Cointegration analysis in the presence of structural breaks in the deterministic trend

Transcription

1 Econometrics Journa (2), voume 3, pp Cointegration anaysis in the presence of structura breaks in the deterministic trend SØREN JOHANSEN,ROCCO MOSCONI,BENT NIELSEN Economics Department, European University Institute, Via Roccettini 9, 516 San Domenico di Fiesoe, Itay E-mai: Dipartimento di Economia e Produzione, Poitecnico di Miano, Piazza L. da Vinci 32, 2133 Miano, Itay E-mai: rocco.mosconi@poimi.it Department of Economics, University of Oxford & Nuffied Coege, Oxford OX1 1NF, UK E-mai: bent.niesen@nuf.ox.ac.uk Summary When anaysing macroeconomic data it is often of reevance to aow for structura breaks in the statistica anaysis. In particuar, cointegration anaysis in the presence of structura breaks coud be of interest. We propose a cointegration mode with piecewise inear trend and known break points. Within this mode it is possibe to test cointegration rank, restrictions on the cointegrating vector as we as restrictions on the sopes of the broken inear trend. Keywords: Break points, Cointegration, Common trend, Deterministic trend, Piecewise inear trend, Stochastic trend, Structura breaks, Vector autoregressive mode. 1. Introduction In the anaysis of economic time series it is often necessary to aow breaks in the deterministic components. When aowing for breaks the timing is important, this coud either be known in advance or an agorithm searching for breaks coud be appied. Whie both issues are discussed in the iterature and mainy in a univariate setting, this paper focuses on cointegration anaysis in a mutivariate setting in the presence of breaks at known points in time. The suggested approach is a sight generaization of the ikeihood-based cointegration anaysis in vector autoregressive modes suggested by Johansen (1988, 1996). There are ony a few conceptua differences and the major issue for the practitioner is that new asymptotic tabes are needed. This paper concerns ony asymptotic anaysis. Finite sampe properties of the rank test and tests for restrictions on the cointegrating vector have been discussed by Johansen (2a,b,c) in the situation of no breaks using an anaytic correction factor to the ikeihood ratio tests. c Roya Economic Society 2. Pubished by Backwe Pubishers Ltd, 18 Cowey Road, Oxford OX4 1JF, UK and 35 Main Street, Maden, MA, 2148, USA.

2 Cointegration anaysis 217 Structura breaks have been discussed intensivey in the context of univariate autoregressive time series with a unit root. An important finding is that a time series given by stationary fuctuations around a broken constant eve is better described by a random wak than a stationary time series, see Perron (1989, 199) and Rappoport and Reichin (1989). Addressing this issue, these authors suggested various univariate modes aowing for breaks in the deterministic term. In particuar, Perron (1989) suggested three modes: (A) crash mode with change in intercept but unaffected sope of the inear trend, (B) changing growth mode with no change in intercept but changing sope of trend function, and (C) where both intercept and sope are changed at the time of the break. The mode presented here generaizes mode (C) and aows for testing hypotheses corresponding to mode (A). A reated concern in econometric modes is parameter stabiity which is investigated by methods reated to those for known break points. These methods typicay aow for structura breaks at unknown times, and have been discussed, for instance, in the specia issues of the Journa of Business & Economic Statistics, voume 1, 199 and the Journa of Econometrics, voume 7, More recenty a test of this type has been suggested by Inoue (1999) in connection with cointegration testing in a vector autoregressive setting. Whie those authors and aso this paper are concerned with breaks in the deterministic terms, some procedures for anaysing breaks in the cointegration parameter have been presented by Kuo (1998), Seo (1998), Hansen and Johansen (1999) and Hansen (2). The approach taken here is to anayse cointegration in a Gaussian vector autoregressive mode with a broken inear trend and known break points. Likeihood anaysis of cointegration is then given in terms of reduced rank regression, a combination of east squares regression anaysis and canonica correation anaysis. The mode and the rank hypothesis are discussed in Section 2. Section 3 presents tests for cointegration rank for modes with broken trend and broken eve. The asymptotic distributions have been simuated and the resuts described by response surface anaysis. Next, in Section 4 various tests for inear restrictions on the sopes for the broken trend are given. Most of these tests are asymptoticay χ 2 -distributed. In Section 5 the suggested procedures are iustrated using data for infation rates, interest rates and exchange rate for Itay and Germany. Throughout the paper the foowing notationa convention is used. For a matrix, a, with fu coumn rank et a = a(a a) 1. Further, et a satisfy a a = and have the property that (a, a ) has fu rank. 2. The Mode The cointegrated vector autoregressive mode with no breaks is anaysed in detai in Johansen (1996). A simpe exampe of the basic mode is X t = X t t + µ + ε t, (2.1) where we have eft out more ags. Cointegration wi appear if has reduced rank in which case we can write = αβ. In that case the process generated by (2.1) has a quadratic trend, which is eiminated if we assume 1 = αγ, as wi be done throughout this paper. Note that the reduced rank invoves the combined matrix (, 1 ) = α(β,γ ). We therefore consider X t = α(β X t 1 + γ t) + µ + ε t, (2.2) c Roya Economic Society 2

