REAL (EFFECTIVE) EXCHANGE RATE IN URUGUAY: A PERIODIC COINTEGRATION APPROACH

Transcription

1 REAL (EFFECTIVE) EXCHANGE RATE IN URUGUAY: A PERIODIC COINTEGRATION APPROACH Elizabeh Bucacos

2 REAL (EFFECTIVE) EXCHANGE RATE IN URUGUAY: A PERIODIC COINTEGRATION APPROACH Elizabeh Bucacos 1 ebucacos@bcu.gub.uy This version: June 1 s, 2007 Absrac When he seasonal componen of a paricular ime series is reaed as if i were a mere deerminisic phenomenon insead of a sochasic one, i may lead o inconsisen esimaions, saisical inference errors and policy biases. This issue is addressed in his paper focusing on he real effecive exchange rae in Uruguay for he 1983:1-2006: period. Keywords: real effecive exchange rae, periodic coinegraion, Uruguay. JEL: F31, C22, N16 1 The opinions expressed herein are hose of he auhor s and do no reflec necessarily hose of he Cenral Bank of Uruguay. 1

3 CONTENTS I. Inroducion. 2 II. Analyical framework and ime series involved. II.1 Definiion of.. II.2 Theoreical framework II.2.1 Produciviy.. 5 II.2.2 Consumpion... 6 II.2.3 Fiscal policy II.2. Capial flows... 6 II.2.5 Ineres raes II.2.6 Terms of rade. 6 II.2.7 Openness degree.. 7 III. Modelling approach.. 7 III.1 Periodic coinegraion: summary... 7 III.2 Time series idenificaion.. 8 III.2.1 Uni roos III.2.2 Periodic auoregressive represenaion IV. Periodic coinegraion: applicaion o. 15 IV.1 Definiions IV.2 The esimaion. 15 V. Concluding remarks.. 19 References. 20 2

4 I. INTRODUCTION I is no a rivial quesion o know wheher he real effecive exchange rae () is saionary or no because, among oher reasons, he effecs of random shocks could be emporary or las forever. This feaure is a crucial one for policy purposes when i is aken ino accoun he role played by he in he resource allocaion mechanism and he profiabiliy among differen economic secors. There are a number of reasons why a paricular ime series could resemble nonsaionary, from he happening of srucural breaks o he misreamen of is seasonal paern. Someimes, srucural breaks affec he long-run relaionship among he coinegraed ime series involved 2 bu he shor-run specificaion sands sill. And if a sable funcional specificaion of he could be found aking ino accoun hose breaks, he effecs of any shock are expeced o fade away in he long run. The same reasoning is rue for he incorrec approach o he acual naure of he seasonal paern of he ime series ied up by a common performace in he long-run, ha is o say, ime series ha are coinegraed. Differen long-run elasiciies (preferences) and differen speeds of adjusmen owards long-run equilibria could be misundersood as insabiliy signals insead of jus logical feaures semming from he combinaion of he sochasic seasonal paerns of periodically coinegraed ime series. Usually, he seasonal componen of many economic ime series is reaed as if i were a oally deerminisic phenomenon which implies ha i is unchangeable and could be perfecly forecased. Bu for many ime series seasonaliy can be sochasic and someimes Spring may become Summer and ohers Auumn. In addiion, coinegraed ime series could share sable long-run relaionships bu could have variable seasonal paerns ha may no coincide in he shor run. Models ha disregard he rue naure of he seasonal componen could lead o inconsisen esimaions, saisical inference errors and economic policy biases. In his paper he for Uruguay is esimaed for he 1983:1-2006: period using a periodic coinegraion approach. In ha way, coinegraed ime series ha have sochasic seasonal paerns could render sable long-run preferences ha neverheless change wih he season (quarer) and differen speeds of adjusmen o long-run equilibrium relaionships according o he season (quarer). The plan of his paper is as follows. In he nex session, he analyical framework is presened and saisical properies of he ime series involved are analyzed. Then, periodical coinegraion approach is briefly explained and performed. Finally, some concluding remarks are drawn. 2 I is well-known ha he coinegraion modelling approach is more prone o suffer from srucural breaks in he long-run relaionships. 3

5 II. ANALYTIC FRAMEWORK AND TIME SERIES INVOLVED In his paper, I will ake a definiion of exernal real exchange rae 3 and concenrae on rying o describe is daa generaing process hrough a periodic coinegraion errorcorrecion approach. II.1 Definiion of RER As he real exchange rae (RER) is jus a relaive price beween wo consumpion bundles, here can be as many definiions of RER as ypes of bundles considered. For insance, i could be defined as he relaionship beween he domesic consumer price index (P) and he foreign consumer price index (P * ), boh expressed in a common currency: * E P ( 1) R E R = P where E is nominal exchange rae (domesic currency per uni of foreign currency). This relaionship can be expressed as a bilaeral or a mulilaeral one and he laer is referred o as real effecive exchange rae. In, each bilaeral relaionship expressed in a common currency such as American dollars (RER cc ) is weighed according o he relaive imporance of each counry in he domesic counry inernaional rade, α i : m cc 2) R E E R = α R E R ( i = 1 where m is he number of commercial parners in he domesic counry inernaional rade. The Purchasing Power Pariy (PPP) heory poins ou ha inernaional rade arbirage guaranees ha he price of he same bundle in differen counries mus be he same expressed in a common currency. Tha leads o a consan uni value for he RER. In is relaive version, PPP esablishes ha hrough inernaional rade arbirage he rae of change of domesic and foreign prices expressed in he same currency mus be he same. This implies ha RER should be consan. Neverheless, should he RER be saionary, ransiory deviaions could be allowed in he shor run for he RER should converge o is fundamenals in he long run. According o he lieraure, here are a number of fundamenals for he : produciviy, consumpion, fiscal policy, capial flows, foreign and domesic ineres raes, erms of rade, openness degree. i i 3 Exernal real exchange rae measures he purchasing power of domesic currency by comparing he same bundle of goods beween wo economies; inernal real exchange rae measures he relaive price of radable goods in erms of he price of nonradable ones boh produced in he domesic economy. The former is beer in order o evaluae he impac of nominal shocks in he economy while he laer is paricularly useful o analyze he effecs of shocks on radable good secor produciviy. See Hinkle and Moniel (1999). I is more appropriae o address error-correcion models as equilibrium-correcion ones. See Ahumada (2006).

