Firs Draf Measuremen of Poenial Oupu for Turkey: Unobserved Componens Model Fehi Öğünç, Dilara Ece * Cenral Bank of he Republic of Turkey Saisics Deparmen 06100 Ulus-Ankara Fehi.Ogunc@cmb.gov.r Phone: 0 (312) 312 43 34 Dilara.Ece@cmb.gov.r Phone: 0 (312) 312 43 34 Absrac This paper specifies a basic univariae and a bivariae unobserved componens models o esimae poenial oupu using informaion from observable aggregaes and presens resuls for he Turkish economy. The mos imporan moivaion behind measuring oupu gap for he Turkish economy is he absence of a cerain measure for he oupu gap ha can be used for boh near-erm and long-erm inflaion forecass especially wihin he inflaion-argeing framework. The firs specificaion used in he paper, i.e. univariae approach, decomposes acual oupu ino poenial oupu ha follows a random walk wih a ime-varying poenial growh rae and a saionary oupu gap. The univariae specificaions commonly ignore some economic conen, which migh be relevan for he measuremen of oupu gap. In his respec, he univariae model is exended by uilizing he relaionship beween inflaion and he oupu gap, namely he Phillips curve. Whereas boh models give similar oupu gap esimaes, signal exracion saisics sugges ha incorporaing he supply side o he sysem reduces he parameer uncerainy and he oal sandard error and improves he gap esimae. JEL Classificaion Numbers: C51, E31 Keywords: Inflaion; Oupu gap; Phillips curve; Unobserved componens models; Kalman filer. * The views expressed in his paper are hose of he auhors and do no necessarily correspond o he views of he Cenral Bank of he Republic of Turkey. We are paricularly indebed o Kenneh N. Kuner for providing us he code and his excellen explanaions and commens. Code wrien in GAUSS can be obained from Kenneh N. Kuner. Also we would like o hank Cevriye Aysoy, Ali Hakan Kara, Zelal Koan and M. Gülenay Ongan for heir helpful commens and suggesions.
1. Inroducion Poenial oupu is he level of producion ha can be achieved wih he exising level of facors of producion wihou puing pressure on inflaion. Therefore level of oupu above poenial will ofen be seen as a source of inflaionary pressures and a signal for he moneary auhoriies. The deviaion of acual oupu from he poenial, namely oupu gap, is a key variable for moneary policy issues in order o undersand he hisorical developmen of he inflaion and assess he exen of curren inflaionary pressures. Unforunaely, here is a lo of uncerainy relaed wih he measuremen issue, as poenial oupu so he oupu gap canno be observed direcly. In general, mehods o esimae hese unobserved variables may be classified as univariae (non-srucural), srucural and mulivariae non-srucural (or mixed) approaches. Univariae approaches perhaps are he mos popular ones since hey require less informaion due o heir solely reliance on he acual daa. Consequenly, hey are merely saisical approaches and suffer from he shorcoming ha hey disregard oher informaion, such as, inflaion, unemploymen, and capaciy uilizaion. On he oher hand, srucural approaches, like producion funcion, use economic heory o esimae poenial oupu. However, daa requiremens for Turkish economy impose igh consrains on he applicaion of hese kinds of approaches. The mixed approach combines he ime series echniques wih he economic heory and can be viewed as an exension o he univariae ones. These kinds of exensions have been applied by Laxon and Telow (1992) for Canada, Kuner (1994) for he U.S., Gerlach and Smes (1999) for EMU-area and Benes and N diaye (2001) for Czech economy. These auhors uilized he relaion beween inflaion and he real oupu (in some cases aggregae demand relaionship is also included) in order o obain a more meaningful measure of poenial oupu. This paper describes a basic univariae and a bivariae 1 unobserved componens models o esimae poenial oupu using informaion from observable aggregaes and presens resuls for he Turkish economy. The mos imporan moivaion behind measuring oupu gap for he Turkish economy is he absence of a cerain measure for he oupu gap ha becomes imporan for boh he near-erm and long-erm inflaion forecass especially wihin he inflaion-argeing framework given ha one of he key issues for he inflaion-argeing framework is he esimaion of poenial oupu. 1 I is called bivariae since oupu and inflaion equaions ogeher form a bivariae unobserved componens model.
