DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń Błażej Mazur Cracow University of Economics

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1 DYNAMIC ECONOMETRIC MODELS Vol. 7 Ncolaus Copercus Uversy Toruń 2006 Cracow Uversy of Ecoomcs Imposg Ecoomc Resrcos a VECM-form Demad Sysem Demad sysems have bee used appled demad aalyss for abou ffy years. I s usually assumed ha boh prces of goods uder cosderao ad he oal expedure are exogeous, whereas quaes demaded are edogeous. Equaos of such a sysem are erpreed as (rasformed) demad fucos. Properes of he fucos ca be derved from he sadard mcroecoomc heory, resulg complcaed (possbly cross-equao) resrcos lkg srucural parameers of he model. Such resrcos correspod o he seg of a sgle uly-maxmzg age. Whe aggregae daa are used, he represeave-age assumpo s mplcly roduced o aalyss. Imposg ad esg of he ecoomc regulary resrcos s herefore val order o maa ecoomc erpreao of he resuls obaed. Tradoal me-seres applcaos he ve assume red-saoary of all he varables. Moder echques of dyamc aalyss accou for osaoary of he macroecoomc me seres. Exogeey resrcos are also que demadg, especally wh respec o he oal expedure. A approach addressg all he ssues has bee developed by Pesara ad Sh (2002). They formulae a sadard model used dyamc macroecoomercs, amely he VECM model, for all he varables. Ecoomc srucure correspodg o a demad sysem s he mposed o coegrag vecors. I he seg, I() o-saoary of he observed me seres s ake o accou ad o arbrary exogeey assumpos are made. Uforuaely, oly lear coegrag relaos ca be modeled wh he sadard VECM framework. The requreme lms he scope of he demad sysem fucoal forms ha ca be employed wh he VECM approach. Copyrgh by The Ncolaus Copercus Uversy Scefc Publshg House

2 270 Wh log-ru srucure erpreed as correspodg o equaos of a demad sysem, all resrcos meoed above have o be mposed o elemes of he coegrag vecors. Mehods of mposg ad esg such cosras (wh maxmum lkelhood ferece mehods) are revewed Boswjk ad Doork (2004). I commo macroecoomc applcaos however, formao cocerg ecoomc srucure of a model s ofe que weak. Coversely, he case dscussed here, ecoomc resrcos are que sophscaed, wh cross-equao or hghly o-lear cosras. The cosras become more complcaed as he umber of goods creases. Applcaos of VECM modelg dscussed here dffer from he sadard oes degree of complexy of he srucure mposed. Oe of he cosequeces of he fac s ha sadard ecoomerc sofware packages cao be used. The paper preses a applcao of he VECM-demad sysem approach wh he emphass o mposg ecoomc resrcos. Number of he aggregae goods cosdered s such ha full VECM model s formulaed for welve varables, whch s more ha mos llusrave applcaos. A covee fucoal form of he demad sysem s employed, amely he Geeralzed Addlog model of Bewley (986). Equaly resrcos are mposed ad esed wh ML mehods, ad some resuls cocerg equaly cosras are also preseed. Sce he laer pose serous heorecal problems for he ML-based sascal ferece, oly some ad-hoc echcal resuls are preseed ha case. The pla of he paper s as follows. Frsly, he Geeralzed Addlog fucoal form s preseed ad a parcular formulao of he ecoomc resrcos he case s dscussed. Secodly, a VECM model s brefly roduced, ogeher wh some mehods of mposg log-ru cross-equao lear ad o-lear resrcos. Thrdly, he VECM-demad sysem approach of Pesara ad Sh s descrbed. Fally, emprcal applcao of he mehods s preseed ad summarzg remarks coclude.. Properes of Demad Sysems ad he GADS Fucoal Form Le w be a -vecor of he observed expedure shares correspodg o (aggregae) commodes, p deoe -vecor of he correspodg prce dces, ad le μ represe correspodg oal expedure. A demad sysem (wh sochasc srucure omed) ca be specfed as: f ( w ) = f ( ω ( p, μ ; θ )), =,, T () where ω (.) s -vecor of fucos represeg heorecal expedure shares, θ s k-vecor of he srucural parameers. The sysem represes a se of demad fucos (rasformed) share form, where vecor fuco f(.) accous for possble rasformao of he observed ad heorecal shares. Demad sysem fucoal forms are usually derved a share form because of he coveece Copyrgh by The Ncolaus Copercus Uversy Scefc Publshg House

