Estimation of a production function with domestic and foreign capital stock

Ziesemer, Thomas Working Paper Estimation of a production function with domestic and foreign capital stock UNU-MERIT Working Papers, No. 2022-002 Provided in Cooperation with: Maastricht Economic and Social Research Institute on Innovation and Technology (UNU-MERIT), United Nations University (UNU) Suggested Citation: Ziesemer, Thomas (2022) : Estimation of a production function with domestic and foreign capital stock, UNU-MERIT Working Papers, No. 2022-002, United Nations University (UNU), Maastricht Economic and Social Research Institute on Innovation and Technology (UNU-MERIT), Maastricht This Version is available at: https://hdl.handle.net/10419/326811 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-sa/4.0/          #2022-002 Estimationofaproductionfunctionwithdomesticandforeign capitalstock  ThomasZiesemer            Published10January2022     MaastrichtEconomicandsocialResearchinstituteonInnovationandTechnology(UNU‐MERIT) email:[email protected]u|website:http://www.merit.unu.edu  Boschstraat24,6211AXMaastricht,TheNetherlands Tel:(31)(43)3884400 UNU-MERIT Working Papers ISSN 1871-9872 Maastricht Economic and social Research Institute on Innovation and Technology UNU-MERIT UNU-MERIT Working Papers intend to disseminate preliminary results of research carried out at UNU-MERIT to stimulate discussion on the issues raised.   1  Estimationofaproductionfunctionwithdomesticandforeigncapitalstock Thomas Ziesemer, Department of Economics, Maastricht University, and UNU-MERIT. Address: P.O.Box616,NL6200MDMaastricht.E‐mail:[email protected].ORCID:0000‐ 0002‐5571‐2238. AbstractWeestimateaCobb‐Douglasproductionfunctiondistinguishingbetweenadomesticanda foreigncapitalstockbuiltfromdataofimportedmachineryandtransportequipmentforBrazil.The preferredregressionusesloglevelsestimatedbyGMM‐HAC.Resultsarethattheelasticityof productionofforeigncapitalisabout40%ofthatofdomesticcapital,thefunctionhasconstant returnstoscaleincapitalandlabourvariables,andhumancapitalandtechnicalchangearealso highlyproductive.JELcodes:C22,C51,E23,F43,O54.Keywords:time‐series,estimation,production function,openeconomy,Brazil.  1. Introduction BardhanandLewis(1970)havemergedthetwo‐gapmodelofCheneryandBruno(1962)withthe growthmodelofSolow(1956).Theresultisaneoclassicalgrowthmodelwithadomesticanda foreigncapitalstock.1Theinvestmentintheforeigncapitalstockhastobepaidfor,byexports, soonerorlaterifforeigndebtisincludedasintheextensionsunderperfectorimperfectcapital mobility(Ziesemer1995,1998).Comparedtotheneoclassicalclosed‐economymodelthestrength ofthemodelisthat(i)itdealswithgrowththroughcapitalaccumulationlinkedtointernational tradeincapitalgoods,consumptiongoodsandforeigndebt,themajoraspectsofglobalization,and (ii)ithasasteady‐stategrowthratewithincomeandpriceelasticitiesofexportdemandbecause worldincomeenterstheexportfunction,besidestechnicalchange.However,therelatedproduction functionwithdomesticandforeigncapitalstockshasneverbeenpresentedasanempirical estimate.Suchanestimateisthecontributionofthispaper.  