Department of Economics School of Social Sciences. Non-linear multivariate adjustment of the UK real exchange rate.

Transcription

1 Dearmen of Economics School of Social Sciences Non-linear mulivariae adjusmen of he UK real exchange rae By Cosas Milas Dearmen of Economics Discussion Paer Series No. 03/08

2 Non-linear mulivariae adjusmen of he UK real exchange rae Cosas Milas* Dearmen of Economics Ciy Universiy, UK Absrac Based on a mulivariae non-linear model, his aer recognises an imoran role for he real exchange rae in affecing UK labour marke condiions. he shor-run real exchange rae adjuss quickly o disequilibrium deviaions of he real exchange rae from is long-run level ouside a raher wide inerval band. When he real exchange rae is undervalued, shor-run unemloymen falls as firms resond o an imrovemen in domesic comeiiveness by increasing heir demand for labour. Furher, here is a srong resonse of shor-run unemloymen o he disequilibrium error ouside a narrow inerval band. o he exen ha he real exchange rae equaion reflecs moneary olicy consideraions, our resuls imly ha unemloymen can be argeed by economic olicy. Furhermore, if economic auhoriies wan o avoid large swings in unemloymen hen hey should be reared o inervene in exchange markes wih he aim of keeing real exchange rae movemens wihin a narrow inerval band. Keywords: Real exchange rae; raded goods; Smooh ransiion Vecor Error Correcion Model. JEL classificaion: C3; C5; C5; F30; F4 *Address for corresondence: Dr Cosas Milas, Dearmen of Economics, Ciy Universiy, Norhamon Square, London ECV OHB, UK. el: +44 ( , Fax: +44 ( c.milas@ciy.ac.uk

3 . Inroducion he use of non-linear models in exlaining economic henomena is moivaed by he idea ha he behaviour of economic variables deends on differen saes of he world or regimes ha revail a any oin in ime. Regime swiching behaviour in real exchange raes can be exlained by he exisence of a band in erms of he coss of rading goods; o he exen ha deviaions of he real exchange rae from is long-run level are small relaive o he coss of rading, hese deviaions are lef uncorreced (Dumas, 99. Over he las few years, he Smooh ransiion Auoregressive (hereafer SAR mehodology has been a oular way of inroducing regimeswiching behaviour in real exchange rae models, where he ransiion from one regime o he oher occurs in a smooh way. For insance, Baum e al. (00 and Michael e al. (997 model he real exchange rae (for a number of counries including he UK as a saionary variable and esimae is dynamics based on Exonenial SAR (ESAR models. Paya and Peel (003 esimae an ESAR model of he dollar yen real exchange rae which incororaes a deerminisic rend as a roxy for he equilibrium level. On he oher hand, Saranis (999 finds ha real exchange raes (including he UK one are non-saionary and roceeds by esimaing real exchange ESAR and Logisic SAR (LSAR models in firs differences. A common feaure of hese aers is ha hey all esimae univariae real exchange rae models. his marks a significan oin of dearure for our aer: while we follow Saranis (999 in reaing he real exchange rae as a non-saionary variable, we find ha he laer coinegraes wih real wages and he real rice of oil. his finding builds uon he heoreical model of Alogoskoufis (990 who derived a linear real exchange rae equaion based on he roducion secor of he economy (see also Chaudhuri and Daniel, 998, who esimae a more resricive model involving only he real exchange rae and he real rice of oil. Noing ha coinegraion ess erform reasonably well when he adjusmen rocess is non-linear (van Dijk and Franses 000, we hen roceed by emloying a mulivariae SAR framework o model he non-linear dynamics of he real exchange rae equaion as ar of a small sysem involving real wages, he unemloymen rae and he real rice of oil. 3 Inclusion of he unemloymen rae in our sysem SAR models were inroduced by eräsvira and Anderson (99 in order o examine non-lineariies over he business cycle, whereas heir saisical roeries were discussed in Granger and eräsvira (993 and eräsvira (994, among ohers. Emirical resuls in suor of a saionary real exchange rae are quie mixed and deend e.g. uon he samle size seleced or he definiion of he rice series used (for a recen survey see e.g. Baum e al., Mulivariae SAR models were recenly discussed in Granger and Swanson (996, Weise (999, Rohman e al. (00 and van Dijk e al. (00 among ohers.

4 could be jusified in erms of an Okun s law channel; for insance, Nakagawa (00 who looks a he relaionshi beween real exchange raes and ineres rae differenials, discusses non-linear effecs in a model where an undervalued real exchange rae raises aggregae demand for ouu relaive o is full emloymen level. Modelling our sysem in a smooh ransiion framework conrass wih he hreshold Auoregressive (AR; see e.g. ong, 990 and he Hamilon (989 Markov regime-swiching models, which assume ha he ransiion beween regimes occurs abruly raher han smoohly. On economic grounds, SAR models seem o be more aroriae han AR or Markov regimeswiching models. Modelling he real exchange rae as a funcion of real wages and he real rice of oil imlies ha real exchange rae movemens are affeced by condiions revailing in he roducion secor of he economy. In his case, a smooh ransiion raher han a shar swich beween regimes could be jusified in erms of fricions in he roduc marke due o roduc heerogeneiy, governmen imosed barriers o rade, or labour marke inflexibiliy disoring he raid adjusmen of wages. Our esimaes sugges he exisence of a long-run real exchange rae equaion, which affecs significanly he shor-run dynamics of he sysem boh in a linear and a non-linear way. In aricular, he shor-run real exchange rae adjuss quickly o disequilibrium deviaions of he real exchange rae from is long-run level ouside an inerval band, which is esimaed o be raher wide. his is no surrising as our samle eriod coincides wih floaing exchange raes being in oeraion. We also find ha when he real exchange rae is above is long-run equilibrium level (i.e. i is undervalued, shor-run unemloymen falls as firms resond o an imrovemen in domesic comeiiveness by increasing heir demand for labour. Furher, here is a srong resonse of shorrun unemloymen o he disequilibrium error ouside an inerval band, which is esimaed o be raher narrow. o he exen ha he real exchange rae equaion reflecs moneary and more generally economic olicy consideraions, our resuls imly ha unemloymen can be argeed by economic olicy. Furhermore, if economic auhoriies wan o avoid large swings in unemloymen hen hey should be reared o inervene in exchange markes wih he aim of keeing real exchange rae movemens wihin a narrow inerval band. Our resuls also sugges ha when he real exchange rae is undervalued, workers resond o an imrovemen in domesic comeiiveness by demanding and geing higher wages. Again, his effec is non-linear. herefore, our findings recognise an imoran role for he real exchange rae in affecing labour 3