3 Imposg Ecoomc Resrcos a VECM-form Demad Sysem 27 of he ecoomc dualy echques (see e.g. Pollak ad Wales (992)). Shares are somemes rasformed wh fuco f(.) order o acheve leary parameers. Aoher reaso for he rasformao s he fac ha shares are resrced o he u erval, whereas addve ormally dsrbued error erms (aag ay real values) are ofe roduced, wh rasformao used o recocle he dscrepacy (see e.g. Fry, Fry ad McLare (996)). The Geeralzed Addlog demad sysem form, descrbed deal by Bewley (986), wh f (.) beg -h eleme of f(.), ca be wre as: ( () ) μ f = + + ω. χ π j l pj θ l, =,,, (2) s P where P s represes Soe prce dex gve by: s l P = w l p, (3) j j ad w deoes vecor of he average shares. Observed shares are rasformed: q ( ) f = w w l, (4) + w where w deoes vecor of he average shares (as above), ad w + s defed as: + = l w w l w, j j wh q represeg observed quay demaded of -h aggregae good. Theorecal ad observed shares rasformed he way descrbed above ca aa ay real vales (o resrced o he u erval). Fucoal form of heorecal shares ω (.) mus sasfy cera properes order o be meagfully erpreed as a demad fuco (see e.g. Pollak ad Wales (992)). These properes are: addg up: heorecal shares mus sum o oe for ay p ad μ, homogeey: demad fucos are homogeous of degree oe w.r.. p ad μ, cosequely shares are homogeous of degree zero w.r.. p ad μ, symmery: correspodg Slusky marx has o be symmerc, egavy: ad egave sem-defe (see e.g. Mas-Collel, Whso, Gree (995)). The frs propery arses solely from he fac ha share form s aalyzed. The secod propery reflecs o moey lluso effec ad s expeced o hold eve aggregae (for heerogeeous cosumers). The las wo properes reflec Copyrgh by The Ncolaus Copercus Uversy Scefc Publshg House I fac w ca represe ay specfed shares. Usually w s defed as a sample mea, bu s also assumed o-radom whch s o fully cohere. Sample mea s used as some proxy for a meagful ypcal suao, whch s mpora because ecoomc characerscs esmaes are calculaed a w.

4 272 he represeave-age assumpo (hough here are some resuls o symmery of he aggregae demad fuco, see Dewer (980)). Addg-up ad symmery geerae cross-equao equaly cosras. Negave semdefeess of he Slusky marx s o ofe addressed appled work. Ths s because whereas he frs hree properes resul equaly paramerc cosras, egavy requres complcaed equaly cosras ha are dffcul o deal wh he coex of ML ferece. I he GADS model π j = 0, χ j = 0, θ j = s requred for addg-up propery. Homogeey s sasfed gve ha: π = 0. j Symmery of he Slusky marx ca be mposed oly locally (for cera values of p ad μ ). I s sasfed a a po correspodg o ω (.) = w f: π j = π j. I such approach, shares gve by w are dsgushed ad parameers of (2) ca be erpreed as ecoomc characerscs evaluaed a w (see Bewley (986)). Parameers π correspod o elemes of he Slusky marx, whereas j θ represe margal shares. Oe of he shares s deermed by values of he remag - shares. I order o avod sgulary of he coemporaeous varace-covarace marx, oe equao s dropped from he sysem, ad s parameers are fully deermed by parameers of he remag equaos by meas of he addgup resrco. Cosequely, oly - equaos are acually esmaed. Basc characerscs of he demads ca be also calculaed; oal expedure elascy ad ow prce elascy evaluaed a w are gve by: θ π ξ =, ξ = w w θ. 2. ML Iferece VECM Model wh Resrced Log-ru Srucure I he seco VECM model s brefly roduced order o esablsh he oao, ad basc facs cocerg esmao of resrced log-ru srucure are preseed. The exposo follows ha of Boswjk ad Doork (2004). Sadard coegraed VECM model for p varables wh k lags he orgal VAR model ad ormal errors ca be wre as: Copyrgh by The Ncolaus Copercus Uversy Scefc Publshg House