2. Themodel TheproductionfunctionthatweestimateinthispaperisoftheCobb‐Douglastype: 𝑌𝑒 𝐾 𝑒𝑇 󰇟𝐿󰇛1𝑢󰇜 𝐾 , (1) Ydenotesoutput,Kdisdomesticcapital,Kfforeigncapital,Hhumancapital,Llabourforce,uthe unemploymentrate,Thlabouraugmentingtechnicalchangetakingintoaccounthumancapitalinits calculation,caconstant,tatimetrend,andvtastochasticterm.Enteringhumancapitalinthesame formasBarroandLee(2013)doresultsinhavingβHinthelog‐linearregression;thisworksslightly betterherethanthelog(H)version. Takinglogsof(1)weget 𝑙𝑜𝑔𝑌  𝑐  𝑏𝑡  𝛼𝑙𝑜𝑔𝐾𝛽𝐻𝛾𝑙𝑜𝑔𝑇 𝜇𝑙𝑜𝑔𝐾 𝑣 (1’) Asthevariablesmayhaveunitroots,weshouldalsoconsiderestimatingthefunctioninfirst differences:  1Importedcapitalgoodsarealsoincreasinglyrecognizedasimportantintheheterodoxliterature(Blecker 2021). 2  𝑑𝑙𝑜𝑔𝑌  𝑏  𝛼𝑑𝑙𝑜𝑔𝐾 𝛽𝑑𝐻  𝛾𝑑𝑙𝑜𝑔𝑇𝜇𝑑𝑙𝑜𝑔𝐾 𝑣 𝑣(2) In(1’)and(2)residualsaredifferent.In(2)wehaveaspecialcaseofamovingaverage.The stochastictermmayhaveserialcorrelationwithlagsjandmovingaverageresiduals𝜀. WewanttolinkthemtoanARMAprocess(seeDavidsonandMacKinnon2004,chapter13) 𝑣∑𝜌   𝑣 𝜀 ∑𝜃𝜀   (3) Theprocesswithonlythefirstsumontheright‐handsideiscalledanautoregressiveprocessof orderj,ar(p),andwithonlythesecondsumitiscalledamovingaverageprocess,ma(q).Together theyarecalledanARMA(p,q)process.𝜌and𝜃willbefoundthroughtheestimation.2Therelated parametershavetobefoundintheestimationtogetherwiththeexponentialparameters,whichare elasticitiesofproduction.Alevelmodelwillthencombine(1’)and(3),andadifferencemodelwill combine(2)and(3)andistypicallycalledARIMAXmodel,wherethe‘I’standsfor‘integrated’and theXfortheregressorsotherthantheconstant.Notethatifthetimetrendisstatistically insignificantinthelevelmodel,thentheinterceptmaybestatisticallyinsignificantinthedifference model.Using(1’)anditslaggedform,weinserttheresidualsintotheautoregressiveprocess(3)and estimatethelevelmodel 𝑙𝑜𝑔𝑌𝑐𝑏𝑡𝛼𝑙𝑜𝑔𝐾 ,𝛽𝐻 𝛾𝑙𝑜𝑔𝑇 ,𝜇𝑙𝑜𝑔𝐾 , ∑𝜌   󰇛𝑙𝑜𝑔𝑌 𝑐𝑏𝑡 𝛼𝑙𝑜𝑔𝐾, 𝛽𝐻  𝛾𝑙𝑜𝑔𝑇 , 𝜇𝑙𝑜𝑔𝐾 ,󰇜𝜀 ∑𝜃𝜀   (4) Similarly,using(2)anditslaggedform,weinserttheresidualsintothedifferencedversionofthe autoregressiveprocess(3)andestimatethedifferencedmodel 𝑑󰇛𝑙𝑜𝑔𝑌 󰇜𝑏𝛼𝑑󰇛𝑙𝑜𝑔𝐾 ,󰇜𝛽𝑑󰇛𝐻 󰇜𝛾𝑑󰇛𝑙𝑜𝑔𝑇 ,󰇜𝜇𝑑󰇛𝑙𝑜𝑔𝐾 ,󰇜 ∑𝜌   󰇟𝑑𝑙𝑜𝑔𝑌𝑏 𝛼𝑑𝑙𝑜𝑔𝐾,𝛽𝑑𝐻 𝛾𝑑𝑙𝑜𝑔𝑇 ,𝜇𝑑𝑙𝑜𝑔𝐾 ,󰇠  𝑑𝜀∑𝜃𝑑𝜀   . (5) Wewillpresentestimatesofspecialcasesofthesemodelsinwhichsomeofthe𝜌,𝜃arezero.The residualin(5)isindifferences,whicharemovingaverages,indicatingoverdifferencing. Overdifferencingisnotaproblemiftheserialcorrelationistakenintoaccount(MaddalaandKim 1998).  3. Thedata WeusedataforBrazil.WetakeoutputasGDPinconstant2010localcurrencyunits,unemployment rates,andlabourforcedatafromWorldDevelopmentIndicators(WDI),(WorldBank2021).As unemploymentrateshavegaps,werunaregressionforOkun’slaw,makeaforecastanduseits valuestofillthegaps.WeusetechnicalchangedatafromZiesemer(2021)selectingtheelasticityof substitutionclosetounitywithaCESparameterof0.99correspondingalmostexactlyequaltothe Cobb‐Douglasfunctionbecauseitiscontainedthereinbothselectionprocedures;theselevelshave fallensinceabout1980inasimilarwayastheTFPdatafromPWT9.1,whichareconstructedslightly differently.Weconstructtheforeigncapitalstockfromimportedmachineryandtransport equipmentincurrent1000$(1989‐2020fromtheWorldIntegratedTradeSolution,WITS),multiplyit by1000andtheofficialexchangerate,dividebytheGDPdeflatorfromWDIandmultiplyitby100. Thenweapplytheperpetualinventorymethodtothisinvestmentvariable.Forthisweusethe averagedepreciationrateof4.25%fromPWT9.1fortheyears1989‐2017,whichalsoentersthe  2Forfirm‐levelpaneldataBlundellandBond(2000)useanar(1)process. 