3 218 S. Johansen et a. as the starting point for this paper. The roe of the deterministic terms, see Johansen (1996, Ch. 5) is briefy summarized as foows. The mode defined by (2.2) wi generate a process with a inear trend and is therefore caed H (r). Evenβ X t has a trend and is hence trend stationary. A number of sub-modes are defined by successivey restricting the parameters γ and µ. If γ = then X t = αβ X t 1 + µ + ε t, (2.3) and the process sti has a inear trend, but β X t does not and becomes stationary with a constant eve. This mode is denoted H c (r). Finay, if γ = and aso µ = αρ then X t = α(β X t 1 + ρ ) + ε t, (2.4) and the process has no inear trend in any direction. This mode is denoted H c (r). In the rest of this section we formuate a mode for the observed time series X t, t = 1,...,T, which is divided into sub-sampes according to the position of break points. For each sub-sampe a vector autoregressive mode is chosen, so the parameters of the stochastic components are the same for a sub-sampes, whie the deterministic trend may change between sub-sampes. In that case the process can be given rather simpe representations and interpretations in each period and statistica anaysis akin to that of the usua vector autoregressive modes Formuation of mode and rank hypothesis The mode aows for any pre-specified number of sampe periods, q say, of ength T j T j 1 for j = 1,...,q and = T < T 1 < T 2 < < T q = T. It foows that the ast observation in the jth sampe is T j whie T j + 1isthefirst observation in sampe period number ( j + 1). A vector autoregressive mode of order k is considered. In anaogy with the usua modes without structura breaks, the mode is formuated conditionay on the first k observations of each subsampe, X Tj 1 +1,...,X Tj 1 +k, and it is given by the equations ( Xt 1 X t = (, j ) t ) k 1 + µ j + Ɣ i X t i + ε t (2.5) for j = 1,...,q and T j 1 + k < t T j. The innovations are assumed to be independenty, identicay normay distributed with mean zero and variance. The parameters vary freey, so, Ɣ i, which reate to the stochastic component of the time series are the same in a subsampes and of dimension (p p) with being symmetric and positive definite, whie the p-vectors j,µ j reate to the deterministic component and coud be different in different sampe periods. A cointegration hypothesis can be formuated in terms of the rank of either aone or in conjunction with 1,..., q, as we saw in the discussion of exampe (2.1). The atter gives nicer interpretations and some advantageous simiarity properties and is given by H (r): rank(, 1,..., q ) r or (, 1,..., q ) = α i=1 β γ 1. γ q, c Roya Economic Society 2

