PPP May no Hold for Agrculural Commodes by Lucano Guerrez Deparmen of Agrculural Economcs Unversy of Sassar, Ialy Absrac We use he well known USDA daase of real exchange raes o address he queson of wheher PPP holds for agrculural commodes. Boh un roo ess and he recenly proposed more powerful class of panel un roo ess, whch ake no accoun cross-secon correlaon across he uns n he panel, are used. Properes of un roos and panel ess are analyzed by Mone Carlo smulaon. Summarzng, our resuls show ha durng he pos-breon- Woods perod of flexble exchange raes, PPP does no hold for agrculural commodes. Key words : Purchasng Power Pary, Agrculural Commodes, Mone Carlo, Un Roo ess, Panel un roo ess. JEL classfcaon: F14 ; F31; C22; C23. Correspondence : Lucano Guerrez Deparmen of Agrculural Economcs Tel.: +39.079.229.256 Unversy of Sassar Fax: +39.079.229.356 Va E. De Ncola 1, Sassar 07100 e-mal: lguerr@unss. Ialy web: hp://www.guerrezlucano.ne
1. Inroducon. Recen work on panel un roo ess has renewed aenon on he purchasng power pary (PPP) hypohess,.e, ha n he absence of rade resrcons and ransporaon coss, he exchange rae beween wo currences mus be equal o he rao of he wo correspondng prces ( he absolue verson of PPP). As s well known, from a sascal pon of vew, f PPP holds, hen he real exchange rae (RER) mus be a saonary varable. Thus evdence on long run properes of PPP can be assessed by esng he presence of un roos n he real exchange raes. If he null hypohess of un roos canno be rejeced, hus RER s no mean reverng and, herefore PPP does no hold eher n he shor or n he long-run. Whle n he las decade an enormous se of unvarae un roo ess have been developed and appled o analyze he PPP hypohess, has become clear ha hese ess suffer from low power when appled o shor me span of daa. Thus any nference based on hese sascs s rreparably compromsed. In an aemp o solve hs problem, researchers have recenly focused her aenon on he use of long-horzon RER, spannng one cenury or more, or adopng panel un roo ess. The use of he frs approach has been crczed because nvolves combnng dfferen perods of fxed and floang nomnal exchange regmes and hus when fndng for example ha durng he las cenury PPP holds, hs does no mean ha he resul wll be vald for he perod of floang or fxed exchange rae regmes. Augmenng he number of observaons, by usng for example monhly daa raher han yearly daa, do no help o solve he problem because he power of un roo ess depends more on he span of daa han on he number of observaons, Shller and Perron (1985). The second approach has been used exensvely, because panel un roo ess have hgher power han unvarae ess. The problem here s ha many of recenly proposed panel ess assume he absence of cross-correlaon among he uns n he panel and, as was sressed by O Connell (1998), hs causes new problems, as RER are usually defned usng he same base counry, and hus crosssecon correlaons arse n a mechancal way. In addon, Karlsson and Löhgren (2000) and Guerrez (2003) show ha he power of panel ess are srcly relaed o he number of saonary (or nonsaonary) seres n he panel. Thus a researcher can erroneously model he panel as nonsaonary even when only some of he seres are acually nonsaonary, and vce versa. In hs paper we address he problem of applyng unvarae and panel un roo ess o a se of agrculural commody monhly real exchange raes publshed by USDA n s web pages, and hus easly avalable o all researchers. The am of he paper s wofold. Frs, when lookng a he 2