5 marke condiions in he UK. he srucure of he aer is as follows. he nex secion discusses briefly he heory of lineariy esing wihin a mulivariae SAR framework. Secion 3 of he aer discusses he economeric secificaion of a real exchange rae model, whereas Secion 4 esimaes he linear and non-linear versions of he model. Secion 5 resens a discussion of our findings and secion 6 rovides some concluding remarks.. Secificaion of mulivariae smooh ransiion models Following Rohman e al. (00, we wrie a k-dimensional Smooh ransiion Vecor Error Correcion Model (SVECM as: y = µ + z + Φ j y j + Φ j x α,, j ( G( s j= j= 0 + µ + α z + Φ j y j + Φ j x 3, 4, j G( s + ε, j= j= 0 ( where y is a (k x vecor of I( endogenous variables, x is an (m x vecor of I( exogenous variables, ε ~ iid(0, Σ, αi, i =,, are (k x r marices, and z = β [y, x ] for some (q x r marix β denoe he error correcion erms, wih q = k + m. Φ,j and Φ 3,j, j =,...,, are (k x k marices. Φ,j and Φ 4,j, j = 0,...,, are (k x m marices and µ i, i =,, are ( k vecors. G(s is he ransiion funcion, assumed o be coninuous and bounded beween zero and one. he SVECM framework can be considered as a regime-swiching model which allows for wo regimes, G(s = 0 and G(s =, resecively, where he ransiion from one o he oher regime occurs in a smooh way. We focus our aenion on he quadraic logisic funcion (Jansen and eräsvira, 996: G ( s ; γ, c, c = { + ex[ ( s c ( s c / σ ( s ]}, γ > 0, γ ( where σ (s is he samle variance of s. his model assumes asymmeric adjusmen o deviaions of s from an inerval band (c, c. he arameer γ deermines he seed of he ransiion from one regime o he oher. If γ 0, he model becomes linear, whereas if γ +, 4

6 G(s is equal o for s < c and s > c, and equal o 0 when c < s < c. In his aer we assume ha he ossible candidaes for he ransiion variable s are he r coinegraing relaionshis in z - = β [y, x ]. More secifically, he nex secion esimaes one coinegraing vecor, which is idenified as a long-run real exchange rae equaion. herefore, model ( above is aricularly aracive from an economic oin of view as i imlies he exisence of an inerval band (c, c ouside which here is a srong endency for he real exchange rae o rever o is equilibrium value. A es of lineariy in model ( using he ransiion funcion (, is a es of he null hyohesis H 0 : γ = 0 agains he alernaive H : γ > 0. By aking a firs-order aylor aroximaion of G(s around γ = 0, he es can be done wihin he rearameerised model (see e.g. he discussion in Saikkonen and Luukkonen, 988: y j= 0 Β = Μ 3, j 0 x + Α j s 0 z + Α + z j= Β s 0, j + y j= j Β + 4, j j= 0 y Β j, j s x + j j= 0 + Α Β 5, j z x s j s + j= + e, Β, j y j s + (3 where e are he original errors ε lus he error arising form he aylor aroximaion. Model (3 is a linear VECM augmened by addiional cross-roduc regressors due o he aylor exansion. Here, he null hyohesis of lineariy is H 0 : A = A = B,j = B 3,j = B 4,j = B 5,j = 0, where j =,..., for he B,j and B 4,j marices and j = 0,..., for he B 3,j and B 5,j marices. For each equaion in he VECM, his is a sandard variable addiion Lagrange Mulilier (LM which follows asymoically he χ disribuion wih r + k( +m degrees of freedom. In small samles, he χ es may be heavily oversized. herefore, i may be referable o use an F version of he es. Boh he χ and F versions of he LM saisic are equaion secific ess for lineariy which are comued from an auxiliary regression of he residuals from each equaion in he linear VECM on all variables enering model (3. o es he null hyohesis of lineariy in all equaions simulaneously, we need a sysem-wide es. Following Weise (999, define Ω 0 and Ω as he esimaed variance-covariance residual marices from he linear VECM and he augmened model (3, resecively. he aroriae log-likelihood sysem-wide es saisic is given by 5