6 Graph 1. Real effecive exchange rae in Uruguay: he arge variable. II.2 Theoreical framework Many heoreical models have been designed in order o explain RER performance relaed o he evoluion of is fundamenals. Mundell (1971), Dornbusch (1980), Edwards (1989), Hinkle and Moniel (1999), among ohers. More recenly and for he Uruguayan economy, Aboal (2002), Gianelli and Mednik (2006) whose paper I will follow more closely. The real exchange rae is perceived as such a crucial raio ha guaranees simulaneous balance of boh domesic and exernal secors. Tha is o say, i is assumed ha here is an underlying relaionship beween absorpion and RER so as o balance boh of hem. Bu he sign of i defers depending on which equilibrium is o be resored. If he economy is in equilibrium, an excess demand of nonradable goods (an increase in absorpion) needs o be compensaed by a negaive excess demand of radable goods hrough a real appreciaion. So, he relaionship beween absorpion and RER is negaive. If he economy is in equilibrium, an excess demand of nonradable goods leads o a curren accoun defici which requires a real depreciaion o resore equilibrium. So, he relaionship beween absorpion and RER is posiive. The obvious way o follow his research is o focus on he fundamenals of absorpion in order o ry o capure an assumed sable long-run relaionship among RER and hose ones. II.2.1 Produciviy (π) Balassa (196) argues ha produciviy growh is differen no only for each economic secor bu also for each economy and ha i is in fac higher in he radable good secors. The Balassa-Samuelson effec poins ou ha a rise in produciviy appreciaes RER. On he one hand, a rise in he radable secor relaive produciviy would increase radable oupu and decrease radable good prices. On he oher, i would increase labor demand and reallocae labor inpus from nonradable good 5

7 secors o radable ones which would increase wages economy-wide. This implies a negaive supply shock o he nonradable good secor which requires a change (rise) in is relaive price in order o resore long-run equilibrium. Those wo effecs would make RER appreciae. II.2.2 Consumpion (CT, CN) A rise in consumpion of radable goods would deeriorae he curren accoun and would need a real depreciaion o resore equilibrium. A rise in consumpion of nonradable goods would cause an excess demand, a rise in nonradable good prices and a real appreciaion. There is compelling evidence ha poins ou ha Governmen consumpion (G) is more inensive in nonradable goods han privae consumpion is. II.2.3 Fiscal policy As long as axes are no disorionary, i seems ha only he level and composiion of fiscal expendiure are relevan in he long-run deerminaion of RER disregarding he sources of financing ha budge. Neverheless, Ricardian equivalence may no hold when a paricular indebedness policy could generae a posiive real effec and a real appreciaion in he shor run. II.2. Capial flows (k) Capial flows are regarded as a loosen of he economy budge consrain which increases absorpion levels a leas in he shor run. Furhermore, nonradable excess demand would rise is price and appreciae RER. Bu he acual effecs of capial flows on RER depend on he ype of flows received by he economy. If hose capial flows are permanen, RER would appreciae; if hose capial flows are only ransiory and mus be reimbursed laer plus heir corresponding ineress, RER could depreciae in he long run. (See Morrisey e al, 200). II.2.5 Foreign, domesic and ne real ineres raes (r *, r, in) Anoher way of loosening he foreign budge consrain is a drop in inernaional real ineres raes. Tha would mean a higher commercial defici wih a real appreciaion. In he domesic secor, a drop in inernaional real ineres raes would push down invesmen coss making new projecs rewarding, fosering nonradable demand and appreciaing RER. Mac Donald and Ricci (2003) sugges ha real ineres rae differenials could capure a Balassa-Samuelson residual lef ou because of measuremens problems in he produciviy proxy used in he calculus. In fac, real domesic ineres rae is he price of capial wihin he economy and a rise in ha wih respec o he inernaional one, could be inerpreed as a capial produciviy rise which would lead o a real appreciaion. II.2.6 Terms of rade (TOT) A rise in he erms of rade is expeced o appreciae RER. A higher exporable price relaive o he imporable one aracs resources o he radable secor similar o a negaive supply shock o he nonradable secor. Wih a posiive wealh effec caused by higher TOT, an excess demand for nonradable goods appears which appreciaes RER. There is a second order effec ha would depreciae RER semming from he 6

8 fac ha he relaive reducion of imporable prices could reduce nonradable demand. In commodiy-expor economies such as Uruguay he effecs of changes of erms of rade are expeced o be significan (Duch disease). II.2.7 Openness degree (OD) An increase in he degree of openness of he domesic economy would cause commercial accoun defici hrough an increase in he imporable demand and would require RER o depreciae o resore equilibrium. Inernally, wo shocks would appear: a posiive supply one, because of he cheapening of impored inpus and a negaive demand one, because of consumpion subsiuion from nonradable goods o impored ones. Boh shocks lead o a reducion in nonradable prices and a real depreciaion. The underlying relaionship among and is fundamenals o be revised in he course of his invesigaion can be expressed as: (3) = f ( π, CT, CN, G, K, r *, r, i n, TOT, OD ) where he sign on he upper righ of each variable reflecs he expeced marginal response of o marginal changes in each fundamenal. The variables ha will be acually considered will depend on daa availabiliy and will be he remaining ones afer coinegraion ess are performed: (3) ' = f ( G, i n, TOT ) III. MODELLING APPROACH III.1 Periodic coinegraion Periodic coinegraion models allow ha boh long-run parameers and adjusmen coefficien vary wih he season. I can be he case ha here is an adjusmen owards he LR relaionship in each season (oally coinegraed) or ha here is no adjusmen in some of he seasons (parially coinegraed). The final specificaion should be: ' ρ ρ ' ( ) y = λ ( y θ z + γ Δ y + β Δ z Δ s s ) i = 1 i i i = 1 i where y is he dependen variable (), z is an independen variable vecor, λ s reflecs adjusmen parameers in season s and θ s is he long-run parameer vecor in season s. In order o perform a periodic coinegraion esimaion, i is imperaive o invesigae he exisence of uni roos in he differen seasons s for each ime series. Franses (1996) proposes o sar from analyzing he periodic auocorrelaion of each ime series, hen wrie i down as a vecor of quarers represenaion (VQ) and finally es wheher he corresponding characerisic equaion has a uni roo (UR). If only one UR is found, and he ohers lie ouside he uni circle hen, i can be said ha here is a common rend and hree coinegraing relaionships. For he explanaory variables Franses advises o use HEGY es. i 7

