A compact open economy DSGE model for Switzerland

Transcription

1 A compac open economy DSGE model for Swizerland Barbara Rudolf and Mahias Zurlinden SNB Economic Sudies 8 / 2014

2 Legal Issues DISCLAIMER Economic Sudies represen he views of he auhor(s) and do no necessarily reflec hose of he Swiss Naional Bank. COPYRIGHT The Swiss Naional Bank (SNB) respecs all hird-pary righs in paricular righs relaing o works proeced by copyrigh (informaion or daa wordings and depicions o he exen ha hese are of an individual characer). SNB publicaions conaining a reference o a copyrigh ( Swiss Naional Bank/SNB Zurich/year or similar) may under copyrigh law only be used (reproduced used via he inerne ec.) for non-commercial purposes and provided ha he source is menioned. Their use for commercial purposes is only permied wih he prior express consen of he SNB. General informaion and daa published wihou reference o a copyrigh may be used wihou menioning he source. To he exen ha he informaion and daa clearly derive from ouside sources he users of such informaion and daa are obliged o respec any exising copyrighs and o obain he righ of use from he relevan ouside source hemselves. LIMITATION OF LIABILITY The SNB acceps no responsibiliy for any informaion i provides. Under no circumsances will i accep any liabiliy for losses or damage which may resul from he use of such informaion. This limiaion of liabiliy applies in paricular o he opicaliy accuracy validiy and availabiliy of he informaion. ISSN X (prined version) ISSN (online version) 2014 by Swiss Naional Bank Börsensrasse 15 P.O. Box CH-8022 Zurich

3 A compac open economy DSGE model for Swizerland* Barbara Rudolf a and Mahias Zurlinden b SNB Economic Sudies 8 / 2014 * Valuable commens by Karin Assenmacher Carlos Lenz and wo anonymous reviewers are graefully acknowledged. The views expressed in his sudy are hose of he auhors and do no necessarily represen hose of he Swiss Naional Bank. a Swiss Naional Bank P.O. Box CH-8022 Zurich; barbara.rudolf@snb.ch b Swiss Naional Bank P.O. Box CH-8022 Zurich; mahias.zurlinden@snb.ch

4 Conens Conens 2 Absrac 3 1. Inroducion 4 2. Model specificaion Domesic households Domesic producers of radable and non-radable goods Domesic imporers Uncovered ineres rae pariy and inernaional prices Moneary policy Foreign economy and general equilibrium Linearised model Esimaion Mehodology Daa and measuremen equaions Calibraion and prior disribuions Esimaion resuls Esimaion of alernaive specificaions Evaluaion Impulse responses o various shocks Variance decomposiion Hisorical decomposiions DSGE-VAR model Forecasing Concluding remark 48 A. Appendix 49 References 58 2 A Compac Open Economy DSGE Model for Swizerland

5 Absrac This sudy describes a compac dynamic sochasic general equilibrium (DSGE) model fied for he Swiss economy wih Bayesian echniques. The model feaures wo economies (small home economy large foreign economy) five ypes of agens (households producers of radables producers of non-radables reailers moneary auhoriy) nominal and real fricions and a number of shocks. The sudy gives deails on he specificaion and he esimaion of he model. The evaluaion is based on impulse responses and variance decomposiions a DSGE-VAR o assess misspecificaions and resuls of forecasing experimens. The model is one of he ools used for policy analysis and forecasing a he Swiss Naional Bank. JEL Classificaion: E30 E40 E50 Keywords: DSGE model open economy Bayesian esimaion forecasing moneary policy. A Compac Open Economy DSGE Model for Swizerland 3

6 1. Inroducion Dynamic sochasic general equilibrium (DSGE) models wih New Keynesian fricions such as price rigidiies have become a sandard ool in quaniaive macroeconomics. This paper describes an open-economy DSGE model fied for he Swiss economy wih Bayesian echniques. The model is par of a suie of models employed by Swiss Naional Bank (SNB) saff for policy analysis and forecasing. Versions of his model have been used a he SNB since The model belongs o he caegory of small-scale New Keynesian open-economy DSGE models. The core of hese models can be raced o Galí and Monacelli (2005) who exended he benchmark New Keynesian DSGE model o a small open economy seing. Monacelli (2005) hen incorporaed price-seing reailers and incomplee exchange-rae pass-hrough. Jusiniano and Preson (2010b) added habi persisence in consumpion and parial indexaion o inflaion. Models of his sor have been esimaed for several counries e.g. by Bäuerle and Menz (2008) and Belran and Draper (2008) for Swizerland. They are all smaller in size han he ypical medium-scale open-economy DSGE models exemplified by Adolfson e al. (2007). While he laer models include sicky wages and capial accumulaion he small-scale models do no. The scope for soryelling and policy analysis increases wih he size of he model. On he downside however large models are ofen less ransparen and idenificaion problems end o be worse. Our model assumes wo economies linked by rade and porfolio flows: he home economy and he foreign economy. Households in he home economy receive uiliy from consumpion and leisure. They spend on goods produced a home and abroad supply labour o domesic producers own he domesic firms and hold domesic and foreign bonds. Consumpion responds slowly due o habi persisence. The firms are of hree ypes: reailers producers of radables and producers of non-radables. The reailers sell impored goods in he domesic marke. The oher firms employ labour o produce eiher non-radables in demand a home or radables in demand a home and abroad. All firms are monopolisically compeiive and se prices in a saggered fashion as in Calvo (1983). In addiion here is parial indexaion o inflaion observed in he previous period. The pass-hrough of he nominal exchange rae o impor prices is incomplee reflecing he assumpion of price sickiness in he reail secor. The foreign economy is modelled along he same lines as he home economy. However inernaional rade and porfolio flows are ignored in modelling he foreign economy because he home economy is assumed o be small relaive o he foreign economy. The model s srucure is similar o ha in Jusiniano and Preson (2010b) wih hree noable differences. Firs a non-raded secor is incorporaed in he home economy following Maheson (2010). This allows us no only o obain a more complee picure of CPI inflaion and is componens bu also o accoun for he cross-secion differences in price sickiness and exchange rae pass-hrough ha we observe in he daa. Secondly he uncovered ineres pariy (UIP) is modified as in Adolfson e al. (2008) o address he forward premium puzzle. Under he modified UIP he exchange rae exhibis a humpshaped paern afer a moneary policy shock. The effecs on real oupu and oher variables 4 A Compac Open Economy DSGE Model for Swizerland