7 LR = { log Ω 0 log Ω }, where is he size of he samle. Under he null hyohesis of lineariy, he es follows asymoically he χ disribuion wih rk + k ( +km degrees of freedom. he equaion secific LM ess and he sysem wide LR es are run for all ossible s candidaes, ha is, all coinegraing relaionshis. he decision rule is o selec as he aroriae ransiion variable he coinegraing relaionshi for which he -value of he es saisic is he smalles one Economeric secificaion of a real exchange rae model he heoreical framework discussed above will now be esed on a small model of he UK real exchange rae. In an earlier aer, Alogoskoufis (990 inroduces a model wih raded ( and non-raded (N goods. 5 he model assumes erfec comeiion in he secor wih firms roducing according o a wo-level CES roducion funcion which is searable ino caial, labour and imored oil. For he N secor, he model assumes rofi maximising monoolisic comeiive firms. he relaive rice of radables o he rice of domesic ouu, is derived as: π = ( τ( w + τ ( O, (4 π where π (0 <π < is he share of value added in gross ouu, τ is he share of radables in oal ouu, (w refers o real roduc wages in he radables secor and ( O is he relaive rice of imored oil. All variables are in logs. Assuming ha he UK is a small oen economy, he rice of domesic radables can be roxied by = * + e, where * is he rice index of UK imors in $, and e is he average /$ exchange rae. In his case, equaion (4 is a measure of he real exchange rae as a negaive funcion of w and a osiive funcion of O. An increase in is equivalen o a real dereciaion or an imrovemen in he real comeiiveness of he domesic economy. Following he noaion in Secion of he aer, our model uses a se of k = 3 endogenous 4 An exension of he Saikkonen and Luukkonen (988 lineariy ess involves a second-order aylor aroximaion of he ransiion funcion as suggesed by Escribano and Jordá (999. his involves adding cubic and fourh ower erms in model (3, which is hardly racical o imlemen since we are faced wih a small samle size. Furher, as van Dijk e al. (00 oin ou, neiher one of he ess in Saikkonen and Luukkonen (988 or Escribano and Jordá (999 dominaes in erms of ower. Given ha he ess are no exac bu aroximaions, some cauion is needed when using he rule of he minimum robabiliy value in order o deermine he aroriae ransiion variable. 5 For oher versions of rice models wih raded and non-raded goods see Marin (997. 6

8 variables: y = [, w, u], (5 condiioning on x = O, ha is, m = exogenous variable. We use quarerly seasonally adjused UK daa over he eriod 973(-000(. he endogenous variables refer o he real exchange rae,, real roduc wages, w, in he manufacuring secor (as a roxy for radables, and he unemloymen rae, u (all variables are in logs; for more deails see he Daa aendix. 6 Wihin our mulivariae sysem, labour marke argumens sugges ha real wages inerac wih he unemloymen rae. Furher, Nakagawa (00 discusses non-linear effecs in a model where an undervalued real exchange rae raises aggregae demand for ouu relaive o is full emloymen level. Hence here should be a negaive correlaion beween he real exchange rae misalignmen and unemloymen hrough an Okun s law channel; when he real exchange rae is undervalued, firms resond o an imrovemen in domesic comeiiveness which induces shifs in aggregae demand by increasing heir demand for labour. As a resul, unemloymen falls. In addiion, firms are assumed o ake he real rice of oil, O, as given and herefore we imose exogeneiy of his variable, which may imrove he saisical roeries of he sysem (see he discussion in Hansen and Juselius, Emirical resuls 4. Long-run behaviour Figure los he logs of he levels and he firs differences of he, w, O and u series. Preliminary analysis using he ADF uni roo ess suggesed ha all series are I( in levels and I(0 in firs differences. o es for coinegraion, we esimae he linear VECM in levels using a lag lengh of = 4 and allowing for a drif arameer o ener he VECM unresricedly. 8 able reors he eigenvalues, λ i, and he λ-max and race saisic ess for 6 As discussed in he Aendix, we use wage coss raher han wage coss er uni of ouu in our emirical resuls. ha said, use of uni wage coss does no make any qualiaive difference o he resuls reored below. 7 Obviously, some oher variables like roduciviy or ax raes can affec wages or unemloymen. Exending he informaion se o his direcion is no ursued here, as we are rimarily ineresed in discussing he non-linear behaviour of he real exchange rae equaion. Furhermore, use of a larger informaion se is imracical because he number of esimaed coefficiens in he lineariy ess and he SVECM rises considerably relaive o he number of esimaed coefficiens in he linear VECM. For hese reasons we sele for a relaively small baseline sysem. 8 he lag lengh is seleced by saring wih 5 lags on each variable, and sequenially esing down using an F-es. 7

9 coinegraion (see Johansen, 988. he 95 ercen criical values are aken from Mackinnon e al. (999 in he conex of he Pesaran e al. (000 linear sysem wih boh endogenous and exogenous I( variables. Boh he λ-max and race saisics indicae he exisence of r = coinegraing vecor. For exac idenificaion, we normalise he esimaed vecor on he real exchange rae,. hen we es one over-idenifying resricion, ha is, long-run exclusion of he unemloymen rae, u. he resricion is acceed as i calculaes χ ( =.73 (-value = 0.0 and he resuling coinegraing vecor is: = (w +0.8 ( O (0.030 (0. where sandard errors are given in arenheses below he esimaed coefficiens. he esimaed coinegraing relaionshi looks like he heoreical real exchange rae equaion (4 wih he share of raded goods in oal ouu (i.e. τ esimaed a 54.3 ercen and he share of value added in gross ouu (i.e. π esimaed a 74.8 ercen (he laer is derived from τ( π /π = Figure los he mean-correced deviaions from he esimaed relaionshi. Movemens of he disequilibrium error above (below he zero line are associaed wih an undervalued (overvalued real exchange rae. We discuss his issue furher in secion 5 of he aer. 4. Lineariy esing and shor-run esimaes Having esimaed he long-run real exchange rae equaion, we es for lineariy in model (3 using he esimaed coinegraing vecor CV - as he ossible ransiion variable s. Lineariy ess he hree endogenous equaions y = [, w, u] in he linear VECM ass he Auocorrelaion es (of u o order 5 and he 4h order Auoregressive Condiional Heeroscedasiciy es and Heeroscedasiciy. hey fail Normaliy and Heeroscedasiciy. Deailed diagnosic ess for he linear and non-linear versions of he shor-run equaions are reored below. 9 Using annual daa over he eriod, Alogoskoufis (990 esimaes τ beween 3 ercen and 39 ercen, and π a 9 ercen. However, he oins ou ha his esimaes for τ are imlausibly low. Our esimae for π a 74.8 ercen is much closer o π a 7.7 ercen in Bruno and Sachs (98. heir esimae is based on a sysem of facorrice fronier, ouu suly, and labour demand equaions using annual daa over he eriod. 0 Chaudhuri and Daniel (998 ado he Engle-Granger wo-se rocedure o es for coinegraion in a bivariae model involving real exchange raes and real oil rices for sixeen OECD counries. Using monhly daa over he 973(-996( eriod, hey find ha he UK real exchange rae coinegraes wih he real rice of oil. he coefficien on he real rice of oil is esimaed a

