A STRUCTURAL VECTOR ERROR CORRECTION MODEL WITH SHORT-RUN AND LONG-RUN RESTRICTIONS

Transcription

1 199 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 A STRUCTURAL VECTOR ERROR CORRECTION MODEL WITH SHORT-RUN AND LONG-RUN RESTRICTIONS KYUNGHO JANG* We consider srucural vecor error correcion models (VECMs) in which permanen shocks are parially idenified wih a se of long-run resricions, and fully idenified wih an addiional se of shor-run resricions. An idenificaion mehod wih a combinaion of shor-run and long-run resricions has been sudied in he vecor auoregressive models lieraure, bu no horoughly applied o he VECM framework. There exiss a separaion in he lieraure; permanen shocks are idenified wih long-run resricions while ransiory shocks are idenified wih shor-run resricions. This paper's innovaion is he idenificaion of permanen shocks using boh horizonal resricions. JEL Classificaion: E3, C3 Keywords: Coinegraion, Idenificaion, Esimaion, Impulse Response, Forecas-Error Variance Decomposiion, Money Demand 8 I. INTRODUCTION This paper develops a mehod of idenifying srucural vecor error correcion models (VECM) using shor-run and long-run resricions. The Received for publicaion: July 5, 006. Revision acceped: Feb., 008. * Deparmen of Social Sudies Educaion, Inha Universiy, Yonghyeon-dong Nam-gu, Incheon, Korea. Tel , Fax: , kjang@inha.ac.kr. This work was suppored by Inha Universiy Research Gran. I hank Masao Ogaki for his guidance and suppor. I hank J. C. Roberson for providing me wih his programs and daa for his sudy. I am also indebed o Paul Evans, and seminar paricipans a he Midwes Economerics Group 11h Annual Meeing, and Ohio Sae Universiy for helpful discussions and commens. This paper has benefied from many valuable commens by wo anonymous referees. I am compleely responsible for any remaining errors and deficiencies.

2 00 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 idenificaion scheme wih a combinaion of shor-run and long-run resricions was firs developed by Gali (199) for vecor auoregressive (VAR) models. However, i has no been horoughly sudied in VECM lieraure. This moivaes us o develop he mehod of idenificaion wih shor-run and long-run resricions for VECM. Impulse response analysis has been used in various empirical sudies wihin a framework of vecor auoregressive models (VAR) since he publicaion of Sims (1980). Blanchard and Wason (1986), Bernanke (1986), and Blanchard (1989) discuss he idenificaion mehod wih conemporaneous shor-run resricions, while Blanchard and Quah (1989) use long-run resricions for idenificaion. Gali (199) combines shor-run and long-run resricions using IS-LM models. I is well known ha VAR models in difference resul in misspecificaion problems in he presence of coinegraion. The IS-LM model of Gali (199) is subjec o his criique as long as money demand is sable or he real ineres rae is saionary. Therefore, i is necessary o develop an idenificaion mehod in VECM wih shor-run and long-run resricions. King e al. (1991; KPSW for shor), firs develop a mehod of idenificaion of VECMs wih long-run resricions. This paper generalizes he sudy of KPSW. Firs, we use block recursive assumpions on he long-run effecs of he srucural permanen shocks. Second, we combine a se of long-run resricions and a se of shor-run resricions o idenify he srucural permanen shocks. Vlaar (004), in his independen work, proposes a wo-sep esimaion of srucural VECMs in he framework of Giannini s (199) C-model. He uses boh shor-run and long-run resricion on he jus- or over-idenified srucural VECMs and performs FIML esimaion of srucural parameers in he second sep using he reduced-form model esimaes in he firs sep. In conras, his paper esimaes he srucural parameers of jus-idenified models vice simple compuaions wihou involving FIML esimaion in he second sep. This mehod is useful for consrucing models ha impose sylized facs in he shor and long run: (i) money does no affec oupu conemporaneously; (ii) money does no affec oupu in he long run, ec.

3 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 01 The res of he paper is organized as follows. Secion II presens he idenificaion problem and develops he mehod of esimaion. Secion III provides empirical resuls. Secion IV concludes. Appendix A describes he daa employed and Appendix B provides he algorihm for compuaion of srucural parameers. II. IDENTIFICATION WITH SHORT-RUN AND LONG-RUN RESTRICTIONS.1 The Model To highligh he mehod of idenificaion wih long-run and shor-run resricions in VECM, we slighly modify he KPSW model. Consider he four-variable model consising of oupu ( y ), nominal ineres rae ( R ), inflaion rae ( Δ p ), and real money supply ( m p ), in which p is he price level. 1 ' Le x = ( y, R, Δ p, m p ). Following KPSW, we assume ha x is a vecor difference saionary process, I (1), and ha he real ineres rae is nonsaionary. We furher assume ha here exiss only one coinegraing vecor represening he sable money demand relaion: m p β y β R + η. (1) = y R In order o inerpre how he srucural shocks are propagaed in he sysem, we also consider he convenional Fisher equaion: R = r E Δp 1, () + + where r is he ex ane real ineres rae and E Δp + 1 is he expeced inflaion rae beween and + 1. The model is consisen wih KPSW 1 y, m, and p are in logarihms. KPSW also consider a model in which he real ineres rae is saionary for heir sensiiviy analysis; ye, we sick o he assumpion ha he real ineres rae is nonsaionary o highligh he generalizaion of recursive assumpions on he long-run effecs. Oherwise, we have wo independen coinegraing vecors and he long-run effecs of he srucural permanen shocks become recursive.