The paper is organized as follows. The following secion briefly discusses he fundamenals of he univariae model. The hird secion exends he univariae model by adding informaion abou inflaion and ries o describe oupu-inflaion relaion. Empirical resuls for Turkish economy are presened in secion 4 and he final par gives some concluding remarks. 2. Basic Univariae Unobserved Componens Model The unobserved componens model is a mehod o esimae he unobserved variables such as poenial oupu, rend growh rae and oupu gap using he informaion from observed variables. Once he model is specified in he sae space form and given he iniial values for he unobserved sae vecor, he unobserved variables can be esimaed by a recursive algorihm known as Kalman filer. Kalman filer uses he iniial values for he unobserved sae vecor in order o predic he unobserved variables and hen updaes he guesses based on he predicion errors. When all he observaions have been processed, he smoohing equaions give he bes esimaors of he unobserved variables based on all he informaion 2. The simples way of measuring poenial oupu is he univariae mehods, in which only he real oupu daa is used. Early specificaion of he rend componen of he oupu is a linear rend, which is based on a srong assumpion ha he supply side of he economy is deerminisic and economic flucuaions usually depend on he changes in he demand-side. Laer, Nelson and Plosser (1982) sugges ha he nonsaionariy in economic aciviy should be removed by firs- differencing, which means he rend componen is a random walk wih drif raher han a sraigh line. I was followed up by Wason s (1986) specificaion, which characerized poenial oupu and oupu gap as random walk wih drif and AR(2) respecively. However in all hese seings, economy s rend growh rae is assumed o be consan and here is no reason for hese componens o be consan over ime, especially when an economy is experiencing considerable srucural change. Hence Clark (1987) and Kuner (1999) consruced a variable growh rae model given he decline of he U.S. produciviy growh in he 1970s, reducion of labor force growh in he 1980s and he apparen increase in he rend growh in he mid-1990s. 2 See Harvey (1990) for he echnical deails.
The general form of he sysem ha encompasses a wide range of possibiliies 3 can be wrien as follows: x = x + z (1) * x = x + µ + η (2) * * 1 µ = (1 ρ ) µ + ρµ + ε (3) 0 1 φ( Lz ) = ξ (4) where x is he log of seasonally adjused real gdp, x * is he poenial oupu, µ is rend growh rae and z is he oupu gap. η, ε and ξ are independen normal whie-noise processes wih sandard deviaions, σ η, σ ε and σ ξ respecively and φ ( L) is he finie auoregressive polynomial wih he lag operaor L. Here, η is he shock o he level of poenial oupu and relaed o capaciy effecs and invesmen, ε is he shocks o he rend s growh rae and relaed o echnology, changing rends in facor inpus and finally, ξ is he gap shock. In he sysem, poenial oupu follows a random walk wih drif and rend growh rae can be shaped wih respec o differen values of ρ. For example, poenial growh rae can be assumed o follow a random walk ( ρ = 1) in which he oupu is I(2) or a serially correlaed poenial growh ( 0 ρ 1) wih nonzero σ ε. The dynamics of poenial oupu and he oupu gap depend on he naure of he shocks, in oher words relaive imporance of he supply and demand shocks. This relaive imporance, which deermines he smoohness of he rend componen, can be denoed by a parameer λ, which is he raio of he variance of he cycle o he variance of rend flucuaions. A small λ implies ha shocks o he economy are mainly supply shocks where poenial oupu moves nearly wih he daa and hence a small oupu gap is expeced. In he opposie case, choosing a higher value of λ, a larger weigh on smoohness in he rend, means shocks o he economy are principally shocks o aggregae demand. All hese poin ou he effecive selecion of his smoohness parameer. Unforunaely, mehodology iself canno provide his informaion. Therefore, i should be eiher seleced a priori, in which he judgmen of knowledgeable exper is essenial or esimaed along wih he oher parameers. 3 For example, as a special case, famous Hodrick-Presco filer can be achieved by imposing he resricions of no level shock ( σ η = 0 ), random walk rend growh rae wih nonzero σ ε ( ρ = 1 ) and serially uncorrelaed 2 oupu gap ( z WN(0, σ ξ ) ). Smoohing parameer of 1600 for quarerly daa is equivalen oσε = 0.025σξ.