10 are run for a differen number of lags of he ransiion variable s -d = CV -d (namely d =,, 3, and 4 lags. hen he aroriae lag is seleced as he one for which he lineariy es is mos srongly rejeced. We reor boosraed -values insead of asymoic -values alhough our resuls are no sensiive o he above choice. o comue he boosraed -values of he equaion secific F ess and he sysem wide LR es reored in able, we followed closely Weise (999. Firs, we esimaed he linear VECM equaions. o conrol for he resence of heeroscedasiciy in he linear models, he VECM residuals were regressed on all RHS variables enering he linear VECM as well as heir squares, and he original residuals were ransformed using he esimaed coefficiens from his auxiliary regression. Draws were aken from he ransformed residuals and one housand arificial daa series were consruced. For each of hese arificial series, F and LR saisics were consruced and hen comared o he corresonding saisics from he acual daa. he boosraed -values were derived as he number of imes he F and LR saisics from he arificial daa exceeded he corresonding saisics from he acual daa, divided by one housand. According o he resuls in able, lineariy is mosly rejeced for CV -. Using he disequilibrium error CV - in he quadraic logisic funcion (c, we herefore roceed by esimaing he non-linear shor-run, w and u equaions. Before esimaing he non-linear models, i is worh menioning ha Granger and eräsvira (993 and eräsvira (994 sress aricular roblems like slow convergence or overesimaion associaed wih esimaes of he γ arameer. For his reason, we follow heir suggesion in scaling he quadraic logisic funcion (c by dividing i by he variance of he ransiion variable σ (CV - (which equals 0.003, so ha γ becomes a scale-free arameer. Based on his scaling, we use γ = as a saring value and values of CV - close o is minimum (which equals 0.5 and maximum (which equals 0.06 as saring values for he arameers c and c, resecively. he esimaes of he linear equaions for, w and u are used as saring values for he remaining arameers in he SVECM equaions (. For comarison reasons, we reor boh he linear and he non-linear versions of he esimaed equaions. ables 3A o 3C reor he OLS esimaes of he arsimonious linear models, whereas ables 4 o 6 reor he non-linear leas squares (NLS esimaes for he arsimonious SVECM equaions (. One could also argue in favour of a srucural raher han a reduced form model by esing he significance of curren, w and u effecs in he esimaed equaions. However, hese effecs were insignifican. 9

11 he main arameers of ineres in he non-linear models are he esimaed values of he hreshold arameers c and c, and he seed of adjusmen, γ. he c and c esimaes reored in ables 4 o 6 are saisically significan in all models. he c and c esimaes indicae he exisence of wo regimes for he, w and u equaions; one characerised by large deviaions of he real exchange rae from is long-run equilibrium and an alernaive one which is characerised by small real exchange rae deviaions from is equilibrium level. he economic imlicaions of hese resuls will be discussed in he following secion. he esimaes of he γ arameer are raher high for all models indicaing ha he seed of he ransiion from G( s γ, c, c 0 o ( γ, c, c ; = G is raid a he esimaed hresholds c and c. Noice, s ; = however, he raher high sandard error of he γ esimaes. eräsvira (994 and van Dijk e al. (00 oin ou ha his should no be inerreed as evidence of weak non-lineariy. Accurae esimaion of γ is no always feasible, as i requires many observaions in he immediae neighborhood of he hreshold arameers c and c. Furher, large changes in γ have only a small effec on he shae of he ransiion funcion imlying ha high accuracy in esimaing γ is no necessary (see he discussion in van Dijk e al., 00. From ables 3 o 6 one can noice a large imrovemen in he diagnosic ess of he non-linear relaive o he linear models. he error variance raio of he non-linear relaive o he linear models (i.e. s NL/s L is less han one, indicaing ha he non-linear models have a beer fi. In aricular, he s NL/s L raio shows a reducion in he residual variances of he non-linear comared o he linear models which ranges from around 6 ercen for he and u equaions in able 4 and able 6, resecively, o around 30 ercen for he w model in able 5. In addiion, he non-linear secificaion of all hree models caures he heeroscedasic and mos of he normaliy failures ha are resen in he corresonding linear models. 5. Discussion of he resuls Looking a he linear shor-run equaions firs, one can noice ha he coinegraing vecor (CV - eners wih he correc sign in he equaion (see able 3A. he CV - effec (i.e suggess a slow adjusmen o disequilibrium deviaions of he real exchange rae from is long-run relaionshi. Bearing in mind ha he real exchange rae equaion caures asecs of he real comeiiveness of he domesic economy ha deend on condiions revailing in he 0