9 In a second sep, i should be analyzed he possibiliy of more han one coinegraing relaionships in a periodic ECM, where all variables are ransformed ino (periodically) saionary ime series hrough he Δ filer. Nex, i should be esed wheher here is parial or oal periodic coinegraion (Wald coinegraion ess) beween and is fundamenals. If he null hypohesis of no coinegraion is rejeced, hen i mus be esed wheher long-run elasiciies are he same (including long-run inerceps) and wheher adjusmen parameers are he same. Finally, weak exogeneiy ess mus be performed in order o rule ou alernaive specificaions. III.2 Time series idenificaion In his secion, I will characerize he ime series involved in his sudy focusing mainly on he seasonal componen of each one. III.2.1 Uni roos Hylleberg, Engle, Granger and Yoo (HEGY, 1990) propose a es o deermine wheher a univariae ime series has uni roos boh a seasonal and a zero frequencies (long run). Tha es follows he sandard Dickey-Fuller approach, wih ransformed series o cover special cases. HEGY use he facorizaion derived from he known Box and Jenkins seasonal filer for quarerly daa: 2 (1 B ) = (1 B ) (1 + B ) (1 + B ), where B is he lag operaor. If he ime series x has uni roos a all seasonal frequencies θ = 0, ¼, ½, ¾ of one cycle (2π), each one of he expressions (x x -1 ), (x + x -1 ), (x + x -2 ) and (x x -2 ) is non saionary. Assuming ha he daa are generaing by a general auoregressive process, φ ( B) x = ε, an expansion around he values θ k = +1, -1, +i, -i defines a procedure for esing he inegraion order of he series hrough an exension of he zero-frequency Dickey-Fuller procedure. The es is based on he auxiliary equaion: y (1 B x = π y + π y + π y + π y + ε = ) where 2 3 y 1 = (1 + B + B + B ) x is he observed ime series adjused by he seasonal uni roos in θ = ¼, ½, ¾ 2 3 y 2 = (1 B + B B ) x is he observed ime series adjused by he seasonal uni roos in θ = 0, ¼, ¾ 2 y 3 = (1 B ) x is he observed ime series adjused by he seasonal uni roos in θ = 0, ½. Tess for UR a frequencies 0, ½, ¼ are based on -values for π 1 and π 2 which are disribued as a Dickey-Fuller disribuion and an F es π 3 π ; when π = 0, i is used -value for π 3. 8

10 Time series Produciviy. Gov Consumpion Foreign in.rae Ne ineres rae Terms of rade Char 1 SEASONAL INTEGRATION TESTS (HEGY) Sample: Auxiliary regression 2 Values in he sample Deerminisic Dependen F par variable lags π 1 π 2 π 3 π π 3 π I 1, 2, ** -.06 ** -2.0 * ** I * * I, SD 1, 2, ** -7.3 ** -.7 ** ** I * ** * I, SD ** ** ** I, T ** ** ** ** * ** -6.9 ** -.7 ** 5.51 ** SD 1, 3 1, * ** -.33 ** -. ** 2.58 ** I 1, * * I, SD 1 1, 2, Criical values for w/o seasonal dummies, a 5% 1% Criical values for π w/o seasonal dummies, a 5% 1% w/seasonal dummies, a 5% 1% Criical values for G w/o seasonal dummies, a 5% 1% w/seasonal dummies, a 5% 1% Criical values for r* w/o seasonal dummies, a 5% 1% Criical values for r w/o seasonal dummies, a 5% 1% w/seasonal dummies, a 5% 1% Criical values for TOT w/o seasonal dummies, a 5% 1% w/seasonal dummies, a 5% 1% Noes: (1) The ime series are: = Real effecive exchange rae, π = produciviy, G= governmen consumpion as a share of GDP, r*= foreign ineres rae, in= ne ineres rae, TOT= erms of rade. (2) The auxiliary regression can be augmened by is deerminisic par: inercep (I), rend (T), deerminisic seasonal dummies (SD i, i=1, 2, 3, ) or by lags in he dependen variable. (3) Criical values correspond o -values of Dickey-Fuller disribuion and o F-values for 5%. They were aken from HEGY (1990) and Fuller (1976) for π1 and π2 and Dickey, Hasza and Fuller (198) for π3. () (*) and (**) indicaes rejecion of uni roo hypohesis a 5% and 1%, respecively. (5) Differen sample sizes were used for each regression, depending on he number of lags needed o obain uncorrelaed errors. 9

