Median-Unbiased Estimation in Panel Data: Methodology and Applications to the GDP Convergence and Purchasing Power Parity Hypotheses

Transcription

1 NÚMERO 407 RODOLFO CERMEÑO Medan-Unbased Esmaon n Panel Daa: Mehodology and Applcaons o he GDP Convergence and Purchasng Power Pary Hypoheses DICIEMBRE

2 Las coleccones de Documenos de Trabajo del CIDE represenan un medo para dfundr los avances de la labor de nvesgacón, y para permr que los auores recban comenaros anes de su publcacón defnva. Se agradecerá que los comenaros se hagan llegar drecamene al (los) auor(es). D.R Cenro de Invesgacón y Docenca Económcas, carreera Méxco-Toluca 3655 (km. 16.5), Lomas de Sana Fe, 01210, Méxco, D.F. Fax: ex.6314 Correo elecrónco: publcacones@cde.edu Produccón a cargo del (los) auor(es), por lo que ano el conendo así como el eslo y la redaccón son su responsabldad.

3 Absrac Ths paper revews he leraure on he economercs of medan-unbased esmaon n panel daa and provdes mehodologcal gudelnes for s mplemenaon. The mehod s hen used o evaluae he well known GDP convergence and Purchasng Power Pary (PPP) hypoheses. In he frs wo examples, panel exacly medan-unbased esmaes wh yearly daa show, respecvely, fas condonal convergence raes of per capa ncome among he 48 USA conguous saes, and exremely slow convergence or non convergence of per capa GDP among he 32 Mexcan saes. In he hrd example, usng monhly daa for a sample of 14 developng counres, he resuls from panel exacly medan-unbased esmaon show ha he dynamcs of real exchange raes s conssen wh he PPP hypohess, alhough hs process s hghly perssen. Key words: Panel Medan-Unbased Esmaon, Dynamc Panel Daa Models, Leas Squares Dummy Varables, Cross Seconal Dependence, GDP Convergence, Purchasng Power Pary. JEL Classfcaon: C33, C15, O40, F31. Resumen Ese arículo revsa la leraura sobre esmacón nsesgada respeco a la medana (medan-unbased) en modelos panel y desarrolla los lneamenos meodológcos para su mplemenacón empírca. Como lusracón, ese méodo es ulzado para evaluar las hpóess de convergenca del ngreso y pardad del poder de compra (PPP). En los dos prmeros ejemplos, los resulados del esmador panel exacamene nsesgado respeco a la medana (exacly medan-unbased) muesran, respecvamene, convergenca condconal relavamene rápda del ngreso por persona para los EE. UU. y convergenca muy lena o ausenca de convergenca en el caso del PIB por persona de los esados de Méxco. En el ercer ejemplo, ulzando el msmo esmador aneror se encuenra que la dnámca del po de cambo real de 14 países en desarrollo es conssene con la hpóess PPP, aunque se raa de un proceso alamene perssene. Palabras clave: esmacón Medan-Unbased en panel, modelos panel dnámcos, esmador de mínmos cuadrados con varables fccas, dependenca de seccón cruzada en panel, convergenca del PIB per cápa, pardad del poder de compra. Códgos de clasfcacón JEL: C33, C15, O40, F31.

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5 Medan-Unbased Esmaon n Panel Daa Inroducon Durng he las decade and a half here have been consderable developmens on medan-unbased Esmaon for auo regressve processes ( AR ). Ths mehod provdes unbased pon and confdence nerval esmaes for he AR parameer, hus offerng useful nformaon o characerze he perssence of a me seres process, whch s parcularly mporan n cases where he evdence from un roo ess may no be conclusve (Andrews, 1993). As s well known, he bas of he OLS esmaor can be que subsanal parcularly when he AR parameer s close o one and/or he sample sze s relavely small, and can produce poenally msleadng nferences. There are a number of mporan economc quesons such as convergence of per capa GDP, prce convergence and purchasng power pary, where a correc measuremen of he perssence of he process s crucal and where medan-unbased esmaon can play an mporan role. Nowhsandng, hs mehod s sll no popular, perhaps because of s compuaonal burden. In he panel daa leraure, he medan-unbased esmaon mehod s even less known despe he fac ha n several emprcal applcaons he me dmenson of he panel s relavely shor. The goal of hs paper s o survey he leraure on he economercs of medan-unbased esmaon n panel daa and o provde mehodologcal gudelnes for s mplemenaon. The mehod s hen used o evaluae () he per capa Income or GDP convergence hypohess n wo dfferen samples, namely, he USA and Mexcan saes, and () he Purchasng Power Pary (PPP) hypohess n a panel of 14 Developng counres. The res of he paper s organzed as follows. Secon 1 brefly surveys he leraure on medan unbased esmaon. Secon 2 descrbes he mehod n deal and dscusses some mehodologcal ssues for s mplemenaon. The emprcal applcaons are provded n secon 3 and, fnally, some conclusons and remarks are offered n las secon. 1. Some Background Orcu and Wnokur Jr. (1969) s probably he frs paper ha focuses on approxmaely mean-unbased esmaes of he AR parameer n saonary models, work ha was movaed by he earler fndngs by Hurwckz (1950), Marro and Pope (1954) and Kendall (1954), who esablshed he (mean) bas of he OLS esmaor n AR models. Laer, Le Breon and Pham (1989) calculaed exac and asympoc bases of he OLS esmaor n saonary and non saonary AR(1) processes. 1 1 Maddala and Km (1998), p DIVISIÓN DE ECONOMÍA 1

6 Rodolfo Cermeño The bas or more precsely he mean-bas of an esmaor s smply defned as he dfference beween he uncondonal expecaon of he esmaor and he rue parameer value. Thus, f he expeced value (mean) of he esmaor s equal o he rue parameer value hen he esmaor s sad o be unbased or more precsely mean-unbased. The concep of medanunbasedness, consders he medan nsead of he mean. If an esmaor s normally dsrbued, here s no dfference beween he mean and medan. However, n suaons were he esmaors have asymmerc dsrbuons he concep of medan-unbasedness becomes more relevan as he medan of he dsrbuon s less sensve o skewness and kuross problems. Ths s precsely he movaon for pursung medan-unbased esmaon as proposed by Andrews (1991, 1993) and Rudebusch (1992) who show he relevance and usefulness of hs mehod o sudy he dynamcs of me seres where he bas of he OLS esmaor can become a serous problem. Essenally he mehod consss of () esmang he AR parameer by OLS and () correcng for he downward bas by machng he OLS esmae o he medan of de dsrbuon of he OLS esmaor and fndng s correspondng rue or unbased value, ha s, he value ofα ha generaes a process for whch he medan of he dsrbuon of he OLS esmaor s equal o he OLS esmae obaned from he acual daa. Obanng he OLS esmae s ceranly a rval ask. However, obanng he medan-unbased esmae requres compuaon of he dsrbuon of he OLS esmaor. More precsely, s necessary compue he medan as well as oher relevan quanles of ha dsrbuon. In pracce hs can be done (approxmaely) by Mone Carlo smulaons, whch wll urn ou o be a relavely easy ask usng acual sorage and processng capacy of personal compuers. Andrews (1993) presens a comprehensve approach o medan-unbased esmaon for frs-order auoregressve-un roo models wh ndependen and dencally dsrbued normal errors. 2 The auhor consders models wh no nercep, nercep only, and nercep and me rend, whch are he well-known specfcaons used n he me seres un roo esng leraure. Generally, he AR parameer value s allowed o le on he nerval (-1, 1], whch ncludes he un roo case. In addon o medan-unbased esmaors and confdence nervals for he AR parameer, he correspondng mpulse response funcon, he cumulave mpulse response, and he half lfe of a un shock can also be obaned wh hs mehod. The papers by Andrews and Chen (1992, 1994) and Rudebusch (1992) consder medan-unbased esmaon n he conex of processes. 3 AR( p) 2 Fuller (1995) develops an esmaor ha s medan-unbased n he un roo case and nearly unbased elsewhere. Ths approach s summarzed by Enders and Folk (1998). Essenally, he OLS esmae s adjused upwards by an amoun commensurae wh he sample sze and he proxmy of he esmae o uny. 3 A relaed approach s gven n Sock (1991) who consrucs unbased asympoc confdence nervals for he larges roo n AR (p) processes when hs roo s close o one. 2 CIDE

7 Medan-Unbased Esmaon n Panel Daa Indeed, he aforemenoned work movaed several emprcal applcaons. I s worhwhle o menon he papers by Ken and Cashng (2003), Cashn, McDermo, and Pallo (2004), Cashn and McDermo (2003), Km (2003), Murray and Papell (2002, 2005a) whch were all appled n me seres conexs. In mos cases, he me seres medan-unbased esmaes mply slow convergence raes and confdence nervals conssen wh shocks ha have permanen effecs. Cermeño (1999) exended Andrews (1993) approach o dynamc panel daa models wh fxed effecs and homogeneous rends, showng ha n ypcal macro panels he bases n he esmaon of he AR parameer usng he LSDV esmaor could be subsanal even for relavely large T dmensons whch may lead o erroneous conclusons f no aken no accoun. In parcular, Cermeño s fndngs show ha (condonal) convergence of per capa GDP s only suppored n he cases of 48 USA saes and 23 OECD counres bu does no hold n wder samples of counres. Furher, Phllps & Sul (2003) generalze hs mehod o cases wh crossseconal dependence as well as full parameer heerogeney. In addon o he orgnal Panel Exacly Medan-Unbased esmaor (PEMU), hey propose he Panel Feasble Generalzed Leas Squares Medan-Unbased (PFGMU) for he case of cross-seconal dependence and he Panel Seemngly Unrelaed Medan Unbased (SURMU) esmaor for he case of complee coeffcen heerogeney of ndvdual cross secons. I s mporan o menon ha he prevous exensons of medanunbased esmaon o panel daa have been made n he conex of AR(1) processes. 4 Is generalzaon o AR( p) processes wh heerogeneous dynamcs and a general error covarance srucure sll remans o be done. 2. Medan-Unbased Esmaon In hs secon we offer a comprehensve summary on medan-unbased esmaon n panel daa, he man focus beng o provde mehodologcal gudelnes for s mplemenaon. In order o address he fundamenal ssues n some deal he me seres approach s also ncluded. As wll become apparen laer, hs approach wll urn ou o be a key componen of he mplemenaon of hs mehod n panel daa. 4 Murray and Papell (2005b) exend Andrews (1993) and Andrews and Chen (1994) medan unbased esmaors o sudy he PPP hypohess n a panel of real exchange raes of 20 developed counres. Alhough hs exenson s nnovave n varous aspecs, sll assumes homogeneous dynamcs for all ndvdual seres n he panel. DIVISIÓN DE ECONOMÍA 3