3  constructionoftheinitialvaluefoundasthe1989importedinvestmentgoodsdividedbytherateof depreciationplusagrowthrateof4.7%,whichisthecapitalgrowthrateforBrazilin1989in Ziesemer(2021).Thedomesticcapitalstockisobtainedinthesameway,basedongrossfixed capitalformationdatafromWDIdiminishedbytheimportedmachineryandtransportequipment; byconstruct,thisvariablealsoincludesinvestmentinbuildings.Humancapitaldataaretakenfrom PWT9.1;theyareconstructedasindexbetween1and5,andthereforecangrowonlytoitsupper limitandactasshiftersoftheproductionfunctionratherthanpermanentlygrowingfactors.  4. Econometrics,estimationresultsandinterpretation InTable1,column1,weshowaleast‐squaresestimateforthelevelmodel(4).Theelasticityof productionoftheforeigncapitalstockisonly40%ofthatofthedomesticcapitalstock,hereandin thenexttworegressionsincolumns2and3.Timetrendsareneversignificantbecauseweincludea technicalchangevariable.Allothervariableshaveelasticitiesofproductionascommoninthe literature.Thelowelasticityofproductionmaybeexplainedbyissuesoftechnologytransferas discussedintheliteratureonappropriatetechnology.Whenconsideringreturnstoscaleweshould notincludehumancapitalbecauseitsindexmaximumvalueof5doesnotallowtakingarbitrary multiples.Returnstoscaleareclosetounitywithap(crs)=0.8inaWaldtest. AllvariablesarenotindependentoftheGDPbecausetheyareendofperiodvaluesincludingcurrent investmentsandaretherebyendogenous.Therefore,weusealsothetwo‐stageleastsquares methodinthesecondcolumnandGMMinthethirdcolumnofTable1.Leastsquaresandtwo‐stage leastsquaresconsiderheteroscedasticityandserialcorrelationonlyinthestandarderrorsand covariances.GMMestimatorstakethemintoaccountalsointhecoefficientestimateshowninTable 1,column3.Theelasticitiesofproductionarenowhigherforhumancapitalandtechnicalchange andslightlylowerfordomesticcapitalandlabourvariablesthanthoseofTable1,column1and2. Theresultisclosetoconstantreturnstoscale. Thear(5)coefficientsprobablyindicateabusinesscycleeffect.3Aswedonotusedataformachine andlabourhours,factorsarefullyusedinbooms,butinotherperiodsthereisloweroutputwhereas theloweruseofthefactors’hoursisnotcapturedinourdata.Intheproductionfunctionasdefined forthesedata,theeconomyisbelowthefunction,notonit,wheneverfactorsrunlesshoursand thismayleadtoar(5)termsindicatingthatevery5yearstheeconomyisinasimilarsituation. Moving‐averagetermsmakeGMMestimatesdependentoninitialvaluesandtherebyunstable.We havedroppedthemfromtheanalysisofthelevelequationbecauseremovingdifferencinginthe formofar(1)terms(seeNau2020)doesnotsolvetheinstabilityproblem. Moreover,havingaGMMresultsuggeststhattheestimateofthecovariancematrixhasconverged. However,withadditionalobservationsitmightincreaseifthereareunitroots(Davidsonand MacKinnon2004,ch.14).Avector‐error‐correctionmodelbasedonaVARwithlaglengthonebased ontheSICcriterion(becauseofthelownumbersofobservations)wouldsuggestthatthemodelhas