leraure on commody rade, s usually assumed ha commody prce arbrage does ake place, or n oher words, PPP mus hold. Usually PPP has been analyzed by lookng a he consumer prce ndexes, and n hs case PPP can be flawed by he presence n he ndex of non-raded goods whch may also no be relaed across counres n he long-run, Rogers and Jenkns (1993). Thus, despe he fac ha rade conrol facors such as arffs or mpor quoas could nroduce sysemac dsoron n he rade of agrculural commodes, when analyzng RER for prmary commodes we presume o have hgher changes o deec PPP for agrculural raded commodes han when focusng on an aggregaed consumpon ndex whch ncludes raded and non-raded goods. Second, usng panel un roo ess ha ake no accoun cross-correlaon beween RER across commodes or across counres and comparng he resuls wh unvarae un roo ess, we should have more chance of hghlghng he sochasc properes of he agrculural commody real exchange raes correcly. Summarzng, our resuls srongly rejec he hypohess ha PPP holds for agrculural commodes. Srong perssence of shocks s deeced boh by unvarae and by panel un roo ess. Thus hese fndngs suppor prevous resuls by Arden (1989), who sressed ha PPP fals o hold boh n he shor as well as n he long run. In he nex secon we brefly nroduce nonsaonary and saonary unvarae un roo ess. In he hrd secon we analyze a recenly proposed panel un roo procedure ha perms crosssecon correlaon across he uns o be aken no accoun. A Mone Carlo analyss s conduced o analyze he sze and he power of hese ess when a small number of uns are nroduced n he panel. Secon fve shows he resuls and furher commens are ncluded n he concludng secon. 2. Unvarae un roo analyss. Before nroducng panel un roo ess, we brefly concenrae on unvarae un roo analyss. As s well known, many sudes have examned wheher he me seres behavor of economc varables s conssen wh a un roo (see for a survey Debold and Nerlove, 1990; Campbell and Perron 1991). In general he analyss has been carred ou usng ess such as he augmened Dckey-Fuller s (ADF) (Dckey and Fuller, 1981) es or sem-paramerc ess, as n he case of he Phllps-Perron ess (Phllps and Perron, 1988). The man problem here s ha, n a fne sample, any un roos process can be approxmaed by a rend-saonary process. For example he smple dfference saonary process y = fy + e wh f = 1 can be well -1 approxmaed arbrarly by a saonary process wh f less han bu close o one. The resul s ha un roo es sascs have lmed power agans he alernave. Generally panel un roo ess 3
show hgher power han unvarae un roo ess bu, as hghlghed by Karlsson and Löhgren s (2000) and Guerrez s (2003), he power of panel un roo ess (or panel conegraon ess) only ncreases when he number of saonary uns n he panel rse. In synhess, for large-t panels, gven he hgher power of panel un roo ess when a small proporon of saonary relaonshps are n he panel, here s a poenal rsk ha he whole panel may be erroneously modeled as saonary when only a fracon of he relaonshps are acually saonary. In addon here s a rsk of modelng he whole panel as non-saonary for small-t panels, gven he low power of he panel ess even when a large number of saonary relaonshps are presen n he panel. In concluson, f nference s based only on panel un roo ess, hen researchers mus be careful when mposng saonary (or nonsaonary) properes on he panel. Ths s why s useful o examne unvarae un roo ess before analyzng panel ess. Le us now defne he log of he real exchange rae q as where s s he nomnal exchange rae, and q = s + p p (1) * p and * p are respecvely he home and foregn log prces. We use wo unvarae un roo ess: he well known he ADF -es and he more powerful DFGLS -es proposed by Ello, Rohemberg, and Sock (1996). The DFGLS es s performed by esng he null hypohess β 0 = 0,.e. real exchange rae nonsaonary, n he regresson p d d d β0 1 β ε = 1 q = q + q + (2) where d d q s he locally derended real exchange rae, obaned for a model wh drf as q = q a0 ˆ and 0 â s compued by a regresson of = 1,1 ( α ) 2,,1 ( α ) ( ) ( ) ' q q L q L qt on z = 1, 1 α,,1 α and α = 1+ c. Ello e al. (1996) argue ha fxng c = 7 n he model T wh drf, he DF-GLS es has greaer power han he ADF es. Crcal values of he es are provded by Ello e al. (1996). Boh DFGLS and ADF ess have nonsaonary as null hypohess. ' 3. Panel un roo ess analyss. Over he las few years, a grea deal of aenon has been pad o he nonsaonary propery of panels. Sarng from he semnal works of Quah (1990, 1994), Breung and Meyer (1991) Levn and Ln (1992, 1993), and Im e al. (1997), many ess have been proposed whch aemp o nroduce un roo ess n panel daa. These show ha, by combnng he me seres nformaon 4