7 are herefore more persisen han under he sandard UIP. Thirdly we assume ha preference (demand) shocks in he foreign economy spill over o hose in he home economy. This provides a shor cu o capure he cross-border effecs of a foreign demand shock on he home economy documened in he empirical lieraure. Wihou his modificaion he cross-border effecs of a demand shock in he foreign economy would be implausibly small similar o he resuls discussed in Jusiniano and Preson (2010a). The presen sudy provides a comprehensive documenaion of he model and is evaluaion. Secions 2 and 3 presen he model in is original and is log-linearised forms. Secion 4 describes he mehodology and he resuls of he esimaion including some resuls for alernaive model specificaions. Secion 5 evaluaes he model s empirical properies. We repor resuls of impulse responses and forecas-error variance decomposiions o gauge he credibiliy of he model. Flucuaions in he rae of inflaion and he oupu gap are decomposed o assess he imporance of various shocks over he las few years. Following Del Negro e al. (2007) a DSGE-VAR version is esimaed in order o sudy misspecificaion of he DSGE model. Finally resuls of a forecasing experimen are repored in order o shed ligh on he forecasing abiliy of he model. Secion 6 concludes. The appendix conains more complee resuls for he alernaive model specificaions. A Compac Open Economy DSGE Model for Swizerland 5

8 2. Model specificaion This secion describes he decision problem of households and firms and he moneary policy funcion of he cenral bank. The various nominal and real rigidiies are inroduced and he shock processes are defined. More deailed accouns of some aspecs of he model can be found in Galí and Monacelli (2005) Monacelli (2005) and Jusiniano and Preson (2010b). 2.1 Domesic households The domesic economy is populaed by infiniely lived households whose preferences are given by he ineremporal uiliy funcion E ( C H ) N 1 s 1+ 0 b ZG = 0 1 s 1+ (2.1) where C is a composie consumpion index and N is labour inpu. Labour is supplied o he raded and he non-raded secor N = N N + N H where he subscrips N and H refer o non-radables and radables produced in he home counry. Consumpion paerns are assumed o change sluggishly (habi persisence) which is refleced in H hc 1. The parameer b denoes he discoun facor s > 0 is he coefficien of relaive risk aversion (or he inverse elasiciy of ineremporal subsiuion in consumpion) and > 0 is he inverse elasiciy of he labour supply. Z G is a preference shifer common o all households. The composie consumpion index C is given by g N (1 g) T C = C + C where C N and C T are he indices of consumpion of non-raded and raded goods respecively g is he share of non-raded goods in he consumpion bundle and > 0 is he elasiciy of subsiuion beween radable and non-radable goods. The index of consumpion of raded goods C T is given by h 1 h 1 1 h 1 h 1 h h h h T a F (1 a) H C = C + C where a is he share of foreign goods in he consumpion bundle of raded goods and h > 0 is he elasiciy of subsiuion beween domesic and foreign raded goods. The indices of consumpion of non-raded goods (C N ) raded goods impored from he foreign economy (C F ) and raded goods produced in he home economy (C H ) are given by he CES aggregaes 6 A Compac Open Economy DSGE Model for Swizerland

9 N () () () 0 N F 0 F H 0 H C C i di C C i di C C i di wih > 1 denoing he elasiciy of subsiuion beween varieies i [01] wihin a given secor and counry. The demand funcions of hese producs are given by PN() i PF() i PH() i CN () i = CN CF () i = CF CH () i = CH PN PF PH (2.2) where N () () () 0 N F 0 F H 0 H P P i di P P i di P P i di are he price indexes of non-raded goods (P N ) foreign raded goods (P F ) and domesic raded goods (P H ). These demand funcions are he resul of he individual household s inraperiod opimisaion problem which deermines he opimal allocaion of expendiure on all ypes of goods. The households allocaion of expendiure on domesic and foreign radable goods is based on he demand funcions h P F P H CF = a CT CH = (1 a) CT P P T T h (2.3) and heir allocaion of oal expendiure on radable and non-radable goods is based on PN PT CN = g C CT = (1 g) C P P (2.4) where P T is he price index of radables defined as 1 1 h 1 h 1 h T a F (1 a) H P P + P (2.5) and P is he consumer price index defined as g N (1 g) T P P + P. (2.6) Assuming ha all households face idenical decision problems and ha markes are complee he aggregae flow budge consrain akes he form PC + D + SB WN +Π +Π +Π + R D + SR Φ() B + T (2.7) N H F where W is he nominal wage rae P N P H and P F are disribued profis from domesic producers and reail firms T are ransfers ne of axes S is he nominal exchange rae defined as he home currency per uni of foreign currency D and B are domesic and foreign one-period nominally riskless bonds held by he home economy s households A Compac Open Economy DSGE Model for Swizerland 7

10 beween ime and +1 R and R are he corresponding gross ineres raes and F ( ) is a risk premium (o be discussed below in deail). Households maximise he ineremporal uiliy funcion Eq. (2.1) subjec o a sequence of budge consrains Eq. (2.7) and he borrowing consrains D +1 I and B +1 I which preven agens from borrowing an unlimied amoun. This yields he firs-order condiions: U = l P (2.8) C U = l W (2.9) N l = bel + 1R (2.10) S l = be[ l S R Φ ] (2.11) where l denoes he Lagrange muliplier U C = Z G (C H ) s is he marginal uiliy of consumpion and U N = Z G N is he marginal disuiliy of labour. We complee he se of opimaliy condiions by adding a ransversaliy condiion and specifying ha he flow budge consrain mus hold wih equaliy in every period. Combining Eq. (2.8) wih Eq. (2.9) yields he inraemporal consumpion/leisure choice: N W = s ( C H ) P (2.12) saing ha he marginal rae of subsiuion beween leisure and consumpion is equal o he real wage. Combining Eq. (2.8) wih Eq. (2.10) gives he ineremporal consumpion choice (consumpion Euler equaion): 1 s ZG + 1( C+ 1 H 1) P + = be s. R ZG ( C H) P + 1 (2.13) 2.2 Domesic producers of radable and non-radable goods Domesic producers produce eiher radables or non-radables. In each group here is a coninuum of monopolisically compeiive firms indexed i [01]. In wha follows we use he subscrip j = NH o describe he decision problem of firms producing non-radables (N) and radables (H) respecively. Each firm i produces differeniaed goods using he producion funcion Y () i = Z Z N () i (2.14) j T Aj j where Z T is a non-saionary echnology process ha is common o all domesic producers Z Aj is a saionary echnology shock common o domesic producers of non-radables and radables respecively and N j (i) is he labour inpu of firm i where he labour marke is assumed o be perfecly compeiive. We can wrie ln ZT = ln ZT 1 + g Z + zt where g Z > 0 is he seady-sae growh rae of oupu and z T is a saionary AR(1) process. Prices are subjec o Calvo-ype price seing. In every period each firm has a probabiliy (1 x H ) of being able o reopimise is price. Firms ha canno reopimise are assumed o parially index heir prices o recen home goods inflaion according o 8 A Compac Open Economy DSGE Model for Swizerland