8 Rodolfo Cermeño 2.1. Exacly Medan-Unbased Esmaon Consder he followng laen-varable model for a me seres: y = dβ + y, = 0, KT (1) * Where d s a deermnsc rend componen ha can ake he values: { 0,1, (1, ) }, and β s a conformable vecor of parameers whch correspondngly can ake he values: {0, β 0. ( β 0, β1)' }. The coeffcens β, β 0 1 are he nercep and rend respecvely. Assume ha he laen varable follows he AR (1) process: y * = αy + ε, = 1, K T * 1, Where he error erms ( ε ) are ndependen and dencally dsrbued, 2 denoedε ~ d.. (0, σ ). The prevous represenaon s que convenen snce f α < 1, y becomes saonary around he deermnsc rend dβ. More * 2 2 precsely ( y dβ ) = y ~ (0, σ /(1 α )), meanng ha he devaon ( y dβ ) s a saonary, zero mean, process. Also, as can be seen n he followng equaons, he prevous formulaon ness he 3 cases consdered n he un roo leraure whch are referred o as cases (or models) 1, 2 and 4. 5 These are also known respecvely as he cases wh no nercep and no rend, nercep only and nercep and rend. For he saonary case, α < 1, from (1) and (2) we can oban he followng models n erms of he observable varable: y = αy + ε y y = αy = αy β (1 α) + ε 0 + β (1 α) + β α + β (1 α) + ε [case (1)] [case (2)] [case (4)] 0 0 The processes n (3) are saonary around, β and ( β0 + β1) respecvely. For he un roo (non-saonary) case, α = 1, he prevous models become: y = y + ε [case (1)] y = y ε [case (2)] y = β1 + y 1 + ε [case (4)] The frs wo processes n (4) are random walks whou drfs whle he hrd one s a random walk wh drf parameer β 1. I s mporan o remark ha (3) and (4) nclude, respecvely, he correspondng processes under he alernave (saonary) and null hypoheses (un roo), as consdered, for (3) (4) (2) 5 See for example Hamlon (1994), pp CIDE

9 Medan-Unbased Esmaon n Panel Daa example, n Dckey and Fuller (1979). 6 For example, he hrd equaon n (3) and (4) represen respecvely he well known rend saonary and dfference saonary processes, whch are compeng represenaons for rendng processes such as GDP, prces or money sock. Andrews (1993) shows ha, under normaly, he fne sample dsrbuon of he OLS esmaor of he parameer α n he prevous AR(1) models s a funcon of a quadrac form n sandard normal varables. Therefore, under hs assumpon, s possble o compue he medan and oher quanles of he dsrbuon of he OLS esmaor ofα, exacly, usng Imohf s (1961) algorhm, mplemened n he FORTRAN sub rounes by Koers and Abrahamse (1971). Specfcally, gven a model and a sample sze T +1, he quanles of he dsrbuon of he OLS esmaor of α are compued exacly for a grd of parameer values on he nerval ( 1,1]. Ceranly, he same can be done by Mone Carlo smulaons, alhough n hs case he quanles of he dsrbuon wll be approxmae. 7 I s well known ha he OLS esmaor of he parameer α s downward based whch mples ha he mean of s dsrbuon s less han he rue value ofα. For example, whenα = 1, he un roo case, and for T +1 =100, he mean (whch s equal o he medan under normaly) of he dsrbuon of he OLS esmaor of α s equal o for model 2 (nercep only) and for model 4 (nercep and rend). 8 Thus he medan and mean bases of he OLS esmaor are no neglgble and can produce hghly msleadng resuls. In boh cases we would conclude ha he processes are sable wh approxmae convergence raes of 4 and 9% respecvely, when n fac hey are random walks. Therefore, he OLS esmaor needs o be correced for s bas and Andrews approach performs hs correcon by elmnang he medan bas. Bu before gvng a defnon of he medan-unbased esmaor s mporan o characerze he OLS esmaor and s dsrbuon. Ths s done n proposons 1 and 2 gven below. Proposon 1: Invarance of he dsrbuon of he OLS esmaor of α In he prevous AR(1) models 2 and 4, he dsrbuon of he OLS esmaor of α s nvaran o he specfc values of he deermnsc componens 2 (nercep and me rend). Also, hs dsrbuon does no depend on σ, he 6 Srcly, he processes n (3) are non-lnear n he parameers. Dckey and Fuller and he foregong un roo leraure consder he unresrced lnear versons of hese models. 7 Ths procedure s descrbed n deal laer n hs paper. 8 See Andrews (1993, p. 149). Ths resul s well know n he un roo leraure and mples ha he dsrbuon of he Dckey-Fuller -es for un roo s lefward skewed. I s also known, more generally, ha for a gven model and sample sze, he bas becomes larger he closer o one s he AR parameer and for a gven AR parameer value he bas becomes larger he smaller s he sample sze. DIVISIÓN DE ECONOMÍA 5

10 Rodolfo Cermeño varance of he error erm, and when α = 1 does no depend on y 0, he nal value of he process. See Andrews (1993) for proofs. The frs par of hs proposon saes ha he dsrbuon of he OLS esmaor does no depend on he specfc values of he nercep and me rend. Ths resul s well known n he me seres leraure. Inuvely, he ncluson of an nercep or an nercep and a me rend n he regresson conrols for he effec of hese deermnsc componens ( d). In fac, from paroned regresson heory, he OLS esmaor of α n a model ha ncludes he aforemenoned deermnsc componens s equvalen o regressng y% on y% 1, whch are resduals from he regressons of y and y 1on d respecvely. As wll become clear shorly, whou hs nvarance propery exacly medan-unbased esmaon of he AR parameer would be mpossble snce would requre pror knowledge of he specfc values of he deermnsc componens. The second par of Proposon 1, ha he dsrbuon of he OLS esmaor of he parameer α s nvaran o he 2 specfc values of σ, s less nuve bu gven ha essenally hs esmaor s obaned by regressng on, her varances cancel ou. y y 1 Proposon 2: Exsence of a monooncally ncreasng medan funcon of αˆ OLS For a gven AR(1) model and sample sze, here exss a monooncally ncreasng relaonshp beween α and he medan of he dsrbuon of he OLS esmaor of hs parameer. Proposon 2 s fundamenal for medan-unbased esmaon snce n he absence of a monooncally ncreasng medan funcon, would be mpossble o map an acual OLS esmae o a unque (medan-unbased) value of he AR parameerα. Unforunaely, here s no formal proof avalable and he exsence of such a relaonshp can only be shown by smulaon. For he medan and oher relevan quanles (.5 h and.95 h ) here s overwhelmng evdence supporng hs proposon for models 2 and 4. However, Andrews (1993) pons ou ha no every quanle exhbs hs behavor n he case of model 1, parcularly when T s relavely small and α akes on values near o one. In pracce s, herefore, advsable o check hs relaonshp by smulaon before proceedng. Le m = f ( α, T ) denoe he medan funcon of he OLS esmaor of α j j n model j and assume ha Proposon 2 s sasfed. The medan-unbased esmaor ˆ α ) mples nverng he prevous funcon n order o fnd he ( MU 6 CIDE

11 Medan-Unbased Esmaon n Panel Daa value ofα for whch he acual OLS esmae ( ˆ α OLS ) corresponds exacly o he medan. Ths s esablshed n he followng defnon, saed along he lnes of Andrews (1993). Defnon 1: Medan-unbased esmaor of α For a gven AR(1) model n (3) and for a gven sample szet, he medanunbased esmaor of he parameer α ( ˆ α MU ) s he value of he AR parameer for whch he medan of he dsrbuon of he OLS esmaor of hs parameer equals he acual OLS esmae ˆ α ). Specfcally, for a ( OLS UR gven AR(1) model j, assume ha 0 < α 1 and le denoe he medan of he dsrbuon of he OLS esmaor of α n he un-roo case,.e. whenα = 1. The medan-unbased esmaor of he parameer α s defned as follows: 1 j j UR f j ( ˆ α OLS ) f ˆ α OLS < m j j ˆ α MU = (5) j UR 1 f ˆ α OLS m j The prevous defnon can be symmercally exended o he cases 1 < α 0. 9 However, we focus here on posve values of he AR parameer for praccal reasons. Frs, mos economc phenomena do no seem o exhb oscllaory dynamcs,.e. cases n whch he AR parameer s negave. Secondly, several economc processes seem o be hghly perssen, whch s conssen wh AR parameer values close o one. Ths wll n fac be he case of he emprcal applcaons consdered n he nex secon. Accordng o Defnon 1, he medan-bas s subraced from he OLS esmae hus provdng an unbased esmae. Acually, he medan-bas s elmnaed compleely. For example, usng he prevous resuls for model 4, f he acual OLS esmae happens o be 0.911, he medan-unbased esmae wll be equal o 1, whch s precsely he value of he AR parameer for whch he medan of he dsrbuon s equal o he acual OLS esmae (0.911). In hs case he medan-unbased pon esmae s equal o 1 whch s conssen wh he un roo hypohess. Implemenng medan-unbased esmaon requres compung he medan of he dsrbuon of he OLS esmaor for each possble value of he AR parameer. More explcly, for a gven model and sample sze we mus oban he medan funcon whch relaes he AR parameer value wh he medan of he dsrbuon of he OLS esmaor of hs parameer. We hen use he medan funcon o fnd he AR parameer value ha corresponds o a gven OLS esmae whch s equaed o he medan. In pracce, he medan m j 9 See Andrews (1993) for deals. DIVISIÓN DE ECONOMÍA 7