fivecointegratingequations(r=5)fromthetracetestortwo(r=2)fromthemaximum‐eigenvalue test,whichwouldimplyoneorfour(K‐r=6‐r)uniteigenvaluesinthesystemofcointegrated equations.ForaVECMestimate,wehaveatoosmallnumberofobservations.Therefore,wealso estimatedthedifferencedmodel(5),although(i)themodelsofcolumn1to3havebeentestedfor ar(1)termsandactuallyincludear(2)andar(5)termsand(ii)thevaluesofDurbin‐Watsonstatistic seemtoprecludefirst‐orderserialcorrelationasinthepresenceofunitrootswithoutcointegration. Asimpleleast‐squaresresultincolumn4ofTable1withoutanyarormatermsleadstovery  3Theeconometricsliteratureoftendiscussesthisunderseasonaleffects,whicharesimilartobusinesscycle effectswithalessclearnumberofperiodsincontrasttothefourseasons. 4  plausibleelasticityandconstant‐returns‐to‐scaleresultsbuttheDurbin‐Watsonstatisticsignalshigh first‐orderserialcorrelation.AddingARMA(p,q)terms(formatermsseenotestocolumn5,Table1) andusinginstrumentsagainstendogeneityagainwegettheresultsincolumn5ofTable1from2SLS (two‐stage‐leastsquares)estimation.Themovingaveragesintheresultsforadifferencedmodel usingGMM(notshown)turnouttobeinfirstdifferencesasinequation(5)andthissuggests removingthedifferencing(Nau2020),referringusbacktothelevelapproachasincolumns1‐3of Table1.Moreover,forthedifferencedmodelestimatedwithGMM‐HACestimatewecannotavoid theproblemofweakinstruments(notshown).  Table1:Estimationresultsforthelevelanddifferencedmodels Variable,MethodLeast sq.(a) 2SLS(b)GMM‐ HAC(c) Leastsquares (differenced)(d) 2SLS (differenced)(e) constant4.26 (2.33) 3.77, (1.68) 4.46 (3.02) ‐0.009, (‐1.00) ‐0.008 (‐2.51) logKd0.2865 (4.7) 0.3, (4.44) 0.283 (6.22) 0.256, (1.834) 0.287 (5.99) H0.278 (6.26) 0.264, (4.90) 0.283 (7.56) 0.363, (3.158) 0.35 (6.23) logTh0.697 (16.5) 0.69, (12.4) 0.71 (25.8) 0.6, (9.60) 0.75 (28.25) Log(L*(1‐u))0.625 (10.7) 0.615, (9.4) 0.61 (16.67) 0.6, (5.83) 0.539 (9.87) logKf(‐2)0.115 (3.61) 0.123, (3.94) 0.12 (5.18) 0.16, (1.83) 0.226 (5.16) ARterms𝜌=0.34 (2.52) 𝜌=‐0.2 (‐2.16) 𝜌 =0.349, (2.49) 𝜌=‐0.196, (‐1.96) 𝜌=0.358 (2.91) 𝜌=0.21 (3.85) ‐ 𝜌=‐0.357 (‐1.87) Adjustedsample1995‐20171995‐20171995‐20171991‐20171997‐2017 Adj.R‐sq., J‐stat.(p(J)) 0.9995, ‐ 0.9995, 11.74(0.3) 0.9995, 8.31(0.6) 0.92, ‐ 0.9877 12,(0.446) Durbin‐Watsonst.2.162.132.171.412.64 Andrewsbandwith,(f)1.17651.0751.17163.582.8121 returnstoscale(g)1.02651.0381.0121.021.052 DependentVariable:LOG(Y);d(log(y))incolumn4and5.T‐valuesbelowcoefficientsinparenthesis.HAC standarderrors&covariance(Bartlettkernel).(a)ARMAConditionalLeastSquares(Gauss‐Newton/Marquardt steps);p≤0.0474.(b)Instrumentspecification:C,LOG(KD(‐1)),(H(‐1)),LOG(TH099(‐1)),LOG(L(‐1)*(1‐U2(‐1))), LOG(KF47(‐2));constantinsignificant,otherp≤0.069;Laggeddependentvariable&regressorsfromar(2)and ar(5)termsaddedtoinstrumentlist;noIVdropped.(c)s.e.heteroscedasticity&autocorrelationconsistent; weightingmatrix:HAC(Bartlettkernel,Andrewsbandwidth=1.23;InstrumentspecificationC,LOG(KD(‐3)), (H(‐3)),LOG(TH099(‐3)),LOG(L(‐3)*(1‐U2(‐3))),LOG(KF47(‐2))isstrongerthanusinglag1;Sequential1‐step weightingmatrix&coefficientiteration;MABackcast:19931994;p≤0.0086;laggeddependentvariable& regressorsfromar(2)andar(5)termsaddedtoinstrumentlist;noIVdropped.