THE APPLICATION OF REGRESSION ANALYSIS IN TESTING UNCOVERED INTEREST RATE PARITY

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1 QUANTITATIVE METHOD IN ECONOMIC Vol. XIV, No., 03, pp. 3 4 THE APPLICATION OF REGREION ANALYI IN TETING UNCOVERED INTERET RATE PARITY Joanna Kselńsa, Kaarzyna Czech Faculy of Economcs cences Warsaw Unversy of Lfe cences GGW e-mal: joanna_selnsa@sggw.pl, aarzyna_czech@sggw.pl Absrac: The am of he paper s o evaluae and compare wo lnear regresson models proposed by Froo and Franel (989) and o show her applcaon n verfcaon of he uncovered neres rae pary (UIP) hypohess n he seleced en exchange rae mares. The paper shows ha boh models proposed by Froo and Franel (989) are formally equvalen, bu hey may gve dfferen regresson resuls. Many researchers clam ha uncovered neres rae pary ends o hold more frequenly n emergng mares han n developed economes. The paper s focused on fve developed and fve emergng economes. I s parly confrmed ha developng counres wor beer n erms of UIP. Keywords: uncovered neres rae pary, exchange rae, lnear regresson model INTRODUCTION Uncovered neres rae pary (UIP) s a ey relaonshp n nernaonal fnance. UIP represens he cornersone pary condon of many fundamenal equlbrum exchange raes models such as capal enhanced equlbrum exchange rae model, behavoral equlbrum exchange rae model and moneary model of exchange rae. Uncovered neres rae pary saes ha hgh-yeld currences should deprecae and low-yeld currences should apprecae. I needs o be emphaszed ha UIP pary holds only when nvesors are rs neural and have raonal expecaons. Very ofen however, we observe he endency of low neres-yeldng currences o deprecae raher han apprecae as UIP suggess. Devaons from UIP s generally called he forward premum puzzle.

2 The applcaon of regresson analyss 33 Froo a and Franel a [989] propose wo lnear models whch may be appled for esng uncovered neres rae pary. The am of he paper s o assess hese models and addonally, o es wheher uncovered neres rae pary holds for chosen en counres. Research s conduced for fve developed and fve emergng economes. The remander of he paper s organzed as follows. econ UNCOVERED INTERET RATE PARITY revews he relevan leraure concernng uncovered neres rae pary. Moreover, hs secon presens wo lnear models proposed by Froo a Franel a [989] for esng uncovered neres rae pary. econ TATITICAL TET FOR REGREION MODEL dscusses possble dfferences n drawng conclusons abou UIP on he bass of hese wo lnear models. The emprcal resuls are descrbed n he nex secon. The las secon provdes concludng remars drawn from he emprcal research. UNCOVERED INTERET RATE PARITY Covered neres rae pary (CIP) saes ha he rao of domesc and foregn neres raes should equal o he rao of forward and spo exchange raes. CIP can be expressed as follows: where ( ) F r * r F ( ) s he prce of foregn currency n uns of domesc currency n me, s he forward value of exchange rae for a conrac agreed n me for an exchange of currences perods ahead, r and r * () are domesc and foregn neres raes respecvely (wh perods o maury). Uncovered neres rae pary (UIP) descrbes he relaonshp beween neres raes and expeced exchange rae changes. r E( ) r * where E( ) s he expeced spo exchange rae a me +, based on nformaon nown a me. Assumng covered neres-rae pary holds he uncovered neres rae pary can be expressed as follows: ( ) E ( ) F (3) Equaon () and (3) canno be drecly esable because mare expecaons of fuure spo exchange raes are hardly observable. Therefore, uncovered neres rae hypohess s esed jonly wh he assumpon of raonal expecaons n ()