4 0 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 excep ha we consider he four-variable sysem wih one coinegraing vecor insead of he six-variable sysem wih hree independen coinegraing vecors. 3. Shor-Run and Long-Run Resricions The presen paper builds on KPSW, who exend Blanchard and Quah s (1989) mehod for VAR models o he VECM framework. Consider he srucural Wold represenaion of he form, 4 Δ x = Γ( L) (3) v and is reduced form Δ x C( L) ε, (4) = where x denoes an n (= 4) dimensional vecor of I (1) ime series, i 1 Γ( L)= Γ 0 + Γ i=1 il, C ( L) = Γ( L) Γ0 and ε = Γ 0 v. We assume ha v is an n 1 vecor of serially uncorrelaed srucural disurbances wih a mean of zero and a covariance marix Σ v, and ha ε is an n 1 vecor of serially uncorrelaed linear forecas errors wih a mean of zero and a covariance marix Σ. Le r denoe he number of coinegraing vecors and k deuoe common rends. In paricular, r = 1 and k = 3 in our specific model. Having represened he model in (3), we can inerpre he long-run resricions imposed by he coinegraing relaions according o Sock and Wason s (1988) common rends represenaion. They show ha he coinegraed model can be represened in erms of a reduced number of common sochasic rends ( k = n r ), and ransiory componens ( r ). These common rends are generaed by permanen shocks so ha v is k' r ' ' decomposed ino ( v, v ), where v k is a k dimensional vecor of 3 The six-variable sysem includes consumpion and invesmen in addiion o he four variables, and he addiional coinegraing vecors are given by he grea raios ha represen he same growh rae of oupu, consumpion, and invesmen. 4 For simpliciy, we drop he deerminisic erms.

5 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 03 permanen shocks and v r is an r dimensional vecor of ransiory shocks. As described in KPSW, his decomposiion ensures ha [ A ] Γ (1) = 0, (5) in which A is an n k marix and 0 is an n r marix wih zeros represening long-run effecs of permanen shocks and ransiory shocks, respecively. The resricions implied by coinegraion separae he srucural permanen shocks and he srucural ransiory shocks. As inerpreed by KPSW, we consider hree srucural permanen shocks: he y p balanced-growh shock ( v ), he long-run neural inflaion shock ( v Δ ), R p and he real ineres rae shock ( v Δ ). Building on Blanchard and Quah (1989), KPSW use he long-run resricions ha he balanced-growh shock has long-run effecs on he p level of oupu, bu he oher wo permanen shocks ( v Δ p and v Δ ) have no long-run effecs on he level of oupu. In conras wih KPSW, we relax he assumpion of no long-run effecs of he real ineres rae shock on he inflaion rae. 5 These long-run resricions idenify he firs permanen shock from oher permanen shocks, bu can no idenify he second or hird permanen shocks. We impose eiher he shor-run resricion ha he conemporaneous price does no ener he money supply rule or he resricion ha he inflaion shock does no affec oupu conemporaneously, in order o idenify hese wo permanen shocks. Gali (199) employs his se of shor-run resricions in order o combine he shor-run and long-run resricions in he framework of vecor auoregressive models. 6 This paper can also be considered as he VECM version of Gali s (199) sudy. 5 As KPSW noed, he sandard model does no predic he responses of oupu o he shock in R p v Δ. Therefore, i is reasonable o relax he assumpions on he effecs of R p v Δ, and o check he sensiiviies. 6 Gali (199) suggess wo more alernaive assumpions: ha conemporaneous oupu does no ener he money supply rule and suggess conemporaneous homogeneiy in money demand. Moreover, he considers a hird se of resricions: ha he money supply and he money demand shocks have no conemporaneous effecs on oupu. Noe also ha he inerpreaion of srucural shocks in Gali (199) is differen from ha in KPSW. In he presen paper, we follow he inerpreaion of KPSW, and consider only one shor-run resricion o make he model jusidenified. R

6 04 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer Idenificaion Following KPSW, decompose Γ 0 and Γ as: = [ H J], = G E Γ0 Γ0 (6) where H, J, G and E are n k, n r, k n, and r n marices, respecively. Noe ha he permanen shocks are idenified once H or G are idenified, and ha hese wo marices have he one-o-one relaion k ' 1 k G = Σ v H Σ, where Σ v is he variance-covariance marix of permanen k shocks, v. 7 Therefore, he above decomposiion of Γ 0 does no generae addiional free parameers. The idenifying scheme of he presen paper basically follows ha of KPSW, bu leaves room o generalize heir model as described below. Our idenificaion uses he resuls of Engle and Granger (1987). Consider a srucural model of he form B( Lx ) = v, (7) where x is an n 1 vecor of ime series ha are coinegraed of order (1,1). To be specific, each elemen of x is difference saionary, and here exis r linear combinaions of x ha are saionary. We assume ha 0 < r < n. The reduced form of (7) is given by A L) = ε. (8) ( x * When he series are coinegraed, from A( L) = A(1) L + A ( L)(1 L), Engle and Granger (1987) show ha here exiss an error correcion represenaion: A L) Δx = A(1) + ε, (9) * ( x 1 7 One can easily derive his relaion from he relaion of 1 ' Γ Σ=Σ Γ. 0 v 0

