasinglereference,whilenotingthatsomeformsmaybeusedforbothpurposes.Wethengoontodescribehowthesemaybecombinedtogiveanage‐profileforacohortofassets.Inpractice,thechoiceforage‐profilesandretirementdistributionscanbesomewhatarbitraryasitcandependontheassetclassunderconsiderationandhistoricalconvention,sowemakenorecommendationsonthepracticaluseofeachdistribution.OECD(2009,p42)makessomepracticalrecommendationsregardingthelinearandgeometricforms.
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2 Age‐efficiency, age‐price and depreciation profiles
TheOECDCapitalManual(OECD,2009)definesage‐efficiency,age‐priceanddepreciationprofilesasfollows:
Age‐efficiencyprofile:Thisdescribesanasset’sproductivecapacityoveritsservicelifeandisrepresentedasanindex.Theindexissettoequaloneforanewassetandbecomeszerowhentheassethasreachedtheendofitsservicelife.Thedeclineinproductivecapacityisaresultofwearandtearand/orobsolescenceoftheasset.
Age‐priceprofile:Thisisanindexofthepriceofacapitalgoodwithregardtoitsage.Theage‐priceprofileshowsthelossinvalueofacapitalgoodasitages,orthepatternofrelativepricesfordifferentvintagesofthesame(homogenous)capitalgood.Theage‐priceprofilecomparesidenticalcapitalgoodsofdifferentageatthesamepointintime
Depreciationprofile:Valuelossofanassetduetoaging,expressedaspercentageofthevalueofanewasset
Age‐efficiencyandage‐priceprofilesareanalogousinmanyrespects,forthisreasonwemayoftenrefertothemcollectivelyas‘age‐profiles’.ThechoiceofwhichisusedinthePIMisdeterminedbywhatwewishtocalculate:theage‐priceprofileisusedforcalculatingnetcapitalstockanddepreciation,whereastheefficiencyprofilegivesproductivecapitalstock.Thesedefinitionsrefertoasingleasset;althoughitisalsoappropriatetorefertotheage‐profileofacohortofassets.Therelationshipbetweenthatofasingleassetandthatofacohortorassetsisconsideredlaterinthisnote.
Wedefineanage‐profilemoreformallyasa(deterministic)function
: →
satisfying:
i) 0 1ii) 0∀ ∈
Inmostcasesthefollowingalsoholds:
iii) lim → 0
Weinterpret astheprice(orefficiency)oftheassetattime relativetoitsprice(orefficiency)attime0.Theconditionscanbeinterpretedassayingthatthepriceoftheassetisnormalisedtoequal1atthebeginningofitslife,thatthepriceoftheassetisalwayspositive,andthatthepriceoftheassettendstowardszerowithtime(thismaynotholdinsomespecialcasesforassetssuchasland).Notethattheage‐profileitselfdoesnotnecessaryrequirethatthepriceeverdoesreachzero;thismaybedealtwithbychoosingsomethresholdvalueafterwhichthevalueoftheassetisreducedtozero.Forinstance,if isanage‐profileandthereexistsnopositiverealnumber suchthat
0;let beourthresholdvaluethenwecanredefine sothat 0forallwhere istheinverseof .
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Alternatively,manytexts,includingOECD(2009),defineage‐profilesasasequencesatisfyinganalogousproperties.Thisisentirelysufficientsinceweonlyeverwishtocalculateefficiencyofvalueatdiscreteintervals,forinstanceannuallyorquarterly.Weusecontinuousfunctions1withinthispaperonlybecausemanyofthemhaveformsusedalsoinretirementdistributionssowemaybesavedpresentingthesameformtwice,thoughwenotethatsomeofthefunctionalformsdescribedbelowaretraditionallydefinedonlyinthediscretecase.Movingbetweendiscreteandcontinuousformsisusuallytrivial:ifasequence canbedescribedbyaclosedform ,wemaysometimesevaluate atnon‐integervaluestogiveacontinuousfunction(forexamplethegeometricform);converselyifacontinuousfunction isgivenwemaydefineasequence by .
Wealsonotethatthepriceandefficiencyprofilesarerelatedbutbynomeansidentical.Thepriceprofilecanbederivedfromtheefficiencyprofileandviceversaiftherateofreturnandassetpriceinflationareknown.TheproceduresfordoingsoforsingleassetsandcohortsareillustratedinOECD(2009)Chapter3andAnnexDrespectively,andinMcLellan(2004,p12‐16).Forthispurposeage‐profilesaredefinedassequences.Let
betheage‐priceprofilesequence, theage‐efficiencyprofile,Tthelifespanoftheasset(whichmaybeinfinite),πtherateofassetpriceinflationandρthediscountrate,thenOECD(2009,p228)gives:
∑ 11
∑ 11
(1)
Theprocesscanbereversedtoderivetheage‐efficiencyprofilefromtheage‐priceprofileonOECD(2009,p229):
1 11 1
(2)
Similarly,thedepreciationprofileisdeterminedbytheage‐priceprofilebythefollowingformula:
1 (3)
whereonceagain isasequencedescribingapriceprofileand isthedepreciationprofile.Acontinuousanalogueisgivenby:
′ (4)
Intheliteratureitisnormaltofirstestimatetheage‐priceorage‐efficiencyprofileandthenderivethedepreciationprofile,thoughinsomecasesitmaybeappropriatetoestimatedepreciationdirectly(seeBlades,1998a).
3 Retirement distributions
Retirementisthe‘actofputtinganassetoutofservicebecauseithasreachedtheendofitsservicelife’,duetoeitherwearandtearorobsolescence,orperhapsduetolegal
1‘Continuous’inthesenseofafunctionwithdomain ,opposedtothemathematicalsenseofafunctionsatisfyinglim →
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requirement(seeOECD,2009).Intermsofage‐efficiencyprofiles,thisisthesameassayingthattheproductivecapacityofanassetiszero.Werecognisethatingeneralnotallassetsofthesametypehaveidenticallifespansandthatthelifespanofanassetisusuallystochastic.Thismeansweareabletodescribethelifespansofacertainclassofassetswithaprobabilitydistribution.