wh ha from he cross-secon, he nference ha un roos exs can be made more sraghforward and precse, especally when he me seres dmenson of he daa relavely shor, and ha smlar daa may be obaned across a cross-secon of uns such as counres or ndusres. However all he panel un roo ess suffer from serous lmaons when he cross-seconal uns are correlaed (see O Connell, 1998). For example, when real exchange raes are defned usng he same base counry, cross-seconal correlaon s mechancal. Forunaely some papers have been presened n recen years ha address hs ssue. For example, Ba and Ng (2001), Moon and Perron (2002) and Phllps and Sul (2002) use common facor componens. In bref, all he above menoned works propose a facor model n whch he panel daa s generaed by one or more facors whch are common o all he ndvdual uns (bu whch may exer dfferen effecs on he ndvdual un) and by uncorrelaed dosyncrac shocks across all he ndvdual uns. Whle Moon and Perron (2002) and Phllps and Sul (2002) sae ha common facor(s) mus be a saonary varable(s), Ba and Ng (2001) allow for non-saonary (or saonary) common componen(s). For hs reason we concenrae our aenon on Ba and Ng s (2003) model. Le us assume ha for each agrculural commody or regon, he logarhm of real exchange rae can be decomposed as q = c + λ f + e = 1,..., N = 1,..., T (4.1) ( ) ( ) I L f = C L v (4.2) ( ρ L) e B ( L) 1 ε where c s a consan or rend varable, = (4.3) f s a ( 1) r consan, when r = 1, or vecor, when r > 1, of common facor(s) and λ s he correspondng vecor of facor loadngs. The error erms v and ε are muually ndependen across and, and Bj ( L) and C( L) are wo polynomal, wh a rank of C ( 1) r1 =. In synhess, when 1 0, C 1 = 0, and (4.2) s over-dfferenced, for r 1 1 he sysem r = ( ) conans one or more common sochasc rends. Noe from (4.3), ha he dosyncrac erm e s saonary when 1 ρ < and non-saonary, or equvalenly, negraed of order one ( 1) In bref, Ba and Ng s (2001) model consss of esmang common facor(s), 5 I, for ρ = 1. f, and dosyncrac componens by applyng he mehod of prncpal componens o he frs dfferenced daa logq (where now logq s he observed ( T N) marx of (sandardzed) dfferenced log of real exchange raes for he N commodes or regons and over T perods), and obanng he
(dfferenced) common facor(s) as he frs r 1 egenvecors wh he larges egenvalues of he marx logq log q'. Facor loadng λ can be easly calculaed as he produc of (ransposed) logq marx and common facor(s) f. Thus, he (dfferenced) dosyncrac erms n (4.1) can be calculaed as ˆ ˆ = ˆ Fnally, he esmae of he level of common facor(s) can be ' log q λ f e. obaned smply by negrang T fˆ = fˆ, k= 2 k and he dosyncrac error erm can be compued as T eˆ = eˆ. k= 2 k Once he f ˆ and e ˆ componens have been compued, we can es f common and/or dosyncrac, or none, of he wo componens have un roos or, n oher words, we can asceran wheher nonsaonary of RER comes from he common or he dosyncrac componens. Because we have N dosyncrac errors e and, by consrucon, hey are no cross-secon correlaed, pooled ess can be effcenly used. In he emprcal analyss, we adop a mehod proposed by Maddala and Wu (1999) who sugges usng a Fsher-ype es o es he null hypohess of ρ = 1, agans he alernave of ρ < 1 for some. They show ha hs es has hgher power han Levn and Ln s (1993) and Im e al. s (1997) ess. The Fsher-ype es consss n compung, for example for he -h ADF es, he p sgnfcance level (p-value). Fsher s sasc ( 2 log p ) has a he sasc ( ) 2 log p 2 N / 4N 2 χ wh 2N degrees of freedom. Cho (2001) show ha for N converges o ( 0,1) N. In secon 5, Ba and Ng s (2001) procedure wll be appled o he agrculural commody real exchange raes. 4. Mone Carlo smulaons Ba and Ng (2001) hghlgh ha her procedure works well when N and T are large. The USDA daabase of real exchange raes consss of less han 20 un. Thus s useful o demonsrae a smple Mone Carlo sudy proposed by Ba and Ng s (2001) n order o show how her procedure works when a small of N uns are nroduced n he panel. Daa are generaed usng X = λf + e, wh e = ρe 1 + ε, and F αf 1 u = +, wh λ N ( 0,1), ε N ( 0,1) and u N( 0,1) n he smulaon he same auoregressve coeffcen s used for all he. Noe ha e. Four pars of ( ρα, ) values were consdered. When ρ = 1 and α = 1 boh componens are nonsaonary (we exclude 6