12 Rodolfo Cermeño j j funcon s a dscree se of pars ( α, m ) and medan-unbased esmaes are obaned by fndng he medan value ha equals he OLS esmae compued j j usng he acual sample ( m = ˆ α ) and locang s correspondng par whch OLS j j j wll be he medan-unbased esmae ( α = ˆ α ). Gven ha wll hardly j concde exacly wh m, nerpolaon wll be needed n mos cases. Inerval esmaon s performed analogously, applyng he prevous conceps o he approprae quanles. Tha s, for quanles oher han he medan, we can use Defnon 1 afer replacng he medan funcon by he approprae funcon for he p h quanle. Thus, he nerval lms wll be gven by he AR parameer values for whch he OLS esmae equals he correspondng nerval quanles. For 90% confdence nervals, he.05 h and.95 h quanles are used. Fgure 1 below llusraes he prevous deas. The horzonal axs represens dfferen values of α over he nerval [0,1] whle he vercal axs represens he 0.95 h, 0.5 h (medan) and 0.05 h quanles of he dsrbuon of he OLS esmaor of hs parameer. 10 MU αˆ OLS FIGURE 1: POINT AND INTERVAL MEDIAN-UNBIASED ESTIMATION Fracles, OLS h 45 Medan.05h AR MU Two mporan pons are worh nong. Frs, he relaonshp beween he quanles of he dsrbuon and he value of α seems o be monooncally ncreasng, supporng Proposon 2. Second, he fac ha he OLS esmaor s downward medan-based can be seen by observng ha he medan T + 1 = Values from Table III, for from Andrews (1993, p.151) are used here and hey correspond o he model ha ncludes an nercep plus a lnear me rend. 8 CIDE

13 Medan-Unbased Esmaon n Panel Daa funcon les below he 45 degree lne. Fnally, he closer o one s α he hgher s he medan bas of he OLS esmaor. 11 Medan-unbased esmaon proceeds as follows. The value of he OLS esmae (uncorreced) whch s assumed o be exacly equal o (vercal axs) s mapped o s medan-unbased esmae usng he medan funcon, whch gves a value of exacly 0.8 on he horzonal axs. Therefore, he medan-unbased esmae wll equal 0.8. The 90% confdence nerval s found by mappng he same OLS esmae o he 0.95 h and 0.05 h quanles respecvely. Ths gves approxmaely he nerval[0.7,0.93] Approxmaely Medan-Unbased Esmaon Andrews and Chen (1994) exend Andrews (1993) medan-unbased esmaon mehod o AR( p ) models. However, n hs case he mehod wll only be approxmae snce reles on smulaon raher han exac compuaon of he quanles of he dsrbuon of he OLS esmaor and also because he unknown nusance parameers mus be esmaed usng an erave algorhm. Consder he followng laen varable model wh an nercep and me rend: 13 * y = β + β + y, for = p+ 1, K T (6) 0 1 Where he laen varable follows he AR( p ) process: * * * * y = φ1y 1+ φ2y φpy p + K ε, for = 1, KT 2 The error process s assumed.. d(0, σ ). The prevous process can equvalenly be expressed as: * * * * y = α y 1+ ψ1 y 1+ + ψ p 1 y p+ 1+ K ε, for = 1, KT * * Andrews and Chen assume ha y s saonary whenever α ( 1,1) whle y s saonary whenα = 1. Ths s equvalen o assumng ha only he larges roo of he characersc polynomal n (7) can be equal o one. From (6) and (8) we can oban: δ δ α ψ K ψ ε y = y 1+ 1 y 1+ + p 1 y p + 1+ Where δ0 = β0 α + β1 α ψ1 ψ p + 1 (1 ) ( K ) andδ 1 = β 1 (1 α). The nvarance properes of he dsrbuon of OLS esmaor of α n model (9) can be summarzed n he followng proposon. 11 Obvously, he smaller he sample szes he larger wh be he downward bases. 12 For smplcy we have aken he closes values o gven n Andrews Tables. These are for he.05 h quanle and for he 95 h quanle. More precse values can be found eher by nerpolaon or by compung hese quanles over a fner grd around 0.65 and We focus here on case 4 only (model wh nercep and me rend). A smlar approach apples o cases 1 and 2. (7) (8) (9) DIVISIÓN DE ECONOMÍA 9

14 Rodolfo Cermeño Proposon 3: Invarance of he dsrbuon of he OLS esmaor of he parameerα The dsrbuon of OLS esmaor of α ˆ α ) n model (9) depends on ( α, ψ 1, K, ψ ) p 1. I does no depend on does no depend on he nal value y. + 1 p ( OLS β β 2 σ and n he case 1 0, 1 or α = Proofs can be found n Andrews and Chen (1994, p.203). The prevous proposon mples ha Andrews (1993) exacly medan-unbased esmaor canno be appled n hs conex snce ( ψ 1, K, ψ p 1) are unknown. Andrews and Chen (1994) propose an erave procedure o oban approxmaely medan-unbased esmaes. Ths s gven n he followng defnon. Defnon 2: Approxmaely medan-unbased esmaor An approxmaely medan-unbased esmaor of he parameer vecor ( α, δ 0, δ1, ψ 1, K, ψ p 1) n model (9), denoed by ( ˆ, α ˆ δ ˆ 0, δ1, ψˆ 1, K, ψˆ p 1) AMU can be deermned usng he followng erave algorhm: Esmae model (9) by OLS o oban ˆ, α ˆ δ, ˆ δ, ψˆ, K, ˆ (1), where he ( ψ p 1) OLS supra ndex (1) denoes he frs eraon. (1) Trea ( ψˆ, K, ) as rue parameer values and oban he exacly medan 1 ψˆ p 1 OLS αˆ MU (1) unbased esmaor. Trea ( ˆ,, ) 1 ψˆ p 1 (1) αˆmu as rue parameer value and compue new values of ψ K from he OLS regresson of (1) ( y ˆ α MU y 1) on ( 1,, y 1, K, y p+ 1).Call hem ( ψˆ ˆ ) (2) 1, K, ψ p 1. If ˆ α ( 1), OLS MU = 1 exclude from he regresson, as mpled by equaon (9). (2) Trea he new values ( ψˆ, K, ) as gven and compue agan he 1 ψˆ p 1 OLS (2) exacly medan unbased esmaor αˆ MU. Repea seps () and (v) unl convergence or for a gven number of eraons. The Andrews-Chen algorhm wll produce he followng vecor of esmaes: ( ˆ, α ˆ δ ˆ ( J ) { ( ) } ( J + ˆ α ˆ ˆ 1) OLS, δ0, δ1, ψˆ 1, Kψ p 1 0, δ1, ψˆ 1,, ψˆ p 1) AMU = ˆ K (10) Once convergence s acheved a eraon J, he approxmaely medanunbased esmaor of he AR parameerα, denoedαˆ, wll be equal o ( J ) he OLS esmae found n sep (v). Tha s, ˆ α = ˆ. Consequenly, he OLS AMU α OLS AMU 10 CIDE

15 Medan-Unbased Esmaon n Panel Daa approxmaely medan-unbased esmaor of he remanng parameers wll be obaned a eraon ( J +1) from he regresson of ( y ˆ α MU y 1) on ( 1,, y, K, y p 1) n he case ˆ α < 1, or from he regresson of y ˆ α y ) on 1 + 1, y, K, ) f ˆ = 1 ( 1 y p+ 1 MU MU ( MU 1 ( α. These are denoed as ( ˆ, ˆ δ, ψˆ,, ) J + 1) ˆ ψ p 1 OLS δ K n equaon (10). ( ) The only dffculy n he prevous procedure s he compuaon of ˆ α j MU n seps () and (v). In order o do hs s necessary o compue he medan funcon whch wll be gven by he se of pars ( α, m), whereα s a gven value on he nerval [ 1,1], and m s he medan of he dsrbuon ofαˆ OLS. Ths can be carred ou hrough Mone Carlo smulaons as follows: Take he resuls of a gven eraon ( ) ( j ˆ δ, ˆ δ, ψˆ, K, ) as well as he ψˆ p 1 OLS sample sze and a gven value of α as fxed and randomly generae a large number of AR( p) processes (say 200,000) as descrbed by equaon (9). The error erm can be obaned from a sandard normal dsrbuon, whch mples assumng σ 2 = 1. For each randomly generaed sample esmae model (9) by OLS and sore he value αˆ OLS n some vecor. Compue he emprcal dsrbuon ofαˆ OLS, parcularly he medan (m) and oher relevan quanles. Ths wll gve he par ( α, m) for he case of he medan funcon. Repeang he prevous procedure for a relevan grd of values of α wll produce he desred medan funcon. The medan-unbased esmae can hen be obaned as shown n Defnon 2. Tha s, by machng he OLS esmae from regresson (9), based on acual daa, o he closes medan value and s correspondng par, whch s he medan-unbased esmae of he AR parameer Panel Exacly Medan-Unbased Esmaon Exacly medan-unbased esmaon can be exended o panel daa. The smples case s consdered by Cermeño (1999). Ths s a sraghforward exenson of Andrews (1993) o panel daa wh homogeneous dynamcs and.. d. dsurbances. Furher, Phllps and Sul (2003) formulae a more general approach and consder a wder varey of models and esmaors ncludng parameer heerogeney and cross-seconal correlaon. Consder he followng panel model wh a laen varable: y µ, = 1, K, N and = 0, K, T (11) = + λ + y Where he ndex refers o ndvduals or cross-secons such as frms or counres, and ndexes me perods. Thus, he panel s made up of ( T +1) DIVISIÓN DE ECONOMÍA 11