3. Bivariae Unobserved Componens Model In general, univariae mehods lack imporan economic conen. Therefore, in his secion poenial oupu and he oupu gap are deermined condiional on he informaion in he sysem (1)-(4), ogeher wih he abiliy of he gap measure o explain inflaion as in Kuner (1994). The supply side of he economy is illusraed by a Phillips curve relaionship. In order o specify his aggregae supply relaionship, general o specific modeling approach is used and he final empirical model is as follows: π = απ + α π + βπ + γz + ψπ + v (5) pub m 1 1 2 2 1 In equaion (5), variableπ is he measure of inflaion and defined as he quarerly logarihmic difference of CPI. In he equaion, he influence of excess demand is capured hrough he oupu gap whereas he oher erms, he public manufacuring price inflaion ( π impor price 4 inflaion ( π m pub ) and he ), are he exogenous facors effecing headline inflaion. Finally, v represens shocks o he inflaion. Coefficiens on he righ hand-side of he inflaion equaion sum up o one in order o saisfy he naural-rae hypohesis (NRH). Noe ha impor price effec is immediae (here is no delay) and oupu gap eners hrough he firs lag. Two lags of quarerly inflaion represen he effecs of hings like conracual lags or oher cos adjusmen ha lead o sickiness in prices. In addiion o ha backward looking behavior dominaes he price expecaions of he agens due o highly inflaionary environmen. However, wih he implemenaion of he IT regime, as he credibiliy of he policies increases, he porion of he agens who end o form heir expecaions in a forwardlooking manner is expeced o increase since he inflaion arge iself is a nominal anchor for moneary policy and inflaion expecaions. We believe ha he change in he public manufacuring prices affecs he CPI inflaion by he following channels: increasing public manufacuring prices increases he cos of privae secor in which he publicly produced inpus are inensively used and he pas experience shows ha his is perceived as a signal for an increase in CPI inflaion hrough he expecaions. I is worh o noe ha here is no guaranee ha his reduced-form equaion specifies a rue oupu-inflaion rade-off. Given ha he daa come predominanly from previous regimes, where inflaion was closely ied o exchange rae movemens and ineria, i is o be expeced ha i will be hard o idenify he effec of he oupu gap, and here is a risk ha economeric 4 I is adjused for he changes in he exchange rae.
esimaes will undersae he value wih he new IT regime. In addiion, wih he floaing exchange rae regime, i is believed ha effec of he oupu gap will increase, and he oupu gap will become a more pronounced variable. In fac, i may be worh o say ha laes released daa may be considered as an early signal. I is believed ha he join esimaion of he univariae and he aggregae supply relaionship will provide more meaningful measure of oupu gap o he exen ha inflaion is relaed o he level of he oupu gap. Kuner (1994) says ha Esimaing he model amouns o choosing he unknown parameers o yield he z mos consisen wih observed inflaion, subjec o he smoohness resricions implicily sochasic rend specificaion for GDP. In his way, he bivariae poenial oupu model adds an elemen of economic conen ha is absen from he univarie derending mehods In oher words, by using mixed approach he srenghs of he saisical models are inegraed wih he srucural ones. Equaions (1)-(5) can be wrien in sae-space form 5 and esimaes of he parameers of he model and he unobserved sae variables can be obained by maximizing he likelihood funcion using he Kalman filer. 4. Empirical Resuls Measuremen of he poenial oupu and oupu gap are shown o be sensiive o he model specificaion (consequenly various assumpions relaed o iniial sae vecor), esimaion period and he mehod of esimaion. Before going hrough he resuls, i is imporan o underscore ha he shor esimaion period and he assumpion of he consan economic srucure in he esimaion period limi he accuracy of our esimaes. In his paper, acual oupu is defined as he logarihmic seasonally adjused gross domesic produc a 1987 consan prices. Sample period covers beween 1987:Q1 and 2002:Q4 6. The issue of seasonaliy is handled wih he commonly used program named TRAMO/SEATS (see Gomez and Maravall, 1998). We believe ha he susainable seady-sae real growh rae for our economy is 4.5 percen on annual basis 7, which corresponds o 1.125 percen per quarer. I is appropriae o assume he 5 See Appendix for he sae-space represenaions. 6 Noe ha forecased values are added for he las hree quarer in order o increase he end poin precision and daa descripion can be found in Appendix. 7 In general, i is hough ha he poenial oupu corresponds o he ideal equilibrium posiion for all he oupu variables; his paricular posiion corresponds o he so-called seady sae. Here, his seady-sae value is se by judgmen in he ligh of he hisorical values.
