Smooth transition vector error-correction (STVEC) models: An application to real exchange rates

Transcription

1 Smooh ransiion vecor error-correcion (STVEC) models: An applicaion o real exchange raes Alenka Kavkler Faculy of Economics and Business Universiy of Maribor, Slovenia alenka.kavkler@uni-mb.si Bernhard Böhm Insiue for Mahemaical Mehods in Economics Universiy of Technology, Vienna, Ausria bernhard.boehm@uwien.ac.a Darja Boršič Faculy of Economics and Business Universiy of Maribor, Slovenia darja.borsic@uni-mb.si Absrac This paper is devoed o sudying economeric models of smooh ransiion characerized by coninuously changing parameer regimes. The process of specifying, esimaing and evaluaing smooh ransiion regression (STR) models is discussed. The firs aemps a exending nonlinear STR echniques o vecor auoregressive (VAR) models have emerged in he las few years. This paper proposes an augmened specificaion procedure for STVAR models ha allows for differen ransiion variables and differen ypes of ransiion funcions in differen equaions of he sysem. As an applicaion of he described modelling approach, a hree-variable linear vecor error-correcion model (VECM) of he componens of he real exchange rae beween Slovenia and Slovakia, namely he consumer price indices and he nominal exchange rae beween he currencies of boh counries is invesigaed. The closely relaed quesion of he validiy of purchasing power pariy (PPP) is also discussed. Keywords: smooh ransiion vecor auoregressive models, smooh ransiion vecor errorcorrecion models, real exchange rae, purchasing power pariy. This research was suppored by a gran from he CERGE-EI Foundaion under a program of he Global Developmen Nework. All opinions expressed are hose of he auhors and have no been endorsed by CERGE- EI or he GDN.

2 INTRODUCTION AND LITERATURE OVERVIEW. Abou Exchange Raes and Purchasing Power Pariy The heory of purchasing power pariy became paricularly ineresing afer he inroducion of flexible exchange rae regimes in he 970s. Since hen here is a number of heoreical as well as empirical sudies dealing wih he phenomena of purchasing power pariy. The purchasing power pariy heory suggess ha exchange rae sysem should provide a mechanism, which would enable a baske of goods being purchased in boh analyzed counries o cos he same amoun of money when recalculaed in one currency. Considering differen views on how he process of economic ransformaion since he beginning of he nineies and is effecs on reforming counries price mechanisms are compaible wih rigorous assumpions of he heory of PPP (see Brada 998), here is an obvious need for furher empirical evaluaion o supply clear-cu evidence on macroeconomic forces ha govern he exchange rae behavior in ransiion economies. Because he majoriy of ransiion counries have undergone several phases of economic resrucuring, hese mos likely also riggered shifs in heir equilibrium real exchange raes. This suggess ha, when comparing developed marke economies wih hose sill under economic reforms, he degree of a counry s similariy, especially in erms of rade paern, level of developmen and he srucure of relaive prices, could imporanly affec he assessmen of PPP. The general model of esing for PPP can be specified in he following form: s = α 0 + α p - α 2 p * + ξ, () where s sands for nominal exchange raes, defined as he price of foreign currency in he unis of domesic currency; p denoes domesic price index and p * foreign price index; while ξ sands for he error erm showing deviaions from PPP. All he variables are given in logarihmic form. In he srices version of PPP, here are he following assumpions: α 0 =0, α =α 2 =. The symmery resricion applies such ha α and α 2 are equal, whereas he limiaion of α and α 2 being equal o one is called he proporionaliy resricion (Froo and Rogoff 995).

3 According o Boršič (2003, 2005) and Boršič and Bekő (2007), we show resuls based on monhly daa series for Slovenia from January 992 December 200, when he euro was pu ino circulaion. Primary daa included monhly averages of nominal exchange raes and consumer price indices gahered from he cenral bank. The exchange rae has been defined as olar (SIT) cos of a uni of foreign currency. Consumer price indices used in his sudy for Slovenia refer o January 992. The radiional empirical analysis sars off wih he mos resricive version of equaion (), α =α 2 =, ha is, wih esing he properies of real exchange raes. In he conex of relaive PPP, he movemens in nominal exchange raes are expeced o compensae for price level shifs. Thus, real exchange raes should be consan over he long run and heir ime series should be saionary (Parikh and Wakerly 2000). Resuls of he augmened Dickey-Fuller es are given in Table. The figures show ha he four ime series of he real exchange raes he olar are inegraed of order one, which means we canno rejec he hypohesis of he presence of he uni roo. Thus, he ADF ess confirm he non-saionariy in he observed ime series. Table : Resuls of he ADF Tes for Real Exchange Raes of he Slovenian olar Variable Level Firs difference AIC -saisic AIC -saisic LRATSSIT LRDEMSIT LRFRFSIT LRITLSIT Noes: L sands for logarihm, R for real; he nex hree leers (ATS, DEM, FRF, ITL) represen he currencies of Ausria, Germany, France and Ialy, respecively, while he las wo leers (CZK, SIT) denoe he currencies of he Czech Republic and Slovenia, respecively. Criical values: (%), (5%) and (0%). The subscrips indicae he ime lag used in he es. Source: Boršič and Bekő 2007 Relaxing he proporionaliy condiion in equaion () allows us o es if nominal exchange raes and relaive prices are coinegraed. PPP holds if he presence of long-run equilibrium relaion is confirmed. In he case of searching for coinegraion among wo variables Engle- Granger es is an appropriae one.

