The Dynamic Effecs of Public Capial: VAR Evidence for OECD Counries Firs draf: November 3 This version: May Chrisophe Kamps* Kiel Insiue for World Economics, Kiel, Germany Absrac The issue of wheher governmen capial is producive has received a lo of aenion in he recen pas. Ye, empirical analyses of public capial produciviy have in general been limied o a small sample of counries for which official capial sock esimaes are available. Building on a new daabase ha provides inernaionally comparable capial sock esimaes, his paper esimaes he dynamic macroeconomic effecs of public capial using he vecor auoregressive (VAR) mehodology for a large panel of OECD counries. The paper adds o he empirical lieraure by presening resuls for many counries for which here is no VAR evidence so far and by proposing a new idenificaion scheme ha exends he approach proposed by Blanchard and Peroi (). Keywords: Public capial; VAR model; Coinegraion; Idenificaion; OECD counries JEL classificaion: C3; E6; H5 * Tel.: +93-88-66, Fax: +93-88-55, E-mail address: kamps@ifw.uni-kiel.de. I would like o hank Kai Carsensen, Annee Kuhn and Joachim Scheide for helpful commens. The usual disclaimer applies.
Inroducion Since Sims (98) firs inroduced vecor auoregressive (VAR) models, hey have become increasingly popular. They are now one of he principal ools of macro-economeric analysis. In heir survey of he VAR mehodology, Sock and Wason () idenify four asks ha have been ackled wih he help of such models: (i) daa descripion, (ii) forecasing, (iii) srucural inference, and, (iv) policy analysis. For our purposes, he las wo asks are especially relevan. While VAR models have been exensively applied o sudy he effecs of moneary policy shocks, applicaion of his mehodology o quesions relaed o fiscal policy in general is a relaively recen phenomenon. In paricular, as discussed in he lieraure survey below, mos VAR sudies on he dynamic effecs of public capial have been published over he pas five years. The VAR approach has a number of advanages over he producion funcion approach pioneered by Aschauer (989): (i) Whereas he producion funcion approach assumes a causal relaionship running from he hree inpus o oupu, he VAR approach does no impose any causal links beween he variables a priori. Raher, VAR models allow o es wheher he causal relaionship implied by he producion funcion approach is valid or wheher here are feedback effecs from oupu o he inpus. (ii) Unlike he producion funcion approach, he VAR approach allows for indirec links beween he model variables. In he producion funcion approach, he long-run oupu effec of public capial is given by he elasiciy of oupu wih respec o capial. In conras, in he VAR approach, he long-run oupu effec of a change in public capial resuls from he ineracion of he model variables. For example, i is conceivable ha public capial does no direcly affec oupu bu ha a change in public capial has an impac on oupu only indirecly via is effecs on he privae facors of producion. The VAR approach allows o capure such indirec effecs. (iii) Unlike he producion funcion, he VAR approach does no assume ha here is a mos one long-run (coinegraion) relaionship among he four model variables. The Johansen (988, 99) mehodology described in Secion 3. allows o explicily es for he coinegraion rank (he number of long-run relaionships) and o impose i in he esimaion of he VAR model. Esimaion of VAR models is based on a reduced form. Wihou he prior soluion of an idenificaion problem, he VAR esimaes canno be given a srucural inerpreaion and can
in general no be used for policy analysis. 6 In his paper, we consider wo soluions o he idenificaion problem. The firs one, known as he recursive approach, was inroduced by Sims (98) and is sandard in he relaed lieraure. This approach is applied in Secion, presening empirical resuls on he dynamic effecs of public capial for OECD counries. The second soluion o he idenificaion problem is, o he bes of our knowledge, an addiion o he lieraure. I exends he idenificaion scheme proposed by Blanchard and Peroi (), who considered he dynamic effecs of axes and aggregae governmen spending, by decomposing aggregae spending ino governmen invesmen and governmen consumpion. Secion 5. presens empirical resuls for his alernaive idenificaion scheme. The paper is organized as follows. Secion briefly reviews recen sudies ha have applied he VAR approach o sudy he effecs of public capial. Secion 3 describes he economeric mehodology underlying our empirical applicaion. Secion presens new empirical evidence on he dynamic effecs of public capial for OECD counries building on capial sock esimaes provided by Kamps (). Secion 5 discusses he robusness of he empirical resuls o alernaive idenifying assumpions. The las secion summarizes he main findings. A Shor Survey of he Lieraure This secion briefly reviews he empirical lieraure having applied he VAR approach o sudy he dynamic effecs of public capial. The only survey of he VAR approach so far, Surm e al. (998a), raced merely four sudies. Insead, Table summarizes informaion on weny VAR sudies, winessing he increased populariy of his approach in he very recen pas. A number of ineresing findings wih respec o he objec of invesigaion and model specificaion emerge from he able: (i) Nearly half of he considered VAR sudies have invesigaed he effecs of public capial for he Unied Saes. Moreover, only wo sudies, Minik and Neumann () as well as Pereira (b), have exended he analysis o a group of OECD counries. (ii) The vas majoriy of sudies has relied on annual daa, due o he resricion ha capial sock daa are no available a higher frequency. (iii) The majoriy of sudies has considered a model in he four variables public capial, privae capial, employmen and oupu. In he remaining cases, in general eiher invesmen has been 6 See Favero () for an insighful reamen of he idenificaion problem, which is by no means special o VAR models bu raher a general phenomenon in economerics.