(d)LeastSquares;Newey‐West HACstandarderrors&covariance(Bartlettkernel);constantinsignificant,otherp≤0.1.(e)MABackcast: 1992‐1996;MAterms𝜃=‐1.18(‐145.9),𝜃=0.274(33.28);instruments:allregressorswithlags1,2,5,6 exceptd(log(Kf47(‐2)))(automaticallyde‐selected).(f)Bartlettkernel.(g)SumofcoefficientsofKd,Kf,labour.  5  Aregressionsufferinglessfromweakinstrumentsisthe2SLSregressionincolumn5ofTable1.The instrumentsfortechnologyandlabourareweak,buttheregressiondoesnotdependoninitial values.Allcoefficientsareabithigherhereandwefindslightlyincreasingreturnstoscale,although withap=0.397fortheconstantreturnshypothesis.AnotherweakpointclearlyistheDurbin‐ Watsonstatisticof2.64.Butthe2SLSregressionforthedifferencedmodelstillshowsthatincaseof unitrootsanddifferencingthereisareasonableestimate.Thedifficultyhereisthechoiceofvalid instruments.DenotingtheregressormatrixasX,ourendogeneityassumptionisE(X’u)>0,implying E(X(‐l)’u(‐l))>0.UsinglagsasinstrumentsimposesE(X(‐l)’u)=0.Theinstrumentselectionpresented innote(e)toTable1takesthisintoaccountbywayofusinginstrumentswithlagq+1formaterms withq=1,5. Insteadofchoosingbetweenlevelanddifferencemodelwecanalsoestimate(4)and(5)asasystem ofequations,wherethecorrespondingdifferenceandleveltermshavethesamecoefficientsexcept fortheconstants.Table2containstheresults.Theweighted‐least‐squaresmethodmultipliesthe equationbyestimatedinverseconditionalvariances.TheSURmethodtakesintoaccountthe contemporaneouscorrelationsoftheresidualsofthetwoequations.The3SLSmethodalsodoesso butusesalsoinstrumentstodealwithendogeneity.THEGMM‐HACmethodusesinstrumentsand usesheteroscedasticityandserialcorrelationconsistent(HAC)estimationforcoefficientsand standarderrorsincludingcontemporaneouscorrelationoftheresiduals.Byandlarge,variableshave similarsizeofcoefficientsacrossmethodsinTable2.Theexceptionsarethelabourcoefficientin column5,whichisabithigherandtheforeigncapitalcoefficientsincolumns3and5,whichareabit lower. TheJ‐statisticfortheIVestimatorsshouldbechi‐squaredistributed.Itshouldnotbetoohighandits p‐valuethereforenottoolowtobeinthechi‐squaredistribution(DavidsonandMacKinnon2004); butitshouldalsonotbetoolowanditsp‐valuetoohigh,becausethatwouldmeanthatinstruments dotoolittle(Roodman2009).Theseresultsjudgeaboutinstrumentsandspecification.AllIV estimationsinTable1haveareasonablep(J)andtherearenoweakinstruments.OnlytheGMM‐ HACestimatestakeheteroscedasticityandserialcorrelationintoaccountinaconsistentway.Ap(J) =0.598incolumn3ofTable1indicatesavoidingatoohighortoolowJ‐statistic.TheGMM‐HAC estimateinlevelsisourpreferredregressionbecauseinstrumentsarewellcorrelatedwith regressors(seeappendix),estimationisheteroscedasticityandserialcorrelationconsistent(HAC), andtheJ‐statisticisneithertoohighnortoolow.Ithasalmostconstantreturnstoscale4anda strongimpactoftechnicalchangeandhumancapitalandtherebydynamicallyincreasingreturnsto scale.Theelasticitiesofproductionofcapitaladdupto0.4,astandardvalueforaggregatecapitalin theliterature(seePerkinsetal.2013).OtherestimatorsinTable1and2donothaveallofthese propertiesbutstillshowsimilarresults.ThesystemversionofGMM‐HACinTable2suffersfroma