5 Imposg Ecoomc Resrcos a VECM-form Demad Sysem 273 k Δ = Π + x x Γ Δx + Ψg +, ~ N ( 0, Ω ε ε ) = where: Π = αβ ', x ( x ', d ')', rak ( Π) = r, 0 < r < m, = ad d are deermsc erms resrced o he coegrao space, whereas g represes uresrced deermsc compoes; α ad β are assumed o have full colum rak equal o r. I he seg s assumed ha all varables x are I() o-saoary, so I(2) ad seasoal o-saoary ad coegrao are ruled ou for smplcy. Ω s assumed o-sgular, al codo marx X 0 s assumed o-radom. Elemes of α ad β represe log-ru dyamcs of he sysem: colums of β are erpreed as coegrag vecors (.e. parameers of he log-ru relaos beg saoary lear combaos of varables x ), whereas elemes of α rasfer mpac of devaos from log-ru equlbrum oo curre dyamcs of he varables. Log-ru weak exogeey of varables he sysem resuls zero resrcos o he correspodg rows of α. Parameers α ad β are udefed whou furher resrcos. Sadard cosras mposg jus-defyg resrcos o α ad β, ogeher wh mehods of esmao ad asympoc ferece o coegrao rak r, kow as Johase s procedure are descrbed e.g. Johase (996). Uforuaely, such cosras are of a purely sascal org, whou ay ecoomc erpreao. Pesara ad Sh (2002) advocae srucural approach, where jus-defyg ad over-defyg resrcos are mposed a way ha allows for ecoomc erpreao of he log-ru srucure. A mehod of mposg ad esg geeral lear resrcos o α ad β s descrbed by Boswjk ad Doork (2004), wh α ad β formulaed as: vec ( β ) = Hφ + h0 vec( α' ) = Gψ, where G, H ad h 0 are kow cosa marces, wh φ ad ψ coag uresrced parameers ad o resrcos lkg α ad β. Maxmum lkelhood esmaes of φ ad ψ, provded ha defcao codos are sasfed, ca be foud usg a swchg-algorhm of Oberhofer-Kmea ype, wh aalyc codoal esmaors of φ, ψ ad Ω evaluaed sequeally a each sep: ψ ( φ, Ω) = [ G' ( Ω β ' S ) ] ( ) ( ˆ β G G' Ω β ' S vec Π' LS ) φ( ψ, Ω) = [ H '( α' Ω α S ) ] '( ' )[ vec( ˆ H H α Ω α S Π' LS ) ( α I ) h0 ] Ω( φ, ψ ) = S 00 S 0βα' αβ ' S0 + αβ ' Sβα' where LS subscrp deoes OLS esmaes obaed from uresrced verso of (5), ad S, S 00, S 0 beg cera sample produc marces defed Johase s procedure. Wh α, β beg fucos of ψ, φ ad mos rece esmaes of Ω, ψ, φ used a each sep, he algorhm coverges a ML esmaes. Copyrgh by The Ncolaus Copercus Uversy Scefc Publshg House (5)