7 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 05 1 * p * i where A ( L) = I p A i i L, A * ' =1 i = A j= i + 1 j, and A(1) = αβ where α and β are n r marices wih full column rank r. Engle and Granger (1987) show ha C (1) in (4) has he forms ' β C(1) = 0 (10) and C (1)α = 0. (11) Following KPSW, le C(1) = AD ˆ and A= AΠ ˆ, where  is a 1 known n k marix, Π is a k k marix, and ˆ' =( ˆ ) ˆ' D AA AC(1). KPSW assume ha Π is a lower riangular marix wih is on he diagonal, bu we relax his assumpion so ha Π is a lower block riangular marix: Π = 1 π 1 π 3. (1) π 31 π 3 1 π 3 Firs, C (1) Γ 0 = Γ(1) implies C(1) H = AΠ ˆ, so ha we have a firs se of resricions of he form: DH = Π, (13) k( k + 1) which gives 1 resricions on H. Noe ha hese long-run y resricions idenify he firs permanen shock ( v ) from he oher permanen shocks separaely, bu he second and he hird permanen shocks are no idenified. 1 Second, (11) can be expressed as Γ (1) Γ 0 α = 0, so ha G α = 0. k ' 1 Since G = Σ Σ, we have a second se of resricions of he form v H

8 06 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 ' α Σ 1 H = 0, (14) which gives kr resricions on H. Third, from C (1) ε = Γ(1) v, ˆ = ˆ k ADε AΠ v. Therefore, we have a hird se of resricions of he form k ' ' ΠΣ Π = DΣD, (15) v k( k 1) which gives resricions on H provided ha Σ k v is diagonal. Finally, he shor-run resricion ha he conemporaneous price does no ener he money supply rule or he inflaion shock does no affec oupu conemporaneously implies G3 = G4 or H 1 =0, (16) Respecively. This addiional resricion idenifies he second permanen p shock ( v Δ p ) from he hird permanen shock ( v Δ ). The above four ses of resricions give nk resricions, and he model is jus-idenified in he sense of solely idenifying he marix H..4 Esimaion Having esimaed he model (9), one can compue all he srucural k ' 1 parameers sequenially. From G = Σ v H Σ, he shor-run resricion ' 1 ' 1 (16) implies ha H,Σ,3 H,Σ,4 = 0, where M, j denoes he j -h column of he marix M. Combining his condiion and he second columns of (13) and (14) yields 1 0 D1:, 1 1 ' H, = α Σ, (17) Σ 3, Σ 4, 0 R

9 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 07 where M 1:, denoe he firs wo rows of he marix M and M i, denoes he i -h row of he marix M. Using H,, we can compue π 3 by π 3 = D3, H,. (18) Having compued π 3, one can compue Π from (15). 8 The oher columns of H ( H,1 and H,3 ) are given by 1 D Π, j H, j =, ' 1 0 α Σ (19) for j = 1,3. Therefore, he permanen shocks and he shor-run dynamics are respecively idenified by v k = Gε (0) and Γ ( L) k = C( L) H. (1) Fisher e al. (1995) exend he KPSW mehod o idenify ransiory shocks. The firs se of resricions comes from convenional orhogonaliy condiions of he form Cov( v r ' ', v ) = [ EΣ G EΣE ] () = [ 0 ], r Σ v r where Σ v is r r a diagonal marix. This assumpion gives r( r +1) nr resricions on J (or E ) provided ha H (or G ) is 8 We use he Newon mehod o ge he soluion. See Appendix B for he deailed algorihm.

10 08 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 idenified. As hese ransiory shocks can no be idenified from long-run resricions, Fisher e al. (1995) sugges a recursive srucure on he conemporaneous effecs of he ransiory srucural shocks. For example, he sub-marix of E ha consiss of he las r rows and he las r columns of E is lower riangular wih 1s on he diagonal. Wih his se r( r +1) of addiional resricions, he model is jus-idenified in he sense of idenifying he marix Γ. Similarly, he ransiory shocks and he shor-run dynamics are respecively idenified by v r = Eε (3) and Γ ( L) r = C( L) J. (4) In our specific model, we do no need addiional assumpions oher han normalizaion condiions for he idenificaion of ransiory shocks since r = 1. Thus, he ransiory shock is idenified and is compued from G. Having normalized E = 1, we can rewrie () as 14 Σ ' 1 G =0 (5) 1:3, E1,1:3 Σ 4, and E 1,1: 3 is given by ' 1 [ Σ G ]. ' E = Σ G (6) 1,1:3 4, 1:3, From Σ = r ' E Σ v J, J is given by ' r 1 J = ΣE ( Σ ), (7) v r ' where Σ v = EΣE from (3). Therefore, he ransiory shocks and he shor-run dynamics are idenified by (3) and (4), respecively.

11 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL The Daa III. EMPIRICAL EVIDENCE The daa are quarerly US observaions over he period 1949:1-000:4. These daa are similar o hose used by KPSW over 1949:1-1988:4, excep we replace gross naional produc wih gross domesic produc. 9 Oupu is measured by he logarihm of per capia privae gross domesic produc ( y ), and he price level is measured by he logarihm of is implici deflaor ( p ). The logarihm of M is used for he money supply measure ( m ), he real balance is measured by m p, and he hreemonh US Treasury Bill rae is used for he ineres rae ( R ). See Appendix A for a deailed descripion of he daa. Since he daa cover he 1990s, we consider he possible breakdown of a sable money demand relaion in 1990s due o financial innovaions. Carlson e al. (1999) show ha he M relaion breaks down in he 1990s according o he sharp increase of M velociy beween 1990 and They illusrae he upward shif in M velociy due o financial innovaions such as growh in sock and bond muual funds ha are close subsiues for ime deposis. In order o capure hese financial innovaions, hey include a dummy variable ha is zero before 1990:1, one afer 1994:4, and which increases linearly beween hem. As long as his dummy variable is included in he money demand regression, hey find a sable relaion. This paper adops he same dummy variable as heirs, and runs he regression over 1954:1-1988:4 and 1954:1-1999: Uni Roos and Coinegraion Univariae properies of he four variables are summarized in Table 1. I shows ha oupu, real balance, and nominal ineres rae are uni-roo processes wih a 5% significance level over he wo sample periods. The inflaion rae is also characerized by he uni-roo process wih a 10% 9 This change does no affec he main resuls. 10 KPSW use daa prior o 1954:1 as iniial observaions in lags o avoid he impac of price conrols during he Korean War. Periods from 1989:1-1989:4 and 000:1-000:4 are used for observaions in leads of regressors in he dynamic OLS.