Everyprobabilitydistributionmaybecharacterisedbythefollowingstatisticalfunctions.TheseparticularformsaregivenastheymaybeusefulinconceptualisingcapitalretirementandinunderstandingtheimpactswithinthePIM.
ProbabilityDensityFunction(PDF):Let : → 0,∞ beafunctionsatisfying0∀ ∈ and 1,then isaprobabilitydensityfunction
(PDF).Wecanthinkof asvalueswhichavariatemaytakeand astheprobabilitythatthevariatetakesthevalueinaninfinitesimallysmallintervalaround .Inthecaseofretirementdistributionswecaninterpret astheageofanassetand astheprobabilitythatanassetretiresatage .Itisconventionthatforacontinuousdistribution , 0∀ ∈ onthebasisthatsincemaytakeuncountablymanyvalues,itsprobabilityofobtaininganyone
particularvalueiszero;ratherwesaythat istheprobabilitythat takesavaluebetween and .
CumulativeDistributionFunction(CDF):Let : → 0,1 beafunctionsatisfying(i)lim → 0andlim → 1;(ii) ismonotonicallydecreasing,i.e. ⇒ ;(iii) iscontinuousfromtheright,i.e.lim →
∀ ∈ .Then isacumulativedistributionfunction(CDF).Foranygivendistribution,thePDFandCDFarerelatedasfollows:
(5)
Thus istheprobabilitythatthevariatetakesavaluelessthan ,orinourcase,theproportionofassetswithlifespanlessthan .
Thesurvivalfunction(alsoreferredtoasareliabilityfunction):ifavariatehasCDF ,thesurvivalfunctionisgivenby
1 (6)
Itshowswhatproportionoftheoriginalmembersofthegroupofassetsarestillinserviceateachpointduringthelifetimeofthelongest‐livedmemberofthegroup(Blades,1983).Inorderwords,theprobabilityofsurvivingatleasttotimepoint .Inpractice,bankruptcyandscrappagecanbedealtwithseparately.
Thehazardfunction:ifavariatehasPDF andsurvivalfunction ,then
thehazardfunctionisdefinedastheratiooftheprobabilitydensityfunctiontothesurvivalfunction:
(7)
Inthestatisticalliteraturethisisalsoreferredtoastheconditionalfailuredensityfunction,intensityfunction,theage‐specificfailurerate,instantaneousfailurerate,ortheforceofmortalitybecauseitreflectstheprobabilitythatthe
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eventoccursatagiventime,conditionalontheassethavingsurvivedupuntilthattime.
ForapplicationofthePIM,onlythePDFandthesurvivalfunctionofaretirementdistributionarenecessarythoughinthefollowingsectionwegivethehazardfunctionforcompleteness.
Manyofthedistributionsbelowtakestrictlypositivevaluesalongthewholerealline,whichistosaythatforanyage thereisanon‐zeroprobabilitythatanassetmaylivetothatage,thoughitmaybesmall.Forpracticalpurposesitisusefultoassumeeveryassetclasshasafinitemaximumlifespan ,andthatassetslivesdonottakenegativevalues.ThismaybedonebycarryingoutthePIMcalculationsonlybetween0and ,whichisequivalenttocuttingoffthetailsoftheretirementdistribution:if istheformoftheretirementdistributionofchoice,thenthedistributionusedinpracticeis
, (8)
(where istheindicatorfunction)whichhastheproperty 1,introducingasmalldownwardbiasintheresult.Alternativelywemaytruncatethedistributionatthepointofuse.Thisisequivalenttocuttingoffthetailsofthedistributionasbeforeanddistributingthelostareabetweenthepointsoftruncation.AdistributionwithPDF andCDF maybetruncatedusingthefollowingformula:
,
0 (9)
whichpreserves 1.
4 Examples of functional forms used within the PIM
Wenowlistsomeofthefunctionalformscommonlyusedforage‐profilesandretirementdistributions.Someformsareusedforbothage‐profilesandretirementdistributionsandinthesecasesthefunctionusedfortheage‐profilewillcorrespondtothesurvivalfunctionofthecorrespondingretirementdistribution.Noticethatage‐profilesandsurvivalfunctionsaresimilarinthatbothareinitiallyequalto1andtendtowards0.Forthisreasonwedonotplottheexampleage‐profilesinadditiontosurvivalfunctions.Itisimportanttorecognisethatage‐profilesarenotthesameassurvivalfunctions:theage‐profileshowsdeterministicallyhowthevalue(orproductivity)ofanassetdeclinesovertimewhereasthesurvivalfunctiongivestheprobabilityoftheassetsurvivingtoagivenageandsaysnothingaboutitsvalue(orproductivity)pathingettingthere.Anyassetclassmayhaveage‐profilesofoneformandretirementdistributionsofanotherdifferentform.Moreover,noteveryage‐profilewillbeabletoberepresentedasasurvivalfunctionsinceage‐profilesmaybeincreasingonpartoftheirdomain(seeBlades,1998a).Intheexamplesbelow, referstothemeanassetlifelengthinthecontextofaretirementdistribution,buthaslessnaturalinterpretationinthecontextofage‐profileswhereitreferstomeanvalueorefficiency.Withregardtoage‐profiles,themaximumassetlifelength, isofgreaterinterest.
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Name Simultaneousexit/Onehossshay
Background Consideredasaretirementdistribution,thisreflects thecasewhereallassetsliveforexactlythesamelengthoftime, ,atwhichpointtheyretire.Asanage‐profile,thisdescribesthecasewhereanassetretainsitsfullproductivecapacityorvalueuntiltheendofitslife, ,whenitsproductivityorvaluedropinstantaneouslytozero.
