Empirical Exchange Rate Models and Currency Risk: Some Evidence from Density Forecasts

Transcription

1 WORKING PAPERS SERIES WP04-0 Empirical Exchange Rae Models and Currency Risk: Some Evidence from Densiy Forecass Lucio Sarno and Giorgio Valene

2 Empirical Exchange Rae Models and Currency Risk: Some Evidence from Densiy Forecass Lucio Sarno! Universiy of Warwick and Cenre for Economic Policy Research (CEPR) Giorgio Valene Universiy of Warwick Firs version: January This revised version: Augus 2004 Absrac A large lieraure in exchange rae economics has invesigaed he forecasing performance of empirical exchange rae models using convenional poin forecas accuracy crieria. However, in he conex of managing exchange rae risk, ineres ceners on more han jus poin forecass. This paper provides a formal evaluaion of recen exchange rae models based on he erm srucure of forward exchange raes, which previous research has shown o be saisfacory in poin forecasing, in erms of densiy forecasing performance. The economic value of he exchange rae densiy forecass is invesigaed in he conex of an applicaion o a simple risk managemen exercise. JEL classificaion: F3; F37. Keywords: foreign exchange; forecasing; erm srucure; densiy forecas; nonlineariy.! Corresponding auhor: Prof. Lucio Sarno, Finance Group, Warwick Business School, Universiy of Warwick, Covenry CV4 7AL, UK. Tel: ; fax: lucio.sarno@warwick.ac.uk Posal address of co-auhor: Dr Giorgio Valene, Finance Group, Warwick Business School, Universiy of Warwick, Covenry CV4 7AL, UK.

3 . Inroducion Exchange rae risk plays a major role in inernaional porfolio diversificaion and in several aspecs of economic policy, including he assessmen of he uncerainy surrounding prices of expors and impors, he value of inernaional reserves and open posiions in foreign currency, and he domesic currency value of deb paymens and workers remiances which, in urn, may affec domesic wages, prices, oupu and employmen. In inernaional financial markes, expecaions of fuure exchange raes affec agens decisions in a number of respecs, including heir invesmen, hedging, and borrowing and lending decisions. I is no surprising, herefore, ha an enormous empirical lieraure has developed which focuses on modelling and forecasing nominal exchange raes. The vas majoriy of he empirical lieraure on forecasing exchange raes has cenered on forecasing he level of nominal exchange raes. This lieraure is highly influenced by he seminal work of Meese and Rogoff (983a,b), who firs documened ha empirical exchange rae models, based on convenional macroeconomic fundamenals suggesed by inernaional macroeconomics heory, canno ouperform a simple no-change or random walk forecas of exchange raes in erms of sandard measures of poin forecas accuracy. Afer over weny years of research since he publicaion of he Meese-Rogoff sudies, heir findings remain, wih a few excepions, very robus (e.g. Mark, 995; Neely and Sarno, 2002). While macroeconomic fundamenals do no appear o be useful in forecasing exchange raes, however, models which exploi he informaion in he erm srucure of forward exchange raes and forward premia have produced saisfacory resuls. Clarida and Taylor (997) firs argued ha, alhough he forward exchange rae is no an opimal predicor of he fuure spo exchange rae (e.g. see Lewis, 995; Engel, 996; and he references herein), forward raes may sill conain valuable informaion for forecasing fuure spo exchange raes. Under he assumpion of saionary expeced foreign exchange excess reurns, Clarida and Taylor (997) derive a linear vecor equilibrium correcion model (VECM) of spo and forward exchange raes. Using his linear VECM, Clarida and Taylor show ha is possible o exrac sufficien informaion from he erm srucure of forward premia o ouperform he random walk model for several exchange raes in ou-of-sample See, for example, he papers published in he special issue of he Journal of Inernaional Economics (May 2003) on Empirical Exchange Rae Models. 2

4 forecasing. Then, following he large lieraure showing he exisence of regime-swiching behavior in exchange rae movemens (e.g. Engel and Hamilon, 990; LeBaron, 992; Engel, 994; Engel and Hakkio, 996), Clarida, Sarno, Taylor and Valene (2003) generalize he linear VECM of spo and forward exchange raes o a nonlinear, hree-regime Markovswiching VECM (MS-VECM) which is found o ouperform a random walk as well as o improve on he linear VECM in erms of ou-of-sample forecasing. While an exensive body of lieraure has invesigaed he performance of exchange rae models in forecasing he level of he exchange rae, surprisingly lile aenion has been devoed o forecasing he densiy of exchange raes. In a decision-heoreical conex, he need o consider he densiy forecas of a variable 2 - as opposed o considering only is condiional mean and variance - seems fairly acceped on he basis of he argumen ha economic agens may have loss funcions ha do no depend symmerically on he realizaions of fuure values of poenially non-gaussian variables (Sachell and Timmermann, 995; Granger, 2003). In his case, agens are ineresed in forecasing no only he mean and variance of he variables in quesion, bu heir full predicive densiies. In various conexs in economics and finance - among which he recen boom in financial risk managemen is an obvious case - here is srong need o evaluae densiy forecass. Several researchers have proposed mehods for evaluaing densiy forecass. These mehods allow us o evaluae a model-based predicive densiy by measuring he closeness of wo densiy funcions or by esing he hypohesis ha he predicive densiy generaed by a paricular model corresponds o he rue predicive densiy (e.g. Diebold, Gunher and Tay, 998; Berkowiz, 200). 3 More recenly, his line of research has also provided a es saisic o formally evaluae he relaive abiliy of compeing models in maching he rue predicive densiy (Corradi and Swanson, 2004a,b). However, a lieraure focusing on he performance of empirical models o forecas he densiy of exchange raes has no emerged o dae, and he 2 A densiy forecas (or predicive densiy) of he realizaion of a random variable a some fuure ime is an esimae of he probabiliy disribuion of he possible fuure values of ha variable. I herefore provides a full descripion of he uncerainy associaed wih a forecas, in conras wih a poin forecas, which conains no descripion of he associaed uncerainy. For a survey of he lieraure on densiy forecasing and a discussion of is applicaions in macroeconomics and finance, see Tay and Wallis (2000). See also Granger and Pesaran (999), Granger (2003) and Sarno and Valene (2004a,b). 3 By rue predicive densiy of he daa we mean he densiy of he daa over he chosen forecas period. Therefore, no forecas is in fac carried ou in his case, and he erm predicive simply refers o he fac ha he densiy in quesion does no refer o he full sample bu only o he forecas period. Also noe ha we use he 3