3 Table : Sudies using he VAR approach Sudy Counry Sample Model Variables Oupu effec of public capial a G D G Cullison (993) Unied Saes 955 99 (A) VAR (FD) I, G, B, Y, M insignifican G P P P McMillin & Smyh (99) Unied Saes 95 99 (A) VAR (L, FD) E,π, K / K, N / K, Y / K insignifican G P Crowder and Himarios (997) Unied Saes 97 989 (A) VECM K, K, Y, N, E n.a. G P Baina (998) Unied Saes 98 993 (A) VECM, VAR (L) K, Y, N, K posiive b G P Pereira & Flores de Fruos (999) Unied Saes 956 989 (A) VAR (FD) K, K, N, Y posiive b G P Pereira () Unied Saes 956 997 (A) VAR (FD) I, I, N, Y posiive b G P Pereira (a) Unied Saes 956 997 (A) VAR (FD) I, I, N, Y n.a. G P Pereira & Andraz () Unied Saes 956 997 (A) VAR (FD) I, I, N, Y posiive b G P Flores de Fruos e al. (998) Spain 96 99 (A) VARMA (L) K, K, N, Y posiive b G P Pereira & Roca Sagales (999) Spain 97 989 (A) VAR (FD) K, K, N, Y posiive b G P Pereira & Roca Sagales () Spain 97 993 (A) VAR (FD) K, K, N, Y posiive b G P Pereira & Roca Sagales (3) Spain 97 995 (A) VAR (FD) K, K, N, Y posiive b G P Oo and Voss (996) Ausralia 959 99 (Q) VAR (L) K, K, N, Y insignifican c G P Everaer (3) Belgium 953 996 (A) VECM K, K, Y n.a. G P Mamazakis (999) Greece 959 993 (A) VECM K, K, N, Y n.a. G P Surm e al. (999) Neherlands 853 93 (A) VAR (L) I, I, Y insignifican c G P Lighar () Porugal 965 995 (A) VAR (L) K, K, N, Y insignifican G P G P Voss () Unied Saes, Canada 97 996 (Q) VAR (FD) Y, p, p, r, I / Y, I / Y n.a. G G P Minik & Neumann () 6 OECD counries 955 99 (Q) VAR (FD), VECM I, C, I, Y insignifican c / posiive G P Pereira (b) OECD counries 96 99 (A) VAR (FD), VECM I, I, N, Y posiive b Noes: A = annual daa. Q = quarerly daa. VAR = vecor auoregression. VECM = vecor error correcion model. VARMA = vecor auoregressive moving P G average model. FD = model in (log) firs differences. L = model in (log) levels. Y = oupu. N = employmen. K = privae capial. K = public capial. I P G G D G = privae invesmen. I = public invesmen. C = public consumpion. G = governmen defense spending. B = governmen deb. M = money supply. G P E = energy price. π = inflaion. p = relaive price of public invesmen. p = relaive price of privae invesmen. r = real ineres rae. a Long-run oupu effec of public capial (public invesmen), measured by he impulse responses of oupu o a shock o public capial (public invesmen). b Sudy does no repor any measure of he saisical significance of he esimaed effec. c Posiive and saisically significan shor-run effec.
subsiued for capial or addiional variables have been included in he model. (iv) There is a wide variey of model specificaions as regards he (non-)consideraion of coinegraion. Some sudies, such as Cullison (993), specify VAR models in firs differences wihou esing for coinegraion. This way of proceeding seems dubious since i neglecs poenial long-run relaionships beween he levels. Oher sudies, such as Lighar (), specify VAR models in levels based on he resul of Sims e al. (99) ha ordinary leas squares esimaes of VAR coefficiens are consisen even if he variables are non-saionary and possibly coinegraed. Unforunaely, he consisency of VAR coefficien esimaes does no carry over o esimaes of impulse response funcions as discussed in he nex secion. Finally, some sudies, such as Pereira (), es for coinegraion using he Engle-Granger approach, hus neglecing he possibiliy ha here may be more han one coinegraion relaionship in higher-dimensional sysems. The las column of Table repors he main conclusions of he considered sudies regarding he long-run oupu effecs of public capial. 63 As can be seen in he majoriy of sudies he long-run response of oupu o a shock o public capial is posiive. In general, he effecs are considerably smaller han hose repored in he lieraure applying he producion funcion approach (see, e.g., Pereira ()). However, almos all of hese sudies fail o provide any measure of he uncerainy surrounding he impulse response esimaes so ha i is impossible o judge he saisical significance of he resuls. For hose sudies for which such measures are provided, he long-run oupu effec is in general insignifican. Anoher imporan resul emerging from his lieraure is ha many sudies find evidence for reverse causaion, i.e., feedback from oupu o public capial and vice versa (see, e.g., Baina (998)). This suggess ha i is indeed imporan o rea public capial as endogenous variable. Our sudy can be viewed as boh a reassessmen of and an addiion o he exising empirical lieraure: (i) We reassess he empirical lieraure by carefully addressing he imporan issue of coinegraion and by providing confidence inervals measuring he uncerainy surrounding he poin esimaes of he impulses responses. (ii) We add o he empirical lieraure by presening resuls for a large sample of OECD counries for many of which here is no VAR evidence so far 6 and by proposing a new idenificaion scheme, exending he approach proposed by Blanchard and Peroi (). 63 Some of he sudies lised in Table 3. do no perform a policy analysis. In hese cases, he las column of he able has an n.a. (no available) enry. 6 Noe ha he wo sudies ha come closes o ours in scope, Minik and Neumann () and Pereira (b), boh use public invesmen as model variable whereas we use public capial.
5 3 Economeric Mehodology This secion presens he vecor auoregressive (VAR) mehodology used in he empirical applicaion laer in his paper. 65 A VAR model is a k-equaion, k-variable linear model in which each variable is in urn explained by is own lagged values, pas values of he remaining k- variables and possibly deerminisic erms such as consans and linear ime rends. Esimaion of unresriced VAR models is sraighforward and is briefly skeched in Secion 3.. However, complicaions arise if some or all of he variables included in a VAR model are non-saionary. In his case, he appropriae esimaion approach depends on wheher he variables are coinegraed or no. 66 If he variables are coinegraed he VAR model is said o have an error-correcion represenaion. As esimaion of coinegraed VAR models is more involved han esimaion of unresriced models, Secion 3. describes a popular esimaion approach in some deail: he Johansen (988, 99) maximum likelihood approach. As he unresriced and coinegraed VAR model used a he esimaion sage are reduced-form models, hey canno direcly be used for srucural inference and policy analysis. An idenificaion problem has o be solved such ha he VAR model can be given a srucural inerpreaion. The idenificaion problem is discussed in Secion 3.3. 3. The Unresriced VAR Model A p-h order vecor auoregressive model, denoed VAR(p), can be expressed as 67 X = A X + A X + K+ Ap X p + ΦD + ε, (3.) where X [ x,, x ] ' is a se of variables colleced in a ( k ) vecor, A j denoes a K k k k marix of auoregressive coefficiens for j =,, K, p, and Φ denoes a k d marix of coefficiens on deerminisic erms colleced in he d vecor D. The vecor ε [ ε,, ε ]' is a k-dimensional whie noise process, i.e., E [ ε ] =, E [ ε ε ' ] = Ω K [ ε ' ] = k E ε s for s, wih Ω a ( k k ) symmeric posiive definie marix., and 65 See Lükepohl () and Sock and Wason () for surveys on he VAR mehodology. 66 See Hendry and Juselius (, ) for an inroducion o he concep of coinegraion, Johansen (995) and Juselius (3) for an exensive reamen of coinegraed VAR models.