12 10 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 significance level, excep he resuls of Phillips-Perron ess. These imply ha he price level and nominal money are I() processes. The resuls of uni-roo ess on he real ineres rae are mixed. Augmened Dickey- Fuller ess show ha he real ineres rae is an I(1) process over wo sample periods, while Phillips-Perron ess, Park s (1990) J(p,q) ess, and Park s (1990) G(p,q) ess show ha i is an I(0) process. Johansen s (1995) maximum likelihood ess show ha he real ineres rae is an I(1) process over 1954:1-1988:4, while an I(0) process over 1954:1-1999:4 wih a 10% significance level. Table shows he resuls of coinegraion ess for he money demand equaion. Even wih inclusion of he dummy variables, Said-Dickey ess and Park s (1990) I(p,q) ess canno rejec he null of no coinegraion, while Johansen s (1995) ess canno rejec he null of coinegraion wih a 5% significance level. Park s (1990) H(p,q) ess show he mixing resuls ha he money-demand relaion is no coinegraed over 1954:1-1988:4 in Panel A, bu is (no) coinegraed over 1954:1-1999:4, when he dummy variable is (no) included in he regression in Panel C (Panel B, respecively). Mos esimaes of he coefficien on oupu are near 1 in Panel A and Panel C, while far from 1 in Panel B. The coefficien on he nominal ineres rae even changes sign in Panel B. These resuls jusify he inclusion of he dummy variable in he model. The resuls of coinegraion ess ogeher wih he univariae properies of he real ineres rae imply ha here migh be, a mos, wo coinegraing relaions among four variables. Table 3 shows he resuls of Johansen s (1995) rank ess. Maximum eigenvalue ess show ha here is one coinegraing vecor in Panel A, and wo (no) coinegraing vecors in Panel C (Panel B, respecively). However, race ess show less evidence of coinegraion han he maximum eigenvalue ess. From he preliminary uni-roos ess and coinegraion ess, we include he dummy variable in he regression over 1954:1-1999:4 o consider he financial innovaions in 1990s. The choice of coinegraion rank is criical o srucural VECMs. In order o compare hese resuls wih KPSW s, we choose one as he rank and he money demand equaion as he coinegraing relaion. As he dynamic OLS coefficiens are no very differen from oher esimaes, hey are chosen as he esimaes of he

13 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 11 coinegraing vecor. 11 [Table 1] Uni Roo Tess Oupu M Ineres Rae Inflaion Rae Nominal Real Nominal Real Sample: 1954:1-1988:4 Null of I(1) ADF -3.1(1) -3.13(8) -.15(1) -1.64(7) -1.58(17) -1.34(15) PP -.61(1) -3.40(1) * -1.81(1) -1.87(1) -5.35(1) *** -4.6(1) *** J(p,q) 0.5(1,3) 11.71(1,3) 0.44(1,3).01(0,3) 0.8(0,3) ** 1.34(0,3) Null of I(0) G(p,q) 4.34(1,3) 9.16(1,3) **.99(1,3) 6.76(0,3) * 3.37(0,3) 6.88(0,3) * LRT( χ ) (3) (0.054) * - Sample: 1954:1-1999:4 Null of I(1) ADF -3.41(1) * -.4(13) -.14(1) -1.95(17) -1.89(17) -1.09(17) PP -.93(1) -1.9(1) -1.86(1) -.0(1) -5.90(1) *** -3.78(1) *** J(p,q) 0.54(1,3) 6.56(1,3) 1.6(1,3) 1.31(0,3) 0.1(0,3) ** 0.91(0,3) Null of I(0) G(p,q) 5.79(1,3) 10.08(1,3) *** 6.64(1,3) ** 7.74(0,3) * 3.4(0,3) 7.3(0,3) * LRT( χ ) (3) (0.135) - Noe: The ADF es and Phillips-Perron saisics are compued by a regression equaion wih consan and rend erms for oupu and M, while wih consan for ineres raes and inflaion raes. The number in each parenhesis of ADF denoes he lag lengh chosen following Campbell and Perron (1991) wih maximum lag lengh as 0, while hose of PP denoe he lag lengh of serial correlaion. The number in each parenheses of J(p,q) and G(p,q) denoes p and q chosen for each regression equaion. LRT denoes Johansen s (1995) likelihood raio es, and p-values are in parenheses. LRT is applied o es coinegraion wih a known vecor. *, **, and *** denoe ha he null hypohesis is rejeced wih he significance levels 10%, 5%, and 1%, respecively. 11 We choose four lags as he lag lengh in he following dynamic OLS: m p = α + βy y β 4 4 R R + i= 4 γ iδ y i + i= 4 δ iδ R i + ε.