5 main focus of he relevan lieraure remains on poin forecasing of he nominal exchange rae. The presen paper conribues o he relevan lieraure in ha we re-examine he forecasing performance of erm srucure models of exchange raes, which were shown o ouperform a random walk in ou-of-sample poin forecasing by Clarida and Taylor (997) and Clarida e al. (2003). However, we assess he abiliy of hese models o forecas ou-ofsample he one-sep-ahead densiy of nominal exchange raes, hence filling, o some exen, he imporan gap in he lieraure described above. Our analysis is carried ou using he recen echniques on evaluaing densiy forecass menioned above as well as on Value-a- Risk (VaR) calculaions. In paricular, using weekly daa for eigh bilaeral dollar exchange raes from January 985 o December 2003, we focus on he abiliy of boh he linear VECM and he MS-VECM o forecas he one-week-ahead exchange rae densiy. To anicipae our main resuls, we find ha Markov-swiching erm srucure models of exchange raes produce saisfacory densiy forecass of exchange raes. In paricular, he MS-VECM of he erm srucure convincingly ouperforms a random walk forecas and a linear erm srucure VECM in our densiy forecasing exercise, suggesing ha he allowance for nonlineariy in hese models may be paricularly imporan o produce saisfacory ou-of-sample densiy forecasing performance. Finally, we illusrae he pracical imporance of our resuls on densiy forecasing wih a simple applicaion o a risk managemen exercise. In recen years, rading accouns a large financial insiuions have shown a dramaic growh and become increasingly more complex. Parly in response o his rend, major rading insiuions have developed risk measuremen models designed o manage risk. The mos common approach employed in his conex is based on he VaR mehodology, where VaR is defined as he expeced maximum loss over a arge horizon wihin a given confidence inerval (Jorion, 200) - more formally, VaR is an inerval forecas, ypically a one-sided 95% or 99% inerval of he disribuion of expeced wealh or reurns. In our simple applicaion we analyze he ou-of-sample forecasing performance of erm srucure models of exchange raes, invesigaing he implicaions of hese forecass for a risk manager who has o quanify he risk associaed wih holding a currency porfolio over a one-week horizon. This applicaion furher illusraes how he MS- VECM capures saisfacorily he higher momens of he predicive densiy of exchange raes, generaing VaRs ha esimae he probabiliy of large losses beer han he oher wo erms predicive densiy and forecas densiy inerchangeably below. 4

6 compeing models. Pu anoher way, our findings indicae ha beer densiy forecass of exchange raes, of he ype recorded by he regime-swiching model considered in his paper, can poenially lead o subsanial improvemens in risk managemen and, more precisely, o beer esimaes of downside risk. The remainder of he paper is se ou as follows. In Secion 2, we briefly review he lieraure employing erm srucure models of forward premia o forecas exchange raes, boh in a linear framework and in a Markov-swiching framework. In Secion 3, we describe he daa and carry ou a preliminary uni roo and coinegraion analysis of he spo and forward exchange raes daa. We also repor lineariy ess and oher ess designed o selec he mos adequae MS-VECM in our conex. We hen forecas from hese models in Secion 4, where we repor ess designed o assess he performance of he random walk model, he linear VECM and he MS-VECM in erms of densiy forecasing. We also explore he implicaions of he densiy forecasing resuls in a risk managemen exercise. A final secion briefly summarizes and concludes. 2. Term Srucure Forecasing Models of Exchange Raes: A Brief Overview Le s and k f be, respecively, he spo exchange rae and he k -period forward exchange rae a ime. Under he assumpions ha (i) each of s and k f are well described by uni roo processes and ha (ii) deparures from he risk-neural efficien markes k hypohesis - namely expeced foreign exchange excess reurns, " # wih respec o a given informaion se expression which implies ha he forward premium, f % E s $ k &', defined ' - are saionary, i is sraighforward o derive an f k % s is saionary (Clarida and Taylor, 997). In urn, his resul implies ha forward and spo exchange raes have a common sochasic rend and are coinegraed wih coinegraing vecor ( * % ). This also implies ha, since his is rue for any k, if we consider he vecor of forward raes of enor o m periods, 2 3 m + ogeher wih he curren spo rae, [ s * f * f * f * * f ], hen his mus be coinegraed wih m unique coinegraing vecors, each given by a row of he marix [%,* I m ], where I m is an m -dimensional ideniy marix and, is an m -dimensional column vecor of ones. Finally, by he Granger Represenaion Theorem (Granger, 986; Engle and Granger, 987) his vecor of forward and spo raes mus possess a VECM represenaion in which he erm srucure of forward premia plays he par of he equilibrium errors. 5