6 Esimaion of he unresriced VAR model is paricularly easy. Condiioning on he firs p observaions ( denoed X, X, K X ) X, p+ p+, and basing esimaion on he sample, X, K X T, he k equaions of he VAR can be esimaed separaely by ordinary leas squares (OLS). Since he se of regressors is idenical across equaions, he OLS esimaor is idenical o he generalized leas squares (GLS) esimaor of he seemingly unrelaed regressions model. Moreover, under he assumpion ha he ε are Gaussian whie noise, i can be shown ha he simple OLS esimaor is idenical o he full informaion maximum likelihood (FIML) esimaor (see, e.g., Hamilon (99: 93-96)). Finally, under general condiions, he OLS esimaor of A [ A,, ] is consisen and asympoically normally K A p disribued. Remarkably, his resul no only holds in he case of saionary variables, bu also in he case in which some variables are inegraed and possibly coinegraed (Sims e al. (99)). Based on his resul many researchers have ignored nonsaionariy issues and esimaed unresriced VAR models in levels. This approach is characerisic, e.g., of he lieraure on he empirical effecs of moneary policy shocks surveyed in Chrisiano e al. (999). A drawback of his approach is ha, while he auoregressive coefficiens in equaion (3.) are esimaed consisenly, his may no be rue for oher quaniies derived from hese esimaes. In paricular, Phillips (998) showed ha impulse responses and forecas error variance decomposiions based on he esimaion of unresriced VAR models are inconsisen a long horizons in he presence of non-saionary variables. In conras, vecor error correcion models (VECMs) produce consisen esimaes of impulse responses and of forecas error variance decomposiions if he number of coinegraion relaions is esimaed consisenly. As impulse response analysis is one of he main ools for policy analysis based on VAR models, a careful invesigaion of he coinegraion properies of he VAR sysem is warraned. The nex secion presens Johansen s (988, 99) maximum likelihood approach for he esimaion of coinegraed VAR processes. 67 This secion builds on he assumpion of a known lag order p. In he empirical applicaion, he opimal lag order is explicily esed for.
7 3. The Coinegraed VAR Model The saring poin of he analysis is ha any VAR(p) model (3.) can always be wrien in equivalen form X = ΠX + Γ X + Γ X + K + Γp X p+ + ΦD + ε, (3.) p A i i= p where Π I + and Γ j Ai ( j =,, K, p ) denoe ( k k ) marices of i= j+ coefficiens, respecively. If no resricions are imposed on Π, hen he k equaions of sysem (3.) can be esimaed by simple OLS. In his case, he esimaion resuls will be idenical o hose obained from he OLS esimaion of he unresriced VAR(p) model, aking ino accoun he relaionship beween Π Γ [ Γ, K Γ ] and A [ A,, ],, p K A p. As we will see, his is a special case, however, arising when none of he variables colleced in he vecor is non-saionary. In he following, we assume ha each of he series in X aken individually is inegraed of order one ( I () ). Under his assumpion he vecor X is said o be coinegraed if for some nonzero ( k ) vecor a he linear combinaion a' X is saionary (see, e.g., Hamilon (99: 57)). Moreover, in a sysem wih more han wo variables, here may be r < k linearly independen vecors a,, a K r such ha X β ' X is a saionary ( r ) vecor, where β ' is he ranspose of he ( k r) marix β [ a,, ]. In his case, here are exacly r coinegraing relaions among he series colleced in X. Furhermore, if he process can be described as a p-h order VAR in levels as in (3.), hen here exiss a ( k r) marix α such ha Π = αβ ' and here furher exis ( k k ) marices Γ K, Γp such ha K a r, X = X + Γ X + Γ X + + Γp X p+ αβ ' K + ΦD + ε. (3.3) This follows from Granger s represenaion heorem, saing ha coinegraed series can be represened by error correcion models (see Engle and Granger (987: 55-56)). The sysem (3.3) differs from he sysem (3.) in ha i imposes a reduced-rank resricion on Π for r < k. The foregoing discussion allows us o disinguish hree ineresing cases: (i) If r =, hen rank( Π ) = and he variables colleced in X are no coinegraed. In his case, here are k
8 independen sochasic rends in he sysem and i is appropriae o esimae he VAR model in firs differences, dropping X as regressor in equaion (3.). (ii) A he oher exreme, if r = k, hen rank( Π ) = k and each variable in X aken individually mus be saionary. Or, in oher words, he number of sochasic rends, given by k r, is equal o zero. As menioned above, in his case, he sysem can be esimaed by applying OLS eiher o he unresriced VAR in levels (equaion 3.) or o is equivalen represenaion given by (3.). (iii) In he inermediae case, < r < k, he variables in X are driven by < k r < k common sochasic rends and rank( Π ) = r < k. In his case, esimaing he sysem given by (3.3) by OLS is no appropriae since cross-equaion resricions have o be imposed on he marix Π. Insead, he maximum likelihood approach developed by Johansen (988, 99) can be applied in order o esimae he space spanned by he coinegraing vecors colleced in β. An addiional asse of Johansen s approach is ha i enables us o es for he number of coinegraing relaions, which in many applicaions is unknown a priori. 68 Thus, i is a unifying framework ha allows us o invesigae which of he hree cases is he relevan one in empirical applicaions. Before urning o he deails of Johansen s esimaion approach i is worhwhile o discuss he role of he deerminisic erms colleced in he vecor D. The specificaion of he deerminisic erms in equaion (3.3) plays an imporan role in he analysis because he asympoic disribuions of he es saisics used for he deerminaion of he number of coinegraing vecors depends on he assumpions made on hese erms (see Johansen (99)). In he following, he deerminisic erm in equaion (3.3) is assumed o be expressible as Φ D µ +, i.e., equaion (3.3) poenially includes k consans (for µ ) as well as k µ linear rends (for µ ), where µ and µ are ( k ) vecors, respecively. The coefficien on hese wo deerminisic erms can be furher decomposed such ha µ αρ + α γ, µ αρ + α γ, (3.) 68 See Hubrich e al. () for a review of sysems coinegraion ess, including Johansen s likelihood raio ess.