3 34 Joanna Kselńsa, Kaarzyna Czech exchange rae mare. Under he assumpon of raonal expecaons, he fuure value of spo exchange rae n me + s equal o expeced spo exchange rae a me + plus a whe-nose error erm a me +. where E ( ) (4) s whe-nose error erm whch s uncorrelaed wh nformaon avalable a me. Uncovered neres rae pary hypohess may be verfed by esmang and esng coeffcens n wo regresson models proposed by Froo, Franel [989]. The frs regresson model (5) refers drecly o equaon (3) and (4). where s s s, s α β fd (5) fd exchange rae a me, f s, s denoes he logarhm of spo s s he logarhm of spo exchange rae a me +, s he logarhm of he -perod forward exchange rae. Under he UIP pary condon, he slope parameer β n equaon (5) should be equal o uny (β = ) and he coeffcen α should be equal zero (α = 0). I needs o be emphaszed ha esng uncovered neres rae pary n hs form s anamoun o esng he jon hypohess ha mare parcpans are endowed wh raonal expecaons and rs-neural. Emprcal sudes based on he esmaon of equaon (5) generally rejec he UIP hypohess. Fama [984], Froo Franel [989], McCallum [994] show ha coeffcen β n equaon (5) s sgnfcanly less han one, and n fac very ofen closer o mnus uny han plus uny. Ths fndng may mply ha he forward exchange rae s based predcor of he spo exchange rae. The volaon of uncovered neres rae pary s radonally called he forward premum puzzle (forward dscoun bas). Leraure provdes several explanaons of he phenomenon. One possble reason s he exsence of a rs premum. Anoher explanaons nvolve nvaldy of he raonal expecaons hypohess, peso problems and mare neffcency [Balle Bollerslev 000]. In he second regresson model proposed by Froo a Franel a [989] forward rae predcon error (a dfference beween fuure spo exchange rae and forward exchange rae forward premum f f s ) s consdered o be a dependen varable and fd s consdered o be an ndependen varable (6). fd s fd (6) where = - and α = - α. Under he UIP pary condon, boh he slope coeffcen β and he coeffcen α n equaon (6) should be equal zero (, 0). 0

4 The applcaon of regresson analyss 35 When coeffcens β and α n equaon (6) are equal zero and coeffcens β and α n equaon (5) are equal one and zero respecvely, hen boh regresson models proposed by Froo a Franel a [989] are equvalen. I needs o be emphaszed however, ha alhough hese models are formally equvalen hey may gve dfferen regresson resuls. TATITICAL TET FOR REGREION MODEL WHICH VERIFY THE UIP Le us consder wo regressve models: y (7) x x η x y (8) Where y s he endogenous varable, x exogenous varable, α and are he srucural parameers of he model, and = - Afer esmang he coeffcens of he models we ge: yˆ a b (9) x b x x yˆ a (0) where ŷ s an esmae of he dependen varable, and a and b are esmaes of he srucural parameers of model (7). Le s nroduce marngs: yy E n n y y, x x, y y x x, y yˆ, R yˆ y xx n n where n s he number of observaons. The basc sascal es based on F sascs, where he number of degrees of freedom for fracon equals and for denomnaor equals n-, verfes he exsence of a lnear relaonshp beween he dependen varable and explanaory varable. We se he null hypohess whch saes ha he dreconal coeffcen n he regressve model s nsgnfcanly dfferen from 0 (whch means no lnear relaonshp beween he varables x and y). In he case of model (7) hs hypohess means ha coeffcen β = 0 whereas n he model (8) β =. For model (7) es sasc, whch we denoe as F s: R F n () E For he model (8) afer ranslaon we oban sasc F : yx n ()

5 36 Joanna Kselńsa, Kaarzyna Czech xx xy F F n n (3) E E F s greaer han F f xx s greaer han xy. In hs case, he sgnfcance of F s greaer han he sgnfcance of F. I may herefore happen ha a ceran level he sgnfcance of he hypohess abou no lnear relaonshp beween he varable x and y canno be rejeced, bu you can rejec he hypohess of no lnear relaonshp beween he varables x-y and x. The hypohess formulaed above can also be verfed by means of sasc. For model (7) hs sasc s deermned by he formula: b b xx E n where (b) s he esmaon error of coeffcen b. sascs for model (8) s equal o: xy b xx xx b E E n ascs xx xy xy xx E n n dfferen from zero whereas xx (4) (5) can verfy he hypohess ha parameer s sgnfcanly ha parameer s sgnfcanly dfferen from. The es based on F sascs s equvalen o he es based on sascs are he followng equales are me: sgnfcance F = sgnfcance If model (7) uses, sgnfcance F = sgnfcance (6) sascs (whch enables he verfcaon of he hypohess ha coeffcen s sgnfcanly dfferen from ) s also necessary o use F sascs, raher han F, because he F sgnfcance wll dffer from sgnfcance. For he ess whch verfy he sgnfcance of nercep, hey are equvalen for boh models, excep ha he emprcal values for es have dfferen sgns. If we wan o verfy he hypohess of uncovered neres rae pary, frs we mus verfy he hypohess ha he parameer =, and he parameer = 0. I s