Theoriginoftheterm‘onehossshay’isdescribedhere:http://stats.oecd.org/glossary/detail.asp?ID=1904
Reading Referencesinclude:
OECD(2009,p114) OECD(1997,p6)
Formula Theprobabilitydensityfunctionisgivenby
∞when
0when
Thesurvivalfunctionandage‐profilearegivenby
1
where0 .
andthehazardfunctionisgivenby
∞when
0when0
Themeanageofretirementis .
Example Thesimultaneousexitfunctioncanbeapproximatedbyeitherusingthelineardistributionandchoosing → ,orconsideringanormaldistributionwherethevarianceparameterapproacheszero.Theareaunderthedensitycurvewillstillequal1and istheaveragelifelength.
Density Survival Hazard
0
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Name Delayedlinearfunction
Background Retirementdistribution:(referredtoastheuniformdistribution)Assetsareequallylikelytoretireatanyagebetweentheminimumage (whichmaybezero)andthemaximumage .
Age‐profile:reinterpreting asthetimeatwhichthevalueoftheassetbeginstofall,thevalueorefficiencyofanassetdeclinesbythesameamounteachperiod.
Setting 0reducestothelinearcase.FunctionsfortheuniformdistributionareavailableinthebasepackageofR(www.r‐project.org).
Reading Referencesinclude:
OECD(2009,p114) OECD(1997,p2) NIST/SEMATECH(2012)withspecificreference
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3662.htm
Formula Let bethemaximumlifelengthand theminimumlifelengththenthePDFisgivenby:
1
where .
Thesurvivalfunctionandage‐profilearegivenby
where0 1
andthehazardfunctionisgivenby
1
where0 1.
Themeanageofretirementisgivenby .
Example Examplewhere istheaveragelifelength,andthereisadelayof whichimplies, 2 ,and 2 .Thisgives 1/ 2 , 1 / 2 ,and
1/ 2 .
Density Survival Hazard
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0.1
0.2
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L+k L+k L+k
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Name Geometricdistribution
Background Thegeometricdistributionisadiscretefunction.FunctionsforthegeometricdistributionareavailableinthebasepackageorR(www.r‐project.org).
Efficiency/valuefallsbyaconstantproportioneachperiod.Lifespanisinfinite;inapplicationsathresholdischosenwhentheefficiency/valueistakentobezero.
Reading Referencesinclude:
http://mathworld.wolfram.com/GeometricDistribution.html OECD(2009p93,96‐97) Biorn(1998,p618)
Formula Thegeneralformulafortheprobabilitydensityfunctionisgivenby
1
where0 1, 1 ,andthemeanis 1 / .
Thedistributionfunctionisdefinedas
1 1
Thesurvivalfunctionisthengivenby
1
andthehazardfunctionisgivenby
1
Supposewewishourthresholdvalueatwhichweretiretheassettobe ,thenwecanselectavalueof suchthattheassetretiresatexactlyage :
1
⇒ 1
Example Geometricdistributionwith: 0.5 andwhere istheaveragelifelength.Noticethatforthisvalueof ,thedensityandsurvivalcurvescoincideandthehazardfunctionisconstantat1.
Notes Blades(1998a,p7)notesthatthegeometricdepreciationiscalculatedbyapplyingaconstantrateofdeclinetotheinitialvalueoftheassetandthatdouble‐decliningbalancedepreciationissimilartogeometricdepreciationexceptthateachyear’sdepreciationissetattwicethatofthepreviousyear’sdepreciatedvalue.Essentially,thiscanbecapturedwithaparameterchangeinthestandardgeometricdistribution,e.g.thechoiceof .Foradetaileddiscussionondouble‐decliningbalancealsoseeStatisticsCanada(2007p13and14).ThisisalsodiscussedinOECD(2009,p97).SNA2008p418paragraph20.22alsocoversgeometricaspects.
Setting givesthedoubledecliningbalance,noticethatlim → exp 2 .
Density Survival Hazard
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0 L 2L 3L 0 L 2L 3L 0 L 2L 3L
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Name Hyperbolicform(alsoreferredtoasbeta‐decay)
Background Thehyperbolicformhasbeenhistoricallyusedforthedevelopmentofarelevantage‐efficiencyfunction.Itdescribesassetswhichdecaygraduallyatfirstandmorequicklywithage.WithinOECD(2009),theconceptofage‐efficiencyisconsideredtobeequivalenttotheformusedforthesurvivalfunction.Botharefunctionswhichgiveasurvivalorfullefficiencyattime0,andthenovertimereducetozero.Ifweacceptthiscomparison,thenequivalentprobabilitydensityfunctionsandhazardfunctionscanbederivedbasedonahyperbolicformforthesurvivalfunction.Thisisgivenbelow.
NotethatmoregeneralstatisticalfunctionsforthehyperbolicdistributionareavailablethroughtheRpackageGeneralizedHyperbolic:
(http://cran.univ‐lyon1.fr/web/packages/GeneralizedHyperbolic/index.html)
Reading Referencesinclude:
OECD(2009,p92) http://www.federalreserve.gov/releases/g17/CapitalStockDocLatest.pdfp9 Blades(1998,p2)
Formula Let bethemaximumservicelifeand assumethatthesurvivalfunctionisofthehyperbolicform
where0 and0 1.
Thisimplies(as 1 )thatthecumulativedensityfunctionisgivenby
1
Derivingtheprobabilitydensityfunctionbydifferentiating meanswecanderive
1
Sothehazardfunctionisgivenby
1
Notethatif 0thenformulaisequivalenttothelineardistribution.
Example Hyperbolicformwith: 0.5, 10 andwhere istheaveragelifelength.