9 where α is a ( k ( k r) ) marix orhogonal o α, i.e. α ' =. The definiions (3.) imply ha he consan and he rend coefficiens can each be decomposed ino one par belonging o he coinegraing space ( ρ i, i =,) and anoher par ha is orhogonal o he coinegraing space ( γ, i =,). Wihou any resricions on he coefficiens, he model given by (3.3), assuming ha i Φ D µ + µ α, is consisen wih linear rends in he differenced process X and, hence, quadraic rends in he process X. Johansen (99) disinguishes five alernaive models, corresponding o alernaive ses of resricions on he deerminisic erms. In he following, we concenrae on he model which seems o be he mos relevan for our problem: he consan is lef unresriced ( ρ, γ ) and he rend is resriced o he coinegraing space ( ρ, γ ). 69 This specificaion eliminaes he poenial for quadraic rends in = X, while allowing for linear rends in X and for rend-saionary coinegraing relaions. The laer may be jusified on he grounds ha he coinegraing space migh conain a producion funcion as one coinegraing vecor (see,e.g., Surm and De Haan (995)). Based on his specificaion of he deerminisic erms, he vecor error correcion model (3.3) may be wrien as X ~ ~ α ' X X K X + ε, (3.5) = β + Γ + + Γp p+ + µ ~ where β ' [ β ', ρ] is an ( r ( k +) ) marix and X [ X ', ] ' is a (( + ) ) ~ k vecor. Having chosen he relevan model, we can now describe Johansen s (988, 99) esimaion approach. The following presenaion of Johansen s algorihm draws on Hamilon (99: 636-637) who presens i for a model wihou rend ( µ = ). Under he assumpion ha he error erms ε are Gaussian whie noise, i can be shown ha he esimaes calculaed wih his algorihm are idenical o he maximum likelihood esimaes (Johansen (988)). Following Hamilon (99: 636-637), Johansen s algorihm can be divided ino hree seps: In he firs sep, a number of auxiliary regressions is carried ou in order o concenrae ou Γ,, Γp K and µ. These parameers are eliminaed by OLS regression of X and X on ~ 69 The resuling model corresponds o model H*(r) in Johansen (99), case IV (wihou exogenous variables) in Pesaran e al. () and case * in Oserwald-Lenum (99). Franses () as well as Pesaran and Smih (998: 83) argue ha he case analyzed here is one of wo cases paricularly relevan in pracice, he oher one being ha of a resriced consan ( ρ, γ ) and no rend ( ρ γ ). =, =
X K and a consan erm. Denoing OLS esimaes by a ha, he firs se of k,, X p+ regressions can be expressed as ˆ K ˆ ˆ uˆ, (3.6) X = Ψ X + + Ψp X p+ + ζ + where Ψˆ i ( i =, K, p ) is a ( k k ) marix of OLS coefficien esimaes, ˆ ζ is a ( k ) vecor of coefficien esimaes of he consan and û denoes he ( k ) vecor of OLS residuals. Noe ha equaion (3.6) is jus a vecor auoregressive model for differences. 7 The second se of ( k +) OLS regressions can be expressed as X in firs ~ X ˆ K ˆ ˆ wˆ, (3.7) = B X + + B p X p+ + ζ + B i K is a (( k ) k) where ˆ ( i =,, p ) + marix of OLS coefficien esimaes, ˆ ζ is a (( k + ) ) vecor of coefficien esimaes of he consan and ŵ denoes he (( k + ) ) vecor of OLS residuals. In he second sep, he canonical correlaions beween û and purpose, firs, he sample variance-covariance marices of û and ŵ are calculaed. 7 For his ŵ are calculaed as T Σˆ ww wˆ wˆ ', Σˆ uu uˆ uˆ ', T T i= T i= (3.8) Σ uw T uˆ wˆ ' T ˆ, ˆ ( ˆ ) ' i= Σ Σ. wu wu λˆ of he (( k + ) ( k +) ) marix ( Σˆ ww ) Σˆ wu ( Σˆ uu ) Σˆ uw Then, he eigenvalues i are calculaed, wih he eigenvalues ordered > ˆ λ ˆ ˆ > K > λ k > λ k + >. The marix holding he 7 If for some reason i were known ha he coinegraing rank was zero, hen (3.6) would be he appropriae model and he esimaion problem would already have been solved. However, in empirical applicaions he coinegraing rank is usually unknown. Johansen s algorihm, a a laer sage, allows o es for he number of coinegraing relaions. 7 See Hamilon (99: 63-635) for an exposiion of canonical correlaion analysis.
ˆ ˆk + eigenvecors associaed wih hese eigenvalues is denoed by V [ ˆ,, ν ] eigenvecors are normalized such ha Vˆ ' Σˆ ww Vˆ = I ν K, where he. The eigenvalues λˆ i can be inerpreed as squared canonical correlaions beween ~ X and X, condiional on X K, X p, + and he consans. Thus, he magniude of λˆ i can be hough of as measuring he ~ saionariy of he corresponding ˆ ν i' X. Inuiively, he larger λˆ i he more confiden we ~ can be ha ˆ ν i' X is saionary (a coinegraing relaion). In conras, a small value for λˆ i is ~ an indicaion ha ˆ ν i' X is only weakly correlaed wih X and, hus, probably nonsaionary. This reasoning suggess ha he number of coinegraing relaions is equivalen o he number of eigenvalues λˆ i ha are significanly differen from zero. A formal es for he number of coinegraing relaions can be based on he following likelihood raio es saisic ofen referred o as race es saisic (see Hamilon (99: 65)): k ( L * L *) = T log( λˆ ) A i= r+ i, (3.9) where L * is he maximum value he log likelihood funcion can aain under he null hypohesis of r coinegraing vecors and where L * A is he maximum value he log likelihood funcion can aain under he alernaive hypohesis ha here are as many coinegraing vecors as here are variables in X. The null hypohesis of he es, hus, is ha he smalles eigenvalues are equal o zero. If his hypohesis can be acceped, hen he process is driven by k r g k r sochasic rends. Since under he null hypohesis sochasic rends are presen, he asympoic disribuion of he race es saisic is non-sandard. Johansen (99: 5-6) shows ha he asympoic disribuion of he es saisic depends on he number of sochasic rends and on he assumpions on he deerminisic erms. Criical values aking his ino accoun have been abulaed by MacKinnon e al. (999), among ohers. In pracice, he coinegraing rank can be deermined by a nesed sequence of hypoheses (see Johansen (: 36)), saring wih he hypohesis ha r = and, if his hypohesis is rejeced, esing r =, and so on, coninuing unil he null hypohesis canno be rejeced anymore. Noe ha we can relae he oucome of his es sequence o he hree model cases discussed above: (i) if he null hypohesis r = canno be rejeced, hen he appropriae model is a VAR for X as X