16 Rodolfo Cermeño observaons for each of he N ndvduals or cross-secons. In he panel daa leraure, µ and λ are referred o as ndvdual and me specfc effecs respecvely and can be reaed as fxed or random. In hs paper, hey are reaed as fxed parameers. The laen varable follows he AR (1) process: y = α y 1 + ε, 1, K, N and = = 1, K, T (12) The error erm ( ε ) s assumed o be ndependen and dencally dsrbued 2 boh across me and ndvduals, denoed ε ~.. d(0, σ ). Ths assumpon mples homoskedasc, non auocorrelaed as well as cross seconally ndependen errors. 14 Explcly: 2 E( ε, ε ) = σ for = j, = s js E( ε, ε ) = 0 js E( ε, ε ) = 0 js for = for j, s j, = s Ceranly, hs assumpon can be que srong n a panel conex and cauon should be exercsed n pracce. We wll urn back o hs pon laer n hs secon. From (11) and (12) we can oban he followng panel model n erms of he observable varable: y 0 0 = + λ + αy 1 µ + ε, = 1, K, N and = 0, K,T 0 0 Where µ = µ (1 α ) and λ = λ αλ 1. In panel regresson, α can be esmaed by OLS eher afer ncludng ndvdual specfc and me specfc dummy varables n he regresson or by performng he so called Whn ransformaon whch s based on paroned regresson. In boh cases, he OLS esmaor yelds he same resul and s generally known as he Leas Squares Dummy Varable ( LSDV ) esmaor. 15 The nvarance of he LSDV esmaor of α n model (14) s esablshed n he followng proposon: Proposon 4: Invarance of he dsrbuon of he LSDV esmaor of α 2 Under he assumpon ha ε ~.. d(0, σ ), he dsrbuon of he LSDV esmaor of α ˆ α ) n model (14) s nvaran o he specfc values ( LSDV 2 of µ, λ, σ and when α = 1 does no depend on he nal values y 0. Ths nvarance propery also holds for he cases () µ 0, λ = 0, () 0, λ = 0 and () µ 0, λ = β. µ (13) (14) 14 The possbly of non conemporaneous cross correlaon s ruled ou elsewhere n he panel daa leraure and s ruled ou here as well. 15 See Balag (2003) or Hsao (2002) for deals on hese panel daa esmaors. 12 CIDE

17 Medan-Unbased Esmaon n Panel Daa Cermeño (1999) offers smulaon evdence ha suppors hs proposon. Phllps and Sul (2003) also show he monooncy of he medan funcon for he cases () µ = 0, λ = 0, () µ 0, λ = 0 and () µ 0, λ = β and for sample szes N 5 and T The nvarance prope ry of he LSDV esmaor n (14) can nuvely be see n by consderng ha he whn ransformaon wpes ou he ndvdual and me specfc effecs. Therefore, he esmaor ˆLSDV α and s dsrbuon are ndependen of hese effecs. The prevous papers also show by smulaon ha, he medan and oher relevan quanles are srcly ncreasng funcons ofα, condon ha s requred for mplemenaon of medan-unbased esmaon. The followng defnon s a sraghforward exenson of Defnon 1 o he panel daa case under he prevous assumpons. Defnon 3: Panel exacly medan-unbased esmaor of α Le m = f ( α, N, T ) denoe he medan funcon of he LSDV esmaor of α UR ( 0 < α 1) for a gven sample sze ( N, T ) n model (14) and le m deno e he medan of he dsrbuon of he LSDV esmaor of α whenα = 1. The panel exacly medan-unbased esmaor of α s defned as f ollows: 1 UR f ( ˆ α ) f ˆ α < m j LSDV LSDV ˆ α MU = (15) UR 1 f ˆ α LSDV m Andrews (1993) was able o compue he quanles of he dsrbuon of he OLS esmaor of he parameer α exacly for he me seres case. For he panel daa case hs can be done approxmaely usng Mone Carlo smulaons as follows: Take µ, λ as well as a sample sze ( T, N ) and a gven value of α as fxed and randomly generae a large number of AR (1) processes as descrbed by equaon (14). The error erm can be drawn from a sandard normal dsrbuon ( σ 2 = 1), assumpon ha s mmaeral gven he nvarance propery n Proposon 4. For each randomly generaed sample, esmae model (14) usng he LSDV esmaor and sore he value αˆ n some vecor. LSDV Compue he emprcal dsrbuo n of αˆ LSDV parcularly he medan (m) and oher relevan quanles. Ths wll gve he par ( α, m) for he case of he medan funcon. The medan funcon can be obaned by repeang he prevous procedure for a relevan grd of values ofα. Once he medan funcon s obaned, he medan-unbased esmaor can hen be compued as shown n Defnon 4. Tha s, by machng he LSDV esmae from regresson (14), based on acual daa, o he closes medan value (m) and fndng s correspondng DIVISIÓN DE ECONOMÍA 13

18 Rodolfo Cermeño par (α ) whch wll be he panel exacly medan-unbased esmae of he AR parameer. As we menoned before, he assumpon of.. d. errors may no be approprae n he panel case. Two possble avenues can be followed o deal wh hs poenal problem: () model explcly any deparures from hs assumpon, and () check he robusness of he medan-unbased esmaor obaned under he.. d. assumpon when n fac hs assumpon s volaed. Cermeño (1999) followed he las approach, showng ha for auocorrelaon, heeroskedascy as well as cross-seconal correlaon paerns of magnudes smlar o hose found n he acual daa, he PEMU esmaor s que robus and re-esmaon of quanles consderng he prevous problems leads o essenally he same conclusons. Phllps and Sul (2003) focus on he frs avenue and develop a more general framework ha explcly consders cross-seconal correlaon as well as heerogeneous AR parameer values. Ths s descrbed n he followng wo secons Panel Feasble Generalzed Medan-Unbased Esmaon Consder now ha he error erm n (12) s no ndependen among cross secons. In hs case, he expecaon n he hrd lne of (13) s dfferen from zero, ha s E( ε, ε ) = σ 0 for j, = s (16) js j Assume for he momen ha for all = s he covarance marx of he vecor τ of dsurbances [ ε 1, ε 2, K ε N ] s known and s gven by he NxN marx V. The supra ndex τ denoes he ransposon operaor. Leε be a vecor of dmensontn ha sacks he N ndvdual vecors of resdua ls, each of dmensont. I s well known ha he covarance marx of ε can be expressed as τ Cov( ε ) = E( εε ) = Ω = ( V I T ) (17) Where denoes he Kronecker produc. From GLS heory s also well known ha 1 ' E( εω ε ) = I NT (18) Ths mples ha for a gven marx V we can perform he well known GLS ransformaon on (14) and compue he GLS esmaor of he parameers. Also, wh a conssen esmae of V he feasble GLS esmaor can be obaned. For a gven V s possble o compue he medan funcon for he GLS esmaor and correc for he bas usng medan-unbased esmaon. For large T and small N panels, compuaon of he FGLS esmaor based upon a general covarance srucure gven by V follows sraghforwardly. 14 CIDE

19 Medan-Unbased Esmaon n Panel Daa However, for small T and large N or large T and large N panels he FGLS esmaor does no exs. Phllps and Sul (2003) consder a specfc paern of cross-seconal correlaon by specfyng an error erm wh a common me effec whch can mpac dfferenly each cross secon. Ths s, u = δ θ + ε Where δ ( = 1,2, K, N ) are parameers, θ ~.. d. N(0,1) over and 2 ε ~.. d. N(0, σ ) over and ε, ε js, θ s are ndependen for all j and for all s,. In hs seng, he covarance among cross secons s gven by E( u u j ) = δ δ j The laen varable model now consss of he followng wo equaons y = µ + λ + y, = 1, K, N and = 0, K, T y = αy 1 + u, = 1, K, N and = 1, K, T (22) As n he prevous subsecon, form (21) and (22) he followng model can be obaned n erms of he observable varable y : 0 0 y = µ + λ + αy 1 + u, = 1, K, N and = 0, K,T 0 0 Where µ µ (1 α ) and λ λ αλ. Assume ha u s defned n equaon = = 1 (19). Consder he cases () (19) (20) (21) (23) µ λ = 0 () µ 0, λ = 0 and () = µ 0, λ 0 and assume ha λ = β. Under hese assumpons, he dsrbuon of he panel GLS esmaor of α has a smlar nvarance propery han he PEMU esmaor. Ths s formulaed n he followng proposon. Proposon 5: Invarance of he dsrbuon of he panel GLS esmaor of α Under he prevous assumpons, he dsrbuon of he panel GLS esmaor of α ( ˆ α PGLS ) n model (23) depends only onα n all cases. Whenα = 1, does no depend on he nal values y 0 n cases () and (). See Phllps and Sul (2003). Ths nvarance propery s remarkable and he nuve explanaon s ha we are regressng y on y 1 and performng he GLS ransformaon only mples re-scalng boh of hem. Unforunaely, n pracce he covarance marx s unknown and mus be esmaed (hs s he panel feasble GLS esmaor) and n hs case he prevous nvarance propery wll no hold n general. However, Phllps and Sul (2003) show ha DIVISIÓN DE ECONOMÍA 15