Density Survival Hazard
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0 L 0 L 0 L
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Name Weibulldistribution
Background TheWeibulldistributionisnamedafterWaloddiWeibull whoformulateditin1951todescribethefatiguelifeofsteel.ItisusedbyanumberofcountriesasaninputintothePIMcalculations.
Retirementdistribution:TheWeibullfunctiontakesallpositiverealvaluesandshouldbetruncatedfor and 0.
Age‐profile:theshapeparameter,γ,maybechosentoreflectwhetherthevalueofefficiencyoftheassetrisesorfallsintheearlyperiodofitslife,andhowlongthisperiodlasts.Setting 1reducestotheexponentialdistributionwhichisthecontinuousanalogueofthegeometricdistribution.StatisticsCanada(2007p17‐18)usestheWeibullfunctiontoempiricallyestimateage‐priceprofiles.
FunctionsfortheWeibulldistributionareavailableinthebasepackageorR(www.r‐project.org),orthroughtheRpackageFAdist(http://cran.r‐project.org/web/packages/FAdist/)whichincludesa3‐parameterversion.
Reading Referencesinclude:
Meinen,G.et.al(1998,p14) OECD(2009,p115)whichdiscussesrangesforWeibullparametersbasedonStatistics
Netherlandsresearch. NIST/SEMATECH(2012)
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm StartisticsCanada(2007,p17‐18)
Formula Fora3‐parameterWeibulldistribution,theprobabilitydensityfunctionisgivenby
/
where =shapeparameter, =scaleparameterand =locationparameter.When 1and 0thisisthestandardWeibulldistribution.Thesurvivalfunctionisgivenby
where 0, 0,andthehazardfunctionisgivenby
where 0, 0.
Themeanisgivenby 1 1 ,whereΓisthegammafunction.
Example Weibulldistributionwith: 1, 1, andwhere istheaveragelifelength.
Weibulldistributionwith: 2.5, 1, .
Density Survival Hazard
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0 L 2L 0 L 2L 0 L 2L
Density Survival Hazard
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0 L 2L 0 L 2L 0 L 2L
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Name Sumofthedigits
Background Blades(1998a,p7)givesthefollowingdescription:
Sumofthedigitsisamethodofdepreciatinganassetbyanamountwhichdeclineslinearlyoverthelifeoftheassetsuchthatthedepreciationaccumulatedoverthelifetimeoftheassetsequalsitsinitialvalue.Foryeart,theamountofdepreciationisobtainedbymultiplyingtheinitialvalueoftheassetby
1 /
Thedenominatoristhe“sumofthedigits”ofL,i.e.,15+14+...+1=120.Thefirstyear’sdepreciationisthereforecalculatedas15/120oftheinitialassetvalue,thesecondyear’sdepreciationis14/120oftheinitialvalueandsoon.
Reading Referencesinclude:
Blades(1998a,p7) StatisticsCanada(2007,p13‐14)
Formula Noticeintheabovedescriptionthatdepreciationisgivenasanabsoluteamountinagivenyear,notasarateofdeclineofvalueasgivenabove.Wederivethecorrespondingage‐priceprofileasfollows.Ifthevolumeofdepreciationoccurringattime isgivenby ,thenthetotalvolumeof
depreciationbytime willbegivenby
2 11
21
1
21
11
22 1
1
Thusthevalueattime willbe
12 1
1
Example Thesumofthedigitsfunctionhasthefollowinggraph:
Sum of Digits
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0 Lmax/2 Lmax
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Name Generalconvex/concavecurves
Background Aconvexfunction : → satisfies
1 1
forany , ∈ and0 1,whichistosaythatthecurveisbowedtowardstheorigin;converselyaconcavefunctionisbowedawayfromtheorigin:
1 1
Reading Referencesinclude:
Biorn(1998,p619‐621)Formula Let betheasset’slifespan,then
1
1
where , 0givesthedegreeofcurvaturefor and respectively. isstrictlyconvexfor 1andstrictlyconcavefor 1.Thereverseistruefor ,whichisstrictlyconcavefor 1andstrictlyconvexfor 1.Both and reducetothelinearcasefor , 1.
Example Thefollowingillustrateshowthefunction maybeeitherconvexorconcavedependingonchoiceofparameter(thefirstandsecondgraphs),andhowthetwofunctionscomparewiththesamechoiceofparameter(thesecondandthirdgraphs)
Notes Althoughboth and maybeeitherconcave orconvex,theyarenotequivalentfor , 1.Inotherwords,afunctionofform cannotbealternativelypresentedasafunctionofform .Otherformsfromstatisticaldistributionscouldbeusedtogenerateconvexorconcavecurves.
g(x;s=1/2) g(x;s=2) f(x;s=2)
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Name Logistic
Background Usedforage‐profiles,valueorefficiencyfallsslowlyatfirst,thenveryquicklyinitsmidlife,slowingdownagainneartheendoflife.Thepointatwhichtheassetdepreciatesofdeterioratequicklyisdeterminedbythelocationparameter,andthedifferencesbetweentheratesofdeclineacrosstheasset’slifearegivenbyashapeparameter.
Aswithsomeoftheotherage‐profilefunctionsdescribedhere,thelogisticfunctionneverreaches0ontherealline,moreoverthereisnovalueforwhichthelogisticfunctionequals1.Thuswehavetoimposetheserestrictionsartificially.
Reading Referencesinclude:
Meinenet.al(1998,p29)Formula Let bethelocationparameterand bethescaleparameter,thelogisticfunctionisgivenby
exp
1 exp
Noticethat foranyvalueof ,thuswerequirethat0 ,where istheasset
lifelength.Adefaultoptionmightbetoset .Theshapeparametercanthenbechosentoensurethat 0 1and 0.