7 I is imporan o noe ha, while assumpion (i) above is unconenious, assumpion (ii) and is key implicaion of a saionary forward premium are more conroversial. In paricular, some auhors have suggesed ha he forward premium may display long memory properies and is well characerized as a nonsaionary fracionally inegraed process, or Id ( ) process, where 0-5. d.. This propery of he forward premium would hen imply ha spo and forward exchange raes are fracionally coinegraed and ha he dynamic relaionship beween spo and forward raes is characerized by a fracional VECM (e.g. see Baillie, 996; Baillie and Bollerslev, 2000; Maynard and Phillips, 200). While i is well documened in he lieraure ha he forward premium is persisen and his propery is likely o plague saisical inference in he conex of esing marke efficiency, he specific long-run properies of he forward premium and of he relaionship beween spo and forward raes are no cenral o he empirical work carried ou in he presen sudy. Our main aim relaes o he invesigaion of he shor-run forecasing abiliy of empirical exchange rae models based on informaion exraced from he erm srucure of forward raes. For his purpose, he assumpion of a saionary (albei poenially persisen) forward premium is no crucial in our analysis, and hence we shall sick o he original assumpions of he Clarida-Taylor framework. The linear VECM used by Clarida and Taylor (997) may be wrien as follows: y 0 / $ 2 y $ 3 y $ u () p% d 0 d % d % where y 0 s * f * f * f * f, wih he superscrip denoing he number of weeks [ ] corresponding o he mauriy of he forward conrac; is he long-run impac marix whose rank deermines he number of coinegraing vecors linking spo and forward raes (equal o four in his specific VECM); and u is a vecor of Gaussian error erms (Johansen, 988, 99). Clarida and Taylor (997) exploi his linear VECM represenaion o show ha sufficien informaion may be exraced from he erm srucure in order o forecas he spo dollar exchange rae during he recen floaing exchange rae regime. Their dynamic ou-ofsample forecass sugges ha he linear VECM is superior o a range of alernaive forecass, including a random walk and sandard spo-forward regressions. Clarida e al. (2003) hen generalize he linear VECM in equaion () o a mulivariae Markov-swiching framework and examine he performance of such a model in ou-of-sample exchange rae forecasing. This generalized erm srucure model was inspired by encouraging resuls previously repored in he lieraure on he presence of nonlineariies (and paricularly by he success of Markov-swiching models) in he conex of exchange rae modelling. Using weekly daa on major spo and forward dollar exchange raes over he 6

8 period 979 hrough 995, Clarida e al. repor evidence of he presence of nonlineariies in he erm srucure, which appeared o be well characerized by a hree-regime MS-VECM ha allows for shifs in boh he inercep and he covariance srucure of he error erms. This MS-VECM may be wrien as follows: where p% y 0 / " z # $ 3 y $ 7 2 y $ 6 * (2) % d % d d 0 z y is as defined in equaion (); / ( ) is a 3 -dimensional column vecor of regime- + dependen inercep erms, / ( z ) 0 [/ ( z )*/ 2 ( z )*/ 3( z )] ; he 3 i s are 38 3 marices of + parameers; 6 0 [ 6 * 6 2 * 63 ] is a 3-dimensional vecor of error erms, 6! NID(0 *9 6 ( z)). The regime-generaing process is assumed o be an ergodic Markov chain wih hree saes z :{ * 2* 3}, governed by he ransiion probabiliies pij 0 Pr( z$ 0 j" z 0i), and 3 p j0 ij 0 ; i* j: { * 2* 3}. This MS-VECM is ermed Markov-Swiching-Inercep-Heeroskedasic- VECM or MSIH-VECM. In order o reflec he fac ha he model has hree regimes and one lag of he dependen variable in each equaion, he model is ermed MSIH(3)-VECM(). Esimaion of an MS-VECM can be carried ou using an expecaions maximizaion (EM) algorihm for maximum likelihood (Dempser, Laird and Rubin, 977; Krolzig, 997). 4 Clarida e al. (2003) use he MSIH(3)-VECM() o forecas dynamically ou of sample over he period 996 hrough o 998. The resuls sugges ha he MS-VECM forecass are srongly superior o he random walk forecass a a range of forecasing horizons up o 52 weeks ahead, using sandard forecas accuracy crieria. Moreover, he MS-VECM also ouperforms a linear VECM for spo and forward raes in ou-of-sample forecasing of he spo rae, alhough he magniude of he gain, in poin forecasing, from using an MS-VECM relaive o a linear VECM is raher small a shor horizons (abou 0% on average a he 4- week forecas horizon). Neverheless, i is possible ha radiional measures of forecas accuracy mask somehow he poenial superioriy of nonlinear models (Sachell and Timmermann, 995; Granger, 2003). Overall, his lieraure suggess ha no only he erm srucure of forward premia conains valuable informaion abou he fuure spo exchange rae, bu also ha he allowance for nonlinear dynamics in he form of regime-swiching behavior enhances somewha his 4 An alernaive way o carry ou specificaion, esimaion and densiy forecasing for his class of regimeswiching models involves conducing robus saisical inference using Markov Chain Mone Carlo (MCMC) mehods (e.g. see Chib, 200, 2004, and he references herein). 7

9 informaion o produce a saisfacory forecasing exchange rae model. While previous research on erm srucure models has analyzed forecasing performance focusing primarily on accuracy evaluaions based on poin forecass, several auhors have recenly emphasized he imporance of evaluaing he forecas accuracy of economic models on he basis of densiy - as opposed o poin - forecasing performance. Especially when evaluaing nonlinear models, which are capable of producing non-normal forecas densiies, i would seem appropriae o consider a model s densiy forecasing performance. This is indeed he primary objecive of he empirical work underaken in his paper, where we carry ou densiy forecasing ess on he linear VECM and he MS-VECM of he erm srucure of forward premia as well as on a random walk exchange rae model. We hen invesigae some of he implicaions of our densiy forecasing resuls for exchange rae risk managemen. 3. Empirical Analysis I: Modelling In his secion, we describe he daa and carry ou a preliminary uni roo and coinegraion analysis of spo and forward exchange raes. We also repor lineariy ess and oher ess designed o selec he mos adequae MS-VECM in our conex, which we hen esimae Daa and Preliminaries Our daa se comprises weekly observaions of eigh bilaeral spo and 4-, 3-, 26- and 52-week forward US dollar exchange raes (vis-à-vis he UK serling, Swiss franc, Japanese yen, Canadian dollar, New Zealand dollar, Swedish krona, Norwegian krona, Danish krona). The sample period spans from January o December Following previous lieraure (e.g. Hansen and Hodrick, 980, p. 852), daa are Tuesdays of every week, aken from Daasream. From his daa se, we consruced he ime series of ineres, namely he logarihm of he spo exchange rae, s and he logarihm of he k -week forward exchange rae, k f for k 0 4* 3* 26* 52. In our empirical work, we carried ou our esimaions over he period January 985-December 995, reserving he las eigh years of daa for ou-of-sample forecasing. 5 Throughou our discussion of he empirical resuls, we employ a nominal significance level of 5% unless explicily saed oherwise. 6 The sar dae was chosen since i was he earlies dae for which daa for all exchange raes examined are available. Full deails on he preliminary empirical analysis discussed in his sub-secion are available from he auhors upon reques, bu are no repored o conserve space. 8