poenial growh rae as a ime varying process in order o capure he slow down and he increases in he rend growh rae. In mos of he early specificaions, he rend growh rae is assumed o follow a random walk, so he poenial and he observed oupu are I(2). However, we consider ha i is more proper o use a mean-revering model where rend growh rae converges slowly o he seady- sae rae of 4.5 percen. Therefore, as in he Czech model (Benes and N diaye, 2001), parameer ρ is se o 0.9 8 and a serial correlaion is assumed in he rend growh rae model. Furhermore he oupu gap is modeled as an AR(1) process 9. By using he iniial guess of he sae vecor and is covariance marix and hen applying he Kalman Filer, maximum likelihood esimaes of he parameers in he models are repored in Table 1. A a firs glance, focusing on he bivariae model esimaes, model is consisen wih he naural-rae hypohesis in oher words; i is super-neural 10. Coefficien of he lagged inflaion erms sum o 0.57, which indicaes high level of inflaion persisence. Here, gap unis are defined as he percenage of he poenial oupu. Therefore, he esimaed coefficien of oupu gap indicaes ha a one-percenage poin increase of oupu over poenial raises he inflaion by 0.16 percen in he nex quarer. Also bear in mind ha oupu gap follows an AR(1) process and he esimaed coefficien is 0.82. For his reason, he fuure inflaion raes will also be influenced due o his persisence. Table 1: Maximum Likelihood Esimaes of Model (1)-(5) (Quarerly Daa : 1987:Q1-2002:Q4) Univariae Model Bivariae Model Noe: -saisics in parenhesis. φ 1 σ ξ α 1 α 2 β γ ψ σ v 0.867 0.027 (4.91) (16.26) 0.824 0.026 0.321 0.253 0.256 0.163 0.161 0.022 (5.81) (15.75) (7.09) (4.67) (4.66) (2.19) (3.26) (9.08) Nex sep is o analyze he gap esimaes. Figure 1 poins ou ha esimae of he oupu gap using only he oupu daa is similar o one esimaed by using he informaion regarding he inflaion. I is imporan o remind ha he aggregae supply relaionship is able o provide more meaningful measures o he exen ha inflaion is relaed o he level of he oupu gap. 8 I means ha, in he absence of shocks, oupu growh would converge o he wihin 1 percen of he seadysae rae in jus abou 10 years. 9 Iniially, following Wason (1986), an AR(2) process for he oupu gap is esed. However, esimaion resuls indicae ha he second erm is insignifican. 10 Neiher he level of he money supply nor he rae of growh of he money supply influences he seady-sae real equilibrium in oher words here is no long-run rade-off beween oupu and inflaion or he level of prices.