given by (3.6), (ii) if < r < k, hen he VECM given by (3.5) under he resricion rank( π ) = r is he appropriae model, and, (iii) if he las null hypohesis of he es sequence, r = k, is rejeced, hen he variables in X are (rend-)saionary and in his case he appropriae model is he unresriced VAR model for X in levels (3.). In he hird and final sep of Johansen s approach, he maximum likelihood esimaes of he model parameers are calculaed. Based on he choice of coinegraing rank r, he maximum likelihood esimae of ~ β is given by he (( k + ) r) marix ˆ~ β = [ ˆ ν, ˆ ν,, ˆ ]. (3.) K ν r where [ ν,, ˆ ] ν r ˆ K are he eigenvecors associaed wih he r larges eigenvalues of he marix ( Σˆ ww ) Σˆ wu ( Σˆ uu ) Σˆ uw. Then, he maximum likelihood esimae of he ( k r) marix of adjusmen coefficiens α is given by ^ ˆ~ ˆ α = u w β, (3.) ~ ~ and he maximum likelihood esimae of he ( k ( k +) ) marix Π αβ ' obains as ˆ~ ˆ ˆ~ Π = αβ '. (3.) Finally, he maximum likelihood esimaes of he ( k k ) marices Γ i and he ( k ) vecor of consans can be calculaed as ˆ~ Γ ˆ = Ψˆ ΠBˆ, for i =, K, p, (3.3) i i i ˆ ˆ ˆ~ µ = ζ Π ˆ ζ. (3.) ~ Under general condiions he esimaors of Π, Γ i and µ are consisen and asympoically normally disribued (see, e.g., Lükepohl and Breiung (997: 37)). Noe, however, ha he
3 same is no rue for he esimaors of ~ α and β. Wihou idenifying resricions only he coinegraion space is esimaed consisenly, bu no he coinegraion parameers β ~. 7 A necessary condiion for he parameers ~ β o be idenified is ha a leas r resricions be imposed on he parameers of each coinegraing vecor. Wihou such resricions i is no possible o give a srucural inerpreaion o he coinegraion parameers. Ye, even if such resricions are imposed, i is in general no possible o infer he long-run effecs of shocks hiing he sysem from he coinegraing vecors alone (see Lükepohl and Reimers (99: 69)). Insead, such effecs can be obained from an impulse response analysis as described in he nex secion. For such an analysis i is no necessary ha he coinegraing vecors be idenified. Thus, he srucural analysis of he coinegraed VAR model can be based on he Π ~ marix ha, as was noed above, is esimaed consisenly in Johansen s approach even if no idenifying resricions are imposed on he marix β ~. Thus, in he empirical applicaion, we will only impose he appropriae rank resricion on Π ~, bu no idenify he individual coinegraing relaions. 3.3 The Srucural VAR Model The previous wo sub-secions have described how he VAR model can be esimaed for alernaive assumpions on he coinegraing rank. As hese models are reduced-form models, lile can be learned abou he underlying economic srucure unless idenifying resricions are imposed. This sub-secion shows how o give VAR models a srucural inerpreaion and, in paricular, shows how o derive impulse response funcions from he reduced-form parameer esimaes. Impulse responses give an insigh ino he reacion of key macroeconomic variables o an unexpeced change in one variable (here, e.g., public capial). The subsequen analysis is based on he following reduced-form model: X = A X + A X + K + Ap X p + µ + µ + ε. (3.5) 7 This idenificaion problem is someimes called he long-run idenificaion problem because i concerns he long-run srucure (he coinegraing relaions). I is disinc from he shor-run idenificaion problem discussed in he nex secion. See Juselius (3) for an insighful reamen of boh idenificaion problems.
Equaion (3.5) is equivalen o he unresriced VAR model (3.), excep ha µ + µ has been subsiued for Φ D. Ye, his model can serve in he srucural analysis irrespecive of wheher he variables in X are non-saionary or no. 73 Pre-muliplying equaion (3.5) by he ( k k ) marix A gives he srucural form A X A X A X K A X A A + Be, (3.6) = * + * + + * p p + µ + µ where * Ai for i =, K, p, and Be = A ε describes he relaion beween he Ai A srucural disurbances e and he reduced-form disurbances ε. In he following, i is assumed ha he srucural disurbances e are whie noise and uncorrelaed wih each oher, i.e. he variance-covariance marix of he srucural disurbances, denoed D, is diagonal. The marix A describes he conemporaneous relaion among he variables colleced in he vecor X. 7 In he lieraure, his represenaion of he srucural form is ofen called he AB model (see Amisano and Giannini (997)). Wihou resricions on he parameers A A*, i and B, model (3.6) is no idenified. Esimaion of he reduced-form model (3.5) yields parameer esimaes for A i, µ, µ and for k ( k +) / disinc elemens of he variance-covariance marix Ω of he reduced-form disurbances. Ye, hese k ( k +) / elemens of Ω do no allow o uniquely deermine he k free parameers in marices beween Ω on he one hand and follows: A, B and D of he srucural form. The relaionship A, B and D on he oher hand can be formalized as [ ' ] = A B E [ e e ' ] B' ( A )' = A B D B' ( A )' Ω = E ε ε, (3.7) 73 While in he esimaion of he VAR parameers i is crucial o disinguish he hree cases analyzed in he previous secion, he analysis can proceed based on he represenaion (3.5) once he esimaion sage has been compleed. Noe ha if he coinegraing rank r is equal o, hen he following relaions can be used o map he parameers Ψ from equaion (3.6) o he A parameer marices in (3.5): i = I k + Ψ i A, A i = Ψi Ψi for i =, K, p, A p = Ψp. If he coinegraing rank is < r < k, he following relaions can be used o map he parameers Π and Γi from he VECM (3.5) o he A i marices: A = Π + I k + Γ, A i = Γi Γi for i =, K, p, A p = Γp. 7 Noe ha he equaions are normalized such ha A is a marix wih ones along is principal diagonal.