20 Rodolfo Cermeño provded a conssen esmaor of he covarance marx s used, he prevous nvarance propery wll hold asympocally and hey propose he followng erave procedure o oban hs esmaor. Defnon 4: Panel feasble GLS medan-unbased esmaor of α The panel feasble medan-unbased esmaor of he parameerα n model (23) denoed αˆ PFGLSMU, can be deermned from he followng erave algorhm: Sar by assumng cross-seconal ndependence and oban he panel exacly medan-unbased esmaor Usng he esmaed resduals from () consruc an esmae of he error covarance marx V ˆ, followng Phllps and Sul (2002), secon 4.2 Usng V ˆ compue he panel feasble GLS Oban he panel feasble GLS medan-unbased usng he medan funcon of hs esmaor, no he panel exacly medan-unbased esmaor Repea seps () o () bu n sep () use updaed resduals from (v) unl he panel feasble GLS medan-unbased esmaor converges As n he prevous cases, compuaon of he medan funcon for he panel feasble GLS esmaor can be done by Mone Carlo smulaons. These smulaons are smlar o he ones explaned for he panel exacly medanunbased esmaor, wh he only dfference beng ha n hs case he resduals n he daa generaon process have a conemporaneous covarance paern gven by Vˆ, whch s he esmaed covarance marx of resduals Panel SUR Medan-Unbased Esmaon The prevous models assume he same AR parameer for all cross secons n he panel. Ths assumpon can be relaxed by consderng he followng laen varable model: y = µ + λ + y y = α y + u 1, = 1, K, N and = 0, K, T, = 1, K, N and = 1, K, T (24) (25) In hs case he followng model s obaned n erms of he observable varable: 0 0 y = µ + λ + α y 1 + u, = 1, K, N and = 0, K,T 0 0 Where µ µ (1 α ) and λ λ α λ. = = 1 = λ = Phllps and Sul consder he cases wh µ 0 (model M1), µ 0, λ = 0 (model M2) and µ 0, λ = θ (model M3), whch are analogous o he models wh no nercep and no me rend, nercep only and nercep and me (26) 16 CIDE

21 Medan-Unbased Esmaon n Panel Daa rend, menoned before. In addon, hese auhors allow for cross-seconal dependence of he form gven n equaon (19) and show ha usng hs nformaon can lead o subsanal effcency gans. The erave procedure for medan-unbased esmaon n hs seng s oulned as follows. Defnon 5: Panel SUR medan-unbased esmaor of The panel SUR medan-unbased esmaor of he parameer α n model (26), αˆ denoed as SURMU, can be deermned eravely as follows: Assumng cross-seconal ndependence, oban exacly medan-unbased ˆ esmaes for each seres αemu, = 1, K, N and use he esmaed resduals o consruc an esmae of he error covarance marx V ˆ as n Phllps and Sul (2003). Usng V ˆ compue convenonal SUR esmaes for he panel αˆ SUR. Oban he panel SUR-MU esmaes by machng each αˆ SUR o s correspondng medan-unbased esmae usng he medan funcon of he αˆ SUR esmaor. Repea seps () o () unl he panel SURMU converges. I s mporan o remark ha n sep () we need o oban he medanfuncon of he αˆ SUR esmaor. As before, hs can be acheved by Mone Carlo smulaons as follows: For a gven sample sze and a parameer value α = α, = 1, K, N generae a large number of processes usng errors wh a covarance srucure gven by Vˆ (obaned a sep ) a each pon n me. For each process, compue he αˆ SUR esmaor and sore n some marx. Compue he emprcal dsrbuon of he sored αˆ SUR esmaes, par cularly he medan as well as oher relevan quanles. Repea () o () for a relevan grd of values for α = α o oban he pars of values ha wll defne he medan funcon How far Panel Medan-Unbased Esmaon has reached? The medan-unbased esmaon echnques have been exended o panel and proved o be useful ools for emprcal analyss. The mos mporan conrbuon from a heorecal and mehodologcal pon of vew has probably been made by Phllps and Sul (2003), as hey generalze he smples panel exacly medan-unbased esmaor (PEMU) o he case ha consders α DIVISIÓN DE ECONOMÍA 17

22 Rodolfo Cermeño coeffcen heerogeney as well as heeroskedasc and cross-correlaed dsurbances (PSURMU). However her approach s focused on AR(1) dynamc process. On he oher hand, as we have menoned n he prevous secon, more recenly Murray and Papell (2005b) apply Andrews and Chen (1994) approxmaely medan-unbased esmaon, whch deals wh AR ( p) processes, o panel daa bu her applcaon s lmed o homogeneous dynamcs for all ndvdual me seres n he panel. Clearly, a model wh heerogeneous dynamcs as well as heeroskedasc and cross-correlaed dsurbances n he panel wll be a naural exenson of he panel medanunbased esmaon leraure. More recenly, Cermeño and Grer (2006) have addressed he modelng of condonal heeroskedascy and cross-seconal correlaon n panel daa. These auhors pon ou ha despe s well known ha mos macroeconomc and fnancal processes are condonally heeroskedasc, he emprcal panel daa leraure n hese areas has praccally gnored hs fac. The leraure on panel medan-unbased esmaon s no an excepon. Thus, consderng a me varyng covarance marx of dsurbances as well as suable non lnear models o capure hs behavor mgh be worh overakng n order o ge more accurae characerzaons of he dynamcs of real phenomena. 3. Emprcal Applcaons In hs secon we presen hree emprcal applcaons of panel medanunbased esmaon. The frs wo applcaons nvesgae he well known hypohess of convergence n per capa GDP for he cases of he USA and Mexcan saes. The hrd emprcal applcaon, evaluaes he PPP hypohess usng daa on real exchange raes of 14 developng counres. In all cases focuses on deermnng f pon medan-unbased esmaes are conssen wh saonary, mean reverng processes or no. I s mporan o remark ha he resuls dscussed n hs secon are prelmnary and hey are based on panel exacly medan-unbased esmaon only. These resuls wll ceranly consue benchmarks for mplemenng he complee se of medan-unbased esmaors dscussed n hs paper under less resrcve assumpons. Ths s ongong research Convergence among he USA saes In hs applcaon we explore, wheher he dynamcs of per capa ncome of he 48 conguous USA saes s conssen wh condonal convergence. The followng panel model s consdered. y = + 1 α) β + αy 1 µ ( + ε (27) 18 CIDE

23 Medan-Unbased Esmaon n Panel Daa Ths model s obaned from (11) and (12) under he assumpon ha λ = β. I s also assumed ha µ are dfferen and sascally sgnfcan, whch s conssen wh exsng emprcal evdence. In he case 0 < α < 1 he per capa ncome process would be conssen wh condonal convergence and wll be characerzed as a rend saonary process. In hs case, sae economes wll grow a he same average rae. On he oher hand, when α = 1 he process wll have a sochasc rend and herefore wll be non convergen. In hs case, he per capa ncome of he dfferen saes wll be growng a dfferen raes. Smlarly han n un roo esng, we also wan o evaluae he null hypohess H 0 : α = 1 (non convergence) agans he alernave H 1 : α < 1 (condonal convergence). Thus, n addon o pon medan-unbased esmaes, we oban 90% nerval unbased esmaes. If he upper bound of he 90% confdence nerval les below uny, he null hypohess H 0 would be rejeced (a he 10% sgnfcance level n hs case) resul ha would agree wh condonal convergence. We use per capa ncome daa for he 48 conguous saes durng he perod 1929 o The LSDV esmae of α s equal o The correspondng panel exacly medan-unbased esmae s equal o and he lower and upper bounds of he 90% confdence nerval are and respecvely. See Table 1 (case N=48, T+1=77) for abulaed quanles of he dsrbuon of hs esmaor. Clearly, he pon and nerval medanunbased esmaes suppor he hypohess of condonal convergence even f we consdered broader confdence nervals. The mpled speed of convergence s approxmaely 15.23% whch s relavely hgh bu acually s no surprsng gven he hgh degree of facor mobly across he USA saes. These resuls are n lne wh prevous fndngs by Evans (1997) bu ceranly dffer sharply from hose based on cross-seconal regressons, such as he 2% convergence rae repored by Barro and Sala--Marn (1991, 1992). 16 Clearly, he fxed effecs bas of he OLS esmaor used n hese sudes produces hghly msleadng resuls. In order o have a beer pcure of he degree of heerogeney n he prevous panel, we have obaned medan-unbased esmaes for each sae separaely, followng Andrews (1993). 16 Evans (1997) shows ha he convenonal cross-secon regresson approach produces nconssen esmaes of convergence raes gven s nably o conrol for all cross economy heerogeney. Usng Sock (1991) approach, Evans fnds an average converge rae of 15.5% for he 48 U.S. conguous saes. DIVISIÓN DE ECONOMÍA 19