11 j P j 1 Pj () i = Pj 1() i P j 2 k where 0 k j 1 is he indexaion parameer. Since all he firms ha can reopimise in a given period face he same decision problem hey will choose a common new price P j. The CES aggregae for he price level P j can hen be rewrien as P (1 ) j j P j j j k 1 H P j 1 = (1 x ) + x P. P j 2 (2.15) When firms are able o reopimise heir prices hey choose he new price in a way ha maximises a weighed sum of discouned curren and expeced fuure profis: k j P T jt 1 x j T jt () P j () jt jt T= P j 1 E Q Y i i P MC subjec o he demand equaion and k H H() i P P HT 1 HT () = ( HT HT ) P HT P + H 1 Y i C C k N P N() i P NT 1 YNT() i CNT P NT P N 1 = respecively depending on he ype of firm considered. Wih a common producion echnology Eq. (2.14) and a fully compeiive labour marke all firms of a given ype face he same real marginal cos MC jt = W T / (Z Z P Aj T TT jt ). The T parameer x j denoes he probabiliy ha he currenly se price will sill be in place periods from now; Q T T 1 1 is he sochasic discoun facor where QT = ( Π j= 0 R+ j) and C HT is he foreign consumpion of goods produced in he home economy. The resuling firs-order condiion for he producer s opimisaion problem is kh P T jt 1 x j T jt P j jt jt T= Pj 1 1 E Q Y () i () i P MC = 0. (2.16) A Compac Open Economy DSGE Model for Swizerland 9

12 2.3 Domesic imporers There is a coninuum of monopolisically compeiive reail firms indexed i [01]. These firms sell impored goods in he domesic marke. Due o heir price-seing power he price hese firms charge in he domesic marke differs from he world marke price. Again we assume ha in each period a fracion (1 x F ) of all firms opimally ses prices while a fracion x F follow he backward-looking rule of humb ha parially indexes prices o he las period s inflaion. The price index of impored goods is k F P (1 ) F 1 F = (1 xf) F + x F F 1. P F 2 P P P (2.17) Reopimising firms choose he price for good i by maximising he expeced presen discouned value of profis kf P T FT 1 xf T FT () P F () T FT T= P F 1 E Q C i i SP subjec o he demand curve k F P F() i P FT 1 CFT() i CFT P FT P F 1 = for all. The resuling firs-order condiion for he reailer s opimisaion problem is kf P T FT 1 xf T FT () P F () T FT 0 T= P F 1 1 E Q C i i SP =. (2.18) The wedge beween he world marke price of foreign goods paid by imporing firms (S P ) and he domesic currency price (P F F ) of hese goods paid by domesic consumers is called he law-of-one-price gap defined as SP Ψ F. PF (2.19) There is a law-of-one-price gap if Ψ A Compac Open Economy DSGE Model for Swizerland

13 2.4 Uncovered ineres rae pariy and inernaional prices From he asse pricing condiions Eq. (2.10) and Eq. (2.11) we obain a UIP condiion of he form S R E l Φ = () ls R (2.20) where F ( ) denoes a risk premium. A posiive (negaive) risk premium implies ha he expeced reurns on he foreign economy bond are smaller (larger) han he expeced reurns on he home economy bond. As poined ou by Schmi-Grohe and Uribe (2003) he inroducion of a risk premium which depends on he scaled foreign asse posiion ensures a well-defined seady sae. Various auhors propose modificaions o accoun for he observaion ha in he daa he risk premia are srongly negaively correlaed wih he expeced change in he exchange rae (forward premium puzzle). Adolfson e al. (2008) le he risk premium depend on he expeced change in he exchange rae beween + 1 and 1. The risk premium F ( ) hen akes he form Φ = + S+ 1 S + 1 A R R Z exp A( A A) SE 1 Z S 1 S 1 (2.21) where A (S B ) / (Z T P ) is he real quaniy of ousanding deb expressed in erms of domesic currency as a fracion of seady sae oupu and Z is he exogenous componen of he risk premium. The modificaion proposed by Adolfson e al. is revered if we se S = 0. I is convenien a his sage o inroduce he real exchange rae S r and he erms of rade X. Boh definiions will be used laer when considering he log-linearised version of he model. The real exchange rae is defined as S SP r (2.22) P while he erms of rade are defined as X P F. (2.23) PH The definiion of he erms of rade corresponds o he relaive price of foreign goods in erms of domesic goods sold in he home economy. We assume producer currency pricing for he home economy s expor secor. The price charged o cusomers abroad is he same as he price charged in he home economy (complee exchange rae pass-hrough). A Compac Open Economy DSGE Model for Swizerland 11