Figure 1 displays ha boh univariae and bivariae esimaes of he oupu gap reach he lowes level a 1994Q2 and indicae an increasing growh unil 1998Q4. Moreover, acual oupu is below he poenial since he firs quarer of 2001 and according o bivariae specificaion, he curren oupu (2002:Q1) is 5.8 percen below he poenial where as he percenage is 6.2 for he univariae one. The oupu gap is esimaed in wo differen ways depending on wha informaion is used. The filered esimae a ime is one-sided and i uses informaion up o ime (z ). According o Kuner (1999), he one sided esimae corresponds approximaely o a real ime esimae 11. On he oher hand, a smoohed value is wo sided and uses informaion from he whole sample, up o ime T (z T where 0 T) i.e. i uses he fuure informaion o compue he curren gap. Figure 2 shows one versus wo-sided gap esimaes according o he bivariae approach. I is worh o noe ha he filered and smoohed values are idenical for he las observaion of he sample. Because gap esimaes are no precise, i is imporan o repor he esimaes wih a measure of uncerainy. The unobserved componens mehod has he advanage ha i allows he consrucion of confidence bands for he poenial oupu and oupu gap. The esimaes of he error covariance marix of he sae variables can be obained by Kalman recursion. Figure 3 and 4 illusrae he wo-sided univariae and bivariae oupu gap esimaes wih ±1.645 sandard error bands, which corresponds o a 90 percen confidence. Firs of all, due o he larger sandard error of univariae gap esimae, here is more uncerainy around univariae gap esimae as compared o bivariae one. In general, error bands indicae ha 1993 and 1997-1998 are he expansions, whereas 1989, 1994 and 2001-2002 are he recessions 12 and also i is obvious from he figures ha he bivariae gap esimaes are more significan. Anoher advanage of he unobserved componen models is o provide a measure of he uncerainy relaed wih he unknown parameers. Toal variance of he sae vecor can be decomposed ino filer variance (variance relaed wih he signal exracion) and he parameer variance (parameer uncerainy) 13. The parameer uncerainies are compued by Mone Carlo simulaion wih 300 draws and he resuls are presened in Table 2. In he able, he average 11 As noed by Kuner (1999), The correspondence is no exac; however, as he parameers are esimaed over he enire sample and final, revised daa are used. 12 The significance of he gap esimaes ruly depends on he saisical crieria used; explicily oher assumpions may designae differen daes. 13 See Kuner (1994) for furher informaion.
size of he error assignable (impuable) o filer and parameer uncerainy appears for one and wo sided esimaes of he univariae and bivariae models. Table 2: Signal Exracion Saisics Univariae Model Bivariae Model One Sided Filer variance (1.54*10-3 ) (8.2*10-4 ) Parameer uncerainiy (1*10-3 ) (3.8*10-4 ) Toal sandard error (4.9*10-2 ) (3.4*10-2 ) Two Sided Filer variance (1.1*10-3 ) (5.1*10-4 ) Parameer uncerainiy (1.2*10-3 ) (3.4*10-4 ) Toal sandard error (4.6*10-2 ) (2.9*10-2 ) Noe: The firs eigh quarers of he sample are excluded in hese oucomes, so ha he influence of he iniial condiions is reduced. For boh of he univariae and bivariae models, he filer variance is larger han he parameer variance in he one sided esimae. While he filer variance in wo sided esimae is 70 percen of wha i was in he one sided case for univariae model, i is only 60 percen of he one sided case for he bivariae model. The oal sandard error is reduced from 4.9 percen o 4.6 percen by moving o he wo-sided esimae in univariae model. In bivariae model, he effec of moving o he wo-sided esimae is more pronounced; he oal sandard error reduces from 3.4 percen o 2.9 percen. Resuls show he noable evidence of using informaion from he whole sample. This finding underlines he imporance of he careful analysis of he end-poin esimae since he filered and smoohed values are he same for he las observaion of he sample. On he oher hand, he comparison of univariae and bivariae models yields 1.7 percenage poins less oal sandard error in he favor of he bivariae one. In addiion, i is worh o noe ha among he four alernaive models he smalles oal sandard error is in wo-sided esimae of he bivariae model.