Example Thefollowinggraphsillustratethelogisticfunctionwithvaryinglocationsandshapeparameters. isthemaximumlifelength.
m=T/2 s=T/10 m=2T/3, s=T/20 m=T/2, s=T/5
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0 T/2 T 0 T/2 T 0 T/2 T
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Name Normaldistribution
Background Thenormaldistributionisoftenusedtodescriberetirements.Itisspecifiedbyameanageofretirement,µ,andthestandarddistribution, ,ofretirements.Thenormaldistributionhastheusefulpropertythatapproximately95%ofretirementsoccurwithintwostandarddeviationsofthemeanandapproximately99%ofretirementsoccurwithinthreestandarddeviationsofthemean.Sincethereisnoimpliedmaximumorminimumlifelength,itisnecessarytousetruncationinapplications.FunctionsforthenormaldistributionareavailableinthebasepackageofR(www.r‐project.org).
Reading Referencesinclude:
OECD(2009,p118) NIST/SEMATECH(2012)
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htmFormula Thegeneralformulafortheprobabilitydensityfunctionisgivenby
/ 2
√2
1∅
where isthestandarddeviationand isthemean.When 1and 0thisisthestandardnormaldistribution.Thefollowingfunctionsareforthestandardform.
Thecumulativedistributionfunctionisdefinedas
Φ1
√2/2
Thesurvivalfunctionisthengivenby
1 Φ
andthehazardfunctionisgivenby
,∅Φ
Example Normaldistributionwith: , 1 where istheaveragelifelength.
Normaldistributionwith: , 2.5.
Density Survival Hazard
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L L L
Density Survival Hazard
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Name Truncatednormaldistribution
Background FunctionsforthenormaldistributionareavailablethroughtheRpackagemsm
(http://cran.univ‐lyon1.fr/web/packages/msm/index.html)
Reading SeeNormaldistributioninformation.
Formula Thegeneralformulafortheprobabilitydensityfunctionisgivenby
1∅
Φ Φ
where =scaleparameter, =locationparameter, =lowerlimit,and =upperlimit.Φ isthecumulativedistributionfunctionforthenormaldistribution,and∅ isdefinedasperthenormaldistribution.
Thecumulativedistributionfunctionforatruncatednormalisthendefinedas
ΦΦ Φ
Φ Φ
Thesurvivalfunctionisthengivenby
1 Φ
andthehazardfunctionisgivenby
,Φ
Example Normaldistributionwith: , 1, withcut‐offsofupperlimitof3,andlowerlimitof‐3timestheaveragelifelength,where istheaveragelifelength.
Normaldistributionwith: , 2.5,withcut‐offsofupperlimitof3,andlowerlimitof‐3timestheaveragelifelength,where istheaveragelifelength.
Density Survival Hazard
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L L L
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Name Log‐normaldistribution
Background Usedasaretirementdistribution.DefineX tobearandomvariablewithanormaldistribution.ThenY=exp(X)willhavealog‐normaldistribution.Similarly,ifYhasalog‐normaldistribution,thenX=log(Y)willhaveanormaldistribution.Thelog‐normaldistributiononlytakespositiverealvalues.Aswiththenormaldistribution,itisnecessarytotruncatethelog‐normaldistributionatsomepre‐defined and ,unlesstheminimumlifelengthiszero.
Functionsforthelog‐normaldistributionareavailableinthebasepackageofR(www.r‐project.org),orthroughtheRpackageFAdist(http://cran.r‐project.org/web/packages/FAdist/)whichincludesa3‐parameterversion.
Reading Referencesinclude:
OECD(2009,p118) NIST/SEMATECH(2012)
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htmFormula Fora3‐parameterlog‐normaldistribution,the probabilitydensityfunctionisgivenby
exp / / 2
√2
where ; , 0.Also =shapeparameter, =scaleparameterand =locationparameter.When 1and 0thisisthestandardlog‐normaldistribution.Thefollowingfunctionsareforthestandardform.
Thesurvivalfunctionisgivenby
1 Φln
where 0, 0,andΦisthecumulativedistributionfunctionofthenormaldistribution.Thehazardfunctionisgivenby
,
1∅
ln
Φln
where 0, 0,and∅istheprobabilitydensityfunctionofthenormaldistribution,andΦisthecumulativedistributionofthenormaldistribution.
Themeanisgivenby exp μ
Otherparameterisationscanbeused,e.g.aswithinOECD(2009,p118).
Example Log‐normaldistributionwith: 1, 1, where istheaveragelifelength.
Density Survival Hazard
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Name Gammadistribution
Background Thegammadistributiontakesallpositiverealvalues,soaswiththelog‐normaldistribution,itisnecessarytotruncateit andat ifthisisdifferentfromzero.FunctionsforthegammadistributionareavailableinthebasepackageofR(www.r‐project.org),orthroughtheRpackageFAdist(http://cran.r‐project.org/web/packages/FAdist/)whichincludesa3‐parameterversion.
Reading Referencesinclude:
OECD(2009,p118) NIST/SEMATECH(2012)
http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htmFormula Fora3‐parametergammadistribution,theprobabilitydensityfunctionisgivenby
Γ γ
where ; , 0.Also =shapeparameter, =scaleparameter, =locationparameter,and
Γ
When 1and 0thisisthestandardgammadistribution.Thefollowingfunctionsareforthestandardform.
Thesurvivalfunctionisgivenby
1ΓΓ
where 0, 0,andΓ istheincompletegammafunctiondefinedas
Γ
andthehazardfunctionisgivenby
Γ γ Γ
where 0, 0.
Themeanisgivenby .
Example Gammadistributionwith: 2, 1, where isthemedianlifelength.
Notes Schmalwasser,O.andSchidlowski,M.(2006,p9) describestheGammadistributionusedwithinGermanyforthemortality(density)functionbutwithadifferentparameterisation.Referringtotheirpapertheparametersare: , μ, .