4 Resuls of he Engle-Granger es for Slovenia are presened in Table 2, which show he esimaed equaions for individual pairs of counries. The firs column conains he independen variable. In all cases he choice of he independen variable does no influence he resuls of coinegraion ess. In he following columns, here are consans, esimaed coefficiens of independen variables, R 2, CDRW saisics and ADF saisics for residuals. There are no -saisics, since he ime series are nonsaionary and, as ess show, noncoinegraed. Consequenly, he esimaed -saisics are no reliable. Table 2 presens he resuls for Slovenia. I can be concluded ha he series are no coinegraed. In he case of Ialy CRDW saisics are above he criical value bu he ADF saisics are below he criical values. Thus, here are no signs of coinegraion among he observed variables and he validiy of purchasing power pariy canno be acceped. Table 2: Resuls of Engle-Granger es for Slovenia Independen variable Consan Coefficien R 2 CRDW ADF Ausria LATSSIT LCPISA Germany LDEMSIT LCPISN France LFRFSIT LCPISF Ialy LITLSIT LCPISIT Noes: L sands for logarihm, he nex hree leers (ATS, DEM, FRF, ITL) represen he currencies of Ausria, Germany, France and Ialy, respecively, while he las wo leers (SIT) denoe he currency of Slovenia. Criical values for Engle-Granger es: (%), -3.7 (5%) and -2.9 (0%). Criical values for CRDW es: (%), (5%) and (0%). Source: Boršič Relaxing he symmery assumpion in equaion () enables esing for coinegraion among nominal exchange raes, domesic price indices and foreign price indices. Johansen coinegraion es is appropriae for esing for coinegraion among hree variables. The resuls of he es are shown in he Table 3, which clearly shows ha here is no evidence for purchasing power pariy in he observed counries. Even when here is evidence of exising

5 coinegraion among he variables in quesion, here is a violaion of signs in he coinegraing coefficiens. In erms of equaion (), we are looking for a normalized coinegraing vecor wih posiive coefficiens α and α 2. Table 3: Resuls of he Johansen Coinegraion Tes for Slovenia Number of coinegraing equaions H 0 : H 0 : H 0 : H 0 : H 0 : H 0 : H 0 : H 0 : Ausria Germany 2 France Ialy 2 r=0 r r 2 r=0 r= r=2 r=0 r r 2 r=0 r= r=2 r=0 r r 2 r=0 r= r=2 r=0 r r 2 r=0 r= r=2 LR r LR max LR r LR max LR r LR max LR r LR max α = (0.302) -α 2 = (2.433) Slovenia Saisic,2 α = (0.3732) -α 2 = (2.7453) α = (0.046) -α 2 = α = (0.3878) -α 2 = (3.5580) α =-.967 (0.4644) -α 2 = (.823) ** * ** * ** ** *2.475 * ** * * * Noes: ** (*) denoes rejecion of he null hypohesis a he % (5%) significance level, respecively; figures in parenheses are sandard errors. Criical values for LR r a he 5% level are (r=0), 5.4 (r ), and 3.76 (r 2); and a he % level are (r=0), (r ), and 6.65 (r 2). 2 Criical values for LR max a he 5% level are (r=0), 4.07 (r=), and 3.76 (r=2); and a he % level are (r=0), 8.63 (r=), and 6.65 (r=2). Source: Boršič and Bekő 2007 The values of LR r and LR max saisics show ha here is coinegraion among he nominal exchange raes and consumer price indices in comparison o Ausria, Germany and Ialy. In

6 all hree cases he coefficiens of domesic prices are proven o be saisically significanly differen from zero, while for coefficiens of foreign prices he sandard errors are oo high o conclude he same. The signs of he esimaed coinegraing coefficiens are again no in accordance wih PPP. Only he coefficien of Ausrian consumer prices ends o have he righ sign. In case of France, here is no proof of coinegraion eiher. In addiion, he esimaed coefficiens are saisically insignifican and only he coefficien of French consumer prices has a sign corresponding o he PPP heory. The discussed resuls of he ime series uni roo ess and coinegraion ess in searching for evidence of purchasing power pariy in Slovenia and Czech Republic failed o find any suppor for he heory. The limiaions of he usage of ime series analysis are: long-run daa unavailabiliy, nonsaionariy and low power of ime series uni roo ess. Thus, he nex sep is o es for uni roos by panel daa echniques, which are supposed o have higher power ha ime series uni roo ess. Table 4 presens resuls of such ess. Despie he fac ha panel uni roo ess are supposed o have a higher power ha ime series uni roo ess, he resuls show no evidence of PPP validiy for he observed counries in he seleced period. Table 4: Summary of Panel Uni Roo Tess for Slovenia Mehod Saisic Prob. Null: uni roo (assumes common uni roo process) Levin, Lin, Chu Breiung Null: uni roo (assumes individual uni roo process) Im, Pesaran, Shin ADF-Fisher PP-Fisher Null: no uni roo (assumes common uni roo process) Hadri Nonlineariies in Real Exchange Raes: Overview of Recen Lieraure Since he real exchange rae in logarihmic form may be viewed as a measure of he deviaion from PPP, he quesion of mean reversion in he real exchange rae is closely relaed o he issue of validiy of purchasing power pariy. In order o circumven he low power problem of convenional uni roo ess, he validiy of PPP is usually invesigaed hrough long-span

7 sudies or panel uni roo sudies. Sarno and Taylor (2002) poin ou he disadvanages of boh of he menioned approaches. As far as he long-span sudies are concerned, he long samples required o generae a reasonable level of power wih univariae uni roo ess may be unavailable for many currencies. Panel sudies, on he oher hand, impose he null hypohesis ha all of he series under observaion are generaed by uni roo processes implying ha he probabiliy of rejecion of he null hypohesis may be quie high when as few as jus one of he series is saionary. For his reason, Sarno and Taylor develop a smooh ransiion auoregressive (STAR) model o sudy he behaviour of he real exchange rae. In heir model, he real exchange rae in he logarihmic form is explained by is lagged values. I is shown ha he four major real dollar exchange raes are becoming increasingly mean revering wih he absolue size of he deviaion from equilibrium, which is consisen wih he recen heoreical lieraure on he naure of he real exchange rae dynamics in he presence of he inernaional arbirage coss. Tradiional empirical analyses of purchasing power pariy validiy and is deviaions are based on linear framework and mosly sugges ha he long run equilibrium is consan. Moreover, hese analyses sugges ha real exchange rae dynamics should be explained by linear auoregressive process wih coninuous and consan speed of adjusmen, no aking ino accoun he size of deviaions from purchasing power pariy (Sarno and Taylor 2002). Using linear framework for a nonlinear daase, he rejecion of a uni roo as a null hypohesis is more likely (Taylor 2006), while he assumpion of consan speed of adjusmen implies downward bias of he resuls. Taylor (2006) presens hree poenial reasons of nonlineariies in real exchange raes: - fricions due o ranspor coss, ariffs or non-ariff barriers; - ineracion of heerogeneous agens in he foreign exchange marke a he microsrucural level; - influence of official inervenion in he foreign exchange marke. Sarno and Taylor (2002), Sarno (2003) and Taylor (2006) provide an overview of nonlinear exchange rae models and assess heir conribuion o explaining he behaviour of he exchange raes. Below we presen some sudies of real exchange rae dynamics providing evidence in suppor of he nonlineariy in exchange rae processes.