14 2.5 Moneary policy The cenral bank follows a Taylor-ype ineres rae rule of he form 1 rr rr p y y 1 / = ZR 1 Y 1/ Y 1 R R P Y Y Y R R P Y (2.24) where R and Y are seady-sae values of gross nominal ineres raes and oupu and Z R is an exogenous moneary policy shock. The rule is fairly general in ha moneary policy responds o conemporaneous inflaion he oupu gap and changes in he oupu gap. In addiion he specificaion allows for policy ineria or ineres-rae smoohing behaviour of he cenral bank. The ineres rule does no feaure feedback from he nominal exchange rae. Exchange rae consideraions are accouned for o he exen ha hey are refleced in he oupu gap and he rae of inflaion Foreign economy and general equilibrium The small open economy assumpion implies ha he home economy is negligible in size relaive o he foreign economy. Expors o and impors from he home economy are assumed o be oo small o maer for he foreign economy. The same holds for porfolio flows from and o he foreign economy. Thus he shares of home economy goods and bonds in he consumpion bundle and porfolio of he foreign economy households are se o zero. Furhermore we make he simplifying assumpions ha he foreign economy produces only raded goods and ha as described in Secion 2.4 he price of home economy goods paid by consumers in he foreign economy corresponds o he price of hese goods se in he home economy divided by he exchange rae (complee exchange rae pass-hrough). Apar from ha he foreign economy is specified as he closed economy varian of he model described above for he home economy. Wih sarred variables denoing he foreign economy variables we can wrie P P Y = Y and C = C. = F F The foreign demand for he raded good produced in he home economy is assumed o be deermined as F C P = Y H H P h where h > 0. 1 In Sepember 2011 he SNB se a minimum exchange rae a 1.20 Swiss francs per euro agains he background of a massive appreciaion of he Swiss franc and he perceived risk of deflaionary developmens. The minimum exchange rae was announced as an addiional operaional arge as he arge for he shor-erm ineres rae could no be lowered furher due o he zero bound. Eq. (2.24) does no model his modificaion of he SNB s moneary policy framework. 12 A Compac Open Economy DSGE Model for Swizerland

15 In equilibrium all markes mus clear. Marke clearing in he home economy requires ha he markes for all individual raded and non-raded goods are cleared: Y () i = C () i + C () i H H H h h PH() i PH PT P H = (1 g) (1 a) C a C P P + P SP H T PN() i PN YN () i = CN () i = g C. P P N (2.25) (2.26) The second equaliies in Eq. (2.25) and Eq. (2.26) make use of Eq. (2.2) and he assumpions ha preferences are symmeric in he home and he foreign economy and ha in equilibrium domesic bonds are in zero ne supply. Plugging Eq. (2.26) and Eq. (2.25) ino he definiion of he aggregae domesic oupu 1 1 ( () ) 0 1 Y Y i di yields h h PN P H PT PH Y = g + (1 g)(1 a) C (1 ga ) C P P +. T P SP (2.27) Marke clearing in he foreign economy requires Y = C. (2.28) A Compac Open Economy DSGE Model for Swizerland 13

16 3. Linearised model In his secion we describe he log-linearised version of he model presened in Secion 2. The model is log-linearised around he saionary seady sae of he derended variables. The resuling equaions are linear in he log-deviaions of he variables from he seady sae. The log-deviaions are denoed by lower case leers. 2 Log-linearising he domesic households Euler equaion Eq. (2.13) yields 1 1 c hc 1 = Ec 1 hc (1 h)( i Ep 1) + (1 h)( zg EzG 1). (3.1) + s + s + Consumpion depends boh on pas consumpion reflecing habi persisence and on fuure consumpion implying consumpion smoohing. We can also see ha moneary policy will have real effecs if i affecs he real ineres rae. CPI inflaion defined by Eq. (2.6) becomes p = gp + (1 g) p N T = gp + (1 g) (1 a) p + ap N H F = gp + (1 g) p + a x (3.2) N H where he change in he erms of rade defined by Eq. (2.23) is x = p p. (3.3) F H The log-linearised versions of he aggregae impor price index Eq. (2.17) and he reail firms firs-order condiion Eq. (2.18) imply he Phillips curve for impored goods: p k p = b( Ep k p ) + + z (3.4) F F F 1 F + 1 F F F mf 1 where = s + p pf is he law-of-one-price gap F = (1 xf)(1 bxf)( xf) and z is mf a shock o he desired mark-up. Similarly aking log-linearisaion of he aggregae price index Eq. (2.15) and he producers firs-order condiion Eq. (2.16) gives he New Keynesian Phillips curves for non-radables and radables produced in he home economy: p k p = b( E p k p ) + mc + z (3.5) N N N 1 N + 1 N N N N mn 2 A echnical appendix conaining more deails on he log-linearisaion is available from he auhors. 14 A Compac Open Economy DSGE Model for Swizerland

17 p k p = b( E p k p ) + mc + z (3.6) H H H 1 H + 1 H H H H mh 1 1 where N = (1 xn)(1 bxn)( xn) and H = (1 xh)(1 bxh)( xh) while z and z are mn m H shocks o he respecive mark-up desired. The real marginal coss mc N and mc H are derived from he log-linearised versions of he producion funcion Eq. (2.14) and he inraemporal consumpion-leisure choice Eq. (2.12) yielding and mc y z z h c hc 1 N = (1 + ) T (1 + ) A (1 ) ( 1) N + s 1 g (1 g)(1 a) + ( sr ) x (3.7) g g mc y z z s h c hc 1 H = (1 + ) T (1 + ) A (1 ) ( 1) H + s + + x (3.8) r respecively. The log-linearised UIP derived from Eq. (2.20) and Eq. (2.21) can be wrien as i i= (1 S) E s+ 1 S s Aa+ z (3.9) while he real exchange rae becomes s = s + p p r where x = p F p H. The flow budge consrain Eq. (2.7) implies = + x p + p (3.10) H where i a = a 1 (1 h + ) + [ a+ h(1 a) (1 h + ) ] x g Z ZT + sr c + ln + y Z T 1 SB a = (1 ga ) ZT P (3.11) i * is he seady-sae ineres rae in he foreign economy g Z is he seady-sae oupu growh and ln( Z T) T / Z is he relaive produciviy rend. In deriving Eq. (3.11) we use ha domesic deb is in zero ne supply in equilibrium and 3 W N +Π +Π +Π T PC = P C S P C. N H F H H F 3 See Jusiniano and Preson (2008). A Compac Open Economy DSGE Model for Swizerland 15