11 According o he resuls presened in Char 1, all ime series analyzed are inegraed of order one a zero frequency a 1%; erms of rade has also uni roos a biannual and annual frequencies and has a uni roo a ¾ frequency. The ess performed have difficuly in separaing a uni roo a he ¼ and ¾ frequencies from a seasonal deerminisic paern. When sandard HEGY ess are applied wihou seasonal dummy variables, produciviy, governmen consumpion and erms of rade seem o be inegraed of order one a all frequencies; on he oher hand, when deerminisic seasonal variables are added, he join es rejecs nonsaionariy of produciviy and governmen consumpion a ¼ and ¾ frequencies (a 1%). Foreign ineres rae and ne ineres rae have only one uni roo each a zero frequency (long run). III.2.2 Periodic auoregressive represenaion A periodic auoregressive model of order ρ, PAR(ρ), is expressed as: ( 5) y = μ + φ y φ y + 1s 1 ρ s ρ ε where μ is an inercepion erm ha varies wih he season s (quarer), and φ 1s,, φ φs are auoregressive parameers up o order ρ ha could change wih he season s, s = 1, 2, 3,. I is assumed ha ε is a sandard whie noise process wih consan variance σ 2, alhough ha assumpion could be relaxed o allow seasonal variance σ s 2. A quarerly ime series is periodically inegraed of order one, PI(1), when i is needed a filer (1-α s B) o remove he sochasic rend from i; α s, he seasonally-variable parameers, verify ha α 1 α 2 α 3 α = 1 and α s α for all s = 1, 2, 3,. This definiion indicaes ha PI ness he usual (1-B) filer and he (1+B) filer oo, which corresponds o he seasonal uni roo a he biannual frequency. As Franses (1996) poins ou, his suggess ha a useful sraegy in he case of hree coinegraing relaions in Y T 5 is firs o check wheher α 1 α 2 α 3 α = 1 and hen o check wheher α 1 α 2 α 3 α = -1. Franses (1996) advises firs o choose he order of periodic auocorrelaion, nex o es periodic variaion of he parameers and finally o es for uni roos. He says, Since a periodic auoregression allows he AR parameers o ake differen values in differen seasons, i seems naural o allow for he possibiliy of periodically varying differencing filer which can be used o remove he sochasic rend. A specific periodic differencing filer corresponds o he noion of periodic inegraion (PI) The implicaion of periodic inegraion is ha he sochasic rend and he seasonal flucuaions are no independen, in he sense ha accumulaion of shocks can change he seasonal paern and ha he ime series canno be decomposed in wo sricly separae rend and seasonal componens. There are wo possible ways of choosing ρ, he order of periodic auocorrelaion. According o one of hem, he analys can concenrae on he residuals coming from a non periodic auoregressive model; according o he oher one, he analys can esimae a PAR(ρ) model, choose ρ using convenional crieria for model selecion and hen check wheher here is periodic variaion in he auoregressive parameers. 5 The ime series y is expressed in is vecor of quarers represenaion. 10

12 Usually, ime series residuals give some guidelines of heir underlying daa generaing process. Auocorrelaion and ime-varying variance of he residuals may be signs of differen coefficiens for each season (quarer) in an auoregressive represenaion of a paricular ime series. In Char 2 i is presened a summary of one of he procedures used in his paper o deermine he number of periodic auoregressive parameers for he ime series assumed o be he real fundamenals of he for Uruguay. Char 2 - ORDER SELECTION IN PERIODIC AUTOREGRESSIVE MODELS, PAR(ρ) Tess based on esimaed residuals from non periodic AR(k) models for convenienly ransformed ime series Variable Produciviy Gov.Consumpion Foreign in.rae Ne ineres.rae Terms of rade Time series 1 Δ Δ π Δ G Δ 1 r * Δ 1 r Δ TOT k ρ Diagnosic saisic values 2 F AR,1-1 F AR,1- F PIAR,1-1 F PIAR,1-2 F SH 0.00(1.00) 0.00(1.00) 1.6(0.22) 2.20 * (0.07) 0.5(0.71) (0.20) 3.29(0.02) 0.56(0.69) 0.9(0.9) 1.05(0.37) (0.01).1(0.00) 0.67(0.61) 0.68(0.71) 0.3(0.73) ** (0.01) 2.59 * (0.0).20 ** (0.00).32 ** (0.00) 1.71(0.17) ** (0.02) 1.33(0.27) 1.5(0.22) 1.0(0.21) 0.7(0.71) ** (0.01) 3.5 ** (0.01) 0.93(0.5) 1.91(0.07) 1.18(0.32) 1 Noes: (1) All ime series are ransformed according o seasonal and non-seasonal UR ess resuls, see Char 1. (2) Diagnosic saisics are: residual auocorrelaion of order 1 and order 1-( lags), F AR,1-1 F AR,1- ; periodic residual auocorrelaion of order 1 and order 1-2 (2 lags), F PIAR,1-1 F PIAR,1-2 and seasonal heeroskedasiciy, F SH. (3) The periodic residual auocorrelaion es is based on he auxiliary regression: v ˆ Σ η x +Σ ( ψ D vˆ D vˆ + u = ψ ) s= 1 i s= 1 1s s, 1 ms s, m applied on he residuals v ha come from an AR(k) model on he x ime series, where x =Δ j z. Under he null hypohesis of no periodic auocorrelaion of order m, ψ 1 =...= ψ ms = 0, he F saisic follows a sandard F disribuion asympoically wih (m, n-k-m) degrees of freedom. Criical values for m=1 are 2.6 (5%) and 3.51 (1%) and for m=2 are 2.03 (5%) and 2.69 (1%). () The seasonal heeroskedasiciy es is based on he auxiliary regression: 2 vˆ = w0 + w1 D1 + w2 D2 + w3 D3 + λ. Under he null hypohesis of seasonal homoskedasiciy, w 1 = w 2 = w 3 = 0, and he corresponding F saisic follows a F sandard disribuion wih (3, n-k) degrees of freedom. Criical values are:, 3.98 (1%). (5) k sands for he order of he auoregressive process AR(k) while ρ sands for he order of he periodic auoregressive process PAR(ρ). * Significan a 5% ** Significan a 1% Anoher way of choosing he order of periodic auocorrelaion is o adjus an equaion such as: ( 6) x = Σ μ D + Σ φ D x Σ φ s= s s s= s s s= ρ D x 1, 1 1,, 1 1. s s, ρ 11