5 where E is he expecaions operaor. Given appropriae resricions on A, B, D, he freely varying parameers of hese marices can be esimaed by full-informaion maximum likelihood or by he generalized mehod of momens (see, e.g., Breiung (: 57-6)). The exisence of a unique maximum of he likelihood funcion necessiaes boh an order condiion and a rank condiion o be saisfied. The order condiion is ha A, B and D have no more han k ( k +) / unknown parameers. 75 Accordingly, a leas k + k( k ) / resricions have o be placed on he parameers of A, B and D in order for he order condiion o be saisfied. In he empirical lieraure, a large number of alernaive idenificaion procedures have been applied. Two of hese will be used in he empirical applicaion: (i) he recursive approach originally proposed by Sims (98) ha resrics B o a k-dimensional ideniy marix and A o a lower riangular marix, and, (ii) he approach of Blanchard and Peroi () discussed in Secion 5. ha places resricions on he A and B marices subsanially differing from hose of he recursive approach. The soluion o he idenificaion problem given by he recursive VAR approach implies ha equaion (3.7) can be rewrien as Ω = A ( )' ( D A = A D D A )' = PP', (3.8) where / P A D and A is lower riangular. This, in urn, implies ha P is a lower riangular marix wih he sandard deviaions of he srucural disurbances on is principal diagonal. Moreover, i can be shown ha P is he (unique) Cholesky facor of he symmeric posiive definie marix Ω (Hamilon (99: 9-9)). Thus, in his idenificaion approach i is paricularly easy o recover he esimaes of he srucural parameers. Noe, however, ha while P is unique for a given ordering of he variables in X, here are k! possible orderings in oal. Hence, i is imporan o check how sensiive he dynamic properies of he model are o alernaive orderings of he variables. Based on hese consideraions, i is possible o disinguish wo ypes of impulse responses (see, e.g., Lükepohl and Reimers (99: 55)): (i) hose impulse responses ha give he dynamic effecs of innovaions in he reduced-form disurbances ε, and, (ii) he impulse 75 As he rank condiion requires a lenghy derivaion, we refer he ineresed reader o is deailed exposiion in Amisano and Giannini (997: 8-57). The srucural VAR models analyzed in Secions 3.3 and 3. all saisfy he rank condiion.
6 responses ha give he dynamic effecs of innovaions in he srucural-form disurbances e. In empirical applicaions, in general, only he second se of impulse responses is of ineres because i allows o sudy he effecs of shocks o one variable in isolaion since he variancecovariance marix of he srucural disurbances is diagonal. In conras, he reduced-form residuals are in general correlaed ( Ω is no diagonal) so ha lile can be learned from he sudy of he effecs of a change in a single elemen of ε if hisorically changes in his elemen have coincided wih changes in oher elemens of ε. Sill, as he srucural disurbances can be inerpreed as linear combinaions of he reducedform disurbances, i is useful o calculae he impulse responses giving he effecs of innovaions in ε in an inermediae sep. These quaniies are given by (see Lükepohl (99: 8)): Ξ n = n Ξ j= n j A j, n =,, K, (3.9) where Ξ = I k and A j = for j > p. The row i, column k elemen of Ξ n gives he response of variable X i o a one-uni increase in he kh variable, n periods ago. Given hese quaniies i is easy o obain he impulse responses o innovaions in he srucural disurbances e. For he general srucural model, hey are given by Θ n = Ξ n A BD, n =,, K, (3.) and for he recursive VAR hey obain as Θ n = Ξ n P, n =,, K. (3.) The elemens of Θ n have an inerpreaion analogous o ha of he elemens of Ξ n, excep ha he size of he impulses is one sandard deviaion here. Impulses of size one uni could easily be obained by pos-muliplying (3.) and (3.) wih D. As he impulse responses are random variables i is useful o provide confidence inervals in order o measure he uncerainy surrounding he esimaed impulse responses. Confidence inervals can be consruced based eiher on analyical derivaives (see Lükepohl (99) for
7 saionary VAR models and Lükepohl and Reimers (99) for coinegraed VAR models), on Mone Carlo simulaion mehods (see, e.g., Sims and Zha (999)) or on he boosrap mehodology (see, e.g., Runkle (987)). In he empirical applicaion, we repor confidence inervals based on he boosrap mehodology. 76 The simple boosrap algorihm can be summarized as follows:. Esimae he parameers of he model (3.5) by he appropriae mehod. 77 * *,, T. Generae boosrap residuals ε K ε by randomly drawing wih replacemen from he se of esimaed residuals ˆ ε, K, ˆ ε. 78 T 3. Condiion on he pre-sample values ( X p+,..., X ) = ( X p+,..., X ) and consruc boosrap ime series X ˆ * X recursively using equaion (3.5), ˆ * * * * = A X + K + Ap X p + ˆ µ + ˆ µ + ε,, K, T * * =.. Re-esimae he parameers A, K, A p, µ and µ from he generaed daa and calculae he impulse response funcions Θˆ *, n =,,, K n. 5. Repea seps a large number of imes (in he empirical applicaion: ) and calculae he α and elemens of α percenile inerval endpoins of he disribuion of he individual Θˆ * n, n =,,, K. In he empirical applicaion, we se α =. 6 and accordingly repor 68% confidence inervals. 79 76 See Hall (99) and Horowiz () for exensive reamens of he boosrap mehodology in general. 77 Kilian (998) proposes a bias-correced boosrap mehod for saionary VAR models based on an adjusmen of he esimaed auoregressive parameers Â. As i is unclear wheher his bias correcion improves he accuracy of esimaed impulse responses also in he case of non-saionary VAR models, we do no pursue his roue here. 78 A major difference beween he boosrap approach and he Mone Carlo simulaion mehod is ha he former builds on random draws from he se of esimaed residuals while he laer in general builds on random draws from a normal disribuion. The boosrap approach, insead, allows for non-normaliy of he residuals. 79 In he empirical VAR lieraure, ypically eiher 68% or 95% confidence inervals are repored. Sims (987: 3) argues agains he use of 95% confidence inervals in VAR sudies on he grounds ha here is no scienific jusificaion for esing hypoheses a he 5 % significance level in every applicaion. He suggess o rea he saisical significance of impulse responses derived from VAR coefficien esimaes differenly from ha of coefficien esimaes in sandard economeric models. I is inheren in VAR models ha mos of he parameer esimaes are insignificanly differen from zero when esed a he 5% level, and his ranslaes ino relaively large confidence inervals for impulse responses. Sill, esimaes from unconsrained VAR models are widely hough o provide a useful daa summary. Agains his background, Sims and Zha (999: 8) recommend he use of 68% confidence inervals for esimaed impulse responses. In he empirical applicaion,
8 Empirical Resuls This secion presens empirical evidence on he dynamic effecs of public capial for OECD counries based on VAR models. Secion. deals wih model selecion and deerminaion of coinegraion rank. Secion. presens he resuls of an impulse response analysis based on a se of benchmark idenifying assumpions. The resuls of a sensiiviy analysis employing alernaive idenifying assumpions are presened in Secion 5.. Model Specificaion and Esimaion Daa The counries considered in his paper are he same as hose considered in Kamps (). 8 Wih a few excepions, he sample periods cover he years 96-. For each counry, we G specify a four-variable VAR model including he public ne capial sock, K, he privae ne P capial sock, K, he number of employed persons, N, and real GDP, Y. 8 Expressing all variables in naural logarihms muliplied by and denoing he ransformed variables by lower-case leers, he vecor of endogenous variables G P [ X k, k, n, y ]'. 8 VAR Order Selecion X can be expressed as The exposiion of he VAR mehodology in Secion 3 was based on he implici assumpion of a known lag order p. In empirical applicaions, however, he lag order is ypically unknown. In he economeric lieraure, a number of selecion crieria have been proposed ha can be used o deermine he opimal lag order. A review of popular selecion crieria can be found in Lükepohl (99: Chapers.3 and..). The saring poin is o esimae VAR(m) models wih orders m =,..., M condiional on M pre-sample observaions X M,..., ) and hen ( X we follow his advice, ye we refrain from drawing srong conclusions abou he saisical significance of he esimaed impulse responses. 8 Ausralia, Ausria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Iceland, Ireland, Ialy, Japan, he Neherlands, New Zealand, Norway, Porugal, Spain, Sweden, Swizerland, Unied Kingdom and he Unied Saes 8 Real GDP and employmen are drawn from he OECD Analyical Daabase, Version June. The public and privae capial socks are aken from Kamps (). The daase is available on reques. 8 Muliplying he variables in logarihms by faciliaes he inerpreaion of he esimaed impulse responses. In his case, he impulse responses give he percenage change in he level of he respecive variable.