Figure 1: 8.0 % Esimaed Oupu Gaps 4.0 0.0-4.0-8.0 Bivariae -12.0 Univariae 87Q2 88Q2 89Q2 90Q2 91Q2 92Q2 93Q2 94Q2 95Q2 96Q2 97Q2 98Q2 99Q2 00Q2 01Q2 02Q2 Noe: Shaded area refers o he forecased values from his poin onward. Boh esimaes are wo-sided. Figure 2: % 8.0 One Versus Two-Sided Gap Esimaes for Bivariae Approach 4.0 0.0-4.0-8.0 Tw o-sided One-Sided -12.0 88Q1 89Q1 90Q1 91Q1 92Q1 93Q1 94Q1 95Q1 96Q1 97Q1 98Q1 99Q1 00Q1 01Q1 02Q1
Figure 3: Basic Univariae Model Resuls (using only he oupu daa) 10.4 log Log Real GDP and Poenial Oupu Esimaed by Univariae Approach 10.3 10.2 10.1 10.0 9.9 9.8 87Q1 88Q1 89Q1 90Q1 91Q1 92Q1 93Q1 94Q1 95Q1 96Q1 97Q1 98Q1 99Q1 00Q1 01Q1 02Q1 % 8.0 Oupu Gap and 1.645*SE Bands ( Univariae Approach) 4.0 0.0-4.0-8.0-12.0 Figure 4: Bivariae Model Resuls (adding informaion concerning inflaion) log 10.4 Log Real GDP and Poenial Oupu Esimaed by Bivariae Approach 10.3 10.2 10.1 10.0 9.9 9.8 87Q1 88Q1 89Q1 90Q1 91Q1 92Q1 93Q1 94Q1 95Q1 87Q2 88Q2 89Q2 90Q2 91Q2 92Q2 93Q2 94Q2 95Q2 96Q2 97Q2 98Q2 99Q2 00Q2 01Q2 02Q2 96Q1 97Q1 98Q1 99Q1 00Q1 01Q1 02Q1 % 8.0 Oupu Gap and 1.645*SE Bands (Bivariae Approach) 4.0 0.0-4.0-8.0-12.0 87Q2 88Q2 89Q2 90Q2 91Q2 92Q2 93Q2 94Q2 95Q2 96Q2 97Q2 98Q2 99Q2 00Q2 01Q2 02Q2 Noe: All esimaes are wo-sided esimaes.
5. Conclusion The absence of a cerain measure for he oupu gap for he Turkish economy is he main moivaion of his sudy. Inensive usage of he oupu gap in he assessmen of he economic aciviy and also is noeworhy role in he near and he long-erm inflaion forecass wihin he inflaion-argeing moneary regime are he main reasons for he developmen of he numerous mehods conaining a variey of judgmen and complexiy levels. Among hese mehods, unobserved componen model is preferred o esimae he poenial oupu or, equivalenly he oupu gap due o is disincive advanages. In he basic univariae seup, we le he daa speak by jus using he ime series properies of he acual oupu daa. On he oher hand, i is known ha building models ha incorporae more economic conen can increase he reliabiliy of he esimaed gaps. For his reason, in he second round, he oupu gap is esimaed condiional on he informaion in he sysem (1)- (4), ogeher wih he abiliy of he gap measure o explain inflaion. Alhough he signal exracion saisics sugges ha incorporaing he supply side o he sysem reduces he parameer uncerainy and he oal sandard error and improves he gap esimae, he figure of univarie and bivariae esimaes does no show a clear disincion. According o our view, he reason for ha is he limied degree of relaion beween he inflaion and he level of he oupu gap due o he dominance of he previous regimes where inflaion is closely ied o exchange rae movemens and ineria. However we hink ha he exen of he relaion is subjec o change wih he floaing exchange rae and he new IT regime. I is imporan o bear in mind ha one of he reasons of he esimaion of he oupu gap is o provide informaion abou excess capaciy in he economy a a paricular poin in ime. However, i is no an easy job o idenify he absolue size of he oupu gap. Assumpions relaed wih he shocks, seady-sae growh rae, and he model specificaion used can place an imporan role in he decomposiion and so in he deerminaion of his absolue size given ha diverse assumpions may bring abou a range of possible soluions ha may differ in heir policy implicaions. All hese issues emphasize he imporance of a careful analysis and exper judgmen. This sudy can be considered as a preliminary sep o measure oupu gap for he Turkish economy. Alhough boh models give similar oupu gap esimaes, signal exracion saisics sugges promising resuls. Along wih his, laes economic developmens may sugges ha furher exensions especially regarding he demand side may deserve aenion.