conegraon), when ρ = 0.9 and α = 1 only he e are saonary whle when ρ = 1 and α = 0.9 he common facor s nonsaonary. In all hree pars of values, X s a nonsaonary varable by consrucon. Fnally when ρ = 0.9 and α = 0.9 boh componens are saonary so as X. Only one facor,.e. r = 1, s ncluded n he model. Table 1 abou here In able 1, we repor he resuls for T=100 and N=10, 50. In he frs wo columns, we nclude he rejecon rae when ADF es and DFGLS es are appled o he esmaed common facor. The augmened regressons have ( NT) 1/4 4 mn, /100 lags. In he followng wo columns, we repor he average rejecon raes and Fsher s c P X es for he orgnal X seres when usng he ADF es. The same sascs are compued for he dosyncrac componens and are repored n he fnal wo columns. Fnally, when compung he rejecon raes, crcal values a he 5% level were used. We sarng analyzng he ADF es appled o X when ρ = 1, α = 1 and for N=10. We noe ha under he null he average rejecon rae s 0.044, whch s close o he correc sze of 0.05. The es over-rejecs he null when ρ = 0.9, α = 1 and ρ = 1, α = 0.9. In addon, he pooled es for X 1 over-rejecs he null when all he N seres are nonsaonary. Ths las resul s n lne wh he prevously ced O Connell (1998) fndngs ha cross secon correlaon leads he sandard pooled es o over-rejec he null hypohess. Turnng o he common componen, boh he ADF and DFGLS ess are close o he nomnal sze of 0.05 when α = 1. The power of DFGLS es s hgher han he ADF es when α = 0.9. Consderng he pooled es, he P c eˆ es shows a small oversze when e s n fac nonsaonary, and he power of he es s near 1.0 for ρ = 0.9. Noe ha he average ADF es appled o e has less power. The mporan pon, n erms of ou objecve, s ha when he number of uns s ncreased o N=50 he prevous resuls are no sensbly alered. Ths means ha Ba and Ng s (2001) procedure can also be frufully used for small panels. 5. The daase and he emprcal resuls 2 Daa for agrculural commody real exchange raes was colleced from he USDA daabase. Monhly weghed real exchange raes are aggregaed for a se of agrculural commodes usng he 7
world, US expor and mpor weghs. The USDA also provdes he real exchange raes for a se of regons. The daase covers he perod 1970.1 2002.12, bu n he analyss we use he perod pos- 1973 when he floang exchange rae was n effec. In able 2 we presen he resuls of he ADF and DF-GLS ess when he USDA s radeworld weghed real exchange raes were used. We nclude a consan n all he regressons. Thus we es for he hypohess of relave raher han absolue PPP. We do no repor, for reasons of brevy, he sascs when he consan s no ncluded because he resuls are smlar o he prevous ones. The lag selecon creron for he auoregressve polynomal was chosen by usng he modfed Akake creron proposed by Ng and Perron (1995). Ths mehod seems o produce beer resuls han he general o specfc Hall s (1994) creron and provdes he bes combnaon of sze and power for boh ess. Table 2 abou here All he ess do no rejec he null hypohess of nonsaonary. Tha s, we fnd dffcul o prove ha here s any convergence oward PPP n he long run for agrculural commodes real exchange raes. As prevously saed, he USDA aggregaes agrculural commodes real exchange raes no only by produc bu also by counry of provenence, and so he same ess were appled o he regonal exchange raes. In able 3 we es f some regons show mean-reverson behavor when analyzed durng he perod 1973.1-2002.12. Table 3 abou here Once more, we fnd ha all he ess rejec he hypohess of mean-reverson of real exchange raes for all he regons. Thus, basng he analyss on unvarae un roo ess, we conclude ha whaever s aggregaed, he agrculural commodes real exchange raes are nonsaonary varables. 