lowJ‐statisticandahighp(J)indicatingthatinstrumentsdotoolittleofbiascorrection. Nevertheless,thedifferencedsingleequationmodelandthesystemmodelsupporttheideaof havingaproductionfunctionwithforeigncapital. ModerndevelopmentsineconometricshavemovedfromAR(I)MAmodelstothemoregeneralcase ofARDLmodelsincludingerror‐correctionmodels(seeChoetal.2021).However,inourcaseofa lownumberofobservationswhere(V)ECMsdonotwork,AR(I)MAXmodelsmaybeagoodwayout becausetheyestimatelessparameters.  4Thep‐valueforthehypothesisofconstantreturnstoscaleis0.8,implyingthatthecrshypothesiscannotbe rejected. 6  Table2:Estimationresultsforthesystemmodel Variable,MethodIterativeLSWeight.LS(a)SUR(b)3SLS(c)GMMHAC(d) Constant (differencedeq.) ‐0.0023 (‐1.09) ‐0.0027 (‐1.4) ‐0.0007 (‐0.95) 0.0002, (0.106) ‐0.003 (‐6.44) Constant(leveleq.)5.935 (1.67) 4.0 (1.79) 5.267 (2.41) 3.71 (1.84) 3.08 (3.04) logKd0.242 (1.94) 0.293, (4.01) 0.246 (3.61) 0.292, (4.62) 0.25 (8.56) H0.274 (4.01) 0.259, (5.14) 0.309 (6.19) 0.26, (5.56) 0.258 (12.22) logTh0.601 (17.29) 0.673, (26.2) 0.693 (27.18) 0.688, (28.76) 0.617 (47.4) Log(L*(1‐u))0.6 (8.84) 0.613, (11.97) 0.6716 (13.21) 0.634, (12.14) 0.8 (40.2) logKf(‐2)0.122 (2.27) 0.128, (3.95) 0.089 (3.05) 0.124, (4.26) 0.089 (5.54) ARtermsdiff.eq.‐ 𝜌‐0.333 (1.85) 𝜌 0.333 (‐2.06) 𝜌‐0.25 (‐1.85) 𝜌0.4 (2.976)  𝜌0.249 (4.16) ARtermsleveleq. 𝜌0.856 (7.64) 𝜌0.377 (3.55) 𝜌‐0.176 (1.7) 𝜌0.31 (3.83) 𝜌‐0.134 (‐1.91) 𝜌0.375 (4.31) 𝜌‐0.17 (‐2.22) 𝜌0.769 (29.16) Adjustedsample1991‐20171992‐20171995‐20171995‐20171992‐2017 Det.resid.covariance2.39E‐105.11E‐101.24E‐102.06E‐105.35E‐10 Adj.R‐sq.:diffeq. Leveleq. 0.92, 0.999 0.913, 0.9995 0.93, 0.9995 0.943, 0.9995 0.898, 0.99925 Durbin‐Watson:diffeq Leveleq 1.41, 1.45 1.985, 1.955 1.75, 1.93 2.31, 2.11 2.03 2.32 returnstoscale(g)0.9651.0341.0071.051.052 (a),(b),(d):Iteratecoefficientsafterone‐stepweightingmatrix.(c)Sequentialweightingmatrix&coefficient iteration.Instruments:C,D(LOG(KD(‐4))),D(H(‐4)),D(LOG(TH099(‐4))),D(LOG(L(‐4)*(1‐U2(‐4)))),D(LOG(KF47(‐ 2)))fordifferencedequationC,LOG(KD(‐1)),(H(‐1)),LOG(TH099(‐1)),LOG(L(‐1)*(1‐U2(‐1))),LOG(KF47(‐2))for levelequation,andlaggeddependentvariablesandregressorsfromartermsaddedtoinstrumentlistoflevel anddifferenceequation;lag4instrumentsarestrongerfordifferenceequationthanlag3.(d)Kernel:Bartlett, Bandwidth:VariableNewey‐West(11),Noprewhitening;J‐statistic0.116,p(J)=1;Instruments:C,D(LOG(KD(‐ 2))),D(H(‐2)),D(LOG(TH099(‐2))),D(LOG(L(‐2)*(1‐U2(‐2)))),D(LOG(KF47(‐2)))forthedifferenceeq.,C,LOG(KD(‐ 2)),(H(‐2)),LOG(TH099(‐2)),LOG(L(‐2)*(1‐U2(‐2))),LOG(KF47(‐2)),andlaggeddependentvariablesand regressorsfromartermsaddedtoinstrumentlistoflevelanddifferenceequation.  5. Conclusionandsuggestionsforfurtherresearch Ourestimatesshowthataproductionfunctionwithforeigncapitalasusedbymodelsbasedon BardhanandLewis(1970)canbedefendedasempiricallyrealisticandthereforethemodelshavea solidempiricalbasis.However,theautoregressiveprocessesemployedhereareundersuspicionof hidingmis‐specificationinsomepartsoftheliterature.Thiswouldbethecaseifactuallythe productionfunctionsareofamoregeneralCESorVEStype.Theelasticitiesofproduction,whichare