10 Having firs confirmed - by using sandard uni roo es saisics - he sylized fac ha each of he ime series examined is a realizaion from a sochasic process inegraed of order one, we hen employed he Johansen (988, 99) maximum likelihood procedure in a vecor auoregression for y 0 s * f * f * f * f and an unresriced consan erm. On he [ ] basis of he Johansen likelihood raio es saisics for coinegraing rank, we could srongly rejec he hypohesis of hree independen coinegraing vecors agains he alernaive of four, bu were no able o rejec he hypohesis of exacly four coinegraing vecors for each exchange rae examined a convenional nominal es sizes. When esing he hypohesis ha he coinegraing vecors linking spo and forward raes are of he form [*% ], we rejeced his hypohesis bu - consisen wih Naka and Whiney (995), Luinel and Paudyal (998) and Clarida e al. (2003) - he deparure from he overidenifying resricions, albei saisically significan a convenional es sizes, was found o be very small in magniude. Following Clarida e al. (2003), we inerpreed he rejecion of he uniy resricions on he coinegraion space as due o iny deparures from he null hypohesis (due, for example, o iny daa imperfecions) which are no economically significan, bu which appear as saisically significan given our large sample size. Given he heoreical economic priors in favor of he uniy resricions and he fac ha, under covered ineres pariy, coefficiens differen from uniy would have he implausible implicaion of a uni roo in inernaional ineres rae differenials, we carried ou our empirical work employing he uniy resricions. We nex esimaed a sandard linear VECM, as given in equaion (), using fullinformaion maximum likelihood (FIML) mehods, assuming a maximum lag lengh of hree, as suggesed by boh he Akaike Informaion Crierion and he Schwarz Informaion Crierion. Employing he convenional general-o-specific procedure, we obained fairly parsimonious models for each exchange rae Lineariy Tess, Idenificaion and Esimaion of he MS-VECM We proceeded o invesigae he presence of nonlineariies furher hrough he esimaion of a general MS-VECM of he form: p% y 0 / " z # $ 3 " z # y $ 7 2 " z # y $ < * (3) % d % d d 0 where all parameers are as defined in equaion (2), excep for he auoregressive parameers 2 s and he long-run marix 3 which are also allowed o be regime-shifing, i.e. " z # d 2 and d 9

11 " z # " z # ; < is a vecor of error erms, <! NID(0 *9 < ( z)). The number of regimes, z - for which we consider a maximum of hree regimes, i.e. z :{ * 2* 3} - is idenified using a likelihood raio es specifically designed for his purpose and described below. The MS- VECM in equaion (3) is indeed slighly more general han he MS-VECM used by Clarida e al. (2003) in ha, excep for he long-run coinegraing marix 5 +, which is resriced o be consisen wih saionary forward premia, every oher parameer of he model is allowed o be regime shifing. In essence, his MS-VECM allows for each of he inerceps, he variancecovariance marix and he auoregressive srucure o be regime dependen. To reflec he fac ha model (3) allows for each of he parameers on he auoregressive lags of y o be regime swiching, in addiion o regime-swiching inercep vecor and covariance marix, his VECM is ermed Markov-Swiching-Inercep-Auoregressive-Heeroskedasic-VECM or MSIAH-VECM. Nex we applied he boom-up procedure designed o deec Markovian shifs in order o selec he mos adequae characerizaion of an MS-VECM for y. The lineariy ess led us, for each exchange rae, o rejec he linear VECM agains an MS-VECM. These ess provide srong empirical evidence ha he linear VECM fails o capure imporan nonlineariies in he daa generaing process, since lineariy is rejeced wih marginal significance levels (p-values) of virually zero - see he las column in Table. For each MS-VECM esimaed, we esed he hypohesis of no regime shifs agains he alernaive hypohesis of regime shifs in each of he vecor of inercep erms, he variancecovariance marix, and he auoregressive srucure respecively. The likelihood raio (LR) ess LR, LR 2 and LR 3, consruced as suggesed by Krolzig (997, p. 35-6) and repored in Table, sugges in each case massive rejecions of he null hypoheses esed, clearly indicaing ha an MS-VECM ha allows for shifs in each of he inercep, he variancecovariance marix and he auoregressive srucure, namely an MSIAH-VECM, is he mos appropriae model wihin is class in he presen applicaion. Finally, in order o discriminae beween models allowing for wo regimes agains models governed by hree regimes we also calculaed an LR es for his null hypohesis. The resuls produced very low p-values (see LR 4, Table ), suggesing ha hree regimes are appropriae in all cases. Therefore, in spie of parsimony consideraions, we allowed for hree regimes in he MSIAH-VECM. Following he resuls from he idenificaion procedure repored in Table, we esimaed he MSIAH-VECM (3) for each of he eigh exchange raes examined. The 0