8 Taylor, Peel, and Sarno (200) ake ino accoun nonlinearly mean revering models of real exchange raes and ransacion coss in inernaional arbirage. By means of Mone Carlo simulaions, he auhors show ha in his case he half lives of real exchange raes imply he fases adjusmen process in comparison o oher echniques. Baum, Barkoulas and Caglayan. (200) apply ESTAR models o deviaions from purchasing power pariy obained by Johansen coinegraion mehod. They find evidence ha mean revering process of deviaions varies nonlinearly wih he size of disequilibrium. On he basis of eleven Asian counries Liew, Chong and Lim (2003) show ha he behaviour of real exchange raes is beer presened by nonlinear STAR models han by linear auoregressive models. Guerra (2003) ess nonlinear adjusmen owards purchasing power pariy by esimaing an ESTAR model for Swiss frank-german mark rae in he period of The resuls imply ha mean reversion is rapid for he whole period as well as for he pos-breon-woods period. Paya, Veneis, and Peel (2003) ake ino consideraion wo differen approaches in solving he purchasing power pariy puzzles: nonlinear adjusmen of real exchange raes induced by ransacion coss and non-consan real exchange rae equilibrium induced by differen produciviy growh raes. Consequenly, he real exchange rae can be described as symmeric, nonlinear dynamics. Addiionally, he auhors show ha he esimaed half-lives of he shocks are much shorer han hose obained by linear models. ESTAR models have also been used o forecas he behaviour of real exchange raes. Kilian and Taylor (2003) find evidence of exchange rae predicabiliy in 2 o 3 years given ESTAR real exchange rae dynamics. Liew, Bahrumshah and Lim (2004) find suppor of ESTAR nonlinear mean revering adjusmen process of nominal Singapure dollar-us dollar rae owards relaive consumer prices. Due o he lack of correc size of saionariy for PPP wihin linear ess, Paya and Peel (2005) employ Mone Carlo experimens o show ha nonlinear ess provide suppor for PPP. They also apply ESTAR models o daa from high inflaion counries and provide furher evidence in suppor of PPP.

9 Peel and Veneis (2005) presen some heoreical limiaions of ESTAR models and propose a new linear model consisen wih raional expecaions, while ESTAR model assumes adapive expecaions. Auhors show fas adjusmen speeds implied by heir model using pos-973 monhly real exchange rae daa. Sollis (2005) uses univariae smooh ransiion models o es for uni roos under he alernaive hypohesis of saionariy around a gradually changing deerminisic rend funcion. They rejec he null hypohesis of a uni roo for real exchange raes of some counries in comparison o he US dollar. Leon and Najarian (2005) check he saionariy of PPP deviaions in he presence of nonlineariy and symmery of adjusmen owards PPP from above and below. Alernaive nonlinear models including STAR models provide evidence of mean reversion and asymmeric adjusmen dynamics. One of he relaively rare papers examining purchasing power pariy deviaions in Cenral European counries is Arghyrou, Boine, and Marin (2005). The auhors analyze he daa from Czech Republic, Hungary, Poland, Slovakia and Slovenia. Among oher resuls i is shown ha he shor run dynamics of he real exchange raes displays nonlinear and asymmeric behaviour while he speed of adjusmen depends on he size and sign of he deviaion. Lahinen (2006) uses a STAR model on he basis of US dollar-euro exchange rae. The auhor disinguishes beween he sudden and smooh adjusmen o he long-run equilibrium and argues ha he adjusmen for he daa under observaion is sudden. Rapach and Wohar (2006) sudy he performance of nonlinear models of he US Dollar real exchange rae monhly daa in he pos-breon Woods period and poin ou ha nonlinear models (ESTAR and hreshold models) provide more accurae poin forecass a long horizons for some counries. 2 SMOOTH TRANSITION REGRESSION Many elemens of economic heory menion he idea ha he economy behaves differenly if values of cerain variables lie in one region raher han in anoher, or, in oher words, follow

10 differen regimes. The firs aemp a modelling such phenomena is represened by discree swiching models, where a finie number of differen regimes is assumed. The cenral ool of his class of models is he so-called swiching variable ha can be eiher observable or unobservable. As smooh ransiion beween regimes is ofen more convenien and realisic han jus he sudden swiches, several scieniss proposed a generalizaion of discree swiching models of he following form: ( ) (, ; ) y = x ϕ+ x θ G γ c s + u, =, 2,, T, (2) where ϕ = ( ϕ0, ϕ,, ϕ p ) and θ = ( θ0, θ,, θ p ) are he parameer vecors, x is he vecor of explanaory variables conaining lags of he endogenous variable and he exogenous variables, (i.e. x = (, x,, x ) p = (, y,, y m, z,, zn) ), whereas u denoes a sequence of independen idenically disribued errors. G sands for a coninuous ransiion funcion usually bounded beween 0 and. Because of his propery no only he wo exreme saes can be explained by he model, bu also a coninuum of saes ha lie beween hose wo exremes. The slope parameer γ > 0 is an indicaor of he speed of ransiion beween 0 and, whereas he hreshold parameer c poins o where he ransiion akes place. The ransiion variable s is usually one of he explanaory variables or he ime rend. The mos popular funcional forms of he ransiion funcion are as follows: LSTR Model: G ( γ, c; s ) = ( s c, + ) e γ LSTR2 Model: G (, c, c ; s ) s c s c 2 γ 2 = ( )( 2) + e γ This is a non-monoonous ransiion funcion ha is paricularly useful in case of reswiching. ESTR Model: (, ; ) s G3 γ c s = e γ 2 ( ) c The funcion is symmeric abou c and very similar o he LSTR2 case wih c = c 2. Therefore i is someimes difficul o disinguish beween an ESTR and an LSTR2 model.