4 Cointegration anaysis 219 where the parameters vary freey so α, β are of dimension (p r) and γ j is of dimension (1 r). The notation H indicates that in each sub-sampe the deterministic component is inear both for non-stationary and cointegrating reations. This feature wi become evident from the Granger representation beow. A reated hypothesis arises in the case of no inear trend but a broken constant eve as in exampe (2.4), H c (r): rank(, µ 1,...,µ q ) r and 1,..., q =. As an aternative to the modes H and H c a rank hypothesis coud be formuated for aone as in exampe (2.3), H c (r): rank r and 1,..., q =. The hypotheses are nested as H c (r) H c (r) H (r). For the purpose of determining the cointegration rank the hypothesis H c is ess attractive than H c, H for two reasons. First, as indicated by the sub-index, the hypothesis H c impies that the non-stationary reations have a broken inear trend whie the cointegrating reations have broken constant eves. Thus under the hypothesis H c the deterministic trend of a component depends on the cointegrating properties, whereas in testing H the deterministic behaviour of the process is the same regardess of the cointegrating properties, namey a inear trend in a directions. Secondy, in Section 3.3 it wi be demonstrated that the asymptotic anaysis is heaviy burdened with nuisance parameters. These issues are discussed in further detai by Niesen and Rahbek (2) Another formuation The above description invoves writing q mode equations of type (2.5). In order to write these as one equation which is more conformabe with standard econometric computer packages some dummy variabes are introduced. Let D j,t = { 1 for t = Tj 1, otherwise, for j = 2,...,q; t =..., 1,, 1,..., so D j,t i is an indicator function for the ith observation in the jth period; that is, D j,t i = 1if t = T j 1 + i. Further, E j,t = T j T j 1 i=k+1 { 1 for Tj 1 + k + 1 t T D j,t i = j, otherwise, is the effective sampe of the jth period. It is convenient to gather the sampe dummies and the drift parameters for the different sampe periods E t = (E 1,t,...,E q,t ), µ = (µ 1,...,µ q ), γ = (γ 1,...,γ q ), of dimensions (q 1), (p q), (q r), respectivey. The mode equation becomes ( ) ( ) β Xt 1 k 1 k q X t = α + µe γ te t + Ɣ i X t i + κ j,i D j,t i + ε t, (2.6) t i=1 i=1 j=2 c Roya Economic Society 2

5 22 S. Johansen et a. where the dummy parameters κ j,i are p-vectors and the observations X 1,...,X k are hed fixed as initia observations. Note, that the effect of the dummy variabes D j,t 1,...,D j,t k corresponding to the observations X Tj 1 +1,...,X Tj 1 +k is to render the corresponding residuas zero thereby essentiay eiminating the corresponding factors from the ikeihood function, and hence producing the conditiona ikeihood function given the initia vaues in each period Interpretation A process satisfying the hypothesis H (r) can be interpreted using Granger s representation theorem. That is, inear combinations of the process, given by β, cointegrate whie the process exhibits a inear trend in each of the sub-sampes. As usua it is necessary to assume that the process is actuay an I(1) process. Assumption 1. Assume that the roots of the characteristic poynomia, A(z) = (1 z)i p αβ k 1 z Ɣ i (1 z)z i, are outside the compex unit circe or at 1 and that the matrices α and β have fu coumn rank r. Further, define = I p k 1 i=1 Ɣ i and assume fu rank of the matrix Theorem 4.2 of Johansen (1996) can be generaized as foows. i=1 α β. (2.7) Theorem 2.1. Granger s Representation Theorem. Suppose Assumption 1 is satisfied. Then, for each period the initia vaues X Tj 1 +1,...,X Tj 1 +k can be given a distribution such that β X t + γ j t and X t are stationary processes. In particuar, X t = C t i=t j 1 +k+1 ε i + Y j,t + τ c, j + τ, j t (2.8) for j = 1,...,q, T j 1 + k < t T j and C = β (α β ) 1 α. The processes Y j,t are stationary, identicay distributed and have zero expectation. The sope parameters, τ, j, can be expressed as τ, j = Cµ j + (C I p )βγ j, whereas the eve coefficient τ c, j depends on initia vaues in such a way that β τ c, j is an identified function of the parameters β τ c, j = α ( C I p )µ j + α ( C )βγ j γ j. For each sampe period the process β X t + tγ E t is stationary and hence it has no trending behaviour. The common stochastic trends are α ts=tj 1 +k+1 ε s and the sopes of the common deterministic trends are α µ j = α τ, j. c Roya Economic Society 2