9 o choose an esimaor of he order p ha minimizes some selecion crierion. The general srucure of he selecion crieria applied here can be expressed as follows: Cr( m) = log Ωˆ ( m) + ϕ( m), m =,, K, M, (3.) T where M is he maximum lag lengh considered. The firs addend in equaion (3.) is he log deerminan of he residual covariance marix, which in general is decreasing in m, while he second addend is increasing in m. The selecion crieria, hus, srike a balance beween goodness of fi and parsimonious specificaion of he model. The selecion crieria considered here differ in he specificaion of he erm ϕ (m) : (i) for he Akaike (97) informaion crierion (AIC) ϕ ( m ) = mk ϕ ( m ) = mk minimizing (3.): ϕ ( m ) = mk, (ii) for he Schwarz (978) informaion crierion (SC) log( T ), and (iii) for he Hannan and Quinn (979) informaion crierion (HQ) log(log( T )). For each crierion, he opimal lag lengh pˆ is found by { Cr( m) m, M } Cr ( pˆ) = min =,...,. (3.3) Asympoically, he AIC overesimaes he lag order wih posiive probabiliy, whereas he wo oher crieria esimae he order consisenly if he VAR process has a finie order and he maximum order M is larger han he rue order p (see Lükepohl (99: 3-3)). These resuls no only hold for saionary variables, bu also in he case of I () variables. Thus, VAR order selecion can be based on he unresriced VAR model (3.) discussed in Secion 3.. Given he small sample size in our empirical applicaion, we choose no o discard he AIC on he grounds of asympoic resuls. The firs hree columns of Table give he opimal lag order seleced by he hree crieria for each of he OECD counries considered. The resuls reveal an ineresing relaion among he crieria ha holds for sample sizes T 6 (see Lükepohl (99: 33)): pˆ ( SC) pˆ( HQ) pˆ( AIC). Whereas he AIC selecs a lag order of for mos counries, he HQ and SC crieria selec a lag order of in mos cases. Given he small sample size, we are ineresed in a parsimonious specificaion of he model. Thus, we choose he lag order seleced by he SC crierion in general. Ye, we also perform specificaion ess ha check
wheher for he lag lengh seleced by he SC crierion he residuals are free from firs-order
Table : Specificaion of VAR orders Counry VAR order minimizing AIC a SC b HQ c Chosen VAR order d Auocorrelaion f Specificaion ess (p-values) e Heeroscedasiciy g Normaliy h Ausralia.6.336.73 Ausria.587.9.85 Belgium.65.83.88 Canada.75.9.59 Denmark 3 3 3..3.* Finland.68.8.35 France.55.9.5 Germany.577.75. Greece 3 3.657.67.3* Iceland.76.96.63 Ireland.5.*. Ialy..5*.5 Japan 3.8.88.3* Neherlands.68.6.* New Zealand 3.5..6 Norway.37.757.36 Porugal.355.9.3 Spain.8.33.* Sweden..57.3* Swizerland.38.39.77 Unied Kingdom.56.5.78 Unied Saes.5.56.5 Noes: The maximum order considered is equal o. The underlying VAR model conains consans and linear ime rends. In he case of Germany, he VAR model also conains a dummy variable (se o in 99 and oherwise) as well as is lagged value. In he case of Denmark, he VAR model also conains a dummy variable (se o in 973, in 97 and oherwise) as well as is lagged value. a Akaike informaion crierion (Akaike (97)). b Schwarz informaion crierion (Schwarz (978)). c Hannan-Quinn informaion crierion (Hannan and Quinn (979)). d The VAR order is chosen on he basis of he informaion crieria and on he basis of specificaion ess. e The specificaions ess are based on he residuals from he esimaion of an unresriced VAR (p), where p is he ineger repored in he column Chosen VAR order. * denoes saisical significance a he 5 percen level. f Mulivariae auocorrelaion LM es (Johansen (995: )). Under he null hypohesis of no serial correlaion of order h (here: h = ) he es saisic is asympoically disribued χ wih 6 degrees of freedom. g Mulivariae exension of Whie s (98) heeroscedasiciy es (Doornik (996)). Under he null hypohesis of homoscedasic residuals he es saisic is asympoically disribued χ wih ( 8 p + ) degrees of freedom, where p is he chosen VAR order. h Mulivariae residual normaliy es (Lükepohl (99: 55 58)). Under he null hypohesis of normally disribued residuals he es saisic is asympoically disribued χ wih 8 degrees of freedom. auocorrelaion, homoscedasic and normally disribued. Since he race es for coinegraion is robus o deviaions from he normaliy assumpion (see Cheung and Lai (993: 3)) and since he asympoic properies of he VAR parameer esimaors do no depend on he normaliy assumpion (see Lükepohl (99: 359)), we do no dismiss he specificaion chosen by he SC crierion if he normaliy es indicaes ha he residuals are non-normal. However, if he auocorrelaion es indicaes ha he residuals are auocorrelaed, we increase he lag
order compared o he one seleced by he SC crierion unil he auocorrelaion es does no rejec he null hypohesis anymore. 83 The las hree columns of Table show he resuls of he hree specificaion ess for he chosen lag order for each of he OECD counries considered. The resuls show ha a he 5% significance level here are no signs of residual auocorrelaion and in general no signs of heeroscedasic residuals. 8 The following seps of he empirical analysis are, hus, based on he lag orders displayed in he middle column of Table. Deerminaion of Coinegraion Rank Neoclassical growh heory suggess ha along he balanced growh pah (seady sae) he socalled grea raios are consan, i.e., variables such as oupu, capial, consumpion and invesmen grow a he same consan rae. King e al. (99) firs invesigaed he coinegraion implicaions of neoclassical growh heory. They showed ha he consancy of he grea raios implies ha if he individual variables are non-saionary hey mus be driven by a single common sochasic rend. Translaed o our problem his implies ha he public capial o oupu raio and he privae capial o oupu raio are poenial coinegraing relaions. In addiion, a hird poenial coinegraing relaion migh be given by a producion funcion of he ype considered, e.g., by Aschauer (989). Ye, his criically hinges on he naure of echnology. If echnology is modeled as a rend-saionary process (see, e.g., Surm and De Haan (995)), hen he producion funcion could be a coinegraing relaion. 