14 1 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 [Table ] Coinegraion Tess and Esimaed Coinegraing Vecors Money Demand, m- p = α + βyy βrr β y β R Tes Saisics Panel A: 1954:1-1988:4 Null of No Coinegraion Said-Dickey (18) I(p,q) (0,3) * Null of Coinegraion CCR, H(p,q) (0,1) ** DOLS LRT( (1) χ ) (0.150) Panel B: 1954:1-1999:4 Null of No Coinegraion Said-Dickey (18) I(p,q) (0,3) Null of Coinegraion CCR, H(p,q) (0,1) ** DOLS LRT( (1) χ ) (0.39) Panel C: 1954:1-1999:4, Dummy Null of No Coinegraion Said-Dickey (18) I(p,q) (0,3) Null of Coinegraion CCR, H(p,q) (0,1) DOLS LRT( (1) χ ) (0.106) Noe: The Said-Dickey es saisics are compued by a coinegraion equaion wih consan erms. The number in each parenheses of Said-Dickey denoes he lag lengh chosen following Campbell and Perron (1991) wih maximum lag lengh as 0. The number in each parenheses of I(p,q) and H(p,q) denoes p and q chosen for each regression equaion. CCR denoes Park s (199) Canonical Coinegraion Regression. LRT denoes Johansen s (1995) likelihood raio es, and p-values are in parenheses. The coinegraing vecors of dynamic OLS are esimaed wih four leads and four lags. *, **, and *** denoe ha he null hypohesis is rejeced wih he significance levels 10%, 5%, and 1%, respecively.

15 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 13 [Table 3] Coinegraion Rank Tess Eigen Value max Trace Number of k (=n-r) Criical Value Coinegraion (r) λ max Trace Panel A: 1954:1-1988: *** Panel B: 1954:1-1999: Panel C: 1954:1-1999:4, Dummy ** * ** Noe: The las wo columns are criical values wih a 5% significance level. *, **, and *** denoe ha he null hypohesis is rejeced wih he significance level 10%, 5%, and 1%, respecively. Finally, we selec nine lags as he lag lenghs in he VECM. The lag lengh is chosen by he robusness of resuls. We check wheher signs of he long-run mulipliers of he VECM change as he lag lengh become smaller saring from 13. The signs are he same hrough lag lenghs of 13 o 9, bu differen as he lag lenghs become smaller han 9. Therefore, a lag lengh of 9 is robus, a leas in erms of properies of long-run mulipliers Impulse Responses Figures 1-4 show responses o he growh shock, he inflaion shock, and he real-ineres-rae shock. The responses are normalized by he 1 The usual AIC and BIC crieria end o selec very small lags in his model (1 or in his paricular model), and resuls based on hese small lags are no consisen wih hose implied by economic heories.

16 14 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 long-run effecs ha he growh shock increases oupu by 1 percen, he inflaion shock raises he inflaion rae by 1 percen, and he real-ineresrae shock raises he nominal ineres rae by 1 percen. The lower and upper bounds are drawn by one sandard-deviaion, calculaed by he Mone Carlo inegraion as described by Jang (001). We consider four cases regarding he resricions on he model. Firs, we examine he responses using he model employed by KPSW (KPSW-I), where all he permanen shocks are idenified wih long-run resricions causally ordered by he growh shock, he inflaion shock, and he real-ineres-rae shock. Second, we swich he causal order of he inflaion shock and he real-ineres-rae shock (KPSW-II) o check he sensiiviy of resricions. Third, we consider he case where one permanen shock is idenified by he long-run resricions and he oher wo permanen shocks are idenified by he shor-run resricion ha price does no ener he money supply (VLR-I). Finally, we consider an alernaive shor-run resricion ha he inflaion shock does no affec oupu conemporaneously (VLR- II). These shor-run resricions are pars of resricions considered by Gali (199). The poin esimaes of responses are very similar over wo sample periods in hree models-kpsw-i, KPSW-II, and VLR-II-while he responses in VLR-I show huge confidence inervals. This implies ha he resricions in VLR-I are no useful for idenifying he las wo permanen shocks. These resuls are consisen wih hose implied by he forecaserror variance decomposiions in he nex secion. 13 Firs, in response o he growh shock, oupu increases by 0.% in he firs quarer, and by 1% afer four years. The nominal ineres rae decreases by 0.15% in he firs quarer and by 1% afer six years, bu no significanly differen from 0 in mos horizons. The growh shock lowers he inflaion rae by 0.4% in he firs quarer, and shows negligible effecs afer wo years. Second, he inflaion shock yields a rise in oupu up o 1% over he firs few quarers, bu he effecs become negligible hereafer. The shor-run responses of he nominal ineres rae are also similar in he hree cases. The nominal ineres rae increases by 0.% in he firs quarer and reaches 1%-1.5% afer one year. The long-run responses 13 Figures for he sample period 1954:1-1988:4 are available upon reques.