Density Survival Hazard
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Name Winfreycurves(symmetric)
Background ThisisnamedafterRobleyWinfrey,whowasaresearchengineerattheIowaEngineeringExperimentationStationduringthe1930s.Winfreycollectedinformationondatesofinstallationandretirementof176groupsofindustrialassetsandcalculated18“type”curvesthatgavegoodapproximationstotheirobservedretirementpatterns.Therewere7symmetrical,6leftskewedand5rightskewed(seeWinfrey,1935;OECD,2009;andBlades,1998b).
Reading Referencesinclude:
Winfrey(1935).Link:http://www.scribd.com/doc/34898535/Statistical‐Analysis‐of‐Industrial‐Property‐Retirements‐Engineering‐Experiment‐Station‐Bulletin‐125‐Revised
OECD(2009,p115) Blades(1998b).Link:http://www.oecd.org/std/nationalaccounts/2662425.pdf Hypergeometricfunction:http://mathworld.wolfram.com/HypergeometricFunction.html
Formula TheformulaisgiveninWinfrey(1935)whileBlades (1998b)alsoprovidesanexcellentsummary.FortheWinfreysymmetricalcurves,theequationderivedbyWinfreyis
1
where istheordinatetothefrequencycurveatage (originatthemeanage), istheordinatetothefrequencycurveatitsmode,and ,and areparameterswhichdeterminethekurtosisandtheskewnessofeachcurve.Blade(1998b)givesexamplesforthechoiceofparameters.
Theexistingliteraturedoesnotgiveaformaldefinitionintermsofstatisticaldistributions,oranequivalentsurvivalorhazardfunction.However,afterinvestigationswecandemonstratethatthehypergeometricfunctionisactuallyequivalenttotheexpressionabovefortheWinfreycurves.Forexample,usingaspecificcaseandnotationofthegeneralisedhypergeometricfunction,thisgives
, ; ; 1
Sowithanappropriatechoiceofparameters,wemaketheconjecturethatthewellknownWinfreycurvesforthesymmetriccaseareactuallyequivalenttoahypergeometricfunctionwiththefollowingparameters
, ; ; 1
Sodefiningthisasourprobabilitydensityfunctionmeansthatthesurvivalandhazardfunctionwillhavetheform
1 , ; ; , ; ; andthen .
Example Examplewith 10, 3.7, 11.911 whichisequivalenttoS2inOECD(2009p117),whereistheaveragelifelength.
Density Survival Hazard
0.0000
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0.0075
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Name Winfreycurves(asymmetric)
Background SeeWinfrey(symmetric)fordetails.Winfrey(1935)givesdetailson18“type”curvesthatgavegoodapproximationstoobservedretirementpatterns.Therewere7symmetrical,6leftskewedand5rightskewed(seeWinfrey,1935;OECD,2009;andBlades,1998b).
Reading Referencesinclude:
Winfrey(1935).Link:http://www.scribd.com/doc/34898535/Statistical‐Analysis‐of‐Industrial‐Property‐Retirements‐Engineering‐Experiment‐Station‐Bulletin‐125‐Revised
OECD(2009,p115) Blades(1998b).Link:http://www.oecd.org/std/nationalaccounts/2662425.pdf Hypergeometricfunction:http://mathworld.wolfram.com/HypergeometricFunction.html
Formula DetailedformulafortheasymmetriccurvesaregiveninWinfrey(1935),whileBlades(1998b)alsoprovidesanexcellentsummaryanddiscussion.Whilesimilartothesymmetricalcurves,theasymmetricformulasaremuchmorecomplicated.Forexample,
1 1 1 1
where istheordinatetothefrequencycurveatage (originatthemeanage), , aretheordinatetothefrequencycurveatitsmean,and , , , , , , , , , areparameterswhichdeterminethekurtosisandtheskewnessofeachcurve.ExamplesforchoiceofparametersaregiveninWinfrey(1935,p98).Inpracticeitseemsthatthe signischangedinanarbitraryway.
Again,theexistingliteraturedoesnotgiveaformaldefinitionoftheWinfreycurvesintermsofstatisticaldistributions,ortheassociatedsurvivalorhazardfunctions.Basedontheapproachforthesymmetriccase,theasymmetriccurvescanberewritteninaformwhichincludescombinationsofthehypergeometricdistribution.However,thecombinationofatleastfourcurveswillleadtoacomplicatedsolution.Forexample,
, ; ; , ; ;
, ; ; , ; ;
Fromthis,thesurvivalandhazardfunctionscanbederivedthroughstandardmethods(e.g.integration)althoughtheywillbecomplicatedandnotreproducedhere.
Example DensityexampleswithdifferentparameterchoicesforL2(red),L3(blue),R3(black)andR4(green).Averagelifelength isalongthex‐axiswherewillbedifferentforeachdistribution.
Notes WhileWinfrey(1935)hasderivedacomplicatedformfortheasymmetriccurves,furtherworkcouldconsiderwherethiscouldbesimplified.Forexample,ratherthantheconvolutionoffourhypergeometrictermsperhapstheuseofjustoneorthemultiplicationoftwohypergeometricfunctionswithappropriateparameterchoicewouldprovideareasonablealternative.
0.000
0.005
0.010
0.000
0.005
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0.020
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5 Examples with cohorts
Onceanage‐profileforasingleandaretirementdistributionhavebeenfoundforaclassofassets,weareabletoderivetheage‐profileforacohortofassets.OECD(2009)definesacohortas:
Cohortofassets:Setofassetsofthesamekindandsameage,butnotnecessarilythesamelifespan.