85 However, if echnology is a non-saionary process (see, e.g., Crowder and Himarios (997)) hen he producion funcion will no describe a saionary relaion beween he variables ~ G P colleced in he vecor X [ k, k, n, y, ]'. To sum up, based on economic heory we expec o find a mos hree coinegraing relaions. We es for he number of coinegraing relaions using Johansen s (988, 99) race es. 83 In he case of Denmark, a dummy variable (se o in 973, - in 97 and oherwise) was included because wihou he dummy variable he null hypohesis of no serial correlaion had o be rejeced a he 5% significance level for all lag orders beween and. In he case of Germany, a dummy variable (se o in 99 and oherwise) was included in order o accoun for he level shif in he variables due o German Reunificaion. 8 Excepions are Ireland and Ialy for which he heeroscedasiciy es saisic is significan a he 5% level. In boh cases, increasing he lag lengh o, as suggesed by he AIC, worsened he performance of he model wih respec o residual auocorrelaion. As auocorrelaion is more derimenal han heeroscedasiciy, we choose he shorer lag lengh in boh cases. 85 This, of course, raises he quesion of where he sochasic rends in he daa come from. Technology is widely viewed o be he prime candidae for a sochasic rend.
3 Table 3: Johansen (988, 99) coinegraion es Counry VAR order Trace saisic H : r = H : r = H : r = H : r = 3 Coinegraion rank a Ausralia 97. 57.73 3.6.3 3 Ausria 89.55 5. 5.8 6. Belgium 63.88 3.59 7. 5.7 Canada 8.59 5.3 7.8.73 3 c Denmark 3.79 58.79 3.6 8.9 3 Finland 7.5 38.76 7.37 8.3 France 8.3 5..89.56 Germany 7.53 38.35 6.3.96 Greece 3 7.93 6.95 3.53 7.58 Iceland 73.3 3. 9.3 6. Ireland 79.85 7.98 3.5. Ialy 98.9 57.7 3.98.56 3 c Japan 3.85 6.3 9.66 8.8 Neherlands 69.85.66. 8.58 New Zealand 58.6 3.98 5.8 5.56 Norway 8.5 9.9 7.67.9 3 Porugal 58. 38.3.7.57 Spain.3 65.8 3.8 9. 3 Sweden 86.35 5.6 3.96.87 3 Swizerland 7.8..66 5.55 Unied Kingdom 89.8 56.76 7. 8.6 3 Unied Saes.3 59.8 6.7.89 3 Criical values b 63.87.9 5.86.5 Noes: The underlying VAR model conains unresriced inerceps and resriced rend coefficiens and is of order p, where p is he ineger repored in he column VAR order. In he case of Germany, he VAR model also conains a dummy variable (se o in 99 and oherwise) as well as is lagged value. In he case of Denmark, he VAR model also conains a dummy variable (se o in 973, in 97 and oherwise) as well as is lagged value. a The es decision is based on he asympoic criical values repored in he boom row of he able. b The asympoic criical values for a 5% significance level for Johansen s log-likelihood based race saisic are aken from MacKinnon e al. (999), Table V. c In he cases of Canada and Ialy, he es resuls sugges ha he model variables are saionary (r=). However, recursively calculaed eigenvalues and race saisics (see Hansen and Juselius (995: 5-63) for deails) sugges ha for boh counries he fourh eigenvalue is no significanly differen from zero. Agains his background, we choose r=3 for boh counries. The es saisics are compued using equaion (3.9), and are hen compared wih he appropriae criical values abulaed by MacKinnon e al. (999). 86 The esing sequence can 86 The MacKinnon e al. (999) criical values are also used in he case of Denmark and Germany. The empirical models for hese wo counries include dummy variables. I is well known ha dummy variables may affec he asympoic disribuion of he race es saisic. This is paricularly rue for sep dummies ha give rise o broken linear rends in he levels of he variables. The dummy variables considered here, insead, are asympoically negligible.
be expressed as follows (Lükepohl ()): H ( r ) : rank( Π ) = r versus H ( r ) : rank(π) = k, r =,, K, 3. (3.) The esing sequence sars wih he null hypohesis ha he coinegraion rank is zero. If his hypohesis canno be rejeced, hen he esing sequence erminaes and a VAR model in firs differences is he appropriae model. A he oher exreme, if all null hypoheses have o be rejeced, hen he variables can be regarded as (rend-)saionary in levels. Table 3 displays he es resuls for each of he counries considered here. The resuls show ha for a large majoriy of counries he number of coinegraing relaions is eiher wo or hree. For he remaining counries, he coinegraion rank is lower; for wo counries, New Zealand and Porugal, i is even zero. As a consequence, for hese wo counries we esimae a VAR model for he variables in firs differences. For he oher counries, we esimae a VECM imposing he appropriae rank resricion.. Impulse Response Analysis This secion analyzes he dynamic properies of he esimaed VAR models for he OECD counries considered in his sudy wih he help of impulse response funcions. As was discussed in Secion 3.3, here is a need o idenify VAR models in order o be able o give he impulse response funcions a srucural inerpreaion. In he empirical lieraure on he effecs of moneary policy shocks, a large number of idenificaion schemes have been proposed. While in principle all of hese idenificaion schemes could also be applied o sudy he effecs of fiscal policy shocks, some of hese schemes do no seem o be useful for our seing. For example, i does no seem o be advisable o impose resricions on he long-run effecs of fiscal policy shocks. In oher seings, long-run resricions can be jusified by neuraliy proposiions derived from economy heory: e.g., Blanchard and Quah (989) impose he resricion ha shocks o aggregae demand do no affec oupu in he long run, and Shapiro and Wason (988) impose resricions such ha shocks o nominal variables such as he money supply or prices have no effec on real variables in he long run. In our seing,