13 wih x he original ime series 6 and s = 1, 2, 3,. Sandard AIC (Akaike Informaion Crieria) and SC (Schwarz Crieria) are used and i aken ino accoun ha parameers are esimaed for differen seasons. I is possible o perform an F es o analyze auoregressive-parameer significance of order greaer han ρ, F PAR. Besides, residuals could be esed in order o invesigae periodic heeroskedasiciy. Franses (1996) advises o use SC o choose he order of ρ as long as i could no be possible o rejec he null hypohesis of Φ ρ+1,s = 0. Char 3- ORDER SELECTION IN PERIODIC AUTOREGRESSIVE MODELS, PAR(ρ) Tess based on esimaed PAR(ρ) models Variable Produciviy Gov.Cons. Foreign in. Ne in. rae Terms of r. Noes: ρ Diagnosic saisic values F AR,1-1 F AR,1- F ARCH,1-1 F ARCH,1- JB F SH F PAR, (1.00) 0.00(1.00) 0.01(0.89) 0.06(0.99) ** (0.00) 0.77(0.52) 1.92(0.11) 0.00(1.00) 0.00(1.00) 0.32(0.57) 0.52(0.72) ** (0.00) 1.85(0.19) 0.60(0.66) 0.00(1.00) 0.00(1.00) 2.33(0.13) 0.76(0.57) 6.18(0.05) 1.6(0.23) 0.22(0.93) 0.00(1.00) 0.00(1.00) 8.2 ** (0.00) 7.16 ** (0.00) ** (0.00) 3.69 ** (0.00) 9.06 ** (0.00) 0.00(1.00) 0.00(1.00) ** (0.00) 5.57 ** (0.00) 1.33(0.51) 0.25(0.86) 1.(0.23) 0.00(1.00) 0.00(1.00) 2.03(0.16) 3.13 * (0.02) 30.7 ** (0.00) 6.17 ** (0.00) 2.51 * (0.00) (1) The order of ρ was chosen according o Schwarz Crieria: SC( ρ ) = nlogσ 2 + logn, wih n = N, he whole sample. (2) The adjusmen for (sandard errors beween brackes) is: ( 257 = ( 253 D ). 11 D ) ε ( D ) ( 256 (3) The adjusmen for produciviy (sandard errors beween parenhesis) is: π.12d D D π D π D π D π + ε = D ) 3 ( 19.1) (9.07) (8.96) (19.76) (20.85) (10.1) () The adjusmen for governmen consumpion (sandard errors beween parenhesis) is: G D G D G D G D G + ε = (182.02) (16.2) (17.95) (179.8) (5) The adjusmen for foreign ineres rae (sandard errors beween parenhesis) is: i 1.002D i D i D i i * * * * * = (50.71) (51.59) (51.67) (50.31) (6) The adjusmen for ne ineres rae (sandard errors beween parenhesis) is: in = D in D in D in D in + ε (.39) (.75) (.76) (.76) (7) The adjusmen for erms of rade (sandard errors beween parenhesis) is: TOT.1068D TOT D TOT D TOT D TOT + ε = (80.83) (75.10) (7.01) (7.29) (8) Columns hree o eigh, es resuls on residuals of PAR(ρ) models are presened: F AR (residual auocorrelaion), F ARCH (heeroskedasiciy), JB (normaliy), F SH (seasonal heeroskedasiciy). (9) Periodic variaion of auoregressive parameers is esed by H O : ф js =ф, for s=1,2,3, 6 Franses (1996) poins ou ha i is no necessary o ake firs differences of he ime series o remove sochasic rends when periodiciy is being analyzed. + ε

14 * ** and j=1,2,, ρ and he saisic F PAR is disribued as a sandard F(3ρ, n-(+ρ)). Criical values are and for ρ=1 and 2.21(%) and 3.0(1%) for ρ=2. Significan a 5% Significan a 1% According o he resuls presened in Char 3, all ime series could be described as periodic auoregressive ime series of order one, excep for governmen consumpion as a share of GDP and erms of rade, which appear o be periodic auoregressive ime series of order wo 7. Diagnosic analyses performed o deec residual auocorrelaion, ARCH paerns, non normaliy and seasonal heeroskedasiciy sugges ha PAR(1) and PAR(2) models give a beer descripion of he daa. The following sep is o es for uni roos. We firs wrie each PAR(1) process 8 as a VQ(1) represenaion: (8) ( 7) X x = Φ = a x s 1 Φ 1 X T 0 1 T 1 and hen analyze wheher he roo of he corresponding characerisic equaion + ε + Φ 1 0 ( 9) Φ0 Φ1 Z = (1 α1α 2α3α ) = 0 is ouside he uni circle, ha is, if g(α) < 1, wih g(α) = α 1 α 2 α 3 α 9. In oher words, he following es is performed: (10) H : g( α) O H : 1 = Πi = 1 g( α) 1 α = 1 is i ε period icall y T saionar y Once H O could no be rejeced we have o invesigae wheher he hypoheses (a)-(b) are rue: (11) H : α = 1 s = 2,3, ( a) O s H : α = 1 s = 2,3, ( b) O s given ha α 1 α 2 α 3 α =1 implies eiher α = 1 or α = -1. The firs one reduces he difference filer o (1-B) while he second one reduces he difference filer o (1+B), which is he corresponding filer o a seasonal roo 1. When he null hypohesis in (a) could no be rejeced, he PAR(1) process has a non seasonal uni roo, ha is o say, i is a PAR(1) process of a I(1) ime series and is usually called PARI. When boh null hypohesis (a) and (b) could be rejeced, i is an AR(1) model periodically inegraed, usually referred o as PIAR(1). According o he resuls presened in Char,, produciviy, Governmen consumpion (as a share of GDP), LIBOR, ne ineres rae and TOT can be described as auoregressive models periodically inegraed, ha is o say, as non saionary auoregressive ime series ha need a filer of he ype (1-α s B) o remove heir sochasic rend, being α s he auoregressive parameers. In he nex session, i will be analyzed he possible exisence of periodic coinegraion among hem. The orders of he periodic AR 7 For G, Schwarz Crierion is used, given ha F PAR is rejeced. 8 For PAR(2) processes, he noaion is similar. See Franses (1996). 9 Some of he values of α may be greaer han uniy. 13