12 roducion secor, his sluggishness could reflec rigidiies in he funcioning of he roduc marke due o roduc heerogeneiy, governmen imosed barriers o rade, or labour marke inflexibiliy disoring he adjusmen of wages. he disequilibrium error also affecs negaively he u equaion (i.e. coefficien on CV - equals 0.36 in able 3C. his resul imlies ha when he real exchange rae is above is equilibrium level, ha is, undervalued, unemloymen falls as firms resond o an imrovemen in domesic comeiiveness by increasing heir demand for labour. We also reor a shor-run negaive effec of as real wage growh (i.e. w - in he u equaion. he coinegraing vecor has a weak osiive effec in he w equaion (i.e. coefficien on CV - equals in able 3B. Hence, here seems o be some weak evidence ha when he real exchange rae is undervalued, workers resond o an imrovemen in domesic comeiiveness by demanding and geing higher wages. he esimaes in able 3B also sugges a significan effec from as changes in unemloymen (i.e. u -, u -, and u -3 in he w equaion bu hese effecs seem o cancel each oher ou. 3 he NLS esimaes sugges ha all hree, w and u equaions exhibi a regime-swiching behaviour according o he variaion of he disequilibrium error. Consider firs he equaion in able 4. he esimae of he disequilibrium error in he second regime (i.e. coefficien α is equal o 0.43 when ( s γ, c, c G is higher han ha of he ; = disequilibrium error in he firs regime (i.e. coefficien α is equal o when G( s γ, c, c 0. his imlies ha when he real exchange rae exceeds an esimaed inerval ; = band of (c, c = ( 0.096, 0.078, he shor-run real exchange rae adjuss faser. From able 6 one can see ha shor-run unemloymen u reacs fas o he disequilibrium error (i.e. coefficien α is equal o 0.9 only in he second regime (i.e. when G( s γ, c, c, ha is, when he disequilibrium error exceeds an esimaed inerval band of ; = (c, c = ( 0.040, In addiion, he esimaed inerval band (c, c for he shor-run unemloymen rae is much narrower comared o ha for he shor-run real exchange rae. aking ino accoun ha our samle covers a eriod of floaing exchange raes (wih he UK See also he discussion in e.g. Johansen and Juselius (99 and Pesaran and Shin (996 in he conex of a sysem involving he UK effecive exchange rae, UK ineres rae, UK rices, foreign rices and foreign ineres rae. 3 Among oher sudies, Manning (993 using annual daa over he eriod, reors a negaive effec from real wages on unemloymen as well as a negaive effec from unemloymen on real wages. However, his model uses he level of u raher han u which is used here.

13 joining he Exchange Rae Mechanism only beween 990 and 99, i is reasonable o exec ha he shor-run real exchange rae will adjus faser when he coinegraing vecor CV - is ouside a raher wide inerval band of hresholds. o he exen ha he real exchange rae equaion reflecs moneary and more generally economic olicy consideraions, he significan effec of he coinegraing vecor in he shor-run unemloymen equaion imlies ha unemloymen can be argeed by economic olicy. Furher, he lower esimaes of c and c for he u equaion sugges ha if economic auhoriies wan o avoid large swings in unemloymen hen hey should be reared o kee real exchange rae movemens wihin a narrow inerval band of hresholds. 4 Our resuls in able 5 sugges ha shor-run wages w are affeced by real exchange rae flucuaions wihin an esimaed inerval band of (c, c = ( 0.070, 0.03 (i.e. coefficien α is equal o 0.0 in he firs regime ( s γ, c, c 0 G. Conrary o our resuls for he ; = and u equaions, we could no find any significan effec from he disequilibrium error on shorrun wages ouside he esimaed band of hresholds. his is raher surrising, as we would exec he imac of he disequilibrium error on he shor-run dynamics o be more eviden when deviaions of he real exchange rae from is long-run level exceed he esimaed inerval band. his finding robably has o do wih he raher wide inerval band esimaes and in aricular he c esimae, which is racically equal o he maximum value of CV - (i.e he relaionshi beween he occurrence of a regime and he disequilibrium error is deiced in Figure 3, which los he values of he ransiion funcion agains CV - for he, w and u equaions. As discussed above, G ( γ, c, c 0 and ( γ, c, c s ; = G are s ; = relaed o small and large deviaions resecively of he real exchange rae relaive o a band of hresholds. In addiion, his Figure hels clarify he discussion abou he seed of ransiion beween he wo regimes. One can see ha he ransiion from one regime o he oher is raid, as he esimaes of γ are raher high for all models. 4 ha said, i is worh oining ou ha olicy makers in he UK were never able o conrol he exchange rae. Describing he main characerisics of macroeconomic olicies in he UK, Andrew Brion, he former direcor of he Naional Insiue of Economic and Social Research, commens: Aems o use he exchange rae as a olicy insrumen misfired; aems o conrol i failed; aems o ignore i were no more successful. he auhoriies never really go on o of he siuaion a all. (Brion, 99,. 98.