11 2. Tesing Lineariy agains STR Le us sar by defining a more convenien noaion: G * i = G 0.5 for i =, 2 and i G = G. * 3 3 Obviously, G = 0 for γ = 0. The null hypohesis of lineariy for model (2) can be expressed * i as H : γ = 0 agains H : γ > 0 or as H : θ = 0 agains H : θ This indicaes an idenificaion problem, since he model is idenified under he alernaive bu no idenified under he null hypohesis. Namely, he parameers c and θ are nuisance parameers ha are no presen in he model under H 0 and whose values do no affec he value of he log - likelihood. Consequenly, he likelihood raio es, he Lagrange muliplier and he Wald es do no have heir sandard asympoic disribuions under he null hypohesis and one canno use hese ess for a consisen esimaion of he parameers c and θ. To overcome his problem, Luukkonen, Saikkonen and Teräsvira (998) replaced he ransiion funcion wih is Taylor approximaion of a suiable order. Le us wrie he firs order Taylor approximaion around γ = 0 for he logisic ransiion funcion s : * G as a polynomial in he ransiion variable T = a (, ; ) 0 + as + R γ c s. (3) Afer replacing * G by T in equaion (2), one obains y = xb + ( xs ) b + u, (4) * 0 where b 0 and b are (p+)-dimensional column vecors of parameers. The null hypohesis of lineariy can be esed as H 0 : b = 0 agains H : b 0 wih a sraighforward Lagrange muliplier es. The es saisic is asympoically χ 2 -disribued wih p+ degrees of freedom. We have o emphasize ha auxiliary regression (4) is suiable only if he ransiion variable s is no an elemen of he vecor x. Oherwise, he variable s appears wice on he righ-hand side of equaion (4). The problem is solved by subsiuing x% = ( x,, x ) in he second erm of (4). p x wih

12 To avoid dealing wih low power in some special cases, he hird order Taylor polynomial is applied. This leads o he following auxiliary regression: y = xb + ( xs ) b + ( xs ) b + ( xs ) b + u. (5) 2 3 * Under he null hypohesis of lineariy, he parameer vecors b, b 2 and b 3 are joinly esed o zero. F-version of he lineariy es is usually preferred because of is beer small sample properies. Comprehensive discussion on hese issues is given in Teräsvira (998) and in Luukkonen, Saikkonen and Teräsvira (998). 2.2 Model Specificaion The choice of he ransiion variable is no sraighforward, since he underlying economic heory ofen gives no clues as o which variable should be aken for he ransiion variable under he alernaive. Teräsvira (998) suggess esing he null hypohesis of lineariy for each of he possible ransiion variables in urn. The candidaes for he ransiion variable are usually he explanaory variables and he ime rend. If he null is rejeced for more han one variable, he variable wih he sronges rejecion of lineariy (i.e. wih he lowes p-value) is chosen for he ransiion variable. This inuiive and heurisic procedure can be jusified by observing ha he es is mos powerful when he alernaive hypohesis is correcly specified, and his is achieved for he "righ" ransiion variable. I has o be emphasized ha one canno conrol he overall significance level of he lineariy es for his heurisic procedure, since several individual ess have o be performed. If he ransiion variable has already been decided upon, he nex sep in he modeling process consiss of choosing he ransiion funcion. The decision rule is based on a sequence of nesed hypoheses ha es for he order of he polynomial in auxiliary regression (5): H04 : b 3 = 0 H : b = 0 b = 0 (6) H : b = 0 b = b = The 3 hypoheses are esed wih a sequence of F-ess named F4, F3 and F2, respecively. If he rejecion of he hypohesis H 03 is he sronges, Teräsvira (998) advises choosing he

13 LSTR2 or he ESTR model. In he pracice, one usually chooses he LSTR2 model and addiionally ess he hypohesis c = c 2 afer esimaion. If i canno be rejeced, i seems beer o selec he LSTR2 model, oherwise ESTR should be seleced. In case of he sronges rejecion of he hypoheses H 04 or H 02, LSTR is chosen as he appropriae model. This heurisic decision rule is based on expressing he parameer vecors b, b 2 and b 3 from auxiliary regression (5) as funcions of he parameers γ, c (or c and c 2 ) and θ and he firs hree parial derivaives of he ransiion funcion * G i a he poin γ = 0. Teräsvira (998) conduced a series of simulaion experimens o invesigae he properies of he proposed heurisic specificaion sraegy for choosing he ransiion variable and he ransiion funcion. The sudy was conduced for smooh ransiion auoregressive (STAR) models in he univariae seing. Differen ypes of STAR models were examined and heir parameers were varied. The "rue" ransiion variable was he lagged endogenous variable y d, where he delay parameer d ran from o 5. For each d he lineariy es was performed for every possible ransiion variable in urn (i.e. for y, y 2, K, y 5) and he variable wih he lowes p-value was chosen. The empirical size of he overall lineariy es was 3 o 4 % when he nominal size was 5 %. The resuls of he simulaion sudy jusified he heurisic specificaion procedure and also showed ha he power of he lineariy es is beer for higher γ values and for lower values of he delay parameer d. The decision rule for choosing he ype of he ransiion funcion was esed for disinguishing beween LSTAR and ESTAR models. I works bes when he number of observaions of he ransiion variable ha lie below c is abou he same as he number of observaions above c. The performance of he rule improves wih he sample size. 2.3 Esimaion of STR models The specified STR model is usually esimaed wih nonlinear leas squares or wih maximum likelihood esimaion under he assumpion of normally disribued errors. Boh mehods are equivalen in his case. Nonlinear opimizaion procedures are used o maximize he loglikelihood or o minimize he sum of squared residuals. The applied STR models from he nex secions are esimaed wih he Gauss package or wih EViews. Several nonlinear opimizaion algorihms are available in Gauss. For example, he Newon - Raphson