6 274 Such procedure s much less compuaoally expesve ha drec maxmzao of he coceraed log-lkelhood fuco w.r.. ψ ad φ. Neverheless, pracce seems ecessary o use some proposal desy o draw sarg pos Ω 0 ψ 0, φ 0 ad ru he algorhm for several hudred mes some cases. Esmaed asympoc varace-covarace marx of he ML esmaor ca be compued as a verse of he formao marx calculaed a ML esmaes accordg o he formula: ψˆ ML G' ( Ω β ' Sβ ) G G' ( Ω α β ' ) H V = ˆ. as φml T H '( α' Ω Sβ ) G H '( α' Ω α S) H Maxmum value of he coceraed log-lkelhood (up o a addve cosa) ca be obaed from: T l c ( φ, ψ ) = l de( Ω), 2 by replacg Ω wh s ML esmae obaed las sep of he algorhm. Sadard lkelhood-rao es sascs ca be used order o es varous esed formulaos (codoally o r ad k), sce follows asympoc chsquared dsrbuo wh degrees of freedom equal o he umber of paramerc resrcos deals are provded e.g. Boswjk ad Doork (2004). Whe olear resrcos o α ad β are cosdered, umercal maxmzao of he above coceraed log-lkelhood s requred, where: Ω ( φ, ψ ) = S 00 S 0βα' αβ ' S0 + αβ ' Sβα', wh α ad β beg o-lear fucos of ψ ad φ. 3. Dyamc Demad Aalyss wh GADS VECM Form VECM-demad sysem approach of Pesara ad Sh (2002) s based o he assumpo ha colums of β ca be erpreed as correspodg o equaos of he sysem (). Geerally, wh uresrced α, r 2 resrcos o β are ecessary o provde defcao of he parameers. As a example cosder β = [-I B] ha s, cossg of (mus) dey marx ad block of varao-free parameers B. Such a srucure geeraes log ru SURE-lke srucure. I equlbrum β x * = 0 whch s parallel o he srucure geeraed by (), (2) ad (3) wh all he varables o he rgh-had sde. Defe x as: μ x ( ) ( ) ' = f w K f w l p L l p l, s P wh d =, x * = (x ) ad arrage β as: Copyrgh by The Ncolaus Copercus Uversy Scefc Publshg House

7 Imposg Ecoomc Resrcos a VECM-form Demad Sysem π L π θ χ π 2 L π 2 θ 2 χ 2 β ' =. M M O M M M 0 0 L π, L π, θ χ Oe (-h) equao s dropped from he sysem o avod sgulary of Ω. If he rue coegrao rak r s equal o ad deermsc compoes are properly specfed, coegrag vecors ca be erpreed as equaos of he GADS sysem. Whe uresrced cosa s roduced o he VECM model, oe row of β coag χ parameers s dropped, esmaes of he laer ca be derved from esmaes of Ψ, see Johase (996). Whe elemes of H ad h 0 are properly arraged, homogeey ad symmery resrcos descrbed above ca be mposed or esed; G s equal o dey marx or cera colums are dropped, f exogey resrcos are mposed. Symmery ca be mposed oly locally a w. I ca be see ha β coas block of π j parameers correspodg o elemes of he Slusky marx. Negavy (mposed locally a w ) requres square block of he parameers wh boh dces ragg from o o be egave defe. Such resrco s cross-equao o-lear equaly cosra. As meoed above, formal reame of equaly cosras wh ML ferece poses que advaced problems. I he paper we descrbe a mehod ha ca be used o mpose he resrco. I pracce obaed esmaes are a a border soluo, so s formal saus s o dscussed here. L. J. Lau (978) proposed a mehod of mposg cocavy cosras usg Cholesky facorzao. Ay (m m) symmerc square marx A ca be facorzed as A = L D L, wh D beg a dagoal marx wh m ozero elemes ad L beg a lower-dagoal marx wh oes o he dagoal. If A s egave defe, all dagoal elemes of D are egave. Wh such facorzao, he meoed above block of β ca be wre as a fuco of m egave elemes of D ad (m 2 - m)/2 varao-free elemes of L. Ths raslaes complcaed o-lear resrcos o a smple cosra o m = ( ) parameers. Numercal opmzao of he log-lkelhood fuco wh respec o free elemes of β (wh o cosras o α, he laer ca be coceraed ou) leads o po esmaes of he GADS parameers ha sasfy (locally) all he regulary codos. Ths ca be useful f some furher applcao requre fully regular po esmaes of he Slusky marx. Copyrgh by The Ncolaus Copercus Uversy Scefc Publshg House 4. Emprcal Applcao: Aalyss of he Aggregae UK Daa The daase aalyzed here was used by Deschamps (2003). I cosss of 72 quarerly deseasoalzed observaos coverg perod of 955: 997:4. Sx