12 esimaion yielded fairly plausible esimaes of he coefficiens, including he adjusmen coefficiens in 4, which were generally found o be srongly saisically significanly differen from zero. For each exchange rae we find ha hree regimes are appropriae in describing he daa. Shifs from one regime o anoher appeared o be due largely o shifs in he variance of he erm srucure equilibrium. On he oher hand, shifs in he inercep erms and, o a lesser exen, in he auoregressive srucure were found o be relaively smaller in magniude, albei srongly saisically significan. This seems consisen wih he large empirical lieraure invesigaing he ime-varying naure of exchange raes risk premia. One enaive inerpreaion of his MSIAH-VECM is, in fac, in erms of shifs in he mean and variance of foreign exchange reurns consisen wih deviaions from he equilibrium levels implied by convenional macroeconomic fundamenals ha may be caused, for example, by peso problems or by oher kinds of deparures from he efficien markes hypohesis (see Engel and Hamilon, 990; Clarida e al., 2003). For each of he counries considered, we clearly idenified one regime characerized by periods in which he average spo rae and is variabiliy were relaively high compared o he remaining wo regimes. We hen invesigaed wheher or no he probabiliy of swiching beween regimes was relaed o macroeconomic fundamenals, in he spiri of recen research by Bansal and Zhou (2002), Bansal, Tauchen and Zhou (2004) and Clarida e al. (2004) - in he conex of ineres raes modelling, hese auhors find ha regime shifs are inimaely relaed o macroeconomic fundamenals. Like in previous research (e.g. Bansal and Zhou, 2002; Clarida e al., 2004), because daa on he explanaory variables we consider are no available a weekly frequency, we use quarerly daa. Hence, from he esimaed MSIAH- VECMs, we convered he weekly smoohed probabiliies ino quarerly probabiliies by averaging. Furher, in order o obain a binary variable from he esimaed average MSIAH- VECM probabiliies we defined a variable which is equal o zero when he average quarerly probabiliy of being in a regime is smaller han 0.5 and equal o uniy when his average probabiliy is greaer han or equal o 0.5. The fundamenals erm we consider in our simple exercise is he deviaion of he spo exchange rae from is equilibrium value as prediced by he moneary model of exchange rae deerminaion (see, iner alia, Mark, 995; Sarno and Taylor, 2003; Sarno, Valene and Wohar, 2004). 7 These deviaions have been squared in 7 Precisely, we focus on he deviaion, say dev, of he nominal exchange rae from is fundamenal value: dev s fund 0 %, where he fundamenals erm, fund is he long-run equilibrium of he nominal exchange

13 order o assign he same weigh o negaive and posiive deparures from equilibrium. We hen calculaed he correlaion beween he binary indicaor variable - capuring he regime of high mean and volailiy of spo exchange rae movemens - and our proxy for he deviaion from exchange rae equilibrium. Our resuls show ha on average, across counries, he probabiliy of being in he regime characerized by high average and volaile spo exchange raes is significanly posiively correlaed wih deparures from heir equilibrium levels. In paricular, he average esimaed correlaion coefficien is abou 8%, which is in line wih he recen lieraure on ineres raes shifs and macroeconomic fundamenals (i.e. Bansal and Zhou, 2002, p. 2020). This posiive relaionship suggess ha when large deviaions (of eiher sign) from he fundamenal exchange rae equilibrium level occur, spo exchange raes are likely o experience periods of high excess reurns and volailiy. In general, we noe ha considerable cauion should be exercised in inerpreing he regime classificaion, as here is considerable error in his classificaion due o parameer esimaion error. However, a firs glance, our resuls sugges ha he shifs in mean and variance of spo exchange raes depiced by our MSIAH-VECMs may be relaed, o some exen, o changes in he sor of economic fundamenals one would expec o play a role in driving exchange raes behavior and regimes. 4. Empirical Analysis II: Forecasing and Risk Assessmen In his secion we compue ess designed o assess he performance of he random walk model, he linear VECM and he MSIAH-VECM in erms of exchange rae densiy forecasing. We hen explore he implicaions of he densiy forecasing resuls in a risk managemen exercise. rae deermined by he moneary fundamenals. The fundamenals erm is given by: fund 0 =! >!? m % % ( g % g ) A B, where m and g denoe he log-levels of money supply and real income respecively; and aserisks denoe foreign variables - daa for narrow money and real Gross Domesic Produc were aken for each counry from he Inernaional Financial Saisics of he Inernaional Moneary Fund. fund may be hough of as a ypical represenaion of he long-run equilibrium exchange rae implied by exchange rae deerminaion heory, and forms of his kind are implied, for example, by he moneary approach o exchange rae deerminaion as well as by general equilibrium models wih cash-in-advance consrains and several new open economy macroeconomic models (see Sarno, 200; Neely and Sarno, 2002; Sarno and Taylor, 2003, Chapers 4-5; Sarno, Valene and Wohar, 2004). Hence, he link beween moneary fundamenals and he nominal exchange rae is consisen wih boh radiional models of exchange rae deerminaion based on aggregae funcions as well as wih more recen microfounded open economy models. 2

14 4.. Ou-of-Sample Densiy Forecass To invesigae he forecasing abiliy of our linear and Markov-swiching VECMs, we aemp o exploi he whole informaion provided by he ou-of-sample predicions of he models in he conex of a densiy forecasing approach. In paricular, i is ineresing o examine wheher he MSIAH-VECM (3), which was chosen on he basis of he idenificaion procedure which rejeced he linear VECM, performs saisfacorily in erms of closeness of he prediced momens of he forecas densiy of he model relaive o he rue momens of exchange rae movemens obained from he daa over he forecas period. This quesion canno be addressed fully by using convenional mehods of poin forecas accuracy evaluaion, since hese mehods consider only he firs wo momens of he disribuion of exchange raes. A large and growing lieraure has recenly focused on evaluaing he forecas accuracy of empirical models on he basis of densiy, as opposed o poin, forecasing performance (see, iner alia, Diebold e al., 998; Granger and Pesaran, 999; Tay and Wallis, 2000; Timmermann, 2000; Sarno and Valene, 2004a,b). Several researchers have proposed mehods for evaluaing densiy forecass. For example, Diebold e al. (998) exend previous work on he probabiliy inegral ransform and show how i is possible o evaluae a model-based predicive densiy. Diebold e al. propose he calculaion of he probabiliy inegral ransforms of he acual realizaions of he variables (i.e. exchange rae changes for each counry under invesigaion) over he forecas period, C n D wih respec o he models forecas densiies, denoed by C " # D n p s $ : 0 s $ " E# s $ 0 w 0 G p de 0 * * n- (4) %F When he model forecas densiy corresponds o he rue predicive densiy, hen he sequence n of Cw D 0 is iid ( 0) U *. The idea is o evaluae wheher he realizaions of he daa over he forecas period come from he seleced forecas densiy by esing wheher he Cw series depars from iid uniformiy. Berkowiz (200) suggess ha raher han working wih he Cw series i may be fruiful o ake he inverse normal cumulaive disribuion funcion (CDF) ransform of he D D series Cw, denoed by Cx. Under he null hypohesis of equaliy of he model densiy and D he rue predicive densiy, Cx is disribued as sandard normal, and Berkowiz proposes an D D 3