14 algorihm, he Broyden - Flecher - Goldfarb Shanno (BFGS) algorihm, he seepes descen algorihm and he Davidon - Flecher - Powell (DFP) algorihm are all implemened in he Gauss library Opmum. An addiional remark should be made on he slope parameer γ of he ransiion funcion. The magniude of he parameer γ depends on he magniude of he ransiion variable s and is herefore no scale-free. The numerical opimizaion is more sable if he exponen of he ransiion funcion is sandardized prior o opimizaion. In oher words, i is advisable o divide γ by he sample sandard deviaion (in case of LSTR models) or by he sample variance (for ESTR and LSTR2 models) of he ransiion variable. In his way he magniude of he slope parameer is brough closer o he magniude of oher parameers. 2.4 Misspecificaion Tess The misspecificaion ess were firs developed by Eirheim and Teräsvira (996) for univariae ime series, i.e. for smooh ransiion auoregressive (STAR) models, bu he generalizaion o STR models is sraighforward. Three ess have o be developed especially for he STAR models, namely he es of no remaining nonlineariy, he es of no error auocorrelaion and he parameer consancy es. For a deailed derivaion of hese ess, see Eirheim and Teräsvira (996) and Lin and Teräsvira (994). Oher ess, like he LM es of no auoregressive condiional heeroscedasiciy of Engle and of McLeod and Li, and he Lomnicki-Jarque-Berra es of he normal disribuion of errors are performed in he same way as in he linear seing. 3 SYSTEMS OF EQUATIONS From recen sudies of univariae models one has learned ha here is much o be gained by allowing a nonlinear specificaion. Represenaions of asymmeric reacions, srucural changes and oher phenomena of economic developmen can be fruifully invesigaed by nonlinear modeling echniques. As many issues in economics require he specificaion of several relaionships, echniques o handle nonlinear feaures in sysems are required. Only

15 during he recen years such mehods have appeared in he lieraure. Mos of he work has been done in he nonlinear VAR framework. Anderson and Vahid (998) devised a procedure for deecing common nonlinear componens in a mulivariae sysem of variables. The common non-lineariies approach is based on he canonical correlaions echnique and can help us inerpre he relaionships beween differen economic variables. The specificaion and esimaion of he sysem of equaions is also simplified, since he exisence of common nonlineariies reduces he dimension of nonlinear componens in he sysem and enables parsimony. This is paricularly imporan in empirical invesigaions involving economic ime series of shorer lengh. Namely, mos of he macroeconomic indicaors are published on a quarerly basis. Weise (999), van Dijk (200) and Camacho (2004) exended he STR modeling approach developed by Teräsvira and coworkers o vecor auoregressive models of smooh ransiion. Their STR specificaion is limied o he case where he ransiion beween differen parameer regimes is governed by he same ransiion variable and he same ype of ransiion funcion in every equaion of he sysem. They argue ha since he economic pracice imposes common nonlinear feaures, all equaions share he same swiching regime. Bu his argumen is no convincing, since such a conclusion canno be derived from economic heory, while applied economeric sudies analyzing nonlinear sysems are scarce. For his reason we shall ry o exend heir approach by allowing differen smooh ransiion funcional forms in differen equaions. The proposed augmened specificaion procedure is explained in secion Smooh Transiion Approach o Vecor Auoregressive Models Whereas here has been exensive research in he field of univariae nonlinear modeling, he saisical heory of mulivariae nonlinear modeling has ye o be developed. The firs aemps a exending nonlinear smooh ransiion regression echniques o a mulivariae seing can be found in Weise (999), van Dijk (200) and Camacho (2004). Similarly, mulivariae Markov swiching models are reaed in Krolzig (997) and mulivariae hreshold models in Tsay (998). Van Dijk (200) applies he STVAR modeling approach o sudy he inraday spos and fuures prices of he FTSE00 index, whereas Camacho (2004) examines he nonlinear forecasing power of he composie index of leading indicaors o predic boh oupu growh and he business - cycle phases of he US economy. Since all hree

16 sudies are similar, while he mos comprehensive descripion of he mehodological approach is given by Camacho (2004), we shall sar wih a shor review of his work. 3.. Specificaion and Esimaion Camacho (2004) considers a 2 - dimensional smooh ransiion vecor auoregressive (STVAR) model ( θ ) y = ϕ X + X G ( s ) + u y y y y y ( θ ) x = ϕ X + X G ( s ) + u, x x x x x (7) where X = (, y, x, K, y p, x p) = (, X% ), ϕ, ϕ, θ, θ are he corresponding parameer x y x y vecors and U = ( u, u ) : N(0, Ω) is a vecor series of serially uncorrelaed errors. The y x difference Di = si ci, i= x, y, in he exponen of he ransiion funcion G i is called he swiching expression. The leers y and x are used for he wo variables in he auoregressive sysem, since he smooh ransiion approach is applied o he rae of growh of he US GDP and he rae of growh of he US composie index of leading indicaors, respecively. The discussion is resriced o he case of s x = s y = s and Gx = Gy, where he same ransiion variable and he same ransiion funcion is used in boh equaions. Afer he linear VAR has been specified, he lineariy es is applied. The problems wih nuisance parameers are solved wih suiable Taylor series expansions, as usually. The auxiliary regression o be performed in case he ransiion variable s belongs o X is 3 h = η y0 + η % yh + y h= y X X s v 3 h = η x0 + η % xh + x h= x X X s v (8) and he null hypohesis of lineariy reads as H : η = η = η = 0, i = x, y. (9) 0 i i2 i3 Consequenly, he null hypohesis can be esed wih he Lagrange muliplier es.