Theage‐profileofacohortofassetsdescribestheproductivityorvalueofthewholecohortatanygivenpointoftime.FollowingthenotationusedinOECD(2009),let denotetheage‐profileofasingleassetand betheage‐profileofthecohort;each isgivenbytakingtheaverageofthe ofalltheindividualassetssurvivinginperiod ,weightedbytheretirementdistribution.Thismaybedonebydefiningadifferentage‐profileforeachretirementageinthedistribution,orbyusingasingleage‐profileforthehighestageintheretirementdistribution.Weexaminethesetwoapproachesinmoredetailbelow.Inbothcasesweneedtospecify:
Amaximumassetlife Afunctionalformfortheage‐profile Aretirementdistribution
Theretirementdistributionmayormaynotbetruncated.
Method1:seeOECD(2009,p118‐121).Foreachlife0 wedefineauniqueage‐profile(noticethatalltheage‐profilesabovearegivenasfunctionslifespan ),
, ,i.e.where , denotestheefficiencyorvalueofan yearoldassetwithlifespan.Thentheefficiencyorvalueofthecohortinyear isgivenbytheaverageof , ,for0 ,weightedbytheprobabilityofanassethavinglifespan ,givenby :
, ∙ (10)
wherewestarttheindexat because , 0forall .
Weillustratetheprocessinthefollowingtable.
year 0 1 2 ⋯ 1 ret.dist.
1 , 1 , 0 1
2 , 1 , , 0 2
⋮ ⋮ ⋮ ⋮ ⋱ ⋮
1, 1 , , ⋯ , 0
1
, 1 , , ⋯ , ,0
, , ,
⋯, 0
Weconjecturethatacontinuousanalogueofthismaybegivenby
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, ∙ (11)
where , isacontinuousage‐profilewith denotingageand ,retirementage.
Method2:seeOECD(2009,p121).Wedefineasingleage‐profileforanassetwithretirementage ,callit .Thistimetheage‐profileforanassetretiringatage
takesthefirst 1termsof andiszerothereafter.Asabove,thetermsofthecohortprofilearetheweightedaverageoftheindividualprofiles.Tocomparethiswiththepreviousmethodweconstructatableanalogoustotheoneabove,thoughthisisnotstrictlynecessary.
year 0 1 2 ⋯ n‐1 n ret.dist.
1 1 1
2 1 2
⋮ ⋮ ⋮ ⋮ ⋱ ⋮
n‐1 1 ⋯ 1
n 1 ⋯0
⋯ 0
Weseeinthelastlinethattheformulasimplifiesto
(12)
Infactthelastterm,∑ ,isadiscreteestimateoftheareaunderthePDFboundedby and ,whichisgivenpreciselybythesurvivalfunction, .Thispaperdoesnotprescribetheuseofoneortheother,howeverwedo,asformethod1postulatethatacontinuousanaloguemaybegivenby
(13)
Wenowpresentanumberofgraphsshowingafewdifferentcombinationsofage‐profileandretirementdistributions.Ineachcaseweassumeminimum,maximumandmeanassetlivesof2,16and9.5yearsrespectively,inlinewiththeexamplein(OECD2009,p118‐121)andillustratetheresultsofbothmethods,whilealsoreplicatingtheexamplewithinOECD(2009).Inallcases,graphaillustratesthecohortprofileresultingfrommethod1(inred)andaselectionofindividualassetprofiles(inblack);graphbcomparesthemethod1andmethod2cohortprofiles(redandbluerespectively)andtheindividualassetprofilecorrespondingtothemaximumlifelength(inblack);graphscanddshowthePDF(usedinmethod1)andsurvivalfunction(usedinmethod2)respectively.
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Linearage‐profile,normalretirement
Fortheage‐profileweassumedecayisnotdelayed,intheretirementdistributionweuseameanof9.5yearsandastandarddeviationof2.021inlinewith(OECD,2009p118‐121).Wenotethatthisgivesrisetoastrictlyconcavecohortprofileundermethod1andalogisticshapedcohortprofileundermethod2.
Geometricage‐profile,normalretirement
Anotherpopular,andindeedtheonlyotherprescribedwithinESA10 forthechoiceofage‐profileisthegeometricprofilewith 0.02 ⇒ 0.22.Thistimeboththecohortprofilesarebroadlygeometricinshapeandmoreconcavethanthepreviouscase.
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Linearage‐profile,Weibullretirement
Thistimewevarytheretirementdistribution,usingaWeibull distributionwithshape,scaleandlocationparameters: 17, 20,and 10respectively.Theoutcomeissimilartousingthenormaldistribution.
Linearage‐profile,normalretirement(specialcase)
Thisexampleisidenticaltothefirstexceptthatwehavereducedthestandarddeviationto0.1.Thisillustratesapotentialissuewiththemethodologysincethemethod1cohortprofilenowdropsalmostimmediatelytozero,lowerthananyoftheindividualprofiles.ThisoccursbecausethePDFis(closeto)zeroeverywhereoutsideasmallintervalaroundthemeanvalue,9.5,whichliesbetweentwointegers.TheconsequenceisthatthePDFiszeroforallthe(integer)values 1 , … , ,(seetableabove),usedinthecalculation.Ifthemeanhappenstofallnearaninteger,theoppositeoccursandthecohortprofileovershootsanyoftheindividualprofiles.ThisproblemisliabletooccurwhereverthePDFhasasteepgradientandmayberesolvedbyreducingthesizeoftheincrementsatwhichthecalculationiscarriedoutorusingthecontinuousversionoftheformula.