6 Cointegration anaysis 221 In some situations it is convenient to have inear combinations of the data representing the non-stationary trends. Exampes are β X t and α X t both of which are combinations that do not cointegrate. The representation shows that in each sub-sampe a inear combinations of the process are aowed to have a inear trend, which generaizes mode (C) suggested by Perron (1989). Tests for inear restrictions on the sope parameter τ = Cµ + (C I p )βγ are discussed in Section 4. The Granger representation shows that the sope for the cointegrating vector, β τ = γ, has to be treated separatey from the sope of the common deterministic trend, α τ = α µ. An exampe is the two-period mode, q = 2, with common sopes, τ,1 = τ,2, corresponding to Perron s mode (A). In genera these hypotheses are of the form H γ (r) : β τ G = γ G = or γ = Gϕ (2.9) for the cointegrating reation and H µ (r) : α τ M = α µm = for the common deterministic trends. Here G and M are known matrices of dimension (q g) and (q m), respectivey, with g, m < q and fu coumn rank. In particuar, Perron s mode (A) is given by G = M = (1, 1), which means that γ 1 = γ 2 and α µ 1 = α µ 2. A more subte question is how the transition happens from one sampe period to the next. The suggested conditioning on k initia vaues in each period aows for great fexibiity and these transition periods can be extended if necessary. An aternative approach woud be to use a specific transition function of some kind. Suppose it is of interest to mode an instantaneous break in the eve. As a simpe exampe of such a probem, with one ag, consider an unobserved components formuation such that for 2 t T, X t = τ c,1 1 (t T1 ) + τ c,2 1 (t>t1 ) + Z t, Z t = αβ Z t 1 + ε t, X t = αβ {X t 1 τ c,1 1 (t 1 T1 ) τ c,2 1 (t 1>T1 )}+τ c,1 1 (t T1 ) + τ c,2 1 (t>t1 ) + ε t. Using the definitions of D j,t and E j,t, in particuar that D 2,t = 1 (t=t1 ), E 1,t = 1 (t T1 ) and E 2,t = 1 (t T1 +2), it foows that X t = αβ (X t 1 τ c,1 E 1,t τ c,2 E 2,t ) +{τ c,2 (I p + αβ )τ c,1 }D 2,t 1 + ε t. Comparing with equation (2.6) or rather (3.6) it is found to be of the form ( ) ( ) β Xt 1 X t = α + κ γ E 2,1 D 2,t 1 + ε t t with γ j = β τ c, j, for j = 1, 2, and κ 2,1 = τ c,2 (I p + αβ )τ c,1 satisfying the restriction β κ 2,1 = (I r + β α)γ 1 γ 2. This restriction on κ 2,1 is reated to ony one observation and is therefore difficut to test. For modes of higher order the conditions for instantaneous breaks woud simiary invove a transition parameters κ j,i. This issue is discussed in further detai for the univariate case by Perron (199). c Roya Economic Society 2