24 Rodolfo Cermeño I should be poned ou, hough, ha gven he relavely shor me span, bases are expeced o be hgher han n he panel daa case. For he same reason, unbased confdence nervals are much wder n he me seres case. 17 A few facs should be remarked. Frs, he medan-unbased pon esmaes are relavely homogeneous and concenrae on he nerval (0.81, 0.94) wh an average value of 0.87 and a sandard devaon of Roughly, he prevous average medan unbased esmae of he AR parameer mples a convergence rae of 13.9 %. Secondly, he medan-unbased pon esmaes are conssen wh he condonal convergence hypohess for all saes. Ths s a remarkable resul snce he pon esmaes ndcae ha all seres are n fac rend saonary, whch s conssen wh he panel resuls and he problem of havng saonary and non saonary seres whn he panel s no presen here. As expeced, he me seres unbased 90% confdence nervals are conssen wh he un roo hypohess n all bu wo ndvdual seres, supporng he non convergence hypohess. 18 Ths resul smply confrms ha wh a relavely small number of me seres observaons s mpossble o dsngush un roo from near o un roo dynamcs. Fgure 2 plos he me seres medan-unbased esmaes and correspondng 90% confdence nervals. Overall, boh he panel daa and me seres medan-unbased esmaes are conssen wh condonal convergence hypohess and ndcae convergence raes of 15.2 and 13.9% respecvely. FIGURE 2: MEDIAN-UNBIASED ESTIMATES AND 90% CONFIDENCE INTERVALS FOR THE USA STATES Lower bound Medan Ubased Upper bound MI IL ID NV OH MN WV PA NE NY CA TN NH CT AR RI ND MA VA NC FL GA LA NM 17 Compare correspondng panels of Table 1 (panel daa) and Table 2 (me seres). 18 Table 3 shows medan unbased and 90% confdence nervals for each USA sae. 20 CIDE

25 Medan-Unbased Esmaon n Panel Daa 3.2. Convergence among he Mexcan saes The prevous convergence hypohess can also be evaluaed n he case of he Mexcan saes. Ths avenue of research s parcularly relevan gven he avalably of a newly consruced yearly GDP daa base for he 32 Mexcan saes over he perod 1940 o 2004 by German-Soo (2005). 19 As was poned ou n Cermeño (2001), he exremely lmed me span of avalable me seres of he GDP for he Mexcan saes, has made dffcul o es convergence hypoheses n he pas, forcng mos researchers o assume a pror some form of convergence, usually absolue convergence. In fac, hs auhor aemps o dsngush beween absolue and condonal convergence fndng no suppor for absolue convergence. Nowhsandng, as Garrdo (2007) pons ou, he avalable nformaon used n all prevous sudes abou convergence n Mexco s serously lmed boh n erms of me span and relably. The presen applcaon aemps o conrbue o he convergence debae by applyng medan-unbased esmaon o he newly avalable yearly daa base. For he case of he Mexcan saes, he LSDV esmae s equal o However, afer he correcon for he medan bas we oban a medanunbased pon esmae of , whch ceranly ndcaes a rend saonary alhough hghly perssen, near o un roo, process. See Table 1 (case N=32, T+1=65) for abulaed fracles of he dsrbuon of he LSDV esmaor for hs case. Roughly, he mpled convergence rae n he case of Mexco s 1.57 per year wh a correspondng half-lfe of 44.1 years. Dfferenly han n he USA case, he 90% confdence nerval ranges beween 0.97 and 1.00, hus ncludng he possbly of non convergence. I s mporan o remark ha even f no form of convergence s suppored when akng all 32 saes s sll possble o fnd some convergence clusers or clubs a he regonal level or by pars of saes. Precsely, Garrdo (2007), usng a me seres approach, fnds some evdence n favor of absolue and condonal convergence among pars of saes n Mexco, alhough he also remarks ha n no case he prevous resuls are favored n more han 50% of he pars. As expeced, he me seres medan-unbased esmaes are conssen wh dfference saonary processes. 20 Dfferenly han n he USA saes case, where all ndvdual medan-unbased esmaes le below uny, n he Mexcan case only n 10 cases (ou of 32) he pon medan unbased esmaes are conssen wh condonal convergence. Also, all bu one unbased 90% confdence nervals do no rejec he mananed null hypohess of non convergence. See Fgure 3 for a plo of hese resuls. To gve a numercal dea, he me seres medan-unbased esmaes are n he range (0.60, 1.15), 19 See Garrdo (2007) for deals on how hs daa base was consruced. 20 See Table 4 for a complee ls of ndvdual medan unbased esmaes and he correspondng plo. DIVISIÓN DE ECONOMÍA 21

26 Rodolfo Cermeño wh an average of 1.00, and sandard devaon of Thus, he average medan-unbased esmae of ndvdual saes does no suppor he condonal convergence hypohess, ponng o a suaon where economes are growng a dfferen raes. FIGURE 3: MEDIAN-UNBIASED ESTIMATES AND 90% CONFIDENCE INTERVALS FOR THE MEXICAN STATES Lower bound Medan Ubased Upper bound Chhuahua Zacaecas Sonora Baja Calforna Durango Nuevo León Morelos Guanajuao Queréaro Areaga Yucaán Colma Snaloa Mchoacán de Ocampo Guerrero Jalsco Méxco Overall, boh he nerval panel esmaes and he me seres pon esmaes are conssen wh he non convergence hypohess n he case of he Mexcan saes. Usng he pon panel medan-unbased esmaes only we can argue n favor of condonal convergence alhough wh a remarkably slow convergence rae The PPP hypohess n 14 Developng Counres. Ths applcaon aemps o evaluae he well known purchasng power pary hypohess (PPP). Consder he followng model: e = p p + u * Where e s he logarhm of he nomnal exchange rae n counry and * p, p are he prce levels n he domesc and foregn counry (USA n hs case) respecvely (also n logarhms). (28) 22 CIDE

27 Medan-Unbased Esmaon n Panel Daa The dsurbance erm u can be vewed as a emporary devaon from pary. One popular es of he PPP hypohess s made by evaluang wheher * here s conegraon among e, p, p, wh conegraon vecor equal o ( 1, 1,1). Essenally, gven ha he prevous varables are negraed o order one, denoed I(1), f hey are n fac conegraed, he devaons u mus be a zero mean saonary process. In hs case, he devaons from he PPP wll only be emporary. An alernave approach s o consder, nsead, * he real exchange rae defned as q = e p + p and o evaluae f hs process s saonary. Tha s, o evaluae wheher he real exchange rae s a mean reverng process or no. Somemes hs approach s referred o as resrced conegraon, snce n hs case he resrcon ha he coeffcens are equal o ( 1, 1,1) s mposed a pror. Ths approach s suable for medanunbased esmaon and wll be used here. In order o es he PPP hypohess consder he model: q = µ (1 α) + αq 1 + ε (29) Ths model corresponds o case 2 menoned n he prevous secon and allows for ndvdual fxed effecs under he alernave hypohess ha he PPP holds H 1 : α < 1. The null hypohess s H 0 : α = 1, n whch case he PPP hypohess wll no hold. For hs applcaon, we use monhly daa on real effecve exchange raes for 14 developng counres over he perod July, 1978 o Sepember, In he presen case he LSDV esmae (uncorreced) s equal o The correspondng medan-unbased esmae of hs parameer s equal o and he unbased 90% confdence nerval has bounds of and See he esmaed quanles of he dsrbuon of hs esmaor n Table 1 (case N=14, T+1=303). Ceranly, boh he pon and nerval unbased esmaes are conssen wh he PPP hypohess, alhough hey ndcae ha he real exchange rae process s hghly perssen and convergence owards seady sae pary levels happens a he slow rae of 1.48%. Ths rae of convergence mples a half lfe esmaed me of 46.8 monhs, abou 4 years. The evdence from me seres esmaes ndcaes ha 5 ou of he 14 developng counres do exhb non mean reverng real exchange rae processes (See Table 5). Fgure 4 llusraes medan-unbased esmaes and 90% confdence nervals for hs sample of counres. For he oher 9 counres he processes are saonary alhough n 2 cases he pon medanunbased esmaes ndcae ha real exchange raes do follow near o un roo dynamcs. In erms of unbased 90% confdence nervals, n all cases he PPP hypohess s no suppored, despe he relavely large me span of he 21 The auhor acknowledges Kevn Grer and Robn Grer for generously provdng hs daa base. DIVISIÓN DE ECONOMÍA 23

28 Rodolfo Cermeño seres. Overall, he panel evdence pons o saonary bu exremely perssen real exchange rae processes whch s conssen wh he mx of un roo and saonary bu near o un roo processes found by examnng ndvdual me seres. FIGURE 4: MEDIAN-UNBIASED ESTIMATES AND 90% CONFIDENCE INTERVALS FOR THE RER IN 14 DEVELOPING COUNTRIES Lower bound Medan Upper bound Venezuela Mexco Peru Argenna Brazl Indonesa Hong Kong Chle Sngapore Korea Thaland Tawan Malaysa Colomba These fndngs are n lne wh oher resuls found n he emprcal leraure on he dynamcs of real exchange raes ha also show eher nonreverng or exremely slow mean reverng processes, as documened for example n Murray and Papell (2002, 2005a). 24 CIDE