17 If he null hypohesis of lineariy is rejeced in favor of he alernaive smooh ransiion vecor auoregressive model, one has o decide which ransiion funcion o use. The decision is based on he sequence of nesed hypoheses ess described in secion 2.2. The parameers of he specified model are esimaed wih he maximum likelihood esimaor under he assumpion of normally disribued errors: U = ( u, u ) : N(0, Ω). (0) y x 3..2 Tesing he Model Adequacy As proposed by Eirheim and Teräsvira (994), hree ess are performed in order o check for he adequacy of he esimaed model, namely he es of no error auocorrelaion, he es of no remaining nonlineariy and he parameer consancy es. The mulivariae generalizaions of he hree ess were developed by Camacho (2004). 3.2 Smooh Transiion Vecor Auoregressive Models wih Differen Transiion Variables and Transiion Funcions As already menioned, Weise (999), van Dijk (200) and Camacho (2004) all assume he same ransiion variable and he same ype of he ransiion funcion in every equaion of a smooh ransiion vecor auoregressive model, wih he inerpreaion ha he economic pracice imposes common nonlinear feaures. Bu his argumen is no convincing, since such a conclusion canno be derived from economic heory, while applied economeric sudies analyzing nonlinear sysems are scarce. For his reason we shall ry o exend he work of Camacho by allowing differen smooh ransiion funcional forms in differen equaions. When performing he sysem lineariy es Camacho posulaes a wo-variable linear VAR(p) model under he null hypohesis and he smooh ransiion vecor auoregressive model (7) wih he same ransiion variable s x = s y = s and he same ype of ransiion funcion G = G = G under he alernaive. Afer esimaing he auxiliary regression x y

18 ( % ) ( % 2) ( % 3) ( % ) 2 ( % ) 3 ( % ) y = X η + Xs η + Xs η + Xs η + v y0 y y2 y3 y x X η X s η X s η X s η v = x0 + x+ x2 + x3 + x, () he null hypohesis H : η = η = η = 0, i = x, y (2) 0 i i2 i3 (wih he alernaive of a leas one of he coefficien vecors differen from zero) is esed. The equaions in () are esimaed as a sysem wih he mehod of maximum likelihood. Single equaion esimaors would also be consisen, alhough no efficien. Null hypohesis (2) can be esed wih he LM es. If he resricion s x = s y = s is no imposed, he sysem lineariy es can be performed by esing he same null hypohesis, his ime based on he auxiliary regression allowing for differen ransiion variables: ( % ) ( % 2 ) ( % 3 ) ( % 2 3 ) ( % ) ( % ) y = X η + Xs η + Xs η + Xs η + v y0 y y y y2 y y3 y x X η X s η X s η X s η v = x0 + x x+ x x2 + x x3 + x. (3) The sysem lineariy es will be rejeced if a leas one of he relaionships under observaion is nonlinear, or more specifically, is characerized by smooh ransiion beween parameer regimes. I is reasonable o believe ha siuaions wih only one of he equaions being nonlinear can occur in he economic pracice. Esimaing boh equaions wih he smooh ransiion specificaion would be inefficien in his case. To solve his problem, single equaion lineariy ess based on he sysem esimaes of auxiliary regression (3) may be applied. For example, o develop he single equaion lineariy es for he firs equaion, he null hypohesis H : η = η = η = 0, (4) 0 y y2 y3 should be verified. Tesing such a null hypohesis corresponds o imposing he model y = ϕ X + u y y ( θ ) ( ) x = ϕ X + X G s + u x x x x x (5)

19 under he null and model (7) wih he STR specificaion in boh equaions under he alernaive. The heurisic procedure for selecing he ransiion variable(s) can be derived similarly o he one explained by Teräsvira (998) in he univariae seing: Perform he sysem lineariy es for each of he pairs of he possible ransiion variables in urn.. Carry ou single equaion lineariy ess for each of he pairs of ransiion variables ha rejec he sysem lineariy es. 2. (i) If here are pairs of ransiion variables for which he single equaion ess rejec he null hypohesis of lineariy for boh equaions, choose he pair wih he sronges rejecion of he sysem lineariy es. (ii) If for each of he pairs rejecing he null of sysem lineariy only one of he single equaion lineariy ess is rejeced, choose he pair of ransiion variables wih he sronges rejecion of he single equaion es. Specify he corresponding equaion as a smooh ransiion regression, while specifying he oher equaion as linear. Afer he ransiion variables (or variable) have been chosen, he decision abou he ype of he ransiion funcion should be made. Camacho proposes a sraighforward generalizaion of he sequence of nesed hypoheses from secion 2.2, namely H H H : η = η = 0, 04 y3 x3 : η = η = 0 η = η = 0, 03 y2 x2 y3 x3 : η = η = 0 η = η = η = η = 0, 03 y x y2 x2 y3 x3 (6) which can be esed wih a sequence of F-ess. The coefficien vecors η ij refer o auxiliary regression (). The decision rule deermines he ype of he ransiion funcion depending on he nesed hypohesis wih he sronges rejecion (see secion 2.2 for deails). Noe ha he same funcional form is seleced for boh equaions. If he assumpion of he same ype of he ransiion funcion in boh equaions, namely G = G = G, is also relaxed, he ransiion funcion is chosen for each equaion separaely. In x y case of he firs equaion, he regressions in (3) are esimaed wih a sysem esimaion mehod and he following sequence of nesed hypoheses is verified:

20 H H H : η = 0, 04 y3 : η = 0 η = 0, 03 y2 y3 : η = 0 η = η = y y2 y3 (7) When he second equaion has a linear specificaion, auxiliary regression (3) should be modified accordingly, wih he coefficien vecors η x, η x2 and η x3 se o zero prior o esimaion. An alernaive specificaion procedure would follow he suggesion of Camacho and specify a model for each pair of he ransiion variables rejecing he null hypohesis of sysem lineariy. The final model can be seleced on he basis of forecasing power or any oher measure of he model adequacy. Of course his sraegy is more ime consuming, especially if he null hypohesis of sysem lineariy is rejeced for several pairs of ransiion variables. Our approach exends he modeling cycle developed by Camacho (2004), since he se of models considered for smooh ransiion specificaion includes all of he models sudied by Camacho. The model adequacy ess for he proposed augmened specificaion procedure, namely he parameer consancy es, he es of no error auocorrelaion and he es of no remaining nonlineariy are sraighforward generalizaions of he ess developed by Camacho (2004). To illusrae he proposed heurisic procedure, we shall apply i o he daa analyzed by Camacho and obained from his web sie ( The y and x ime series denoing he rae of growh of he US GDP and he rae of growh of he US composie index of leading indicaors, respecively, include quarerly observaions from 959: o 2002:. Le us firs derive he final model of Camacho. Using he informaion crieria, a linear VAR() model is specified and subjeced o he sysem lineariy ess. In addiion o y and x, he variables 2 y and x 2 are also regarded as candidaes for he ransiion variable. The resuls are given in he corresponding rows of Table 5. As he null of sysem lineariy is rejeced in all four cases, Camacho esimaes four smooh ransiion auoregressive models. The ype of he ransiion funcions for each of he ransiion variables is seleced using he decision rule explained on page 9 and is he same for boh equaions. The final model is decided upon on he basis of predicive accuracy (see Camacho (2004) for he descripion of

21 he employed measures of predicive accuracy). The maximum likelihood esimaes of he final model given below are slighly differen from hose in he paper by Camacho: y = x ( y 0.6) 2 + e (8) (0.092) (0.200) (.359) (0.393) x = y x + ( y ) 4.43( ) e y + (0.26) (0.54) (0.060) (0.252) (0.23) (0.00) (0.343) Sandard errors of he parameer esimaes are given in brackes. As poined ou by Teräsvira (998), precise join esimaion of he slope parameer and he hreshold can be problemaic, since i requires a high number of observaions in he close neighborhood of he hreshold parameer c. When he resricions regarding he ransiion variable and he ype of he ransiion funcion are omied, he lineariy ess have o be carried ou for every pair of he possible ransiion variables. The resuls of he sysem lineariy ess as well as single equaion ess are given in Table 5. I can be observed ha he single equaion lineariy es is srongly rejeced for every pair of he ransiion variables when esing he second equaion, while his holds rue only for he pair x 2, y in case of he firs equaion. Thus he rejecion of sysem lineariy is due mainly o he nonlinear feaures in he second relaion. Following he previously explained heurisic procedure, we choose he ransiion variables x 2 and y for he firs and second equaion, respecively. Nex, he quesion of he ype of he ransiion funcion in each of he equaions has o be invesigaed. The sequence of nesed hypoheses (7) is performed wih he resuls given in Table 6. The F2 es yields he lowes p-value for boh equaions hus indicaing he LSTR ransiion funcion in boh cases.

22 Table 5: Lineariy Tes Resuls (p-values) Tvar in. LINEARITY TESTS (p-values) and 2. eq. Sysem es Firs eq. es Second eq. es x, x x, x x, y x, y x 2, x x 2, x x 2, y x 2, y y, x y, x y, y y, y y 2, x y 2, x y 2, y y, y Table 6: Tess for Choosing he Type of Transiion Funcion (p-values) NESTED TESTS (p-values) Tvar in. Firs equaion Second equaion and 2. eq. F4 F3 F2 F4 F3 F2 x, y The maximum likelihood esimaes of he specified smooh ransiion vecor auoregressive model are given by y = x ( x 0.053) 2 + e (9) (0.03) (0.37) (0.900) (0.523) x = x.680 y + (0.89) (0.059) (0.9) y 2.592( 0.64) e y + + (0.580) (3.288) (0.994)

23 I can be deduced from Table 7 ha models (8) and (9) are comparable in erms of fi, if model (9) is no even slighly beer. Therefore i would be wise o consider also (9) when searching for he model wih he bes forecasing properies. Table 7: Comparing he Fi of Models (8) and (9) Model Firs equaion Second equaion Value logl R 2 S.E. R 2 S.E. Model (8) Model (9) The modeling procedure proposed by Camacho has several drawbacks. Firsly, he sysem lineariy es based on auxiliary regression () is rejeced if a leas one of he equaions includes nonlinear erms. Specifying every equaion as nonlinear based only on he rejecion of he sysem lineariy es hus neglecs he possibiliy of a sysem involving linear and nonlinear equaions and can yield inefficien esimaes. Secondly, he limiaions in he specified funcional form can be jusified neiher by he relaionships posulaed wihin he economic heory nor by aking ino accoun he very few exising applied sudies. Thus, he specificaion wih he same ransiion variable and he same ype of he ransiion funcion in every equaion of a smooh ransiion vecor auoregressive model is oo resricive and should no be imposed a priory. 4. Empirical analysis In his secion, a hree-variable smooh ransiion vecor auoregressive model of he consumer price index for Slovenia, consumer price index of Slovakia and he nominal exchange rae beween he currencies of boh counries is discussed. The invesigaion applies he proposed augmened specificaion procedure o a small model of he real exchange rae, decomposed ino is hree componens, domesic prices (P ), foreign prices (P * ) and he nominal exchange rae (S ). Monhly daa for he period from January 992 ill May 2006 were obained from he Vienna Insiue for Inernaional Economics Sudies (WIIW), he adminisraor of he Monhly Daabase on Cenral and Easern Europe. Due o he fac ha Slovenia declared independence