3 Gven he hgher power of panel un roo ess, n able 4 we presen he resul when Ba and Ng s (2001) procedure and ess are used. Naurally we do no apply he panel un roo ess o all he seres presened n he prevous ables bu only o a sub-sample of hem. Table 4 s self-explanaory and shows whch real exchange raes were ncluded n he panel. Table 4 abou here The frs ask before compung mulfacor analyss, as n (4), s o correcly specfy he number of facors r. For he panel of agrculural commodes and he panel of regonal exchange raes, he frs prncpal componen explans respecvely 59% and 52% of he varance, and he second componen only 20% and 11% of he oal varance. Values of r=1 and r=2 were used n he 8
emprcal analyss. We smply repor sascs for r=1 o conserve space. The same resuls, avalable on reques, were found when usng r=2. 4 In able 4 es values for he mehodology proposed by Ba and Ng (2001) are presened boh for he common componen and for he dosyncrac componen. Generally all he ess do no rejec he null of nonsaonary. Thus boh he common componen as well as producs or regon specfc shocks have permanen effecs on he real exchange raes. The only excepon s he commody poulry, where he ADF es rejecs he null of nonsaonary for he dosyncrac componen. 5 Thus our resuls ndcae ha agrculural commody real exchange raes are no mean reverng or n oher erms hey are no saonary varables. Summarzng, durng he pos-breon- Woods sysem of flexble exchange rae PPP does no hold for agrculural commodes. 6. Conclusons Afer more han a decade we have addressed he ssue rased by Arden (1989) of wheher purchasng power pary holds for agrculural commodes. We analyze hs ssue by usng he well known USDA daabase on a wde sample of agrculural real exchange raes aggregaed by produc and by regon. Usng more powerful un roo ess and recenly proposed panel un ess we were able o rejec he hypohess ha PPP holds for agrculural commodes n he long-run. Thus our resuls renforce Arden s (1989) conclusons ha researchers mus be aware ha he hypohess of PPP may no hold n rade models when agrculural commodes are analyzed. Changes n nernaonal prces are no fully refleced n domesc prces neher n he shor-run nor n he long-run. Agrculural commodes prces are probably nfluenced by mpor quoas, arffs and oher rade conrols, whch nroduce sgnfcan and permanen devaons from PPP. Noes 1 To calculae he percenles of he ADF ess used for her p-values, we followed he smulaon mehod of MacKnnon (1994). 2 All he ess presened n he emprcal analyss were mplemened n GAUSS 3.2 and are freely avalable upon reques. 3 Noe ha he same resuls hold when usng mpor or expor weghs o aggregae RER 4 Ba and Ng (2002) sugges welve dfferen crera o address hs pon. Unforunaely, her sascs oversae he correc number of common facors when N s smaller han 20. 5 We also apply he DF-GLS es o he dosyncrac componens. The resuls, no repored for brevy, do no rejec he null of nonsaonary. 9
Reference Arden, P.G. (1989). Does he Law of One Prce Really Hold for Commody Prces? Amercan Journal of Agrculural Economss, 73:1, 661-669. Ba, J., Ng, S. (2002). Deermnng he Number of Facors n Approxmae Facor Models. Economerca, 70:1, 191-221. Ba, J., Ng, S. (2001). A PANIC Aack on Un Roos and Conegraon, mmeo, Boson College. Breung, J. and Meyer, W. (1991). Tesng for Un Roos n Panel Daa: are Wages on Dfferen Barganng Levels Conegraed? Insue für Wrschafsforschung Workng Paper, June. Campbell, J. and Perron, P. (1991). Pfalls and Opporunes: Wha Macroeconomss Should Know abou Un Roos. NBER Macroeconomcs Annual, MIT Press. Cho, I. (2001). Un Roo Tess for Panel Daa. Journal of Inernaonal Money and Fnance, 20, 249-272. Dckey, D. A. and Fuller, W. A. (1981). Lkelhood Rao Sascs for Auoregressve Tme Seres wh a Un Roo. Economerca, 49: 1057-1072. Debold, F.X. and Nerlove, M. (1990). Un Roos n Economc Tme Seres: A Selecve Survey. In Advances n Economercs: Conegraon, Spurous Regressons, and Un Roos, eded by T.B Fomby and