18 Marke clearing of he goods marke implies y = [ g+ (1 g)(1 a)] c (1 ga ) ( h) + (1 g) a[ h a( h ) ( h )] x + (1 g) as r ZT + (1 ga ) ln y + Z T (3.12) where we make use of he log-linearised demand funcions from he opimal allocaion of expendiures beween raded and non-raded goods and beween domesic and foreign radables. Finally he moneary policy rule described by Eq. (2.24) becomes i = rr i 1 + (1 rr)( p p + yy) + y y + zr (3.13) which is a Taylor-ype rule wih ineres-rae smoohing. Turning o he foreign economy he log-linearised equaions are y 1 1 hy = Ey hy 1 1 (1 h )( i E p 1) (1 h )( z Ez + + ) G G + 1 s + s (3.14) p kp = b( E p kp) + mc + z (3.15) m mc = y + z + h y h y (3.16) 1 (1 ) s (1 ) ( 1) T i = rr i 1 + (1 rr)( p + y) + y + z. (3.17) p y y R These are closed-economy counerpars of he log-linearised equaions derived for he home economy. The sysem of log-linear equaions described above is driven by 13 exogenous shocks. There are hree echnology shocks in he domesic economy a uni-roo echnology shock (z T ) and wo saionary echnology shocks one in he radable goods secor (z AH ) and one in he non-radable goods secor (z ). Furhermore we have a preference shock (z ) a AN G counry risk premium shock ha affecs he relaive riskiness of foreign o domesic asses (z ) hree mark-up shocks one for each ype of good (z z z mh m N m F ) and a moneary policy shock (z R ). In he foreign economy block here are a uni-roo echnology shock (z T ) a preference shock (z G ) a mark-up shock in he pricing of foreign goods (z m ) and a moneary policy shock (z R ). These shocks are defined as AR(1)processes wih i.i.d. innovaions. Excepions are he wo moneary policy shocks which are assumed as i.i.d. In addiion i should be emphasised ha we specify he home economy s preference shock d z G as a weighed average of a domesic preference shock z G and he foreign economy s preference shock z G. This specificaion allows he preference shock in he home economy o co-move wih he corresponding shock in he foreign economy. The empirical evidence 16 A Compac Open Economy DSGE Model for Swizerland

19 srongly suggess a common facor in inernaional business cycles (see Kose e al. (2008) among ohers). The case wih no spillover (a G = 0 in Eq. (3.18)) is considered in Secion 4.5. The shock processes can be wrien as zt = rtzt 1 + T za H = ra z H AH 1 + A H za N = ra z N AN 1 + A N d zg = (1 ag) zg + agzg d d d d zg = rgzg 1 + G z = rz 1 + zm H = rm z H mh 1 + m H zm F = rm z F mf 1 + m F zm N = rm z N mn 1 + m N zr = R zt = r T zt + 1 T zg = r z G G + 1 G z r z m = m m + 1 m z = (3.18) R R where he AR(1) parameers are 0 < r i < 1 for all i. We adop hree addiional assumpions o esimae he model. Firs he saionary echnology shock in he raded goods secor is idenical wih he one in he non-raded goods secor (z = z = z AH A N A ). This assumpion is inroduced because he parameers are hard o idenify separaely from he daa. Secondly he subsiuion elasiciy beween domesic and foreign raded goods is he same in boh economies (h = h ). Thirdly an equaion for inflaion of oil producs p oil is added o he model and CPI inflaion is calculaed as a linear combinaion of CPI inflaion ex oil producs and oil produc inflaion. Alernaively we could specify he presence of an impored inpu of producion whose demand would be opimally chosen by he raded goods firm. We ake a shor cu for reasons of simpliciy and convenience. The prices of oil producs are nooriously hard o predic. They have a subsanial effec on he volailiy of he CPI inflaion rae while heir effec hrough he producion funcion is hard o pin down. In addiion he model simulaions underlying he inflaion forecas published by he SNB are ypically based on he assumpion ha he oil price says consan a is curren level. Thus oil produc inflaion in he model is assumed o follow an AR(1) process of he form p = r p + z (3.19) oil oil oil 1 oil where z oil = oil is a whie noise innovaion. A Compac Open Economy DSGE Model for Swizerland 17

20 4. Esimaion The raional expecaion soluion of he log-linearised model is esimaed wih Bayesian Maximum Likelihood on daa for Swizerland. 4 Following Smes and Wouers (2003) Bayesian Maximum Likelihood has become he sandard mehod for esimaing DSGE models. The Bayesian approach requires choosing prior disribuions of he parameers o be esimaed. These priors represen our previous knowledge. The priors are updaed wih observed daa using Bayes rule. The resuling poserior disribuions are hen used o compue he parameer esimaes. 4.1 Mehodology The esimaion of DSGE models wih Bayesian mehods is described in a series of papers by Frank Schorfheide and various co-auhors. This secion gives a brief ouline based on An and Schorfheide (2007) and Schorfheide e al. (2010). Eq. (3.1) o Eq. (3.19) form a linear raional expecaions sysem. The soluion of his sysem can be wrien as s =Φ ( ) s +Φ ( ) (4.1) 1 1 where he vecor s conains he sae variables he vecor conains he innovaions o he exogenous processes and he coefficiens of he marices F 1 and F are funcions of he model parameers colleced in vecor. The sae variables are linked o observed daa via a se of measuremen equaions (o be discussed in Secion 4.2). As some variables in he observed daa se are in growh raes we augmen he se of saes s wih lagged values of sae variables o allow for lagged sae variables in he measuremen equaions. The augmened vecor of sae variables akes he form V = [ s s M ( )] (4.2) 1 s where M s () is a suiably chosen marix. Eq. (4.1) can be rewrien as V =Φ 1 ( ) V 1+Φ ( ) (4.3) and he measuremen equaions can be wrien in compac form as 4 We use he gensys procedure described in Sims (2002) o compue he raional expecaions soluion of he DSGE model and one of he Gauss rouines provided on Frank Schorfheide s homepage o perform he Bayesian esimaion. 18 A Compac Open Economy DSGE Model for Swizerland