6 The applcaon of regresson analyss 37 beer han o use model (8), as he sandard regresson analyss enables us o deermne he value of he relevan sascs. If we use model (7) he resuls mus be ransformed accordng o formulas (3) and (5). In addon, f xx s greaer han xy, may be ha for a ceran level of sgnfcance, we canno rejec he hypohess ha he coeffcen n model (7) s nsgnfcanly dfferen from zero, whle n model (8) we fnd ha s sgnfcanly dfferen from. EMPIRICAL REULT everal explanaons for he UIP falure have been pu forward n he leraure. Flood and Rose [00] have shown ha uncovered neres rae pary wors sysemacally beer for counres n crss, whose exchange raes are consderably more volale. Clarda e al. [009] have obaned conssen resuls. In her opnon he sgn of slope coeffcen β depends on mare volaly. In hgh volaly regmes, coeffcen β occurs o be posve. Bansal and Dahlqus [000] have noced ha UIP performs beer n developng compared o developed counres. mlar researches have been conduced by Io and Chnn [007], Franel and Poonawala [00]. Accordng o Bansal and Dahlqus [000] counry-specfc arbues such as per capa ncome, neres raes, nflaon and counry rs rang are mporan n explanng devaons form uncovered neres rae pary. The am of he paper s o es uncovered neres rae pary hypohess for chosen fve developed and fve emergng economes and addonally o chec wheher resuls are smlar o hose obaned by Bansal and Dahlqus [000]. The UIP hypohess s verfed on he bass of regresson models (5) and (6) proposed by Froo a Franel a [989]. The models are bul for en exchange raes: UD/GBP, UD/AUD, JPY/UD, CAD/UD, CHF/UD, BRL/UD, MXN/UD, MYR/UD, THB/UD, RUB/UD. Counres were seleced on he bass of Ghoshray a and Morley a [0]. The uncovered neres rae pary hypohess s verfed for fve developed economes such as Uned Kngdom (Brsh Pound, GBP), Ausrala (Ausralan Dollar, AUD), Japan (Japanese Yen, JPY), Canada (Canadan Dollar, CAD), wzerland (wss Franc, CHF) and for fve emergng economes such as Brazl (Brazlan Real, BRL), Mexco (Mexcan Peso, MXN), Malaysa (Malaysan Rngg, MYR), Thaland (Tha Bah, THB), Russa (Russan Ruble, RUB). We use monhly daa from Bloomberg on spo exchange raes ( s ) and forward exchange raes ( f ) for UD/GBP, JPY/UD, CAD/UD, CHF/UD from January 995 o December 0, for THB/UD from epember 995 o December 0, for MXN/UD from November 997 o December 0, for BRL/UD from February 999 o December 0, for UD/AUD from epember In hs model he dependen varable y s fd s, whle he ndependen x s fd

7 38 Joanna Kselńsa, Kaarzyna Czech 000 o December 0, for RUB/UD from Augus 00 o December 0 and for MYR/UD from Aprl 005 o December 0. Tme perods for exchange raes are dfferen because of daa lmaons. Table presens he obaned resuls of he UIP es on he bass of regresson model (5). Models are mared as M (UD/GBP), M (UD/AUD), M3 (JPY/UD), M4 (CAD/UD), M5 (CHF/UD), M6 (BRL/UD), M7 (MXN/UD), M8 (MYR/UD), M9 (THB/UD) and M0 (RUB/UD). Table dsplays coeffcens, her sandard errors, es sascs and her correspondng sgnfcance levels. Table. OL esmaon resuls of equaon (5) for en exchange raes Model M - UD/GBP M - UD/AUD M3 - JPY/UD M4 - CAD/UD M5 - CHF/UD M6 BRL/UD M7 - MXN/UD M8 - MYR/UD M9 - THB/UD M0 - RUB/UD b (b) -,5 (,784) 0,477 (,68) -,75 (,40) -,858 (,854) -,83 (,675) 0,53 (0,805) -0,045 (0,49) 0,339 (0,98),780 (0,489),59 (0,99) ource: own esmaons a (a) -0,0008 (0,00) 0,0056 (0,007) -0,0038 (0,0040) -0,007 (0,007) -0,0067 (0,0037) -0,0047 (0,0083) 0,008 (0,0035) -0,004 (0,000) -0,0038 (0,007) -0,0069 (0,007) F g. F g. 0,493 0,483-0,70-0,39 0,696 0,03 0,859 0,78 0,770 0,443 0,898 0,345-0,948-0,948 0,344,376 0,5 -,54 -,047 0,97,858 0,093 -,69 -,79 0,075 0,4 0,57 0,649-0,560 0,577 0,0 0,95-0,07 0,798 0,46 0,36 0,73 0,369 -,87 0,39 3,53 0,0003 3,64 -,397 0,64 8,376 4,E-07 5,37 -,55 0,03 The slope coeffcen s sgnfcanly dfferen from zero n models M5, M9 and M0 wh a leas 90 percen level of confdence. Furhermore, only n M5 and M0 esmaes for α are sgnfcanly dfferen from zero. For exchange raes UD/GBP, UD/AUD, JPY/UD, CAD/UD, BRL/UD, MXN/UD and MYR/UD boh slope coeffcen β and nercep coeffcen α are nsgnfcan. Only for CHF/UD, THB/UD and RUB/UD slope coeffcen s sgnfcan,