24 in June 99 and inroduced is own currency (olar) in Ocober of he same year, only he daa for he period from January 993, when Tolar was already an esablished currency, were used in he sudy. Figure : Consumer Price Indices in Slovenia and Slovakia and Tolar/Koruna Nominal Exchange Rae 320 CPI_SLO 360 CPI_SK SIT_SKK Noes: CPI_SLO sands for consumer price index in Slovenia, CPI_SK for consumer price index in Slovakia and SIT_SKK for nominal exchange rae of Slovenian olar in Slovak koruna. Source of daa: WIIW Monhly Daabase on Cenral and Easern Europe. Figure presens he graphs of consumer price indices in Slovenia and Slovakia and olar nominal exchange rae, defined as a price of Slovak koruna in Slovenian olar. I can be seen ha he developmens of consumer prices in he wo observed economies are raher similar, alhough here are a few pariculariies. In he firs years of he observed period inflaion in Slovenia was higher han in Slovakia. Slovenian consumer prices show similar seady increases hroughou he observed period wih an excepion of las wo years when he increases were lower. This is due o he fac ha Slovenia enered ERMII in June 2004 and

25 has kep inflaion low since hen in order o fulfil he Maasrich crieria. In Slovakia seady increases in consumer price index fade away in 998. Since hen he consumer prices evidenced much bigger flucuaion. The hird par of Figure illusraes he movemens in olar/koruna nominal exchange rae. I can be clearly seen ha olar experienced nominal depreciaion agains koruna. Falling from 3.43 o 6.38 olars for koruna in he whole observed period resuls in 86% of nominal depreciaion. (Taking ino accoun he inflaion in he period in boh economies he real depreciaion amouns o 04%.) The seady depreciaion of olar agains koruna was inerruped in 998, when koruna depreciaed. Bu he olar appreciaion agains koruna was shor-lived since Bank of Slovenia coninued wih he policy of seady depreciaion hroughou he exisence of olar while Slovak Naional Bank reaced o depreciaion of koruna in 998 wih a change in exchange rae regime by inroducing managed floa (Amerini 2003) and kep he koruna raher sable hereafer. The economeric model employs variables expressed in growh raes wih he help of he logarihmic ransformaion. Therefore small leers are used o denoe he ransformed variables, where s sands for he logarihm of nominal exchange rae of olar, p for logarihm of consumer price index in Slovenia and p * represens logarihm of consumer price index in Slovakia. 4.. Does he purchasing power pariy hold? We sar our empirical invesigaion by esing for he purchasing power pariy beween Slovenia and Slovakia. The resuls of he augmened Dickey Fuller (ADF) uni roo ess for each of he hree variables and heir firs differences are given in Table 8. The null hypohesis of uni roo is rejeced (a he 5 % level) only for he differenced variables, herefore s, p and * p are all inegraed of order. Table 8: ADF Uni Roo Tes Resuls for Slovenia and Slovakia Variable Tes saisic (p-value) Variable Tes saisic (p-value) s (0.067) s (0.0000) p (0.2437) p * p (0.599) p (0.0000) * (0.0000)

26 Boh he race saisic and he max-eigenvalue saisic of he Johansen coinegraion es indicae one coinegraing equaion a he 5 % level. The normalized coinegraion coefficiens yield he following coinegraing equaion: s p p ξ * = (.464) (.4693) (20) The sandard errors of he coefficien esimaes are given in brackes. Noe ha he parameers α and α 2 (as denoed in equaion ()) are significanly differen from zero and he signs are in accordance wih purchasing power pariy. The main advanage of he Johansen coinegraion es over he Engle-Granger es when checking he purchasing power pariy lies in he possibiliy o es linear resricions imposed on he parameers of he coinegraing vecors. We shall es boh he symmery and he proporionaliy condiion wih he help of he likelihood raio es. The proporionaliy condiion α = α2 = (p = ) and he symmery condiion α = α2 canno be rejeced (p = ), hus PPP holds in is srices form. 4.2 Real exchange rae model In a preliminary specificaion, he linear vecor error-correcion model was specified. This simplifies he search for an appropriae nonlinear specificaion. As negleced auocorrelaion srucure may lead o false rejecions of he lineariy hypohesis (Teräsvira, 994), he order of auoregression was chosen on he basis of he serial correlaion ess. Thus, a VECM model wih lag and coinegraing equaion (given in Equaion (20)) was indicaed as he bes choice. The Schwarz informaion crierion also seleced VECM(), whereas according o Akaike informaion crierion 2 lags should be specified. The sysem lineariy ess and he single equaion lineariy ess are performed in he nex sep. The variables s, p, * p and ce are regarded as candidaes for he ransiion variable. Since in he augmened specificaion procedure differen ransiion variables are allowed in differen equaions of he sysem, here are 4 3 = 64 possible ransiion variable riples. The resuls of he sysem lineariy ess as well as single equaion lineariy ess are shown in Table 9. Noe ha for he hird equaion, he null hypohesis of lineariy is never rejeced. Therefore

27 we shall specify a linear relaionship when * p is he dependen variable. The sysem lineariy es and he firs equaion lineariy es are mos srongly rejeced (i.e. p-val.<0.000), if is seleced for he ransiion variable in he firs equaion. In his case here are 2 s candidaes for he ransiion variable in he second equaion, namely and ce. We have chosen ce o examine differen regimes in he of he deviaion from PPP. Table 9: Lineariy Tes Resuls (p-values) s p equaion associaed wih differen levels LINEARITY TESTS (p-values) Transiion variables Tes resuls s eq. 2nd eq. 3rd eq. Sysem s eq. 2nd eq. 3rd eq. s - s - s s - s - p s - s - p* s - s - ce s - p - s s - p - p s - p - p* s - p - ce s - p* - s s - p* - p s - p* - p* s - p* - ce s - ce s s - ce p s - ce p* s - ce ce p - s - s p - s - p p - s - p* p - s - ce p - p - s p - p - p p - p - p* p - p - ce p - p* - s p - p* - p p - p* - p* p - p* - ce p - ce s p - ce p p - ce p* p - ce ce p* - s - s p* - s - p