15 LR es for zero mean and uni variance, under he mainained hypohesis of iid normaliy. We rely on he es of Berkowiz (200) in our empirical work, since Berkowiz shows ha working wih he inverse normal ransform of he series Cw drasically increases he power of he es relaive o he version based on uniformiy. While, under general condiions, he linear VECMs forecas densiies are easy o calculae analyically (hey are in fac mulivariae normal disribuions wih means and variances given by funcions of he esimaed parameers), he MSIAH-VECM forecas densiies can, in general, be obained analyically only for one-sep-ahead forecass. The MSIAH-VECM forecas densiies are mixures of mulivariae normal disribuions wih weighs given by he prediced regime probabiliies. In general, he MSIAH-VECM forecas densiies are non-normal, asymmeric and heeroskedasic. In his paper we focus on he onesep-ahead forecas densiy of he MSIAH(3)-VECM(), which is given by: 3 3 H I = > p $ " y $ # 0 J pij Kp $? y $ z$ 0 A B j0 L i0 M D 77 P (5) where p 0 Pr( z 0 j z 0 i) are he ransiion probabiliies; P is he ransiion marix ij $ condiional on he informaion se a ime, = > ' ; and p $? y $ z$ A B 0 *' is he regimecondiional forecas densiy. We now urn o he evaluaion of he probabiliy inegral ransforms. The null of iid normaliy is a join hypohesis and, in he spiri of Diebold e al. (998), we consider each par of he hypohesis in urn. The iid assumpion is esed by calculaing he Ljung-Box (978) es for no serial correlaion. In order o ake ino accoun he dependence occurring in he higher momens, we consider " x # x N % for N up o four. In our forecasing exercise, we compare he densiy forecasing performance of he linear VECM in equaion () and he MSIAH-VECM in equaion (3) wih he sandard benchmark in he lieraure on exchange rae forecasing, namely he random walk model. 8 The p-values from carrying ou hese ess are repored in Panel a) of Table 2. The resuls sugges ha, for each exchange rae, we are unable o rejec he null hypohesis of no serial correlaion a convenional significance levels. This finding holds for each of hree models examined, namely he random walk model, he linear VECM, and he MSIAH-VECM. We hen carry ou ess for normaliy o verify 8 Obviously, he random walk model refers o he level of he exchange rae, implying ha in he case of exchange rae changes he model becomes a model of normally disribued exchange rae changes wih mean equal o he drif erm and variance equal o he residual variance esimaed over he sample period available. 4

16 D wheher he Cx series displays any saisically significan skewness or excess kurosis. The p-values from hese ess, repored in Panel b) of Table 2, sugges ha he null hypoheses of no skewness and no excess kurosis canno be rejeced for any of he exchange raes and any of he hree compeing models considered. Taken ogeher, he resuls in Panel a) and Panel b) of Table 2 indicae ha he null hypohesis of iid normaliy of he Cx D series canno be rejeced. Given hese findings, he mainained hypohesis which is required o carry ou he LR es proposed by Berkowiz (200) is validaed by he daa. Hence, we calculae he LR ess of zero mean and uni variance proposed by Berkowiz (200), which we repor in Panel a) of Table 3. The resuls are ineresing. On he basis of hese LR ess, he only model for which we canno rejec he null hypohesis is he MSIAH-VECM. Indeed, he forecas densiies of he random walk model and he linear VECM lead o rejecion of he null hypohesis for each exchange rae excep for he New Zealand dollar. In oher words, excep for he New Zealand dollar (where each of hree compeing models performs saisfacorily), he random walk model and he linear VECM produce densiy forecass ha are saisically significanly differen from he acual densiy of exchange raes daa over he forecas period. On he oher hand, he MSIAH-VECM generaes, for each exchange rae considered, densiy forecass ha are saisically idenical o he rue predicive densiies. One of he limiaions of he esing procedure employed above is he fac ha, while i allows us o measure how well a model s predicive densiy approximaes he rue predicive densiy of he daa, i does no allow a formal es of which of he compeing models considered performs bes in erms of densiy forecasing performance. Heurisically his can be invesigaed by inspecing he p-values of he Berkowiz ess for each individual model (i.e. higher p-values presumably reflec a beer densiy forecas) bu i is no possible o use he Berkowiz esing procedure o obain a p-value for he null hypohesis ha wo models perform equally well in forecasing he rue predicive disribuion. A soluion o his problem has recenly been proposed by Corradi and Swanson (2004a), who derive a es saisic for he null hypohesis ha wo models have equal densiy forecas accuracy. In some sense, his es evaluaes compeing forecasing models in erms of densiy forecasing in he same spiri of he Diebold and Mariano (995) es for equal poin forecas accuracy of compeing models. Using a measure ha may be seen as he analogue of he mean square error in he cones of densiy forecass, Corradi and Swanson (2004a) es a null hypohesis ha can be expressed as 5