17 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 15 [Figure 1] Impulse Responses: KPSW-I, A. Impulse Responses o he Balanced-Growh Shock B. Impulse Responses o he Long-Run Neural Inflaion Shock C. Impulse Responses o he Real-Ineres-Rae Shock

18 16 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 [Figure ] Impulse Responses: KPSW-II, A. Impulse Responses o he Balanced-Growh Shock B. Impulse Responses o he Long-Run Neural Inflaion Shock C. Impulse Responses o he Real-Ineres-Rae Shock

19 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 17 [Figure 3] Impulse Responses: VLR-I, A. Impulse Responses o he Balanced-Growh Shock B. Impulse Responses o he Long-Run Neural Inflaion Shock C. Impulse Responses o he Real-Ineres-Rae Shock

20 18 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 [Figure 4] Impulse Responses: VLR-II, A. Impulse Responses o he Balanced-Growh Shock B. Impulse Responses o he Long-Run Neural Inflaion Shock C. Impulse Responses o he Real-Ineres-Rae Shock

21 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 19 [Figure 5] Impulse Responses o he Money-Demand Shock A. KPSW-I, B. KPSW-II, C. VLR-I,

22 0 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 [Figure 5] (Coninued) D. VLR-II, depend on he resricions. In KPSW-I and VLR-II, he nominal ineres rae rises by less han 1% in he long run, implying a decrease in he real ineres rae. In conras, he real ineres rae is no affeced by he inflaion shock in he long run by he resricions in VLR-I. The inflaion rae responds o is shock by rising % in he firs quarer and by abou 1% afer a few quarers. The real balance increases over he firs few quarers, bu decreases afer one year by 1%. Third, he real-ineres-rae shock raises oupu by 1.5% in he firs period, he effecs become negaive afer five quarers, and become negligible afer hree years. This shock has he larges effec on oupu in he shor run among he permanen shocks. The nominal ineres rae rises by 1.1% in he firs quarer. The response of he inflaion rae depends on he model. The inflaion rae decreases by 0.5% in he firs quarer, bu becomes posiive up o 0.6% afer one quarer in KPSW-I and VLR-II, while he negaive effecs are more persisen in KPSW-II. The response of real balance is very similar. I is negligible over he firs few quarers and decreases 1.5%-% afer wo years. Figure 5 shows he responses o he ransiory shock: he money demand shock. The shock is normalized so ha i increases he real balance by 1% in he firs quarer. The responses are similar over he four models. The money demand shock lowers oupu significanly up o 5%- 8% over he firs few quarers. This shock has much greaer larger effecs

23 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 1 han he real-ineres-rae shock in he shor run. I also raises he nominal ineres rae by %, and he inflaion rae by 1% in he firs quarer, bu hose responses are no significanly differen from 0. Increases of real balance over wo years are no significanly differen from Forecas-Error Variance Decomposiions Tables 4-7 show he forecas-error variance decomposiions. As in he previous secion, he resuls are similar over wo sample periods in hree models-kpsw-i, KPSW-II, and VLR-II-while he resuls in VLR-I imply ha he shor-run resricion is no useful in idenifying he second and he hird permanen shocks. For insance, he inflaion shock has he larges conribuion o he forecas-error variance of he nominal ineres rae, while he real-ineres-rae shock has he larges conribuion o ha of he inflaion rae. This implies ha he model VLR-I is misspecified and is no able o idenify he srucural shocks. 14 Firs, he growh shock makes a small conribuion o he variaion of oupu, bu he larges conribuion o ha of he real balance in he shor run. Is conribuion o oupu becomes dominan afer eigh quarers. Second, he inflaion shock has he larges conribuion o he forecaserror variance of he inflaion rae, bu he smalles conribuion o ha of oupu. Third, he real-ineres-rae shock has he larges conribuion o is variaion. I also has he larges conribuion o ha of oupu in he shor run among he permanen shocks. Fourh, he ransiory shock (he money demand shock) has he larges conribuion o he forecas-error variance of oupu over he firs four quarers Tables for he sample period of 1954:1-1988:4 are available upon reques. 15 The fracion of he forecas-error variance aribued o he ransiory shock is no lised in he able, bu can be easily calculaed from he able by noing ha he sum of he fracions is 1.

24 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 [Table 4] Forecas-Error Variance Decomposiions: KPSW-I, Fracion of he forecas-error variance aribued o balanced growh shock Horizon y R Δ p m p (0.1) (0.08) (0.14) (0.1) (0.14) (0.07) (0.1) (0.1) (0.13) (0.08) (0.10) (0.13) (0.10) (0.09) (0.07) (0.13) (0.08) (0.11) (0.08) (0.13) (0.07) (0.1) (0.09) (0.13) (0.06) (0.13) (0.10) (0.13) B. Fracion of he forecas-error variance aribued o inflaion shock Horizon y R Δ p m p (0.06) (0.06) (0.17) (0.05) (0.09) (0.1) (0.13) (0.04) (0.05) (0.13) (0.11) (0.05) (0.04) (0.13) (0.09) (0.06) (0.03) (0.13) (0.09) (0.06) (0.03) (0.13) (0.10) (0.06) (0.0) (0.14) (0.11) (0.06) C. Fracion of he forecas-error variance aribued o real-ineres-rae shock Horizon y R Δ p m p (0.18) (0.14) (0.06) (0.06) (0.16) (0.14) (0.04) (0.11) (0.10) (0.14) (0.04) (0.11) (0.06) (0.14) (0.04) (0.11) (0.05) (0.15) (0.04) (0.10) (0.04) (0.15) (0.04) (0.10) (0.04) (0.16) (0.04) (0.10) Noe: Numbers in parenheses denoe sandard errors.