14 Figure 4 los he esimaed ransiion funcion for each model agains ime in order o illusrae he succession of regimes over he samle eriod. From Figure 4A, he and eriods are classified ino he second regime of real exchange rae deviaions ouside he esimaed inerval bands. One can noice from Figure, ha during he eriod, he esimaed ransiion funcions ick a highly overvalued real exchange rae. Suored by he grea areciaion of he dollar vis a vis he currencies of mos of he indusrialised counries during he firs half of he 980s (see e.g. Engel and Hamilon, 990, he real exchange rae consequenly revers o is long-run equilibrium. During he eriod, he esimaed ransiion funcions ick a highly undervalued real exchange rae, which hen began revering o is longrun level. Figure 4C shows ha swiches from one regime o he oher are aricularly acive for he unemloymen equaion where a much narrower inerval band was esimaed. I is noable ha he firs eriod, which caures he economic recession, follows he second OPEC oil rice hike (an increase in oil rices of around 5 ercen in June 979 and coincides wih imoran changes in economic olicies following he elecion of he hacher governmen in May 979. In aricular, 979 saw he aboliion of exchange rae conrols, which was no aimed a any aricular effec on he exchange rae, as well as ublic sending cus and an increase in indirec axaion. he new governmen encouraged he use of cheaer labour, esecially female labour, which led o more ar-ime emloymen. A he same ime, a very igh moneary olicy aiming a a raid decrease in he rae of inflaion, led o a more overvalued real exchange rae (see Figure, a raid increase in unemloymen (see Figure and a severe recession (see e.g. Brion, 99; Mizon, 995. aking ino accoun he slow adjusmen of he real exchange rae reored in he revious secion of he aer, i is no surrising ha afer he UK's exi from he ERM in Seember 99, he real exchange rae exerienced a ah of ersisen dereciaion, which eaked beween 995 and 997 (see Figure. Indeed, his is he non-lineariy in he shorrun real exchange rae caured by Figure 4A. 6. Conclusions his aer examined non-lineariies in a mulivariae model of he UK real exchange rae. Afer esimaing a long-run real exchange rae equaion as ar of a small sysem involving real roduc wages, he unemloymen rae and he real rice of oil, we found evidence ha deviaions of he real exchange rae equaion from is long-run equilibrium level affec in a nonlinear way no only he shor-run real exchange rae equaion, bu also he shor-run 3

15 unemloymen and wage equaions. According o our esimaes, he shor-run real exchange rae adjuss faser when he coinegraing relaionshi is ouside a wide inerval band of hresholds. his is no surrising as our samle covers a eriod of floaing exchange raes. On he oher hand, he shor-run unemloymen rae adjuss fas when he coinegraing relaionshi is ouside a narrower inerval band. o he exen ha he real exchange rae equaion reflecs economic olicy consideraions, our resuls sugges ha olicy makers should aim a a narrow band for he real exchange rae if hey wan o avoid large swings in unemloymen. Daa aendix : GDP value added rice deflaor (995 = 00. Source: Economic rends Annual Sulemen (EAS. = * + e: he rice of domesic radables, where * is he $ rice index of UK imors (990 = 00 and e is he average /$ exchange rae. Source: IMF, Inernaional Financial Saisics. w = w NE + : average roduc wages in he manufacuring secor (990=00, where w NE refers o average weekly wages in manufacuring (ne of emloyers axes and is he ax rae on labour aid by emloyers, consruced as: (emloyers conribuions / (oal wage bill. Source: EAS. O : rice index of maerials and fuels urchased by manufacurers (995 = 00. Source: EAS. u : UK unemloymen rae. Source: EAS. All variables are in logs. 4

16 References Alogoskoufis, G., 990. raded goods, comeiiveness and aggregae flucuaions in he Unied Kingdom, Economic Journal 00, Baum, C.F., J.. Barkoulas and M. Caglayan, 00. Nonlinear adjusmen o urchasing ower ariy in he os-breon Woods era, Journal of Inernaional Money and Finance 0, Brion, A.J.C., 99. Macroeconomic Policy in Briain Naional Insiue of Economic and Social Research, Economic and Social Sudies 36. Cambridge Universiy Press, Cambridge. Bruno, M. and J.D. Sachs, 98. Inu rice shocks and he slowdown in economic growh: he case of UK manufacuring, Review of Economic Sudies 49, Chaudhuri, K. and B.C. Daniel, 998. Long-run equilibrium real exchange raes and oil rices, Economics Leers 58, Dumas, B., 99. Dynamic equilibrium and he real exchange rae in a saially searaed world, Review of Financial Sudies 5, Engel, C. and J.D. Hamilon, 990. Long swings in he dollar: are hey in he daa and do markes know i? American Economic Review 80, Escribano, A. and O. Jordá, 999. Imroved esing and secificaion of smooh ransiion regression models, in Rohman, P. (Ed., Nonlinear ime Series Analysis of Economic and Financial Daa, Boson, Kluwer, Granger, C.W.J. and N.R. Swanson, 996. Fuure develomens in he sudy of coinegraed variables, Oxford Bullein of Economic and Saisics 58, Granger, C.W.J. and. eräsvira, 993. Modelling nonlinear economic relaionshis. Oxford Universiy Press, Oxford. Hamilon, J.D., 989. A new aroach o he economic analysis of nonsaionary ime series and he business cycle, Economerica 57,

17 Hansen, H. and K. Juselius, 994. Manual o coinegraion analysis of ime series: CAS in RAS. Mimeo, Universiy of Coenhagen. Jansen, E.S. and. eräsvira, 996. esing arameer consancy and suer exogeneiy in economeric equaions, Oxford Bullein of Economics and Saisics 58, Johansen, S., 988. Saisical analysis of coinegraion vecors, Journal of Economic Dynamics and Conrol, Johansen, S. and K. Juselius, 99. esing srucural hyoheses in a mulivariae coinegraion analysis of he PPP and UIP for UK, Journal of Economerics 53, -44. Mackinnon, J.G., A.A. Haug and L. Michelis, 999. Numerical disribuion funcions of likelihood raio ess for coinegraion, Journal of Alied Economerics 4, Manning, A., 993. Wage bargaining and he Phillis curve: he idenificaion and secificaion of aggregae wage equaions, Economic Journal 03, Marin, C., 997. Price formaion in an oen economy: heory and evidence for he Unied Kingdom, 95-99, Economic Journal 07, Michael, P., A.R. Nobay, and D.A. Peel, 997. ransacions coss and nonlinear adjusmen in real exchange raes: an emirical invesigaion, Journal of Poliical Economy 05, Mizon, G.E., 995. Progressive modeling of macroeconomic ime series. he LSE mehodology, in Hoover, K.D. (Ed., Macroeconomerics: Develomens, ensions and rosecs. Kluwer Academic Press, Dordrech, Nakagawa, H., 00. Real exchange raes and real ineres differenials: imlicaions of nonlinear adjusmen in real exchange raes, Journal of Moneary Economics 49, Paya, I., and D.A. Peel, 003. Purchasing ower ariy adjusmen seeds in high frequency daa when he equilibrium real exchange rae is roxied by a deerminisic rend, he Mancheser School 7, (Sulemen. Pesaran, M.H. and Y. Shin, 996. Coinegraion and seed of convergence o equilibrium, Journal 6