21 y y = A ( ) + A V + ε (4.4) 0 1 where he vecor y conains he observables and he vecor ε y collecs he measuremen errors. Eq. (4.3) and Eq. (4.4) form he sae space represenaion of he DSGE model. Assuming ha he innovaions are i.i.d. realisaions of a normal disribuion he likelihood funcion p(y T ) where Y T = [y y T ] can be evaluaed using he Kalman filer. The Kalman filer also generaes a sequence of esimaes of he sae vecor V : V ( ) [ = E V Y ]. (4.5) The Bayesian esimaion of he DSGE model combines a prior densiy p() wih he likelihood funcion p(y T ) o obain a join probabiliy densiy funcion for daa and parameers. The poserior disribuion is given by T p( Y ) p( ) T p( Y ) = p( Y) (4.6) where p( Y T ) = p( Y T ) p( ) d p( Y T ) is he marginal daa densiy. We employ Markov-Chain-Mone-Carlo (MCMC) mehods described in An and Schorfheide (2007) o implemen he Bayesian inference. More specifically a random-walk Meropolis-Hasings algorihm is used o generae draws from he poserior disribuion p( Y T ). The poserior momens are compued from he poserior draws. 4.2 Daa and measuremen equaions Daa for Swizerland are aken from wo sources: he Swiss Federal Saisical Office (SFSO) and he Swiss Naional Bank (SNB). Oupu is measured by real GDP (SFSO) he price level by he CPI (SFSO) he ineres rae by he hree-monh CHF Libor (SNB) he erms of rade by he raio of CPI foreign goods o CPI home goods (SFSO) and prices of oil producs by he CPI componen of gasoline diesel and fuel oil (SFSO). The real exchange rae is defined as a weighed average of he EUR/CHF and USD/CHF real exchange raes where he weighs are 0.7 and 0.3 respecively. The bilaeral real exchange raes are compued using he corresponding nominal exchange raes and CPI daa for Swizerland he euro area and he US. All foreign variables are weighed averages of he euro area and US daa (i.e. euro area and US real GDP for foreign oupu euro area and US CPI for he foreign price level and EUR and USD hree-monh Libor for he foreign ineres rae). The weighs correspond o hose on he bilaeral exchange raes in he calculaion of he effecive exchange rae. Prior o esimaion we ransform real GDP and he erms of rade ino quarer-on-quarer growh raes and he price levels ino annualised quarer-on-quarer growh raes (compued as firs differences in he naural logarihm of he seasonally adjused variable and muliplied by 100 and 400 respecively). Furhermore poenial labour hours (in logs) are subraced from he Swiss real GDP (in logs) o remove he ime varying rend in he growh rae of labour inpu in Swizerland. A smooh HP rend (smoohing parameer 10000) is removed from he log of foreign real GDP he log of he real exchange rae and he log of he erms of rade. To accoun for disinflaion in he early 1990s he mean A Compac Open Economy DSGE Model for Swizerland 19

22 he pos-1994 period (by subracing from he raes in he period he difference beween he mean of he period and he mean of he pos-1994 period). This The resuling se of saionary observables (measuremen variables) includes Δ y i s Δ x Δ y i. These en variables consiue vecor y. Char 1 shows he ime series of he variables for he period 1983Q2 o 2013Q2. The en measuremen equaions are Δ y = 100( + y y + z + ) Z 1 = 400( + (1 ) + + ) = 400( + + ) i = 400( i+ i) s = 100( s + s + ) Δ x = 100( Δ x+ x x + ) 1 Δy = Z + y y 1 + z + 100( ) = 400( + + ) i = 400( i + i ) = 400( + + ). (4.8) 4.3 Calibraion and prior disribuions Mos parameers of he model are esimaed bu some parameers are calibraed. The calibraed parameers are displayed in Table 1. The seady-sae discoun facor is se o and he risk premium parameer is se o The weigh of non-raded goods and services in he CPI (ex oil producs) is se o 0.6 he weigh of foreign goods in he raded goods componen of he CPI (ex oil producs) is se o 0.23 and he share of oil producs in he CPI is se o These values for and are calculaed as muli-year averages based on CPI weighs provided by SFSO. The spillover from foreign o home preference shocks is se o G 0. seady-sae values Z i s x * Z i and are se o heir sample means. The measuremen errors have zero mean and he variance is se o 0.05 in and 0.1 in. xdaa The assumpions abou he prior disribuions of he esimaed parameers are summarised in Table 2. Our choice is guided by he evidence gahered from micro-sudies and by he prior disribuions of he home economy parameers. The corresponding foreign economy assumpions are he same excep ha prior parameers governing he moneary policy rule are less concenraed in he foreign economy block han in he home economy block of he model. 20 A Compac Open Economy DSGE Model for Swizerland

23 Char 1 Daa Oupu growh (quarer-on-quarer (qoq) in %) Inflaion CPI excluding oil producs (qoq annualised in %) Impored gooods inflaion (qoq annualised in %) Oil produc inflaion (qoq annualised in %) Ineres rae (in %) A Compac Open Economy DSGE Model for Swizerland 21

24 Char 1 coninued Real exchange rae (% dev.) Terms of rade change (qoq in %) Foreign oupu growh (qoq in %) Foreign inflaion (qoq annualised in %) Foreign ineres rae (in %) A Compac Open Economy DSGE Model for Swizerland

25 Table 1 Calibraed parameers Srucural parameers Discoun facor b Share of non-raded goods and services in CPI (ex oil producs) g 0.6 Share of foreign goods in raded goods componen of CPI (ex oil producs) a 0.23 Share of oil producs in CPI a oil 0.04 Risk premium f A Spillover from foreign o home preference shock a G 0.4 Seady sae calibraions Seady sae oupu growh g Z Seady sae inflaion rae p Seady sae inflaion rae impored goods p F Seady sae nominal ineres rae i Seady sae real exchange rae s r Seady sae change in erms of rade D x Seady sae oupu growh in foreign economy g * Z Seady sae inflaion rae in foreign economy p * Seady sae nominal ineres rae in foreign economy i * Seady sae oil produc price inflaion p oil Disribuion of measuremen errors e y Daa e p Daa e pf Daa e Sr Daa e x Daa e y *Daa e p *Daa Daa N(00.1) N(00.05) N(00.05) N(00.05) N(00.05) N(00.1) N(00.05) oil N(00.1) Table 2 Prior disribuions Parameer ype mean sdv. Domesic behavioural parameers Calvo: Home x H Bea Calvo: Foreign x F Bea Calvo: Non-raded x N Bea Indexaion: Home k H Bea Indexaion: Foreign k F Bea Indexaion: Non-raded k N Bea Habi formaion h Bea Inverse elasiciy: Cons/Labour s Gamma Inverse elasiciy: Labour j Gamma Elasiciy: Home/Foreign h Gamma Elasiciy: Traded/Non-raded n Gamma A Compac Open Economy DSGE Model for Swizerland 23