15 models do no necessarily have o be equal, nor do he lenghs of he auoregressive polynomials in each of he seasons have o be equal. Time Series 1 π G i * in TOT 6 Char PERIODIC UNIT ROOT TEST Two-sep procedure H O : g(α) = Πα s = 1 H 1 : g(α) < 1 Sample Saisics 3 value 2 LR j LR j N(g(α)-1) Sample value 2.37 * 3.65 * 9.2 * 2.83 *.89 ** 63.6 ** H O : α s = 1 H O : α s = -1 Criical Value F 1-B Sample value ** ** * ** ** ** Criical Value 5 F 1+B Type PIAR(1) PIAR(1) PIAR(2) PIAR(1) PIAR(1) PIAR(2) Noes: (1) Original ime series, wihou any previous ransformaion., is real effecive exchange rae, in logs; π is produciviy, a difference of log variables; G, Governmen consumpion as a raio of GDP; i * is foreign ineres rae (LIBOR); in is ne ineres rae; TOT is erms of rade, in logs. (2) I is calculaed as he produc of esimaed parameers shown in Char 3, excep for G and TOT. (3) The saisics are: 3.1 Likelihood raio (LR): LR j = n ln (SSR O /SSR j ), obained by comparing SSR (sum of squared x Σ α D x + ε = 1 s s 1, wih respec o he SSR of a LS residuals) of a LS esimaion on 3 1 esimaion imposing H O, x = Σ1 α s Ds x 1 + ( α1α 2 α3 ) D x 1 + ε. In boh cases, inerceps and rend variables are added when needed. 3.2 LR j = (sgn(g(â)-1)lr 1/2 1, wih g(â) evaluaed a H O. The 5 and 10 per cen asympoic criical values are 9.2 and 7.52 (LR i ) and (LR i ). 3.3 N(g(â)-1) is similar o he one used for non periodic AR models scaled by N in order o compare wih values repored by Fuller (1976), chars and () Saisics o he parameer resricions relaed o he filer (1-B) are disribued F(3, n-k). (5) Saisics o he parameer resricions relaed o he filer (1+B) are disribued F(3, n-k). (6) As TOT is a PAR(2) process i can be wrien in is periodically differenced form as: ( T OT α T OT ) = μ β ( T OT α T OT ) + ε s 1 s 1s 1 s 1 2, where a s-k = a s and β s-k =β s for kєn and s=1, 2, 3,. The invesigaion of he presence of uni roos amouns o invesigae he soluion o he characerisic equaion, which is reduced o: ( 1 β 1 β 2 β 3 β z) (1 α1 α 2 α 3 α z ) = 0 Hence, TOT has a UR eiher when β 1 β 2 β 3 β =1 or α 1 α 2 α 3 α = 1 and a mos wo UR when boh producs equal uniy. (See Franses (1996), pp 99). The same reasoning applies o G. (7) When H O : α s = 1 canno be rejeced, i is a PARI(ρ) process, ha is, a PAR(ρ) process for an I(1) ime series; when boh H O : α s = 1 and α s = -1 are rejeced, i is an AR process of order ρ for a periodically inegraed ime series, PIAR(ρ). * ** Significan a 5% Significan a 1% 1

16 IV. PERIODIC COINTEGRATION APPROACH: APPLICATION TO In his session, I will ry o find a sable represenaion of he wih respec o is fundamenals (given by he lieraure) ha akes ino accoun he saisical characerisics of he ime series ha were depiced in he previous session. Tha is o say, I will ry o find an equaion ha combines he variable seasonal paerns of he ime series involved in a way ha renders sable bu changing long-run relaionships and adjusmen parameers ha change wih he season. IV.1 Definiions Definiion. Two periodically inegraed processes y and x are coinegraed when he linear combinaion y -θ s x can be described using a periodically saionary process, where θ s is a seasonally varying parameer.( Franses, 1996, pp. 181). As Franses (1996) argues, he common rend propery of wo PIAR series implies ha here is a single uniy soluion o he characerisic equaion for a VQ model for vecor processes Z. And a naural and simple es procedure seems o be given by evaluaing he residuals from he OLS regression ( 12) y = ˆ μ + ˆ θ x + uˆ Franses (1996) suggess a wo-sep procedure similar o he Engle-Granger approach for he non periodic analysis which is he one ha will be performed in his paper. s s IV.2 The esimaion In he firs sep of he esimaion procedure followed here, four long-run relaionships are found, one for each quarer (season) s: ( 13) + θ TOT + θ in + θ s 0 s 1s s 2s s 3s = θ G ) + ε s s Nex, a periodic error correcion equaion is presened: (1) Δ =Σ D λ ( θ θ TOT θ in θ G ) s= 1 s s 0s 1s 2s 3s +Σ Δ + n p q r +Σ0 β1 ΔTOT i +Σ0 Δin i +Σ0 ΔG 1 +Σ0 ΔX i + ε where X is a vecor of exogenous variables. Then, he exisence of oal or parial periodic coinegraion is esed The variables ha sand before each long-run relaionship, λ s, are sricly negaive as needed in order o verify coinegraing relaionships. m 1 i 10 Franses (1996) disinguishes beween periodic coinegraion models (PCM) and periodic errorcorrecion models (PECM): a PCM has seasonally varying coinegraion relaionships while in a PECM only he adjusmen parameers are seasonally variable. 11 There could be conemporaneous variables, provided hey are weakly exogenous. 15

17 Char 5- PERIODIC COINTEGRATION ANALYSIS Dependen variable: Δ Sample: 1983:1-2006: Parial coinegraion es H O : d s = 0, H 1 : d s 0 wih d s = (λ s, λ s -δ is ), i = 1, 2, 3, s = 1, 2, 3, Season 1 W s Criical value 3 Resul H O of no periodic coinegraion in Q1 is rejeced a 5% H O of no periodic coinegraion in Q2 is rejeced a 5% H O of no periodic coinegraion in Q3 is rejeced a 10% H O of no periodic coinegraion in Q is rejeced a 5% Toal coinegraion es H O : d s = (d 1,, d ) = 0, H 1 : d s 0 wih d s = (λ s, λ s -δ is ), i = 1, 2, 3, s = 1, 2, 3, W 2 Criical value 3 Resul H O of no periodic coinegraion in all quarers is rejeced a 5% Noes: (1) Wald es for parial coinegraion for season s: SSR SSR Os 1 W = ( n l) s SSR1 where n is he number of observaions, l is he number of esimaed parameers under H 1, SSR Os and SSR 1 are he sum of squared residuals from he LS regressions under H Os and H 1 respecively, for s = 1, 2, 3,. (2) Wald es for oal coinegraion: SSR SSR O 1 W = ( n l) SSR1 where n is he number of observaions, l is he number of esimaed parameers under H 1, SSR O and SSR 1 are he sum of squared residuals from he LS regressions under H O and H 1 respecively. (3) Assympoic disribuions for W s and W saisics are no sandard and hey were derived by Boswijk and Franses (1995). I repor he corresponding criical values for wo (one weakly exogenous) variables and a significance level of 5% and 10%. The empirical evidence for Uruguay in he 1983:1-2006: period allows us o rejec he absence of parial and oal periodic coinegraion hypohesis for he real effecive exchange rae and is fundamenals. Besides, i is no possible o rejec some resricions on he adjusmen parameers and long-run elasiciies as well (See Char 6). Finally, we arrive a he following PCM specificaion (sandard deviaions beween parenhesis) 12 : 12 Dummy variables were added o he long-run relaionships in: 1990:1 and 1990:3 (regional crisis) and (abandonmen of nominal exchange rae arge zone in Uruguay) and (domesic crisis ha followed). Also, in order o improve shor-run adjusmen, ouliers were aken care of when he corresponding error was greaer han hree imes he sandard error of he regression. Tha happened in: 1985:, 199:, 2002:3, 2003:3, 2003: and 200:3. 16