17 G CP " # " # " # " # D 2 2 S * $ $ $ QT PS * $ $ $ QT O " # H R E F s & u % F s & u % F s & u % F s & u u du* 0 U where u: U U V, " u# 0 " # q* $ $ O W is a possible unbounded se on he real line; " u# du U (6) O 0 ; F s & u is he predicive cumulaive densiy funcion (CDF) implied by model q 0 2 * a ime $ for a given u ; and F " s u# & is he value of he rue CDF for a given u. The 0 $ es saisic for he null hypohesis (6) akes he form of where 7 C T " # G U T * uo (7) Z 0 Z u du* " # " # " # " # n% # 2 # 2 T * u 0 P n 0 * $ $ & % 0 $ & Q % P Q 0 2* $ $ & % 0 $ & Z S F s u F s u T S F s u F s u T and F # q *$ is he esimaed counerpar of F q * $ for q 0 2 *. The es saisic Z T is hen calculaed by averaging Z Tu * over u: U. Following Corradi and Swanson (2004a), his can be done by generaing a fine grid of u whose values are equally spaced across he range deermined by he minimum and maximum value of s $ over he sample period. Hence, if we assume a grid wih S poins, he es saisic (7) becomes: Z 0 F s & u % F s & u % F s & u % F s & u - S H n% # 2 # 2 I X H I XP = > = > Q P = > = > Q XX T 7 J 7 Z Y Z KK S X XS * $ A $! B A $! BT S * $ A $! B A $! B T XX! 0 L n 00 L M M Differenly from he previous esing procedures based on he probabiliy inegral ransform, he limiing disribuion of he es saisic (8) is a funcional of a Gaussian process whose covariance kernel is no a maringale difference sequence in he presence of model misspecificaion. This implies ha is limiing disribuion is no nuisance-parameer free and herefore canno be abulaed. Corradi and Swanson (2004a) show how o obain criical values for he disribuion of he es saisic (8) by boosrap. 9 0 The resuls from calculaing he Corradi-Swanson (2004a) es are repored in Panel b) of Table 3. In general, he null hypohesis of equal densiy forecas accuracy is rejeced for D ; (8) 9 In our densiy forecasing exercise we implemen he boosrap designed o calculae criical values using subsampling, as shown in Poliis, Romano and Wolf (999, Ch. 3) and recenly employed by Linon, Maasoumi and Whang (2003). 0 Noe ha anoher es for equal densiy forecas accuracy available in he lieraure is due o Sarno and Valene (2004a). While his es has he advanage of being easily applicable since i has a known limiing disribuion, we prefer o use he Corradi-Swanson es since his is more general and works under less sringen assumpions and regulariy condiions. 6

18 mos exchange raes a he 5% significance level. In paricular, here is clear evidence ha he MSIAH-VECM is able o generae predicive densiies which are beer approximaions of he rue predicive densiy han he ones implied by he random walk model and he linear VECM. Indeed, by inspecing he second and hird columns in Panel b) of Table 3, he Corradi-Swanson es saisics are all posiive and saisically significan. This means ha he disance beween he rue predicive densiy implied by he daa and he predicive densiies generaed by he random walk model (column 2) or he linear VECM (column 3) is significanly larger han he disance beween he rue predicive densiy and he predicive densiy generaed by he MSIAH-VECM. Furher, while i is sraighforward o esablish, on he basis of hese resuls, ha he MSIAH-VECM is he bes model in erms of densiy forecasing performance, i is difficul o discriminae beween he linear VECM and he random walk model since here is no clear paern in he sign of he es saisics. Summing up, he forecasing resuls in his secion sugges ha, in erms of densiy forecasing performance, he MSIAH-VECM performs beer han he linear VECM and he random walk model in erms of explaining he ou-of-sample behavior of exchange rae movemens. Clearly, his finding, obained by boh esing procedures employed - namely he Berkowiz es and he Corradi-Swanson es given in Table 3 - is due o he allowance for muliple regimes in he MSIAH-VECM, which enhances he informaion embedded in he forward premia and generaes densiy forecass ha are closer o he rue predicive densiies, providing a beer characerizaion of he uncerainy surrounding he exchange rae forecass han he oher wo compeing models The Economic Value of Densiy Forecass: A Simple Example of VaR Analysis In his secion we furher invesigae he pracical implicaions of he densiy forecasing resuls repored in he previous sub-secion in he conex of a simple risk managemen exercise. Given he predicions of he hree compeing models examined here, assume ha a US risk manager wishes o quanify he one-week-ahead risk associaed wih holding a posiion in foreign currency. Assume ha he posiion in quesion is a naive diversified porfolio comprising a domesic asse and N foreign asses which are idenical in all respecs excep for he currency of denominaion, e.g. euro-deposi raes. Each asse delivers yields in local currency, and given ha diversificaion of his porfolio is assumed o be naive, he weigh on each asse is and N $. Define i he domesic (US) one-period ineres rae c i he foreign one-period ineres rae associaed wih counry c. Then, given a cerain 7

19 level of wealh a ime invesed in his porfolio, say follows: P N Q Y = c c > Z $ 0 Y exp " # $ exp? $ Z Y A B N c N Z S $ 0 $ T W, he law of moion of wealh is as W i 7 i s W (9) where s c $ is he firs difference in he dollar log-exchange rae vis-á-vis he currency of counry c (e.g. Mark, 200; Elon, Gruber, Brown and Goezmann, 2003). Given ha i and c i are known a ime, he only source of risk o be aken in consideraion by he risk c manager a ime is he fuure nominal exchange rae for c 0 * * N. Normalizing W 0 for simpliciy, he risk manager can use a specific model of exchange raes a ime o produce one-week-ahead densiy forecass of s c $, which, in urn, imply one-week-ahead densiy forecass for he one-week-ahead wealh W $. On he basis of hese densiies he risk manager calculaes, for each of he hree compeing models - he random walk, he linear VECM and he MSIAH-VECM - he VaR as a confidence inerval for losses such ha " # $ $ s $ Pr W. VaR 0 % O- (0) In our example he VaR is a 99% confidence level for losses (i.e. O ), for all models; and N 0 8, corresponding o he eigh dollar exchange raes sudied in his paper. Equaion (0) simply saes ha he probabiliy ha one-week-ahead wealh, W $ is less han he VaR is equal o he significance level " % O #. Summary saisics are repored in Panel a) of Table 4. In order o assess he relaive size and relaive variabiliy of he VaR esimaes across he compeing models we use he mean relaive bias saisic ( MRB ) and roo mean squared relaive bias saisic ( RMSRB ), suggesed by Hendricks (996). The MRB saisic is calculaed as: MRB g n VaRg * $ h % VaR $ h 0 7 () n VaR h0 g*$ h where VaR g * is he esimaed VaR from he specific model g used a ime, VaR is he cross-secional average VaR a ime over he hree compeing models, and n is he number of ou-of-sample observaions. This saisic gives a measure of size for each esimaed VaR relaive o he average of all compeing models. For his sub-secion, we obained weekly observaions of -week euroraes for each counry examined during he sample period January 996-December 2003 from Daasream. 8