25 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 3 [Table 5] Forecas-Error Variance Decomposiions: KPSW-II, A. Fracion of he forecas-error variance aribued o balanced growh shock Horizon y R Δ p m p (0.1) (0.08) (0.14) (0.1) (0.14) (0.07) (0.1) (0.1) (0.13) (0.08) (0.10) (0.13) (0.10) (0.09) (0.07) (0.13) (0.08) (0.11) (0.08) (0.13) (0.07) (0.1) (0.09) (0.13) (0.06) (0.13) (0.10) (0.13) B. Fracion of he forecas-error variance aribued o inflaion shock Horizon y R Δ p m p (0.10) (0.1) (0.18) (0.05) (0.1) (0.14) (0.14) (0.05) (0.07) (0.15) (0.11) (0.07) (0.05) (0.15) (0.09) (0.06) (0.04) (0.14) (0.10) (0.06) (0.03) (0.14) (0.11) (0.05) (0.03) (0.14) (0.11) (0.05) C. Fracion of he forecas-error variance aribued o real-ineres-rae shock Horizon y R Δ p m p (0.16) (0.18) (0.11) (0.06) (0.13) (0.15) (0.06) (0.10) (0.08) (0.15) (0.05) (0.10) (0.05) (0.15) (0.05) (0.09) (0.04) (0.15) (0.06) (0.09) (0.03) (0.15) (0.06) (0.09) (0.03) (0.15) (0.06) (0.09) Noe: Numbers in parenheses denoe sandard errors.

26 4 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 [Table 6] Forecas-Error Variance Decomposiions: VLR-I, A. Fracion of he forecas-error variance aribued o balanced growh shock Horizon y R Δ p m p (0.1) (0.08) (0.14) (0.1) (0.14) (0.07) (0.1) (0.1) (0.13) (0.08) (0.10) (0.13) (0.10) (0.09) (0.08) (0.13) (0.08) (0.11) (0.08) (0.13) (0.07) (0.1) (0.09) (0.13) (0.06) (0.13) (0.11) (0.13) B. Fracion of he forecas-error variance aribued o inflaion shock Horizon y R Δ p m p (0.0) (0.) (0.18) (0.05) (0.18) (0.0) (0.17) (0.10) (0.11) (0.1) (0.19) (0.13) (0.07) (0.0) (0.17) (0.13) (0.06) (0.0) (0.15) (0.13) (0.05) (0.1) (0.15) (0.1) (0.05) (0.1) (0.15) (0.1) C. Fracion of he forecas-error variance aribued o real-ineres-rae shock Horizon y R Δ p m p (0.10) (0.17) (0.19) (0.05) (0.1) (0.17) (0.16) (0.05) (0.07) (0.19) (0.16) (0.06) (0.05) (0.19) (0.16) (0.06) (0.04) (0.19) (0.14) (0.06) (0.03) (0.19) (0.14) (0.06) (0.03) (0.19) (0.15) (0.06) Noe: Numbers in parenheses denoe sandard errors.

27 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 5 [Table 7] Forecas-Error Variance Decomposiions: VLR-II, A. Fracion of he forecas-error variance aribued o balanced growh shock Horizon y R Δ p m p (0.1) (0.08) (0.14) (0.1) (0.14) (0.07) (0.1) (0.1) (0.13) (0.08) (0.10) (0.13) (0.10) (0.09) (0.07) (0.13) (0.08) (0.11) (0.08) (0.13) (0.07) (0.1) (0.09) (0.13) (0.06) (0.13) (0.10) (0.13) B. Fracion of he forecas-error variance aribued o inflaion shock Horizon y R Δ p m p (0.00) (0.16) (0.18) (0.05) (0.03) (0.14) (0.17) (0.06) (0.03) (0.16) (0.16) (0.07) (0.03) (0.17) (0.15) (0.07) (0.03) (0.16) (0.13) (0.07) (0.0) (0.16) (0.13) (0.07) (0.0) (0.16) (0.13) (0.07) C. Fracion of he forecas-error variance aribued o real-ineres-rae shock Horizon y R Δ p m p (0.19) (0.17) (0.10) (0.06) (0.18) (0.16) (0.11) (0.11) (0.11) (0.18) (0.13) (0.1) (0.07) (0.19) (0.13) (0.1) (0.06) (0.0) (0.1) (0.11) (0.05) (0.1) (0.11) (0.11) (0.04) (0.1) (0.1) (0.11) Noe: Numbers in parenheses denoe sandard errors.

28 6 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 IV. CONCLUDING REMARKS The presen paper generalizes he exising srucural VECM lieraure by developing a mehod of idenifying srucural VECM wih a combinaion of shor-run and long-run resricions. We consider a generalized block recursive VECM in which permanen shocks are parially idenified wih a se of long-run resricions, and fully idenified wih an addiional se of shor-run resricions. As an applicaion, we slighly modify he KPSW model and adop a se of shor-run resricions ha are used by Gali (199). The resuls show ha he shor-run resricion, ha he price does no appear in he moneary policy rule, does no help idenify he inflaion shock (or he moneary policy shock, in a sense). In conras, he oher shor-run resricion, ha he inflaion shock does no affec oupu conemporaneously, does well o idenify he shock. Wih he laer resricion, he VECM wih shor-run and long-run resricions are well specified, and give resuls ha are consisen wih hose of KPSW.

29 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 7 Daa Sources Appendix A All daa for 1948:1-000:4 excep he earlier M are obained from FRED, The Federal Reserve Bank of S. Louis. GDP : Nominal gross domesic produc. Seasonally adjused. Billions of dollars. GDP 96 : Real gross domesic produc in chained 1996 dollars. Seasonally adjused. Billions of dollars. GGE : Nominal governmen consumpion expendiures and gross invesmen. Seasonally Adjused. Billions of dollars. GGE 96 : Real governmen consumpion expendiures and gross invesmen in chained 1996 dollars. Seasonally Adjused. Billions of dollars. M : M money sock. Seasonally Adjused. The earlier M daa for are obained from KPSW series ha are repored in Banking and Moneary Saisics, (Board of Governors of he Federal Reserve Sysem, 1976). The monhly daa are averaged o obain he quarerly daa. Billions of dollars. R : The hree-monh Treasury Bill rae in he secondary marke. Averages of business days. An annual percenage rae. The monhly daa are averaged o obain he quarerly daa. N : Civilian noninsiuional populaion, 16 years and older. The monhly daa are averaged o obain he quarerly daa. Thousands. y : Log real privae oupu per capia formed by y = log( GDP96 GGE96) log( N). Millions of dollars.