29 Medan-Unbased Esmaon n Panel Daa Conclusons In he las decade we have seen mporan developmens on panel medanunbased esmaon, from he smples case of panel exacly medan-unbased esmaon o he more general case of panel SUR medan-unbased esmaon, whch allows for parameer heerogeney as well as heeroskedasc and cross-seconally correlaed dsurbances. Ye, hese developmens can sll be aken furher, for example by consderng hgher order AR processes. In addon, gven ha several dynamc processes, parcularly n macroeconomcs and fnance mgh be subjec o volaly cluserng, seems necessary o model explcly a me varyng covarance marx. The paper provdes hree emprcal applcaons ha use boh he panel as well as he me seres exacly medan-unbased esmaor. Roughly speakng, he resuls obaned n he frs wo examples are conssen wh condonal convergence of per capa ncome among he USA saes and wh non convergence or exremely slow condonal convergence of per capa GDP among he Mexcan saes. In he hrd example, he panel medan-unbased esmaes are conssen wh he PPP hypohess n a sample of 14 developng counres, alhough he real exchange processes are exremely perssen and, hence, hey exhb que slow convergence raes. In all prevous applcaons, remans o see how robus are hese resuls when consderng oher panel medan-unbased esmaors under less resrcve assumpons. DIVISIÓN DE ECONOMÍA 25

30 Rodolfo Cermeño References Andrews, Donald, (1991), Exacly Unbased Esmaon of Frs Order Auoregressve-Un Roo Models, Cowles Foundaon, Yale Unversy, Cowles Foundaon Dscusson Papers: 975. Andrews, Donald, (1993), Exacly Medan-Unbased Esmaon of Frs Order Auoregressve/Un Roo Models, Economerca, 61(1): Andrews, Donald and Chen Hong Yuan, (1992), Approxmaely Medan-Unbased Esmaon of Auoregressve Models wh Applcaons o U.S. Macroeconomc and Fnancal Tme Seres, Cowles Foundaon, Yale Unversy, Cowles Foundaon Dscusson Papers: Andrews, Donald and Chen Hong Yuan, (1994), Approxmaely Medan-Unbased Esmaon of Auoregressve Models, Journal of Busness and Economc- Sascs, 12(2): Balag, Bad, (2005), Economerc Analyss of Panel Daa, Thrd Ed. John Wley. Barro, R. J. and X. Sala--Marn, (1991), Convergence across Saes and Regons, Brookngs Papers on Economc Acvy, 1, Barro, R. J. and X. Sala--Marn, (1992), Convergence, Journal of Polcal Economy, 100, (Apr.), Cashn, Paul; McDermo, C. John and Caherne Pallo, (2004), Terms of Trade Shocks n Afrca: Are They Shor-Lved or Long-Lved? J. of Developmen Economcs 73(2): Cashn, Paul and McDermo C. John (2003), An Unbased Apprasal of Purchasng Power Pary, IMF Saff Papers, 50(3): Cermeño, Rodolfo, (1999), Medan-Unbased Esmaon n Fxed-Effecs Dynamc Panels, Annales d'econome-e-de-sasque, 55-56: Cermeño, Rodolfo, (2001), Decrecmeno y convergenca de los esados mexcanos. Un análss de panel, El Trmesre Económco, LXVIII(4), núm. 272: Cermeño, Rodolfo and Kevn B. Grer, (2006), Condonal Heeroskedascy and Cross-Seconal Dependence n Panel Daa: An Emprcal Sudy of Inflaon Uncerany n he G7 Counres, n Balag, Bad H. (Edor), Panel Daa Economercs, Theorecal Conrbuons and Emprcal Applcaons, Conrbuons o economc analyss, 274, Chaper 10. Elsever. Davd A. Dckey, Wayne A. Fuller, (1979), Dsrbuon of he Esmaors for Auoregressve Tme Seres Wh a Un Roo, Journal of he Amercan Sascal Assocaon, vol. 74, no (Jun.), pp Enders, Waler and Falk Barry, (1998), Threshold-Auoregressve, Medan- Unbased, and Conegraon Tess of Purchasng Power Pary, Inernaonal Journal of Forecasng, 14(2): Paul, Evans, (1997), How Fas Do Economes Converge?, The Revew of Economcs and Sascs, vol. 79, no. 2, (May), pp Fuller, Wayne, (1995), Inroducon o Sascal Tme Seres, 2 nd Ed., John Wley. Garrdo, D., (2007), Convergenca a nvel esaal: Un enfoque de conegracón para el caso de Méxco, Tess de Lcencaura, Cenro de Invesgacón y Docenca Económcas, Méxco. 26 CIDE

31 Medan-Unbased Esmaon n Panel Daa Hamlon, J., (1994), Tme Seres Analyss, Prnceon Unversy Press. Hsao, Cheng, (2003), Analyss of Panel Daa, Second Ed., Cambrdge Unversy Press. Hurwcz, L., (1950), Leas-squares Bas n Tme Seres, n Sascal Inference n Dynamc Economc Models, ed. by T. C. Koopmans, Wley. Imhof, J. P., (1961), Compung he Dsrbuon of Quadrac Forms n Normal Varaes, Bomerka, 48, Kendall, M. G., (1954), Noe on Bas n he Esmaon of Auocorrelaon, Bomerka, vol. 41, no. 3/4. (Dec.), pp Ken, Chrsopher and Cashn Paul, (2003), The Response of he Curren Accoun o Terms of Trade Shocks: Perssence Maers, Inernaonal Moneary Fund, IMF Workng Papers: 03/143. Km, Jae H., (2003), Forecasng Auoregressve Tme Seres wh Bas-Correced Parameer Esmaors, Inernaonal Journal of Forecasng, 19(3): Koers, J. and A. P. J. Abrahamse, (1971), On he Theory and Applcaon of he General Lnear Model, Roerdam Unversy Press. Le Breon A. and D. T. Pham, (1989), On he Bas of he Leas Squares Esmaor for he Frs Order Auoregressve Process, Annals of he Insue of Sascal Mahemacs, 41, Maddala, G. S. and In-Moo Km, (1998), Un Roos, Conegraon and Srucural Change, Cambrdge Unversy Press. Marro, F. H. C. and J. A. Pope, (1954), Bas n he Esmaon of Auocorrelaons, Bomerka, vol. 41, no. 3/4. (Dec.), pp Murray, C. J. and D. H. Papell, (2002), The purchasng power pary perssence paradgm, Journal of Inernaonal Economcs, 56: Murray, C. J. and D. H. Papell, (2005a), The purchasng power pary puzzle s worse han you hnk, Emprcal Economcs, (2005) 30: Murray, C. J. and D. H. Papell, (2005b), Do Panels Help Solve he Purchasng Power Pary Puzzle?, Journal of Busness and Economc Sascs, Ocober 2005, Orcu, G.H. and H. S. Wnokur Jr., (1969), Frs Order Auoregresson: Inference, Esmaon and Predcon, Economerca, 37, Rudebusch, Glenn, (1992), Trends and Random Walks n Macroeconomc Tme Seres: A Re-examnaon, Inernaonal Economc Revew, 33(3): Sock, James H., (1991), Confdence Inervals for he Larges Auoregressve Roo n U. S. Macroeconomc Tme Seres, Journal of Moneary Economcs, 28 (Dec.), DIVISIÓN DE ECONOMÍA 27

32 Rodolfo Cermeño TABLE 1: FRACTILES OF THE LSDV ESTIMATOR IN A DYNAMIC PANEL DATA MODEL WITH FIXED EFFECTS LSDV N=14, T+1=303 N=32, T+1=65 N=48, T+1= Fracles were abulaed usng Mone Carlo smulaons wh 20,000 replcaons ( ) model ncludes fxed effecs n he saonary case and becomes a random walk whou drf when a α =1 (b ) model ncludes fxed effecs and a lnear me rend n he saonary case and ndvdual specfc drfs when α = CIDE

33 Medan-Unbased Esmaon n Panel Daa OLS TABLE 2: FRACTILES OF THE OLS ESTIMATOR IN A TIME SERIES AR (1) MODEL T+1=303 a b T+1=65 b T+1= Fracles were abulaed usng Mone Carlo smulaons wh 20,000 replcaons ( ) model ncludes an nercep n he saonary case and becomes a random walk whou drf when a α =1 (b ) he model s rend saonary under he alernave and a random walk wh drf under he null. DIVISIÓN DE ECONOMÍA 29

34 Rodolfo Cermeño TABLE 3: MEDIAN-UNBIASED POINT ESTIMATES AND 90% CONFIDENCE INTERVALS FOR THE 48 USA STATES Sae OLS Unbased Esmaes Unbased Esmaes Sae OLS Lower Medan Upper Lower Medan Upper MI NH IA WA IL CT DE AL ID AR VT NJ NV RI WI ME OH ND MS MD MN MA SC WY WV VA IN MT PA NC CO OR NE FL KY OK NY GA UT KS CA LA MO TX TN NM AZ SC CIDE

35 Medan-Unbased Esmaon n Panel Daa TABLE 4: MEDIAN-UNBIASED ESTIMATES AND 90% CONFIDENCE INTERVALS FOR THE 32 MEXICAN STATES Sae OLS Lower bound Unbased Esmaes Medan Upper bound Chhuahua Coahula de Zaragoza Zacaecas Baja Calforna Sur Sonora San Lus Poosí Baja Calforna Dsro Federal Durango Qunana Roo Nuevo León Veracruz de Ignaco de la Llave Morelos Aguascalenes Guanajuao Tamaulpas Queréaro Areaga Puebla Yucaán Hdalgo Colma Campeche Snaloa Oaxaca Mchoacán de Ocampo Nayar Guerrero Tlaxcala Jalsco Tabasco Méxco Chapas DIVISIÓN DE ECONOMÍA 31