18 of Economerics 7, Pesaran, M.H., Y. Shin, and R.J. Smih, 000. Srucural analysis of vecor error correcion models wih exogenous I( variables, Journal of Economerics 97, Rohman, P., D. van Dijk, and P.H. Franses, 00. A mulivariae SAR analysis of he relaionshi beween money and ouu, Macroeconomic Dynamics 5, Saikkonen, P. and R. Luukkonen, 988. Lagrange Mulilier ess for esing non-lineariies in ime series models, Scandinavian Journal of Saisics 5, Saranis, N., 999. Modeling non-lineariies in real effecive exchange raes, Journal of Inernaional Money and Finance 8, eräsvira,, 994. Secificaion, esimaion, and evaluaion of smooh ransiion auoregressive models, Journal of he American Saisical Associaion 89, eräsvira,., and H.M. Anderson, 99. Characerizing nonlineariies in business cycles using smooh ransiion auoregressive models, Journal of Alied Economerics 7, S9-S36. ong, H., 990. Non-linear ime series: a dynamical sysem aroach. Oxford Universiy Press, Oxford. van Dijk, D., Franses, P.H., 000. Nonlinear error-correcion models for ineres raes in he Neherlands, in: Barne, W.A., Hendry, D.F., Hylleberg, S., eräsvira,., josheim, D., Würz, A., (Eds., Nonlinear Economeric Modelling in ime Series Analysis, Cambridge Universiy Press, Cambridge, van Dijk, D.,. eräsvira, and P.H. Franses, 00. Smooh ransiion auoregressive models a survey of recen develomens, Economeric Reviews, -47. Weise, C.L., 999. he asymmeric effecs of moneary olicy: a nonlinear vecor auoregression aroach, Journal of Money, Credi, and Banking 3,

19 able Eigenvalues, es saisics and criical values λ i λ-max race 0.3 H 0 H Sa. 95% H 0 H Sa. 95% 0.4 r = 0 r = r = 0 r r r = r r r r = r r = Noes: r denoes he number of coinegraion vecors. Criical values are from Mackinnon e al., (999. able Lineariy ess ransiion variable model Lagrange Mulilier F saisics for: w model u model Sysem wide es LR CV -. (0.375 CV (0.684 CV (0.5 CV ( ( ( ( ( ( ( ( ( ( ( ( (0.094 Noes: Boosraed -values in arenheses. he -values for he equaion secific Lagrange Mulilier F saisics and he sysem wide LR es saisic are derived from boosraing wih one housand relicaions. CV is he ransiion variable: CV = (w 0.8 ( O, in mean-correced form. he null hyohesis is lineariy. he alernaive hyohesis is he SVECM reresenaion. 8

20 able 3 Esimaed linear models Panel A: Linear model = CV w - (0.00 (0.044 (0.7 ( O O O - (0.03 (0.04 (0.06 s L = 0.03, AR(5 = 0.39[0.856], ARCH(4 = 0.[0.980], HE =.99[0.035], NORM( =.7[0.000] Panel B: Linear w model w = CV u - (0.00 (0.060 (0.097 ( u u O O - (0.0 (0.083 (0.3 (0.6 s L = 0.05, AR(5 = 0.8[0.93], ARCH(4 = 0.60[0.66], HE = 4.60[0.000], NORM( = 34.[0.000] Panel C: Linear u model u = CV w - (0.00 (0.055 (0.87 ( u u O (0.063 (0.058 (0.07 s L = 0.04, AR(5 =.0[0.364], ARCH(4 =.6[0.77], HE =.9[0.04], NORM( = 7.99[0.08] Noes: Sandard errors are given in arenheses below he esimaes. s L : sandard error of he linear regression. AR(5: F-es for u o 5h order serial correlaion. ARCH(4: 4h order Auoregressive Condiional Heeroscedasiciy F-es. HE: F-es for Heeroscedasiciy. NORM(: Chi-square es for normaliy. Numbers in square brackes are he -values of he es saisics. 9

21 able 4 Esimaed non-linear model he able reors he NLS esimaes of he following SVECM equaion: = ( µ + ( µ 4,3 + α CV + α CV O, 3, w G( CV ; γ, c, c, 4, O O ( G( CV 4, where G(CV - ; γ, c, c = { + ex[ γ (CV - c (CV - c / σ (CV - ]} -, O ; γ, c, c is he quadraic logisic ransiion funcion, wih CV - as he ransiion variable. Values of 0 and of he ransiion funcion are associaed wih he wo alernaive regimes. he dynamics in he firs regime, when G(CV - ; γ, c, c = 0, are: = ( µ + α CV ( G( CV ; γ, c, (,, O c. In he second regime, when G(CV - ; γ, c, c =, is dynamics are: = ( µ + α CV w 4,3 O 3, G( CV ; γ, c, c 4, O 4, O For inermediae values of G(CV - ; γ, c, c, i.e. 0 < G(CV - ; γ, c, c <, dynamics are a weighed average of he wo equaions. he seed of ransiion beween he wo regimes is deermined by he arameer γ. = ( CV (0.00 (0.059 (0.08 where O - ( G(CV - ; γ, c, c (0.096 ( CV w O (0.007 (0.09 (0.36 ( O O - G(CV - ; γ, c, c (0.49 (0.43 G(CV - ; γ, c, c = {+ ex[ 7.06(CV (CV / σ (CV - ]} - (3.93 (0.003 (0.006 s NL = 0.0, s NL/s L = 0.834, AR(5 =.85[0.00], ARCH(4 = 0.3[0.97], HE = 0.9[0.999], NORM( = 8.4[0.06] Noes: Sandard errors are given in arenheses below he esimaes. s NL : sandard error of he non-linear regression. he diagnosic ess are discussed in he noes of able 3. 0