26 Table 2 coninued Parameer ype mean sdv. UIP risk premium: Modificaion Adolfson e al. f S Bea UIP risk premium: Modificaion Chrisiano e al. f i Gamma Policy: Ineres rae smoohing r R Bea Policy: Inflaion p Gamma Policy: Oupu y Gamma Policy: Oupu growh Dy Gamma Foreign behavioural parameers Foreign Calvo x * Bea Foreign indexaion k * Bea Foreign habi formaion h * Bea Foreign inverse elasiciy: Cons/Labour s * Gamma Foreign inverse elasiciy: Labour j * Gamma Foreign policy: Ineres rae smoohing r R * Bea Foreign policy: Inflaion y p * Gamma Foreign policy: Oupu y y * Gamma Foreign policy: Oupu growh y Dy * Gamma AR(1) coefficiens and sandard deviaions Non-saionary echnology shock r T Bea Saionary echnology shock r A Bea Preference shock r d G Bea Risk premium shock r f Bea Mark-up shock: Home r mh Bea Mark-up shock: Foreign r mf Bea Mark-up shock: Non-raded r mn Bea Foreign echnology shock r T * Bea Foreign preference shock r G * Bea Foreign mark-up shock r m * Bea Oil price shock r oil Bea Sd dev: Non-saionary echnology shock s T InvGamma Sd dev: Saionary echnology shock s A InvGamma Sd dev: Preference shock s d G InvGamma Sd dev: Risk premium shock s f InvGamma Sd dev: Mark-up shock: Home s mh InvGamma Sd dev: Mark-up shock: Foreign s mf InvGamma Sd dev: Mark-up shock: Non-raded s mn InvGamma Sd dev: Moneary policy shock s R InvGamma Sd dev: Foreign echnology shock s T * InvGamma Sd dev: Foreign preference shock s G * InvGamma Sd dev: Foreign mark-up shock s m * InvGamma Sd dev: Foreign moneary policy shock s R * InvGamma Sd dev: Oil price shock s oil InvGamma A Compac Open Economy DSGE Model for Swizerland

27 Parameers bounded by heory beween 0 and 1 are given sandardised bea disribuions. This perains o he Calvo and indexaion parameers consraining he price-seing decisions of producers and reailers he habi formaion parameer he Adolfson e al. modificaion of he UIP risk premium he degree of ineres-rae smoohing in he moneary policy rule and he AR(1) coefficiens of he shock processes. The prior means for he Calvo parameers are se o 0.75 implying an average price spell duraion of four quarers. This corresponds wih he average duraion of price rigidiy for Swizerland repored in Kaufmann (2009). The sandard deviaion is se o 0.05 implying a relaively small uncerainy abou he degree of price rigidiy. The prior means for he indexaion parameers are se o 0.5 wih a sandard deviaion of 0.1 reflecing a lack of knowledge on his parameer. For he UIP modificaion we se a prior mean of 0.4 wih a sandard deviaion of 0.1. For he degree of habi persisence in consumpion he prior mean is se o 0.7 and he sandard deviaion o 0.05 broadly in line wih assumpions made in oher DSGE models. The ineres-rae smoohing parameer is cenred around 0.8 wih a sandard deviaion of 0.05 consisen wih esimaed values obained in sandard Taylor rule equaions wih he hree-monh Libor CPI inflaion and various measures for he oupu gap. Priors for he AR(1) coefficiens of he exogenous shock processes are assumed o have a mean of 0.5 (0.8 in he case of uni-roo echnology shocks and preference shocks). Parameers resriced o be posiive are given eiher gamma or inverse gamma disribuions. The priors of he invered elasiciy of ineremporal subsiuion in consumpion (prior mean: 1.5) he invered labour supply elasiciy (1.0) he elasiciy of subsiuion beween radables and non-radables (1.0) and he elasiciy of subsiuion beween raded home goods and impored goods (1.0) are all specified as gamma disribuions. Wih sandard deviaions of 0.1 hey are se o be relaively non-informaive allowing he poseriors o be primarily influenced by he daa. For he parameers on he cenral bank s response o inflaion and oupu movemens he prior mean is se o 1.5 and 0.5 respecively in line wih he values proposed by Taylor (1993). The prior mean of he cenral banks responses o oupu growh is se o 0.2. The prior disribuions for he sandard deviaions of he srucural shocks are modelled as inverse gamma disribuions wih 2 degrees of freedom and a common mode of 0.10 reflecing he fac ha here is lile prior informaion on hese parameers. 4.4 Esimaion resuls The model is esimaed for he sample period 1983Q2 o 2013Q2. The poserior momens are compued from poserior parameer draws afer he firs have been discarded. Table 3 shows he resuls in erms of poserior means and 90 percen credible inervals. In he home economy he poserior mean of he Calvo parameer is in he neighbourhood of 0.9 for all hree goods caegories significanly above he prior mean of The indexaion parameer is highes for non-raded goods (k N = 0.58) followed by impored goods (k F = 0.46) and domesic raded goods (k H = 0.40) wih fairly large sandard deviaions. The habi persisence is esimaed a h = 0.48 which is smaller han he prior and he esimaes in Smes and Wouers (2003) or Adolfson e al. (2007). The ineremporal elasiciy of subsiuion (1/s) is below 1 and in line wih he lieraure. The inverse of he labour supply elasiciy is esimaed a = 0.98 while he subsiuion elasiciies beween raded and non-raded goods and beween home and foreign raded goods are = 1.02 and h = 0.88 respecively. The parameer governing he UIP modificaion S = 0.38 is somewha lower han wha Adolfson e al. (2007) found for Sweden. A Compac Open Economy DSGE Model for Swizerland 25