7 222 S. Johansen et a. Another type of restriction of interest is co-breaking, see Hendry (1997) or Cements and Hendry (1999, pp ). In the mode H (r), the sope and the intercept of the inear trend of the cointegrating reation β X t are in genera different from period to period. An r-vector ω is an equiibrium sope co-breaking vector if the sope of the deterministic trend ω β X t does not change from period to period; that is, ω must satisfy (ω β τ ) span(1,...,1). Granger s representation Theorem 2.1 shows that β τ = γ and hence co-breaking is a inear restriction on the row space of the (q r)-matrix γ. When q < r or when the rank of γ otherwise is smaer than r then there are at east (r rankγ) co-breaking vectors, given by the orthogona compement, (γ ), to the matrix γ. Note, that the hypothesis in (2.9), γ = Gϕ, is formuated in terms of the coumn space of γ and is therefore not directy inked to co-breaking, athough, it gives an upper bound for the rank of γ and thereby a ower bound for the number of co-breaking vectors. When g is smaer than r there are at east (r g) co-breaking vectors given by (ϕ ). 3. Test for Rank The cointegration rank can be tested by modifying the procedures suggested by Johansen (1996). Whereas the statistica anaysis is hardy changed the asymptotic resuts are reated but different. New asymptotic distributions arise. First, these are described formay for the three different cases: H, H c, and H c. The anaysis of the atter hypothesis is burdened with nuisance parameters and ess usefu than the first two. Secondy, the asymptotic distributions reated to H and H c are described by response surface anaysis which can easiy be programmed. For the suggested mode the ikeihood function can be maximized using canonica correation methods as deveoped by Hoteing (1936), Bartett (1938), Anderson (1951), and impemented in cointegration anaysis by Johansen (1996, Ch. 6). In particuar, in the case of mode H inference is based on the squared sampe canonica correations, 1 > ˆλ 1 > > ˆλ p >, of X t and (X t 1, te t ) corrected for the regressors E t, X t i (i = 1,...,k 1), D j,t i,(i = 1,...,k; j = 2,...,q). These wi be denoted { ( ) } Xt 1 CanCor X t, F te t, t where F t is shorthand notation for the σ -fied generated by the regressors. The ikeihood ratio test statistic for the hypothesis of at most r cointegrating reations, H (r), against H (p) is given by p LR{H (r) H (p)} = T og(1 ˆλ i ). (3.1) i=r Asymptotic distribution: a broken inear trend Inference shoud ideay be based on the exact distribution of the test statistic (3.1). Unfortunatey this is not feasibe so some kind of asymptotic distribution approximation is needed. In order to ensure a good approximation the breaks need to be treated with care. The approach taken here is that the reative break points given by v j = T j /T are fixed whie an asymptotic argument in T is made. c Roya Economic Society 2

8 Cointegration anaysis 223 For the asymptotic resuts the foowing notation is convenient. The reative break points v j = T j /T satisfy = v <v 1 < <v q = 1. Let v j = v j v j 1 and define a q- dimensiona vector of indicator functions for the sampe periods, e u ={...,1 (v j 1 <u v j ),...}. This function is the imit of E [Tu] as T increases. For any two vector vaued continuous functions f u and g u on [,1] we use the notation 1 ( f u g u ) = f u for the residua of f after correcting for g. ( 1 ) 1 f s g s ds g s g s ds g u, Theorem 3.1. Suppose H (r) and Assumption 1 are satisfied. Then the asymptotic distribution of the ikeihood ratio test statistic for H (r) against H (p) is given by { 1 ( 1 ) 1 } 1 tr dw u F u F u F u du F u dw u (3.2) as T and for fixed reative break points, v j. Here W is a standard Brownian motion of dimension (p r) and F is a (p r + q)-dimensiona process, ( ) Wu F u = ue u e u. (3.3) The asymptotic distribution has been simuated and the resuts anaysed by a response surface anaysis presented in Section 3.4. For anaytic reasoning and for computer simuations it is convenient to rewrite the representation of the distribution given by (3.2). This is based on two ideas. First, the distribution is invariant with respect to inear transformations of the vector process F. Thus if the first (p r) components are regressed on the ast q components giving F,u = (W u ue u, e u ), the transformed version of the matrix ( 1 F u F u du) 1 is bock diagona. It foows that expression (3.2) can be rewritten as the sum of two terms which do not invove the eves of the Brownian motion, see (3.4) beow. We find { 1 tr ( 1 ) 1 } 1 dw u F,u F,u F,u du F,u dw u [ 1 { 1 } 1 ] 1 +tr dw u (ue u e u ) (ue u e u )(ue u e u ) du (ue u e u )dw u. The second term is χ 2 {q(p r)}-distributed since 1 (ue u e u )dw u D = N q (p r) {, 1 } (ue u e u )(ue u e u ) du I p r. Secondy, when regressing a Brownian motion on the eve in each of the two or more sub-sampes, the sub-sampe Brownian motions wi be independent. These considerations ead to a second representation of the asymptotic distribution (3.2). c Roya Economic Society 2