8 276 cosumpo caegores are cluded: food, drks, foowear ad clohg, eergy, oher o-durables, re. w Case A: Esmaed β wh k = 2, r = 5, logl = f (w ) f 2 (w 2 ) f 3 (w 3 ) f 4 (w 4 ) f 5 (w 5 ) l p l p l p l p l p l p l (μ/p s ) Case B:Esmaed β wh k = 2, r = 5, logl = 0966 l p l p l p l p l p l p l (μ/p s ) (shaded eres correspod o resrced parameers) Case A Esmaed come ad ow prce elascy, FOOD DRINK CLOTH ENE OTH RENT ξ Sd err ξ Sd err Case B Esmaed come ad ow prce elascy FOOD DRINK CLOTH ENE OTH RENT ξ ξ Copyrgh by The Ncolaus Copercus Uversy Scefc Publshg House Properes of he daase wh respec o coegrao ad aggregae demad aalyss are subjec o dealed aalyss Mazur (2005). Here ca be meoed ha he varables aalyzed seem o be I() o-saoary wh lear

9 Imposg Ecoomc Resrcos a VECM-form Demad Sysem 277 reds, ad he umber of coegrag relaos ca be assumed o be 5, whch s accordace wh VECM-demad erpreao. Wh k = 2 ad uresrced cosa VECM, wh jus defed parameers, maxmzed log-lkelhood s 02. Wh homogeey oly mposed (5 cosraed parameers) becomes 004, wh symmery ad homogeey (5 cosraed parameers) s All he resrcos are herefore rejeced asympocally wh LR es (codoally o seleced values of r ad k). Balcombe (2004) repors smlar resuls, arbug he rejeco o small sample properes of he es procedures. Below are preseed resuls of he esmao wh homogeey ad symmery mposed, wh o exogeey resrcos (Case A) ad wh addoal mposo of egavy (Case B). I ca be see ha mposg egavy resuls esmaes of ow-prce elasces ha are more accordace wh ecoomc uo ha case A. Esmaed come elasces have que plausble values case A; mposg egavy seems o fluece esmaed demad elascy of he oher goods. Esmaed values of he characerscs he resrced case are dffcul o erpre, sce here s o heory supporg calculao of he sadard errors. Soud ecoomc erpreao would requre cosderg oher fucoal forms ad broader rage of deermsc specfcaos. All he calculaos were coduced usg auhor s roues wre Ox (Doork (2002)). I should be oed ha wh 6 aggregae goods, srucure of he resrco marces G ad H s oo complcaed o be coformable wh sadard ecoomerc packages. Numercal opmzao ecessary o mpose egavy cosra was coduced wh BFGS algorhm beg par of he Ox sysem. 5. Coclusos The paper dscussed problems of mposg ad esg ecoomc regulary resrcos VECM-demad sysems. I radoal demad aalyss, regulary resrcos were ofe sascally rejeced bu everheless mposed (see Keuzekamp ad Bare (995)). I he moder, dyamc approach, smlar ssues seem o arse. Such a resul s perhaps due o small sample dsoros, so exac ferece echques should be employed a example usg boosrap s provded by Balcombe (2004). Aoher sascal problem s ha of esg equaly cosra (lke he egavy cosra arsg demad aalyss). ML ferece echques seem usasfacory he case. Iferece he VECM model s sequeal s separaely decded wha lag legh should be used, wha s he coegrao rak, wha specfcao of he deermsc compoes s plausble ad fally srucural ad exogeey resrcos are roduced. I would be desrable o be able o coduc jo Copyrgh by The Ncolaus Copercus Uversy Scefc Publshg House