28 In he moneary policy reacion funcion we find evidence for srong ineres rae smoohing (r R = 0.90). The poserior means and sandard deviaions for he responses o inflaion ( p = 1.49) he oupu gap ( y = 0.49) and he change in he oupu gap ( Dy = 0.24) are all near heir priors. Wih few excepions he esimaes for he foreign economy are reasonably close o he esimaes obained for he home economy. The larges differences emerge in he indexaion parameer (esimaed a k = 0.25) and habi persisence (h = 0.71). The poserior means for he shock parameers show a similar paern in boh he home and he foreign economy. Preference shocks are more persisen han echnology shocks which in urn are more persisen han mark-up shocks. Table 4 compares he observed and model-implied momens (means sandard deviaions and auocorrelaions) of seleced variables. The model-implied momens are calculaed from 2500 Meropolis-Hasings draws from he poserior parameer disribuion. The variables considered are oupu growh CPI inflaion (ex oil producs) impored goods inflaion (ex oil producs) he shor-erm ineres rae he real effecive exchange rae he changes in he erms of rade and oil produc inflaion all in he home economy. The model appears o reflec he momens of he observed daa quie well. Poserior means are reasonably close o hose in he observed daa. For all variables he sample mean falls inside he uncerainy band compued for he model-implied daa. Poserior sandard deviaions end o be larger han hose observed in he daa. The differences are significan for he changes in he erms of rade. Wih respec o auocorrelaions we find significan differences for CPI inflaion and he changes in he erms of rade. Char 2 displays he smoohed shock processes over he period 1988Q1 o 2013Q2. Overall he shocks are saionary and heir variances do no increase over ime. However several large shocks are idenified by he model in he las few years of he sample period. In here are large negaive uni-roo echnology shocks boh in he home economy and in he foreign economy indicaing a sudden decrease of seady-sae oupu afer he collapse of Lehman Brohers. A abou he same ime large posiive moneary policy shocks occur indicaing ha cenral banks were no expansionary enough from he model s poin of view. 5 This is followed by negaive risk premium shocks in 2010 and 2011 when he Swiss franc srenghened subsanially unil he SNB se a minimum exchange rae of 1.20 francs per euro in Sepember Furhermore negaive mark-up shocks for impored goods in 2011 sugges a emporarily higher pass-hrough following he srong appreciaion of he Swiss franc. 5 We should remember ha he ineres rae was approaching he zero lower bound afer The ineres rae is he only moneary policy insrumen in his model. The model does no accoun for unconvenional moneary policy acions such as hose underaken by cenral banks afer he Lehman collapse. See he discussion of he hisorical decomposiion of he oupu gap and he inflaion gap in Secion A Compac Open Economy DSGE Model for Swizerland

29 Table 3 Poserior esimaes Prior Poserior Domesic behavioural parameers x H 0.75 [ ] 0.89 [ ] x F 0.75 [ ] 0.91 [ ] x N 0.75 [ ] 0.89 [ ] k H 0.50 [ ] 0.40 [ ] k F 0.50 [ ] 0.46 [ ] k N 0.50 [ ] 0.58 [ ] h 0.70 [ ] 0.48 [ ] s 1.50 [ ] 1.32 [ ] j 1.00 [ ] 0.98 [ ] h 1.00 [ ] 0.88 [ ] n 1.00 [ ] 1.02 [ ] f S 0.40 [ ] 0.38 [ ] r R 0.80 [ ] 0.90 [ ] p 1.50 [ ] 1.49 [ ] y 0.50 [ ] 0.49 [ ] Dy 0.20 [ ] 0.24 [ ] Foreign behavioural parameers x * 0.75 [ ] 0.86 [ ] k * 0.50 [ ] 0.25 [ ] h * 0.70 [ ] 0.71 [ ] s * 1.50 [ ] 1.47 [ ] j * 1.00 [ ] 0.97 [ ] r R * 0.80 [ ] 0.89 [ ] y p * 1.50 [ ] 1.53 [ ] y y * 0.25 [ ] 0.36 [ ] y Dy * 0.20 [ ] 0.25 [ ] AR(1) coefficiens and sandard deviaions r T 0.80 [ ] 0.63 [ ] r A 0.50 [ ] 0.49 [ ] r d G 0.80 [ ] 0.72 [ ] r f 0.50 [ ] 0.72 [ ] r mh 0.50 [ ] 0.29 [ ] r mf 0.50 [ ] 0.34 [ ] r mn 0.50 [ ] 0.26 [ ] r T * 0.80 [ ] 0.71 [ ] r G * 0.80 [ ] 0.81 [ ] r m * 0.50 [ ] 0.23 [ ] r oil 0.50 [ ] 0.53 [ ] s T 0.25 [ ] 0.18 [ ] s A 0.63 [ ] 0.53 [ ] s d G 0.63 [ ] 3.39 [ ] s f 0.63 [ ] 0.55 [ ] s mh 0.63 [ ] 0.16 [ ] s mf 0.63 [ ] 0.17 [ ] A Compac Open Economy DSGE Model for Swizerland 27

30 Table 3 coninued Prior Poserior s mn 0.63 [ ] 0.14 [ ] s R 0.63 [ ] 0.18 [ ] s T * 0.25 [ ] 0.18 [ ] s G * 0.63 [ ] 2.00 [ ] s m * 0.63 [ ] 0.20 [ ] s R * 0.63 [ ] 0.16 [ ] s oil 0.63 [ ] 4.11 [ ] Noes: Poserior means and 90% inervals in parenheses from draws wih he firs draws discarded. 90% inervals for priors are aken from draws. Table 4 Uncondiional momens Mean Daa Model y Daa [ ] Daa [ ] Daa F [ ] Daa i [ ] s [ ] Daa r x Daa [ ] Daa oil [ ] Sandard deviaions y Daa [ ] Daa [ ] Daa F [ ] Daa i [ ] s [ ] Daa r x Daa [ ] Daa oil [ ] Auocorrelaions y Daa [ ] Daa [ ] Daa F [ ] Daa i [ ] s [ ] Daa r x Daa [ ] Daa oil [ ] Noes: For he model he mean and he 90% uncerainy inervals (in parenheses) are calculaed from simulaing he model 2000 imes (using individual draws from he poserior disribuion of model parameers) wih 122 periods. 28 A Compac Open Economy DSGE Model for Swizerland

31 Char 2 Hisorical shocks 0.15 Uni roo echnology shock 0.16 Saionary echnology shock Preference shock 2.0 Risk premium shock Mark-up shock domesic raded goods 0.4 Mark-up shock impored goods A Compac Open Economy DSGE Model for Swizerland 29