36 Rodolfo Cermeño TABLE 5: MEDIAN-UNBIASED ESTIMATES AND 90% CONFIDENCE INTERVALS FOR THE RER PROCESS IN 14 DEVELOPING COUNTRIES UNBIASED ESTIMATES Counry OLS Lower bound Medan Upper bound Venezuela Mexco Peru Argenna Brazl Indonesa Hong Kong Chle Sngapore Korea Thaland Tawan Malaysa Colomba CIDE

37 Novedades DIVISIÓN DE ADMINISTRACIÓN PÚBLICA Cejudo, Gullermo, Crcal Juncures or Slow-Movng Processes? The Effecs of Polcal and Economc Transformaons, DTAP-186 Sour, Laura, Un repaso de concepos sobre capacdad y esfuerzo fscal, y su aplcacón para los gobernos locales mexcanos, DTAP-187 Sanbañez, Lucreca, School-Based Managemen Effecs on Educaonal Oucomes: A Leraure Revew and Assessmen of he Evdence Base, DTAP-188 Cejudo, Gullermo y Sour Laura, Cuáno cuesa vglar al goberno federal?, DTAP-189 Cejudo, Gullermo, New Wne n Old Boles: How New Democraces Deal wh Inhered Bureaucrac Apparauses, DTAP-190 Arellano, Davd, Fallas de ransparenca: haca una ncorporacón efecva de polícas de ransparenca en las organzacones públcas, DTAP-191 Sour, Laura y Munayer Lala, Aperura políca y el poder de la Cámara de Dpuados durane la aprobacón presupuesara en Méxco, DTAP-192 Casar, Ma. Amparo, La culura políca de los polícos en el Méxco democráco, DTAP-193 Arellano, Davd y Lepore Waler, Economc Growh and Insuons: The Influence of Exernal Acors, DTAP-194 Casar, Ma. Amparo, Los gobernos sn mayoría en Méxco: , DTAP-195 DIVISIÓN DE ECONOMÍA Casañeda, Alejandro y Vllagómez Alejandro, Ingresos fscales peroleros y políca fscal ópma, DTE-382 Dam, Kanska, A Two-Sded Machng Model of Monored Fnance, DTE-383 Dam, Kanska, Gauer Axel y Mra Manpushpak, Effcen Access Prcng and Endogenous Marke Srucure, DTE-384 Dam, Kanska y Sánchez Pagés Sanago, Depos Insurance, Bank Compeon and Rsk Takng, DTE-385 Carreón, Vícor, D Gannaale Sona y López Carlos, Mercados formal e nformal de crédo en Mexco: Un esudo de caso, DTE-386 Vllagómez, Alejandro y Roh Bernardo, Fscal Polcy and Naonal Savng n Mexco, , DTE-387 Sco, John, Agrculural Polcy and Rural Povery n Mexco, DTE-388 Hogan, Wllam, Rosellón Juan y Vogeslang Ingo, Toward a Combned Merchan- Regulaory Mechansm for Elecrcy Transmsson Expanson, DTE-389 Roa, Ma. José y Cendejas José Lus, Crecmeno económco, esrucura de edades y dvdendo demográfco, DTE-390 Krsansen, Tarje y Rosellón Juan, Merchan Elecrcy Transmsson Expanson: A European Case Sudy, DTE-391

38 DIVISIÓN DE ESTUDIOS INTERNACIONALES Schavon, Jorge y Velázquez Rafael, El 11 de sepembre y la relacón Méxco- Esados Undos: Haca la securzacón de la agenda?, DTEI-150 Velázquez, Rafael, La paradplomaca mexcana: Las relacones exerores de las endades federavas, DTEI-151 Meseguer, Covadonga, Do Crses Cause Reform? A New Approach o he Convenonal Wsdom, DTEI-152 González, Guadalupe y Mnushkn Susan, Líderes, opnón públca y políca exeror en Méxco, Esados Undos y Asa: un esudo comparavo, DTEI-153 González, Guadalupe y Mnushkn Susan, Leaders, publc opnon and foregn polcy n Mexco, he Uned Saes, and Asa: a comparave sudy, DTEI-154 González, Guadalupe y Mnushkn Susan, Opnón públca y políca exeror en Méxco, DTEI-155 González, Guadalupe y Mnushkn Susan, Publc opnon and foregn polcy n Mexco, DTEI-156 Orz Mena, Anono, El Traado de Lbre Comerco de Amérca del Nore y la políca exeror de Méxco: lo esperado y lo aconecdo, DTEI-157 Orz Mena, Anono y Fagan Drew, Relang o he Powerful One: Canada and Mexco s Trade and Invesmen Relaons wh he Uned Saes, DTEI-158 Schavon, Jorge, Políca exeror y opnón públca: Méxco ane el mundo, DTEI- 159 DIVISIÓN DE ESTUDIOS JURÍDICOS Fondevla Gusavo, Esudo de percepcón de usuaros del servco de admnsracón de jusca famlar en el Dsro Federal, DTEJ-14 Pazos, Ma. Inés, Consecuenca lógca derroable: análss de un concepo de consecuenca falble, DTEJ-15 Posadas, Alejandro y Hugo E. Flores, Análss del derecho de conar con un juco juso en Méxco, DTEJ-16 Posadas, Alejandro, La Responsabldad Cvl del Esado /Análss de un caso hpoéco, DTEJ-17 López, Sergo y Posadas Alejandro, Las pruebas de daño e nerés públco en maera de acceso a la nformacón. Una perspecva comparada, DTEJ-18 Magalon, Ana Laura, Cómo esudar el derecho desde una perspecva dnámca?, DTEJ-19 Fondevla, Gusavo, Cumplmeno de normava y sasfaccón laboral: un esudo de mpaco en Méxco, DTEJ-20 Posadas, Alejandro, La educacón jurídca en el CIDE (Méxco). El adecuado balance enre la nnovacón y la radcón, DTEJ-21 Ingram, Mahew C., Judcal Polcs n he Mexcan Saes: Theorecal and Mehodologcal Foundaons, DTEJ-22 Fondevla, Gusavo e Ingram Mahew, Deencón y uso de la fuerza, DTEJ-23

39 DIVISIÓN DE ESTUDIOS POLÍTICOS Lehoucq, Fabrce E., Srucural Reform, Democrac Governance and Insuonal Desgn n Lan Amerca, DTEP-188 Schedler, Andreas, Paerns of Represson and Manpulaon. Towards a Topography of Auhoraran Elecons, , DTEP-189 Benon, Allyson, Wha Makes Srong Federalsm Seem Weak? Fscal Resources and Presdencal-Provncal Relaons n Argenna, DTEP-190 Crespo, José Anono, Culura políca y consoldacón democráca ( ), DTEP-191 Lehoucq, Fabrce, Polcymakng, Pares and Insuons n Democrac Cosa Rca, DTEP-192 Benon, Allyson, Do Invesors Assess he Credbly of Campagn Commmens? The Case of Mexco s 2006 Presdenal Race, DTEP-193 Nacf, Beno, Para enender las nsucones polícas del Méxco democráco, DTEP-194 Lehoucq, Fabrce, Why s Srucural Reform Sangnang n Mexco? Polcy Reform Epsodes from Salnas o Fox, DTEP-195 Benon, Allyson, Lan Amerca s (Legal) Subnaonal Auhoraran Enclaves: The Case of Mexco, DTEP-196 Hacker, Casano y Jeffrey Thomas, An Anrus Theory of Group Recognon, DTEP-197 DIVISIÓN DE HISTORIA Ppone, Ugo, Aperuras chnas (1889, 1919, 1978), DTH-34 Meyer, Jean, El conflco relgoso en Oaxaca, DTH-35 García Ayluardo Clara, El prvlego de perenecer. Las comundades de feles y la crss de la monarquía caólca, DTH-36 Meyer, Jean, El crujano de herro ( ), DTH-37 Sauer, Mchael, Clock Wachers and Sargazers: On Tme Dscplne n Early- Modern Berln, DTH-38 Sauer, Mchael, The Enlghenmen on Tral, DTH-39 Ppone, Ugo, Oaxaca prehspánca, DTH-40 Medna Peña, Lus, Los años de Salnas: crss elecoral y reformas, DTH-41 Sauer, Mchael, Germans n Space: Asronomy and Anhropologe n he Egheenh Cenury, DTH-42 Meyer, Jean, La Iglesa caólca de los Esados Undos frene al conflco relgoso en Méxco, , DTH-43

40 Venas El Cenro de Invesgacón y Docenca Económcas / CIDE, es una nsucón de educacón superor especalzada parcularmene en las dscplnas de Economía, Admnsracón Públca, Esudos Inernaconales, Esudos Polícos, Hsora y Esudos Jurídcos. El CIDE publca, como produco del ejercco nelecual de sus nvesgadores, lbros, documenos de rabajo, y cuaro revsas especalzadas: Gesón y Políca Públca, Políca y Goberno, Economía Mexcana Nueva Época e Isor. Para adqurr alguna de esas publcacones, le ofrecemos las sguenes opcones: VENTAS DIRECTAS: Tel. Dreco: Tel: Ex y 6091 Fax: Ex Av. Consuyenes 1046, 1er pso, Col. Lomas Alas, Del. Álvaro Obregón, 11950, Méxco, D.F. VENTAS EN LÍNEA: Lbrería vrual: Dudas y comenaros: publcacones@cde.edu Nuevo! Adquera el CD de las coleccones compleas de los documenos de rabajo de la Dvsón de Hsora y de la Dvsón de Esudos Jurídcos. Próxmamene! los CD de las coleccones compleas de las Dvsones de Economía, Admnsracón Públca, Esudos Inernaconales y Esudos Polícos.