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Whatisclearfromthecohortprofilesaboveisthattheseoftenexhibitsimilarshapestothesingleassetprofiles,andweaskwhetheritmightbemorefeasibletoestimatecohortprofilesdirectly,ratherthanestimatingandcombiningseparateassetprofilesandretirementdistributions.Besidesbeingsimplertoimplement,thiswouldavoidanotherproblem:ifaretirementdistributionhasasmallstandarddeviation(orthelengthofperiodisgreat)andameanneartheminimumlifelength,itispossiblethatthecohortprofilewillbegreaterthan1evenwheresingleassetprofilesaremonotonicallydecreasing.Thisimpliesthattheefficiencyofacohortofassetscanexceedthatofeventhemostefficientassetinthecohort,whichisclearlyunrealistic,andarisesfromthediscretenatureofthecalculation.Similarlyitispossiblethatthereversecanhappenandtheefficiencyofthecohortislowerthaneventheleastefficientassetinthecohort.
6 Conclusion
Thispaperhasdescribedarangeofdistributionswhichareusedinpracticeforcalculatingestimatesofcapitalconsumption.Pointstohighlightinclude:
Inpractice,thechoiceofretirementdistributioncanbesomewhatarbitrary.Therearesomeconventionsandrecommendedpracticesbutthisdoesdependontheassetclassunderconsideration.
Priceandefficiencyprofilessharemanycommoncharacteristics.Theycannotbedefinedindependently,butonecanbederivedfromtheother.
Itisoftennecessarytotruncateretirementdistributionsinpracticalapplications.
Itmaybefeasibletoestimatetheage‐profileofawholecohortinonestep,ratherthancombiningasingleassetprofileandaretirementdistribution.
Usingasteepretirementdistributionatalowresolution(e.g.largeincrementsoftime)canproduceunreasonableresultsincalculatingcohortprofiles.
Furtherworkcouldexploretheimpactofthechoiceofretirementdistributionandsingleassetprofileondifferentcohortprofiles,andfromapracticalperspectiveofwhattypesofdistributionsareappropriatefordifferentassets,seeforinstance(Meinenet.al,1998p28‐34).
7 References and further reading
Biorn,E.(1998),Survivalandefficiencycurvesforcapitalandthetime‐age‐profileofvintageprices,EmpiricalEconomics,23,p611to633.Partialpdflink:http://link.springer.com/article/10.1007%2FBF01205997
Blades,D.(1983),Servicelivesoffixedassets,OECDWorkingpaper,March1983.Link:http://www.oecd.org/economy/productivityandlongtermgrowth/1915571.pdf
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Blades,D.(1998a),MeasuringDepreciation,SecondmeetingoftheCanberraGrouponCapitalStockstatistics.Link:http://www.oecd.org/std/na/2662743.pdf
Blades,D.(1998b),MortalityFunctionsforEstimatingCapitalStocks,SecondmeetingoftheCanberraGrouponCapitalStockstatistics.Linkhttp://www.oecd.org/std/na/2662425.pdf
Dey‐ChowdhuryS.(2008),Perpetualinventorymethod:methodsexplained,EconomicandLabourMarketReview,Vol2,No.9,September2008,OfficeforNationalStatistics.
McLellan,N.(2004),MeasuringProductivityusingtheIndexNumberApproach:AnIntroduction.NewZealandTreasury,Workingpaper04/05,June2004.Link:http://www.treasury.govt.nz/publications/research‐policy/wp/2004/04‐05/twp04‐05.pdf
Meinen,G.,Verbiest,P.,anddeWolf,P.(1998)PerpetualInventoryMethod:Servicelives,discardpattersanddepreciationmethods,OECD.Link:http://www.oecd.org/std/na/2552337.pdf
NIST/SEMATECH(2012)e‐HandbookofStatisticalMethods,Link:http://www.itl.nist.gov/div898/handbook/
OECD(1997)MortalityandSurvivalFunctions,CapitalStockConferenceMarch1997,Link:http://www.oecd.org/std/nationalaccounts/2666812.pdf
OECD(2009)MeasuringCapital:OECDManual,Secondedition.Link:http://www.oecd.org/dataoecd/16/16/43734711.pdf,Chapter13.1andalsoAnnexA.
OfficeforNationalStatistics(2014)CapitalStocksandConsumptionofFixedCapitalpublication,link:http://www.ons.gov.uk/ons/rel/cap‐stock/capital‐stock‐‐capital‐consumption/capital‐stocks‐and‐consumption‐of‐fixed‐capital‐‐2013/index.html
Oulton,N.,andSrinivason,S.(2003)Capitalstocks,capitalservices,anddepreciation:anintegratedframework,BankofEngland.
StatisticsCanada(2007)EconomicDepreciationandRetirementofCanadianAssets:AComprehensiveEmpiricalStudy,Link:http://www.statcan.gc.ca/pub/15‐549‐x/15‐549‐x2007001‐eng.pdf
Schmalwasser,O.andSchidlowski,M.(2006)MeasuringCapitalStockinGermany.Link:https://www.destatis.de/EN/Publications/Specialized/Nationalaccounts/MeasuringCapitalStockWista1106.pdf
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UnitedNations,SystemofNationalAccounts(SNA2008).Link:http://unstats.un.org/unsd/nationalaccount/sna2008.asp
West,P.andClifton‐Fearnside,A.(1999).ImprovingtheNon‐financialBalanceSheetsandCapitalStocksEstimates,OfficeforNationalStatistics.
Winfrey,R.(1935).StatisticalAnalysisofIndustrialPropertyRetirements,EngineeringExperimentStationBulletin125,Revised.Link:http://www.scribd.com/doc/34898535/Statistical‐Analysis‐of‐Industrial‐Property‐Retirements‐Engineering‐Experiment‐Station‐Bulletin‐125‐Revised
8 Acknowledgements
TheauthorswouldliketothankErikBiørn,SteveDrew,WesleyHarrisandJimO’Donoghuefortheirfeedbackandcommentsinpreparingthisarticle.Ofcourse,anyerrorsandomissionsareourresponsibility.
9 Disclaimer
Theviewsinthispaperdonotnecessarilyreflecttheofficialviews,endorsedorcurrentlyusedmethodsoftheOfficeforNationalStatistics.