8 The applcaon of regresson analyss 39 s whch mples ha here s lnear relaonshp beween dependen varable ( ) fd and ndependen varable ( ). For CHF/UD coeffcen equals -,833, for THB/UD equals,7796 and for RUB/UD equals,5999. These resuls confrm he fndngs of Bansal and Dahlqus [000]. The slope coeffcens for emergng economes (Thaland and Russa) are posve and for developed economes (wzerland) negave. As was menoned before, UIP s sasfed f he parameer β n he frs model (5) s posve, and unsasfed when β s negave. The esmaon sgns of coeffcens confrm he resuls obaned by Bansal and Dahlqus [000]. For he currences of emergng counres (Brazl, Malaysa, Thaland, and Russa) parameers β are posve, whle for developed counres (Uned Kngdom, Japan, Canada and wzerland) negave. Only for Ausrala and Mexco Bansal and Dahlqus s [000] hess has no been confrmed. I needs o be emphaszed, however, ha for mos of he models coeffcen β was nsgnfcan. I suggess ha presened fndngs should be reaed wh some cauon. Table dsplays regresson resuls of equaon (6). Models are mared as M (UD/GBP), M (UD/AUD), M3 (JPY/UD), M4 (CAD/UD), M5 (CHF/UD), M6 (BRL/UD), M7 (MXN/UD), M8 (MYR/UD), M9 (THB/UD) and M0 (RUB/UD). The able below conans slope coeffcen esmaes, sandard errors, es sascs and her correspondng sgnfcance levels (p-value). Moreover, he value of xx, xy and EE are provded, whch enable o apply formulas (3) and (5). The slope coeffcen s nsgnfcanly dfferen from zero n models M, M, M6, M8 M9 wh a leas 0% sgnfcance level. Ths s equvalen wh saemen ha n equaon (5) s nsgnfcanly dfferen from one, whch mples ha UIP hypohess canno be rejeced. In oher models slope coeffcen s sgnfcan whch means ha forward rae predcon error s fd can be explaned by forward premum fd. Moreover, when we ncrease he sgnfcance level o 0,5, he coeffcen n model M9 wll be also sgnfcan. Esmaon resuls mply ha for JPY/UD, CAD/UD, CHF/UD, MXN/UD and RUB/UD exchange raes, uncovered neres rae hypohess s rejeced. However, here are no foundaons o rejec β = 0 null hypohess for UD/GBP, UD/AUD, BRL/UD, MYR/UD and THB/UD wh a leas 0% sgnfcance level.