18 (1) Δ = 0.77 ( TOT+ 0.01in G) (0.1) (0.31) (0.07) (0.003) (0.20) 1 (0.09) 0.5( TOT in) (0.33) (0.07) (0.003) 2, + 0.5( TOT in G) (0.09) (0.37) (0.06) (0.003) (0.21) 3, 0.77( TOT in) (0.1) (0.39) (0.08) (0.00), Δ ( Δ E 3 Δ E ) ( Δ p Δ p ) ( Δ p (0.0) (0.001) (0.05) (0.13) * * 1 1 Δ p 2 ) ( Δ TOT Δ TOT (0.06) 1 ) +ε where E is nominal exchange rae (Uruguayan pesos per US dollars), p * is foreign price aggregae index and p is (domesic) Consumer Price Index. The corresponding diagnosics are: R 2 = 0.93; σ = 3.1%; F AR 1, 1 = 0.00; AR,1 = 0.00; FARCH 1, 1 = 3.37; FARCH,1 = 2.25; JB =.23 which indicae a relaively good adjusmen wih a raher high sandard deviaion, hough. Residuals are saionary, homoskedasic and normally disribued. As he only conemporaneous variable wih he is TOT, i is esed weak exogeneiy of TOT wih respec o he parameers of he long-run relaionship. To do so, an AR() model for Δ TOT is regressed adding he coinegraing vecors from he PCM jus found. The F saisic is lower han able values a 1% so i could no be rejeced he hypohesis of weak exogeneiy of TOT o long-run parameers deerminaion. Tha is why he singleequaion approach used in his sudy seems o be appropriae. Also, Granger noncausaliy from TOT, in and G o was rejeced in he long-run relaionships. As a resul, his specificaion allows he invesigaor o make inferences and make forecass as well. F Residual Acua l Fied Graph 2. Periodic coinegraion adjusmen o he daa. 17

19 In Char 6 he resuls of resricion ess on parameers are presened. I can be seen ha: (a) adjusmen coefficiens reflec variable cos adjusmens according o he quarer, bu are he same for he firs and fourh quarer and he second and hird one; (b) long-run elasiciies are no he same, excep for TOT in Q1 and Q3; (c) TOT and in are always real fundamenals for he while G appears o have a sable long-run relaionship wih only in he firs and hird quarers, being greaer in he laer one. This means ha he objecive relaionships are changing according o he quarer. Char 6 PARAMETER RESTRICTIONS Dependen variable: Δ Sample: 1983:1-2006: Resricion Sample Criical Resul value 1 values 2 H O : λ 1 = λ The H O of equal adjusmen coefficiens a Q1 and Q could no be rejeced a 1% H O : λ 2 = λ The H O of equal adjusmen coefficiens a Q2 and H O : λ 1 = λ λ 2 = λ H O : θ 1 = θ H O : λ 1 = λ λ 2 = λ θ 1 = θ 3.20 Q3 could no be rejeced a 1% The H O of equal adjusmen coefficiens a Q1 and Q and of Q2 and Q3, could no be rejeced a 1% The H O of equal long-run elasiciy of TOT o a Q1 and Q3 could no be rejeced a 1% The H O of equal adjusmen coefficiens a Q1 and Q and of Q2 and Q3 and ha long-run elasiciies of TOT o a Q1 and Q3 are equal, could no be rejeced a 1% Noes: (1) I corresponds o a sandard F-es on he periodically coinegraed equaion. (2) A 5% and 1%, respecively. I seems as if i is relaively simple o adequaely describe he for Uruguay hrough a periodic coinegraion approach. The final periodic coinegraion model has four long-run arge relaionships: (15).1 (15).2 (15).3 (15) in 0.98G = 630 TOT e e in he firs quarer in = 565 TOT e in he second quarer in 1.80G = 1307 TOT e e in he hird quarer in = 322 TOT e in he fourh quarer I means ha, ceeris paribus, a change in erms of rade has is major impac in he hird quarer despie he fac ha he higher long-run elasiciy occurs in he second one. The speed of adjusmen of misalignmens is differen depending on he quarer. The effecs of any shock ha occurs in Q1 and Q dissipae faser han if i ook place in Q2 and Q3. In fac, i could ake eiher four quarers o reurn o equilibrium (Q1 an Q) or en ones (Q2 and Q3). I seems as if Ocober-March period (Q and Q1) is he highspeed ime for disequilibrium adjusmens. 18