G.F. Rhodes. Greenwch, CT:JAI Press, 3-70. Ello, G., Rohemberg T.J. and Sock J.H. (1996), Effcen Tess for an Auoregressve Un Roo, Economerca, 64, 813-836. Guerrez L. (2003). On he Power of Panel Conegraon Tess: A Mone Carlo Comparson. Economcs Leers, 80(1): 105-111. Hall, A. (1994) Tesng for a Un Roo n Tme Seres wh Prees Daa-Based Model Selecon. Journal of Busness & Economc Sascs, 12: 461-470. Im, K.S., Pesaran, M.H. and Shn, Y. (1997). Tesng for Un Roos n Heerogeneous Panels. Deparmen of Appled Economcs, Unversy of Cambrdge. Karlsson, S., Löhgren M., (2000). On he Power and Inerpreaon of Panel Un Roo Tess. Economcs Leers, 66, 249-255. Levn, A. and Ln, C.F. (1992). Un Roo Tess n Panel Daa: Asympoc and Fne-Sample Properes. Dscusson Paper Seres 92-23, Deparmen of Economcs, Unversy of San Dego. Levn, A. and Ln, C.F. (1993). Un Roo Tess n Panel Daa: New Resuls. Dscusson Paper Seres 93-56, Deparmen of Economcs, Unversy of San Dego. MackKnnon, J.G. (1994) Approxmae Asympoc Dsrbuon Funcons for Un-Roo and Conegraon, Journal of Busness and Economc Sascs, 12, 167-176. 10
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Table 1. Rejecon raes for he null hypohess of a un roo T=100, N=10 Common facor Seres Idosyncrac comp. ρ α ADF es DF_GLS es X c c P X ê P eˆ 1.00 1.00 0.044 0.057 0.044 0.076 0.048 0.061 1.00 0.90 0.133 0.254 0.064 0.127 0.049 0.055 0.90 1.00 0.048 0.083 0.010 0.395 0.286 0.973 0.90 0.90 0.220 0.413 0.211 0.859 0.306 0.984 T=100, N=50 Common facor Seres Idosyncrac comp. ρ α ADF es DF_GLS es X c c P X ê P eˆ 1.00 1.00 0.041 0.069 0.044 0.153 0.045 0.040 1.00 0.90 0.176 0.368 0.065 0.304 0.045 0.040 0.90 1.00 0.040 0.067 0.102 0.601 0.281 1.000 0.90 0.90 0.202 0.395 0.211 0.991 0.294 1.000 Table 2. Agrculural commodes world rade-weghed real exchange raes: un roo ess, perod 1973.1 2002.12 Commodes ADF es DF-GLS es U.S. Markes Toal Trade -1,370-0,595 U.S. Markes Agrculural Trade -1,458-0,622 Bulk Commodes -0,441 0,421 Corn -1,715-1,194 Coon -1,333-0,244 Rce -0,423 0,798 Soybeans -1,131-0,402 Raw Tobacco -0,928-0,019 Whea -1,154-0,728 Hgh-value Producs -1,565-0,755 Processed Inermedaes -1,152-0,249 Soymeal -1,870-1,720 Soyol -1,878-1,528 Produce and Horculure -1,523-0,160 Frus -1,918 0,324 Vegeables -1,598-0,799 Hgh-value Processed -1,630-0,654 Fru Juces -1,586-1,285 Poulry -1,402-0,428 Red Meas -1,347-0,264 5% crcal values -2,874-1,950 Source: Auhor s calculaon based on USDA daase 12
Table 3 Regonal real exchange raes: un roo ess, 1973.1 2002.12 Regon ADF es DF-GLS es Cenral Amerca and Carbbean -0,910-0,361 Oher Souh Amerca -1,029 0,733 Oher Wesern Europe -2,488-0,927 Oher Subsaharan Afrca -0,097 0,003 Oher Norh Afrca and Mddle Eas -2,377 1,026 Oher Asa and Oceana -1,073-0,240 UE -1,882-1,755 Afrca -0,312 0,012 Norh Afrca -0,764-0,450 Lan Amerca -2,198-0,093 Asa -1,756-1,224 Souheas Asa -1,841-1,139 Souh Asa -1,441 1,390 5% crcal values -2,874-1,950 Source: Auhor s calculaon based on USDA daase 13
Table 4. Ba and Ng (2001) panel un roo resuls. Produc and regonal aggregaed real exchange raes, 1973.1 2002.12 ADF Tes ADF Tes Produc dosyncrac Regon dosyncrac Componens Componens Idosyncrac componens Idosyncrac componens Indvdual ADF ess Indvdual ADF ess Corn -0,186 Cenral Amerca and Carbbean -0,259 Coon -1,421 Souh Amerca 0,861 Rce -1,371 Wesern Europe -1,451 Soybeans -0,482 Subsaharan Afrca -0,411 Raw Tobacco -1,227 Oher Norh Afrca and Mddle Eas 1,725 Whea -0,371 Oher Asa and Oceana -0,486 Soymeal -0,093 UE -1,142 Soyol -0,070 Norh Afrca -0,555 Frus -1,171 Souheas Asa -1,853 Vegeables -2,396 Souh Asa 1,563 Fru Juces -1,302 Poulry -2,455 Red Meas -1,123 5% crcal value -1,95 5% crcal value -1,95 Pooled ADF es 33.52 Pooled ADF es 13.57 2 2 26 χ 20 31.40 5% crcal value χ ( ) 38.90 5% crcal value ( ) Common facor componen : Common facor componen : ADF es (a) -1,188(-2.87) ADF es (a) -1,826(-2.87) DFGLS es (a) -0.385(-1.95) DFGLS es (a) -1.767(-1.95) (a) n parenheses 5% crcal value Source: Auhor s calculaon based on USDA daase 14