9 40 Joanna Kselńsa, Kaarzyna Czech Table. OL esmaon resuls of equaon (6) for en exchange raes Model M - UD/GBP M - UD/AUD M3 - JPY/UD M4 - CAD/UD M5 - CHF/UD M6 BRL/UD M7 - MXN/UD M8 - MYR/UD M9 - THB/UD M0 - RUB/UD b (b),5 (,784) 0,53 (,68),75 (,40) 3,859 (,854) 3,83 (,675) 0,478 (0,805),045 (0,49) 0,66 (0,98) -0,780 (0,489) -0,59 (0,99) ource: own esmaons F sg. F xx xy EE,595 0,08,63 0,000-0,000 0,6 0,038 0,846 0,95 0,000 0,000 0,3 3,077 0,08,754 0,0007-0,0008 0,8 4,330 0,039,08 0,000-0,0005 0,5 5,34 0,03,88 0,0004-0,000 0, 0,35 0,554 0,594 0,0043 0,00 0,456 6,30 0,04,496 0,005-0,000 0,6 0,59 0,473 0,70 0,0004 0,000 0,034,354 0, -,595 0,0050 0,0088 0,43 3,93 0,050 -,98 0,0087 0,038 0,04 Esmaons of equaon (6) do no provde unambguous resuls. Uncovered neres rae pary holds for wo developed economes (Uned Kngdom and Ausrala) and wo emergng economes (Brazl and Malaysa). However, he UIP hypohess s rejeced for hree developed counres (Japan, Canada and wzerland) and for hree developng counres (Mexco, Thaland and Russa). The conclusons were drawn on he bass of evaluaon of slope coeffcen. The nercep coeffcen has been nsgnfcanly dfferen from zero for Uned Kngdom, Ausrala, Japan, Canada, Brazl, Mexco, Malaysa and Thaland. I may mply he exsence of ransacon coss or non-zero rs premum. Only for wzerland and Russa has occurred o be sgnfcan. From a sascal pon of vew, model (6) seems o be beer han model (5). In he frs regresson model, mos slope coeffcens β was nsgnfcanly dfferen from zero. Hence, was no sensble o es wheher s dfferen from. The esmaon of equaon (6) gves nsgnfcan slope coeffcen only n four cases. A dfference n sgnfcance of F-es sascs n models (5) and (6) resuls from he es srucure. Ths s because he probably dsrbuon of he es sasc s deermned by he null hypohess bu no by real varable s dsrbuon.

10 The applcaon of regresson analyss 4 UMMARY The paper verfes he hypohess of uncovered neres rae pary for he currences of fve developed and fve developng counres. The resuls do no gve a clear confrmaon or denal of he UIP heory. Bansal and Dahlqus [000] have found ha he UIP wors sysemacally beer for developng counres whle for developed counres he UIP hypohess s generally rejeced. The pary s sasfed f he parameer β n he frs model (5) s posve, and unsasfed when β s negave. The esmaon sgns of parameer n he frs model confrm he resuls obaned by Bansal and Dahlqus [000]. For he currences of developng counres (Brazl, Malaysa, Thaland, and Russa), parameer β has posve values, whle for developed counres (Uned Kngdom, Japan, Canada and wzerland) negave. Only for he exchange rae of Ausralan and Mexcan currency Bansal and Dahlqus s hess has no been confrmed. I should be noed, however, ha parameer β s nsgnfcanly dfferen from zero for mos of he models, whch suggess ha hese fndngs should be reaed wh some cauon. The resuls of calculaons presened n he paper confrm he advanage of he second regresson model (6) over he frs model (5). The frs model shows ha for mos exchange raes parameer β s nsgnfcanly dfferen from 0, and herefore s no advsable o es wheher s dfferen from. However, he resuls of calculaons on he bass of he second model, show ha hs parameer s nsgnfcanly dfferen from only n four cases. REFERENCE Balle R. T., Bollerslev T. (000) The forward premum anomaly s no as bad as you hn, Journal of Inernaonal Money and Fnance, No. 9, sr Bansal R., Dahlqus M. (000) The forward premum puzzle: dfferen ales from developed and emergng economes, Journal of Inernaonal Economcs, No. 5, sr Clarda R., Davs J., Pedersen N. (009) Currency carry rade regmes: Beyond he Fama regresson, Journal of Inernaonal Money and Fnance, No. 8, sr Fama E. F. (984) Forward and spo exchange raes, Journal of Moneary Economcs, No. 4, sr Flood R. P., Rose A. K. (00) Uncovered Ineres Pary n crss, Inernaonal Moneary Fund aff Papers, No. 49 (), sr Franel J., Poonawala J. (00) The forward mare n emergng currences: Less based han n major currences, Journal of Inernaonal Money and Fnance, No. 9, sr Froo K.A., Franel J.A. (989) Forward dscoun bas: s an exchange rs premum, The Quarerly Journal of Economcs, No. 04, sr

11 4 Joanna Kselńsa, Kaarzyna Czech Ghoshray L D., Morley B. (0) Measurng he rs premum n uncovered neres pary usng he componen GARCH-M model, Inernaonal Revew of Economcs and Fnance, No. 4, sr Io H., Chnn M. (007) Prce-based measuremen of fnancal globalzaon: a crosscounry sudy of neres rae pary, Pacfc Economc Revew, No. (4), sr McCallum B. T. (994) A Reconsderaon of he Uncovered Ineres Pary relaonshp, Journal of Moneary Economcs, No. 33, sr erwa D, Rubasze M, Marcnowsa-Lewandowsa W. (red. nau.) (009) Analza ursu waluowego, Wydawncwo C.H. Bec, Warszawa, sr