30 8 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 m : Log money sock per capia calculaed by m = Log( M ) Log( N). Millions of dollars. p : The GDP deflaor calculaed by p = log( GDP GGE) log( GDP96 GGE96) = 1. Δ p : An annual percenage inflaion rae. I is annualized by Δp 400 ( p p ). = 1

31 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 9 An Algorihm o Solve he Models Appendix B Le C π and Σ v is diagonal, π 1, π 3 and ' = DΣD. As 1 = π 13 = 0 Σ v,11 are given by Σ v,11 = C 11, π 1 C = C 1 11, π 31 C = C Provided ha π 3 is known, we can rewrie (15) by ΠΣ v Π ' Σ v, π 1 π 31 = 1 1 x 1 0 x π π 3 = C. π 31 π 3 1 x x 3 0 x1 1 x 1 ' Le f (x) be he las hree elemens of vech( ΠΣ vπ C) and g (x) be f ( x) ' he gradien of f (x),, where x = ( x, x, x ) ' 1 3. Then x and f ( x) = π 1 π 1Σ v,11 + x + x1 x3 C π 31Σ v,11 + π 3 x + x1 (1 x1 ) x3 C π Σ + π + (1 x ) x C 31 v, = 0 x1x3 g( x) = (1 x1 ) x3 (1 x1 ) x 3 π π x 1 x1 (1 x1). (1 x1 ) Having specified f and g, we can apply he Newon mehod o ge he soluions of (15) by ieraing

32 30 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 x = x + g( x) f ( x ( i) ( i 1) 1 ( i 1) ) wih an arbirary iniial vecor (0) x unil changes in (i) x become 0.

33 KYUNGHO JANG: A STRUCTURAL VECTOR ERROR CORRECTION MODEL 31 References Bernanke, B. S. (1986), Alernaive Explanaions of he Money-Income Correlaion, Carnegie-Rocheser Conference Series on Public Policy, Vol. 5, Blanchard, O. J. (1989), A Tradiional Inerpreaion of Macroeconomic Flucuaions, American Economic Review, Vol. 79, No. 5, Blanchard, O. J. and D. Quah (1989), The Dynamic Effecs of Aggregae Supply and Demand Disurbances, American Economic Review, Vol. 77, Blanchard, O. J. and M. W. Wason (1986), Are Business Cycles All Alike? R. J. Gordon, eds., The American Business Cycle: Coninuiy and Change, vol. 5 of Naional Bureau of Economic Research Sudies in Business Cycles, 13-18, Chicago: Universiy of Chicago Press. Campbell, J. Y. and P. Perron (1991), Pifalls and Opporuniies: Wha Macroeconomiss Should Know abou Uni Roos, O. J. Blanchard and S. Fischer, eds, NBER Macroeconomics Annual, , Cambridge, MA: MIT Press. Carlson, J. B., D. L. Hoffman, B. D. Keen, and R. H. Rasche (1999), Resuls of a Sudy of he Sabiliy of Coinegraing Relaions Comprised of Broad Moneary Aggregaes, Working Paper No , Federal Reserve Bank of Cleveland. Engle, R. F. and C. W. J. Granger (1987), Co-Inegraion and Error Correcion: Represenaion, Esimaion, and Tesing, Economerica, Vol. 55, Fisher, L. A., P. L. Fackler and D. Orden (1995), Long-run Idenifying Resricions for an Error-Correcion Model of New Zealand Money, Prices and Oupu, Journal of Inernaional Money and Finance, Vol. 14, No. 1, Gali, J. (199), How Well Does he IS-LM Model Fi Poswar U.S. Daa? Quarerly Journal of Economics, Vol. 107, No., Giannini, C. (199), Topics in Srucural VAR Economerics, New York: Springer Verlag. Jang, K. (001), Impulse Response Analysis wih Long Run Resricions on Error Correcion Models, Working Paper No , Deparmen of Economics, Ohio Sae Universiy, 001. Johansen, S. (1995), Likelihood-Based Inference in Coinegraed Vecor Auoregressive Models, Oxford: Oxford Universiy Press. King, R. G., C. I. Plosser, J. H. Sock and M. W. Wason (1991), Sochasic

34 3 THE KOREAN ECONOMIC REVIEW Volume 4, Number 1, Summer 008 Trends and Economic Flucuaions, American Economic Review, Vol. 81, No. 4, Park, J. Y. (1990), Tesing for Uni Roos and Coinegraion by Variable Addiion, Advances in Economerics, Vol. 8, (199), Canonical Coinegraing Regressions, Economerica, Vol. 60, No. 1, Sims, C. A. (1980), Macroeconomics and Realiy, Economerica, Vol. 48, Sock, J. H. and M. W. Wason (1988), Tesing for Common Trends, Journal of he American Saisical Associaion, Vol. 83, No. 404, Vlaar, P. J. G. (004), On he Asympoic Disribuion of Impulse Response Funcions wih Long Run Resricions, Economeric Theory, Vol. 0, No. 5,