9 224 S. Johansen et a. Theorem 3.2. Let W (1),...,W (q) be independent (p r)-dimensiona standard Brownian motions and define { 1 J j = (u 1) 2 du K j = L j = 1 1 {W ( j) } 1/2 1 {W ( j) u 1, u}{dw ( j) u }, u 1, u}{w u ( j) 1, u} du. Then the imiting variabe (3.2) can be expressed as (u 1){dW ( j) u }, ( q { q tr K j v j) 2} 1 ( q ) ( q ) L j ( v j ) K j v j + J j J j. (3.4) j=1 j=1 j=1 j=1 From representation (3.4) of the imit distribution it is seen that the asymptotic distribution ony depends on the reative ength of the sampe periods, not on their ordering. For instance, in the case of one break point, the asymptotic distribution is the same if T 1 = T/3asifT 1 = 2T/3. Moreover, the first term in (3.4) is the trace of a (p r)-dimensiona square matrix whie the second term is the sum of inner products of (p r)-dimensiona vectors. This refects the degrees of freedom arising from the matrix and the vectors j, respectivey. This representation aso shows another feature of the imit distribution. In the case of q + 1 sampe periods et DF q+1 (v 1,...,v q+1 ) denote the asymptotic distribution. When the engths of one of the sampe periods, v j, tends to zero it foows that the contributions K j, L j vanish in the first term of (3.4). In the second term we can isoate J j J j and find im v j DF q+1(v 1,...,v q+1 ) = DF q (v 1,...,v j 1,v j+1,...,v q+1 ) + J j J j, (3.5) where DF q and J j J j are independent and J j J j is χ 2 (p r)-distributed. The additiona χ 2 term arises because the dimension of the vector (X t 1, te t ) is preserved athough one of the reative sampe engths vanishes, and hence the dimension of the restrictions imposed by the rank hypothesis is unatered. If the dummies with the vanishing sampe ength are taken out of the statistica anaysis the additiona χ 2 -distributed eement disappears. The asymptotic distribution given above does not depend on the parameters for the deterministic component. The test is therefore asymptoticay simiar with respect to these parameters provided that Assumption 1 is satisfied, see aso Niesen and Rahbek (2). In order to estimate the rank a sequentia testing procedure is necessary. One suggestion is to test the hypotheses H (), H (1),...,H (p 1) sequentiay against the unrestricted mode H (p). If H (r) is the first hypothesis to be accepted then the cointegrating rank is estimated by r. For consistency properties of this procedure see Johansen (1996, Section 12.1). c Roya Economic Society 2

10 Cointegration anaysis A broken constant eve In some appications the eve of the data may change from time to time but the data do not exhibit a inear trend. Then the mode is given by ( ) Xt 1 X t = (, µ) E t k 1 + Ɣ i X t i + i=1 k i=1 j=2 q κ j,i D j,t i + ε t. (3.6) The hypothesis of reduced cointegration rank is given by H c (r): rank (, µ) r, or equivaenty that (, µ) can be written as α(β,γ ), whie the ikeihood ratio test statistic for H c (r) against a genera aternative, H c (p), is of form (3.1). The resut of Theorem 3.1 appies with F repaced bya(p r + q)-dimensiona process with components F u = ( Wu e u ). (3.7) 3.3. Modes with unrestricted parameters for the broken trend The hypothesis H c (r) in mode (3.6) imposes rank restrictions on the first-order autoregressive parameter as we as the parameter for the broken deterministic trend. In some situations it may seem reasonabe to anayse a rank hypothesis which ony invoves the autoregressive parameter for eves H c (r) : rank r or = αβ, whie µ is eft unrestricted. This hypothesis is anaysed by correcting X t and X t 1 for the remaining components in the mode and subsequenty performing a canonica correation anaysis of the residuas. The ikeihood ratio test statistic for H c (r) against H c (p) = H c (p) is of form (3.1). Its asymptotic distribution is given as foows. Theorem 3.3. Suppose H c (r) and Assumption 1 are satisfied. Let W be a (p r)-dimensiona standard Brownian motion and F the (p r +q)-dimensiona process given in Theorem 3.1. The asymptotic distribution as T of the ikeihood ratio test statistic for H c (r) against H c (p) depends on α µ, in particuar, et n = rank (α µ) min(p r, q). (i) Suppose n = (p r) q. Then the asymptotic distribution is χ 2 {(p r) 2 }. (ii) Suppose n = q <(p r). Then the asymptotic distribution is given by { 1 ( 1 ) 1 } 1 tr dw u F u N N F u F u dun N F u dw u (3.8) where N is the {(p r + q) (p r)}-matrix N = ( ) I p r q (p r q) q. (3.9) I q c Roya Economic Society 2