10 278 ferece o eresg aspecs of he VECM demad sysem. All he sascal problem meoed above seem o sugges ha applcao of he bayesa ferece would be very promsg he area. Aoher lmao of he aalyss preseed here s leary of he fucoal form used. I radoal demad aalyss, lear forms are abadoed favor of he complcaed o-lear specfcaos. Uforuaely, o-lear coegrao echques are o ye well developed. The (quarerly) daa used sugges ha here s some possbly of seasoal coegrao also. However, should be oed ha GADS form of demad sysem seems o be que a eresg alerave wh he VECM approach, domaed by Almos Ideal model of Deao ad Muellbauer (980). I s of parcular eres ha allows for mposo of he egavy cosras whe Cholesky facorzao s used as proposed by Lau (978). Refereces Balcombe K. (2004), Reesg symmery ad homogeey a coegraed demad sysem wh boosrappg: he case of mea demad Greece, Emprcal Ecoomcs, 29. Bewley R.A. (986), Allocao Models, Ballger, Cambrdge, MA. Boswjk H.P., Doork J.A. (2004), Idefyg, Esmag ad Tesg Resrced Coegraed Sysem: A Overvew, Sasca Neerladca, 58. Deao A.S., Muellbauer J. (980), A Almos Ideal Demad Sysem, Amerca Ecoomc Revew, 70. Deschamps, P. (2003), Tme-Varyg Ierceps ad Equlbrum Aalyss: A Exeso of he Dyamc Almos Ideal Demad Model, Joural of Appled Ecoomercs, 8. Dewer W. E. (980), Symmery codos for marke demad fucos, Revew of Ecoomc Sudes, 47. Doork J.A. (2002), Objec-Oreed Marx Programmg Usg Ox, 3rd ed. Lodo: Tmberlake Cosulas Press ad Oxford: Faell L., Mazzocch M. (2002), A coegraed VECM demad sysem for mea Ialy, Appled Ecoomcs, 34. Fry, J. M., Fry, T. R. L., McLare, K. R. (996), The Sochasc Specfcao of Demad Share Equaos: Resrcg Budge Shares o he U Smplex, Joural of Ecoomercs, 73. Johase S. (996), Lkelhood-based ferece coegraed vecor auoregressve models, Oxford Uversy Press, Oxford. Karagas G., Karads S., Velezas K. (2000), A error-correco almos deal demad sysem for mea Greece, Agrculural Ecoomcs, 22. Keuzekamp H. A., Bare A. P. (995), Rejeco Whou Falsfcao. O he Hsory of Tesg he Homogeey Codo he Theory of Cosumer Demad, Joural of Ecoomercs, 67. Lau L. J. (978), Tesg ad Imposg Moocy, Covexy ad Quas Covexy Cosras. Appedx 4A Produco Ecoomc: A Dual Approach o Theory Copyrgh by The Ncolaus Copercus Uversy Scefc Publshg House

11 Imposg Ecoomc Resrcos a VECM-form Demad Sysem 279 ad Applcaos, vol. I, eded by M. Fuss ad D. McFadde, Amserdam, Norh Hollad. Mas-Colle A., Whso M.D., Gree J. R. (995), Mcroecoomc Theory, New York, Oxford, Oxford Uversy Press. Mazur B. (2005), Log-ru srucural modelg of aggregae demad: a applcao of alerave fucoal forms, Aca Uversas Lodzess, Fola Oecoomca, 90. Ng S. (995), Tesg for homogeey demad sysems whe he regressors are osaoary, Joural of Appled Ecoomercs, 0. Pesara M.H., Sh Y. (2002), Log-Ru Srucural Modelg, Ecoomercs Revews 2. Pollak R.A, Wales T.J. (992), Demad Sysem Specfcao ad Esmao, Oxford Uversy Press, New York, Oxford. Copyrgh by The Ncolaus Copercus Uversy Scefc Publshg House