Appendix: A.1. Daa Descripion x : Logarihmic seasonally adjused gross domesic produc a 1987 consan prices π : Quarerly log difference of consumer price index, (1994=100) pub π : Quarerly log difference of public manufacuring price index, (1994=100) m π : Quarerly log difference of impor price index, adjused for exchange rae, (1994=100) Figure 1: GDP and Growh Rae % 16.0 Grow h Rae Logarimic Seasonal Adjused GDP log 10.4 12.0 10.3 8.0 10.2 4.0 10.1 0.0-4.0 10.0-8.0 9.9-12.0 9.8 88Q1 89Q1 90Q1 91Q1 92Q1 93Q1 94Q1 95Q1 96Q1 97Q1 98Q1 99Q1 00Q1 01Q1 02Q1
A.2. Sae Space Represenaions The vecor of observed variables (oupu and inflaion) is denoed as x, while he vecor of unobserved sae variables (poenial oupu, rend growh rae and he oupu gap) is denoed by z. Then he measuremen equaion where he evoluion of he observed variables is described as a funcion of he unobserved sae variables and ransiion equaion are: x = Cz + Du + Hw z = Az + Be + Gw 1 A.2.1 where he u and e denoes vecors of normally disribued iid shocks which are assumed o be uncorrelaed and w is he vecor of exogenous variables. Equaions (1)-(5) can be wrien as: = µ 1 [ x ] [ 1 0 0 0 1 0 0] * x * x 1 µ z z 1 z 2 A.2.2 * * x 1 0 1 0 0 0 0 0 0 x σ 1 η 0 * * 1 0 0 0 0 0 0 0 0 0 x 1 x 2 0 µ 0 0 ρ 0 0 0 0 0 0 µ σ 1 ε 1 (1 ρµ ) 0 µ = 0 0 1 0 0 0 0 0 0 0 1 µ + 2 1 + 0 1 z 0 0 0 0 φ1 φ2 0 0 0 z σ 1 ξ 1 0 z 0 0 0 0 1 0 0 0 0 0 1 z 2 0 z 0 0 0 0 0 1 0 z 0 0 0 0 2 3 [] A.2.3 and when inflaion-oupu relaionship is included in he model, represenaion is: * x * x 1 µ 1 π 1 x 1 0 0 0 1 0 0 0 0 0 0 0 0 1 [] µ 1 π 2 π = 0 0 0 0 0 γ 0 + σ + ν 0 α1 α2 β ψ pub z π m z π 1 z 2 A.2.4 * * x 1 0 1 0 0 0 0 0 0 x σ 1 η 0 0 0 0 0 * * 1 0 0 0 0 0 0 x 0 0 0 1 x 2 1 0 0 0 0 0 µ 0 0 ρ 0 0 0 0 0 0 µ σ 1 ε 1 (1 ρµ ) 0 0 0 0 0 π 1 µ = 0 0 1 0 0 0 0 0 0 0 1 µ + 2 1 0 0 0 0 0 π + 2 z 0 0 0 0 φ1 φ2 0 0 0 z σ pub 1 ξ 1 0 0 0 0 0 π m z 0 0 0 0 1 0 0 0 0 0 1 z 2 0 0 0 0 0 π z 0 0 0 0 0 1 0 z 0 0 0 0 0 0 0 0 2 3 A.2.5
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