20 The RMSRB saisic is calculaed as: RMSRB g n = VaRg * $ h % VaR > $ h (2) n? h0 A g*$ h B This measure provides us wih informaion abou he exen o which he esimaed VaR ends o vary around he average VaR a ime. The resuls from calculaing he MRB and RMSRB, repored in Panel a) of Table 4, sugges ha he MSIAH-VECM produces higher VaRs (compared o he average VaR produced by he oher wo compeing models) and i also produces more volaile VaRs (around he average VaR). The higher VaR produced by he MSIAH-VECM is indicaive ha his model is less conservaive, on average, han he oher wo compeing models. We also repor in Panel a) of Table 4 he average disance ( AD ) beween he realized daa and he VaR implied by he random walk model, he linear VECM and he MSIAH- VECM, sandardized by he average disance of he random walk model; and, finally, we repor he correlaion coefficien beween he esimaed VaR from each model and he realized daa, calculaed as in Hendricks (996) and ermed 2 corr VaR* W. These calculaions indicae, on he basis of he AD, ha he MSIAH-VECM produces he closes VaR o he realized daa by some 3% relaive o he random walk model and 2% relaive o he linear VECM. Also, on he basis of corr VaR* W, he MSIAH-VECM generaes he VaR mos highly correlaed wih he realized daa across he hree models considered. 2 In Panel b) of Table 4, we repor some VaR backess. In paricular, we repor (as V ) he number of imes ha W$. VaR$, and he implied esimae of he violaion rae (i.e. V divided by he number of ou-of-sample observaions), say VR, for each of he random walk model, he linear VECM and he MSIAH-VECM respecively. We also es of he null hypohesis ha he violaion rae VR does no exceed he heoreical % violaion rae considered in his VaR applicaion, calculaed as in Kupiec (995). Finally, we repor he Chrisoffersen and Diebold (2000) es for he sample firs-order auocorrelaion of a binary variable which is equal o uniy if a violaion occurs and zero oherwise. The resuls in Panel b) indicae ha he random walk model and he linear VECM are oo conservaive in ha hey boh exhibi one violaion, whereas he number of violaions under he VaR esimaed for he MSIAH-VECM is hree, which implies an esimaed violaion rae of 0.72%. Such esimaed 2 The resuls show he poor performance of he oher wo models, which boh display saisically insignifican correlaion coefficiens and, for he random walk model, a negaively signed one. 9

21 violaion rae is indeed insignificanly differen from he heoreical violaion rae of %, as confirmed by he Kupiec es saisic. On he oher hand, he esimaed violaion rae for he oher wo compeing models is, in each case, equal o abou 0.23%, and is saisically significanly differen from he heoreical violaion rae of %. The CD es also suggess ha here is no firs-order serial correlaion in he VaR violaions for each of he hree models under examinaion, indicaing ha hese violaions are, for each model, non-sysemaic. 3 Summing up, his simple applicaion furher illusraes he saisfacory ou-of-sample forecasing performance of he MSIAH-VECM relaive o he random walk benchmark and o he linear VECM. We find ha he MSIAH-VECM provides a violaion rae which is saisically insignificanly differen from he heoreical violaion rae of %, whereas he oher wo compeing models provide esimaed VaRs which are oo conservaive and, generally, poor esimaes of he risk of he porfolio under examinaion. This may be seen as evidence ha he mos general model, he MSIAH-VECM does beer han he oher wo compeing models a maching he momens of he predicive disribuion of exchange rae changes, generaing VaRs ha are more in line wih he desired violaion rae, confirming he findings of he previous sub-secion on densiy forecasing performance and illusraing he pracical imporance of such resuls. 5. Conclusion This aricle has re-examined he performance of some empirical exchange rae models in erms of ou-of-sample forecasing of nominal exchange raes. In paricular, inspired by he success of recenly developed models of he erm srucure of forward exchange raes in erms of poin forecasing, we have carried ou a densiy forecasing analysis, applied o boh linear and regime-swiching versions of hese erm srucure models. This exercise was aided by he recen developmens of sophisicaed economeric echniques which allow us o formally evaluae he performance of ime series models in erms of densiy forecasing. Our main resul, using weekly daa for eigh US dollar exchange raes during he recen floaing exchange rae regime, is ha a Markov-swiching VECM for spo and forward exchange raes ha explicily akes ino accoun he mouning evidence ha he condiional disribuion of exchange raes is well characerized by a mixure of normal disribuions 3 In erms of inerpreaion, a significan auocorrelaion coefficien denoes a persisen series of violaions, which in urn implies unsaisfacory performance of a model in esimaing he VaR. Noe ha, while our resul of no significan serial correlaion is obvious for he random walk model and he linear VECM since hese models only imply one violaion, i is no obvious for he MSIAH-VECM. 20

22 produces very saisfacory one-week-ahead densiy forecass. This model was found o ouperform is more parsimonious linear counerpar as well as he sandard benchmark in he exchange rae forecasing lieraure, namely he random walk model. The implicaion of our findings were furher invesigaed in he conex of a simple applicaion of Value-a-Risk mehods. In our applicaion we specifically examined he implicaions of our exchange rae densiy forecass for a risk manager who has o quanify he risk associaed wih holding a currency porfolio over a one-week horizon. This applicaion furher illusraed how he Markov-swiching VECM capures saisfacorily he momens of he predicive densiy of exchange raes, generaing VaRs ha measure he probabiliy of large losses more accuraely han he oher wo compeing models. Overall, he pecking order implied by our densiy forecasing resuls - Markovswiching erm srucure model, linear erm srucure model, random walk model 4 - is he same as recorded in previous work based on convenional poin forecasing crieria. However, when evaluaing hese models in erms of densiy forecasing resuls, he superioriy of he Markov-swiching VECM relaive o he linear VECM becomes much clearer han previously recorded in he lieraure using poin forecas evaluaion. Overall, our findings highligh how beer densiy forecass of exchange raes, of he ype recorded in his paper using Markov-swiching models of he erm srucure, can poenially lead o subsanial improvemens in risk managemen and, more precisely, o beer esimaes of downside risk. 4 Specifically, his is he resul suggesed by he Berkowiz es saisic, while using he Corradi-Swanson es we were no able o discriminae beween he linear VECM and he random walk model. All ess indicaed, however, ha he bes performing model is he Markov-swiching VECM. 2