20 V. CONCLUDING REMARKS In his paper, for Uruguay and is long-run fundamenals have been viewed as periodically inegraed ime series ha could combine heir changing seasonal paerns in such a way ha sable bu quarerly-changing objecive relaionships could be achieved and adjusmen parameers ha change wih he season (quarer) could be found. This approach, by recognizing he sochasic naure of he seasonal paern of he ime series involved, avoids inconsisen esimaions, errors in saisical inference and also biases in economic policy decisions. The periodic coinegraion model finally esimaed shows ha in Uruguay for 1983:1-2006: can be reaed as a saionary ime series in fourh differences, ha is o say, shocks ha affec he annual change of have only ransiory effecs and in he long run would achieve he value given by is fundamenals. Besides, i seems as if has differen fundamenals, depending on he quarer (season). In addiion, he impac of changes in hose fundamenals on he long-run arge relaionships depends on which quarer hose changes ake place. And las bu no leas, he adjusmen of equilibrium misalignmens are faser if hey ake place in he Ocober- March period han in he res of he year. In effec, a 77-percen disequilibrium dissipaion would ake eiher one quarer (Q1 and Q) or wo quarers and a half (Q2 and Q3); a 90-percen disequilibrium dissipaion would ake eiher hree and ½ quarers (Q1 and Q) or eigh quarers (Q2 and Q3) and a oal disequilibrium dissipaion would ake eiher four quarers (Q1 and Q) or en quarers (Q2 and Q3). Real exchange rae recen hisory plays a very imporan role in shor-run dynamics as well as annual inflaion boh domesic and foreign. Terms of rade is he only long-run fundamenal ha is also presen in he deerminaion of in he shor-run. Some oucomes of his invesigaion are useful for economic policy. There are jus a few variables ha are significan in explaining he evoluion of and ha he policy maker can conrol. They are G, E and π, ha is, Governmen consumpion as a share of GDP, nominal exchange rae (depending on he exchange rae policy) and domesic inflaion (o a cerain exen 13 ). I seems ha reducions in Governmen size could increase in he long run as well as reducions in ne ineres rae 1 ; changes in nominal exchange rae or in domesic inflaion could only affec shor-run dynamics bu hey could no influence fundamenal deerminans of, which seems o be a saionary process. 13 The inflaionary process depends on variables ha are no under he direc conrol of he policy maker. 1 Induced by reducions in domesic ineres rae. 19

21 REFERENCES Aboal, Diego (2002). Tipo de cambio real de equilibrio en Uruguay. XVII Jornadas Anuales de Economía, Banco Cenral del Uruguay, Ahumada, Hildegar (2006). Banco Cenral de la República Argenina. Aizenman, Joshua and Riera-Cachon Daniel (2006). Real exchange rae and inernaional reserves in he era of growing financial and rade inegraion. NBER WP 12363, Baffes, John; Elbadawi, Ibrahim and O`Connel, Sephen (1997). Single-equaion esimaion of he equilibrium real exchange rae. WPS 1800, Policy Resarch Working Paper, The World Bank Policy Research Group. Boswijk, H. Peer, 199. Tesing for an unsable roo in condiional and srucural error correcion models Journal of Economerics, 63: Boswijk, H. Peer y Philip Hans Franses, Uni roos in periodic auoregressions, Journal of Time Series Analysis. Bucacos, Elizabeh (2005). Acerca de la esacionalidad esocásica. Una aplicación para la demanda real de dinero en Uruguay. Banco Cenral del Uruguay, Revisa de Economía, Vol. 12, Nº 2, Segunda Época, Noviembre Capurro, Alfonso; Davies, Guillermo and Oonello, Pablo (2006). El ipo de cambio real y los precios relaivos en un enfoque de res bienes, Banco Cenral del Uruguay, Revisa de Economía, Vol. 13, Nº 2, Segunda Época, Noviembre Cashim, Paul and Mc Dermo John (200). Pariy reversion in real exchange raes: fas, slow or no a all. IMF Working Paper 0/128. Washingon DC, Inernaional Moneary Fund. Dickey, David A., D. P Hasza y Wayne A. Fuller, 198. Tesing for uni roos in seasonal ime series, Journal of American Saisical Associaion 76, Engle, Rober F. y Clive W.J. Granger, Co-inegraion and error correcion: represenaion, esimaion, and esing, Economerica, 55: Engle, Rober y B. S. Yoo, Forecasing and esing in coinegraed sysems, Journal of Economerics 35, Engle, Rober F., Clive W.J. Granger, Svend Hylleberg y H.S. Lee, Seasonal coinegraion. The Japanese consumpion funcion, Journal of Economerics 55, Edwards, Sebasian (1989). Real exchange raes: devaluaion and adjusmen. MIT Press, Cambridge, Massachuses. 20

22 Edwards, Sebasian (1998). Exchange raes misalignmens in developing counries. The World Bank Occasional Papers /Nº2, New series. Fernández, Adrián; Ferreira, Mariana; Garda, Paula; Lanziloa, Bibiana and Manero, Rafael (2005). Tipo de cambio real compeiivo y oras soluciones desajusadas, in XX Jornadas Anuales de Economía, BCU. Finger, Harald (2006). Assessing compeiiveness in Uruguay. Mimeo (IMF). Franses, Philip Hans (1996). Periodiciy and sochasic rends in economic ime series. Oxford Universiy Press. Fuller, Wayne. A., Inroducion o saisical Time Series, New York: John Wiley. Gianelli, Diego and Mednik, Maias (2006). Un modelo de corrección de errores para el ipo de cambio real en el Uruguay: 1983:I-2005:. Banco Cenral del Uruguay, Revisa de Economía, Vol. 13, Nº 2, Segunda Época, Noviembre Hamilon, James D., 199. Time series analysis, Princeon Universiy Press. Princeon, New Jersey. Harberger, Arnold C. (200). The real exchange rae: issues of concep and measuremen. Universiy of California, Los Angeles, 200. Hinkle, Lawrence and Moniel, Peer (1999). Exchange rae misalignmen: conceps and measuremens for developing counries. The World Bank Research Publicaion, Oxford Universiy Press. Hylleberg, Svend; Engle, Rober F.; Granger, Clive y Yoo, Seasonal inegraion and coinegraion, Journal of Economerics, Mc Donald, Ronald and Ricci, Luca (2003). Esimaion of he equilibrium real exchange rae for Souh Africa, IMF Working Paper 03/. Washingon DC, Inernaional Moneary Fund. Morrisey, Oliver; Lloyd, Tim and Opoku-Afari, Maxwell (200). Real exchange response o capial inflows: A dynamic analysis for Ghana, CREDIT Research paper 0/12, School of Economics, Universiy of Noingham. Mundell, Rober (1971). Devaluaion in his Moneary Theory: inflaion, ineres and growh in a world economy, Pacific Palisades, 1971, ch