22 able 5 Esimaed non-linear w model he able reors he NLS esimaes of he following SVECM equaion: w = ( µ + ( µ 4, + α CV 3, O, G( CV 3, u ; γ, c, c, 3,3 O u 3 ( G( CV where G(CV - ; γ, c, c = { + ex[ γ (CV - c (CV - c / σ (CV - ]} -, ; γ, c, c is he quadraic logisic ransiion funcion, wih CV - as he ransiion variable. Values of 0 and of he ransiion funcion are associaed wih he wo alernaive regimes. he w dynamics in he firs regime, when G(CV - ; γ, c, c = 0, are: w = ( µ + α CV ( G( CV ; γ, c,. (,, O c In he second regime, when G(CV - ; γ, c, c =, is dynamics are: w = ( µ u u 4, 3, O G( CV 3, ; γ, c, c. 3,3 For inermediae values of G(CV - ; γ, c, c, i.e. 0 < G(CV - ; γ, c, c <, w dynamics are a weighed average of he wo equaions. he seed of ransiion beween he wo regimes is deermined by he arameer γ. 3 w = ( CV (0.00 (0.06 (0.083 where 0.33 O - ( G(CV - ; γ, c, c (0.098 ( u - (0.0 (0.306 ( u O - G(CV - ; γ, c, c (0.00 (0.98 G(CV - ; γ, c, c = {+ ex[.89(cv (CV / σ (CV - ]} - (.657 (0.007 (0.04 s NL = 0.0, s NL/s L = 0.705, AR(5 = 0.84[0.55], ARCH(4 = 0.64[0.635], HE = 0.7[0.83], NORM( = 3.57[0.00] Noes: Sandard errors are given in arenheses below he esimaes. s NL : sandard error of he non-linear regression. he diagnosic ess are discussed in he noes of able 3.

23 able 6 Esimaed non-linear u model he able reors he NLS esimaes of he following SVECM equaion: u = ( µ, + ( µ, O + α CV ( G( CV 3, u, w 3, ; γ, c, c u 3 4,,3 u O G( CV ; γ, c, c where G(CV - ; γ, c, c = { + ex[ γ (CV - c (CV - c / σ (CV - ]} -, is he quadraic logisic ransiion funcion, wih CV - as he ransiion variable. Values of 0 and of he ransiion funcion are associaed wih he wo alernaive regimes. he u dynamics in he firs regime, when G(CV - ; γ, c, c = 0, are: u = ( µ w u,, O, ( G( CV ; γ, c, c In he second regime, when G(CV - ; γ, c, c =, is dynamics are: u = ( µ + αcv 3, u 3, u 3 4, O G( CV; γ, c, c.,3 For inermediae values of G(CV - ; γ, c, c, i.e. 0 < G(CV - ; γ, c, c <, u dynamics are a weighed average of he wo equaions. he seed of ransiion beween he wo regimes is deermined by he arameer γ. u = ( w u - (0.003 (0.68 (0.46 (0.07 where 0.74 O - ( G(CV - ; γ, c, c (0.7 ( CV u u -3 (0.004 (0.063 (0.087 ( O G(CV - ; γ, c, c (0.69 G(CV - ; γ, c, c = {+ ex[.00(cv (CV / σ (CV - ]} - (8.96 (0.003 (0.008 s NL = 0.0, s NL/s L = 0.840, AR(5 =.43[0.3], ARCH(4 = 3.6[0.00], HE =.30[0.00], NORM( =.78[0.00] Noes: Sandard errors are given in arenheses below he esimaes. s NL : sandard error of he nonlinear regression. he diagnosic ess are discussed in he noes of able 3.

24 Figure : Plos of he levels and he firs differences of he series (A (B w (C O (D u (E (. (F (w (G (O (H u

25 Figure : Deviaions from he esimaed long-run real exchange rae relaionshi

26 Figure 3: Esimaed ransiion funcions (verical axis agains CV - (horizonal axis (A model (B w model (C u model Noes: Panels (A, (B and (C reor he esimaed ransiion funcions for, w and u agains he ransiion variable CV - from he corresonding SVECM equaions as reored in ables 4 o 6. he esimaed ransiion funcions are: (A G(CV - ; γ, c, c = { + ex[ 7.06(CV (CV / σ (CV - ]} - (B G(CV - ; γ, c, c = { + ex[.89(cv (CV / σ (CV - ]} - (C G(CV - ; γ, c, c = { + ex[.00(cv (CV / σ (CV - ]} - 5

27 Figure 4: Esimaed ransiion funcions agains ime (A model (B w model (C u model Noes: Panels (A, (B and (C reor he esimaed ransiion funcions for, w and u agains ime from he corresonding SVECM equaions as reored in ables 4 o 6 and Figure 3. Exreme values of 0 and of he ransiion funcions are associaed wih he wo alernaive regimes. 6