11 226 S. Johansen et a. (iii) Suppose n < min(p r, q). Then there exist matrices θ,η of rank n and dimensions {(p r) n}, (q n), respectivey, so α µ = θη. The asymptotic distribution is then given by (3.8) where N now depends on η ( ) N I = p r n (p r n) n η. (3.1) The test is not as attractive as the previousy considered tests. The imit distribution is a compicated function of α µ. The test is therefore not asymptoticay simiar with respect to the sope parameters for the broken trend. Athough the third situation in Theorem 3.3 ony occurs on a nu subset of the parameter space, the issue ought to be addressed in the statistica anaysis. Had there been no breaks in the trend the test strategy suggested by Johansen (1996, Section 12.2) coud be used. A generaization of that idea is not simpe for two reasons. First, the imit distribution depends continuousy on the nuisance parameter. Secondy, in many appications it coud be of interest subsequenty to test hypotheses corresponding to rank restrictions on α µ. Such tests are discussed in Section Critica vaues for rank tests Exact anaytic expressions for the asymptotic distributions are not known and the quanties have to be determined by simuation. The asymptotic distributions depend on a number of factors: the number of non-stationary reations, the ocation of break points and the trend specification. The moments of these distributions have been approximated using a arge number of simuations and a subsequent response surface anaysis based on these factors. Then, the quanties can be approximated using the empirica observation that the shape of rank test distributions typicay are approximated rather we by Ɣ-distributions, see Niesen (1997) and Doornik (1998). Since the parameters of a Ɣ-distribution are given by the first two moments, it suffices to report adequate approximations to the asymptotic mean and variance of the trace test distributions. The quanties can then be determined using a numerica routine for the incompete Ɣ-integra or a χ 2 -distribution with non-integer degrees of freedom which is avaiabe in most statistica computer packages. In the foowing the cases with a broken trend or a broken eve are considered with up to three sampe periods, q = 3. The cases with q = 1, 2, 3 can be described jointy. Let v j = T j /T denote the break points as a percentage of the fu sampe. For the case q = 3 there are three reative sampe engths, v 1,v 2 v 1, 1 v 2. Let a and b denote the smaest and the second smaest of these. For the case q = 2 there are two reative sampe engths v 1, 1 v 1. Let b denote the smaest of these and et a =. Finay, for q = 1 et a = b =. The moments of the asymptotic distributions are unknown functions of (p r), a, b. We have found that such functions are very accuratey approximated by og(moment) f moment (p r, a, b, T ) (3.11) = α m + β im x i + γ ijm x i x j + δ ijkm x i x j x k d m m= i=1 i=1 j i i=1 j i k j where x 1 = (p r), x 2 = a, x 3 = b, x 4 = T 1, d m = (p r) m. This function is essentiay a third-order poynomia in (p r), a, b and T 1, where the terms in (p r) 1 and (p r) 2 pay c Roya Economic Society 2

12 Cointegration anaysis 227 b Figure 1. Vaues of a and b used in the simuations. a the same roe as the dummies for dimensions 1, 2 and 3 used in Doornik (1998), but give a better fit. Note that the regression incudes the inverse of the sampe size, T 1. The roe of the sampe size in fitting response surfaces for the trace test is discussed in Doornik (1998). The asymptotic moments are easiy cacuated from (3.11) by etting T. Note aso that x 1 d 1 = d = 1 and x 1 d 2 = d 1. Some of the parameters in (3.11) are therefore not identified, and are set to zero. The remaining 75 parameters have been estimated by ordinary east squares, adding an error term to (3.11) and minimizing the sum of squared residuas. The moments of the asymptotic distribution were simuated for various vaues of (p r), a, b and T. The invoved Brownian motions can be discretized in severa ways. One possibiity is to mimic the representation (3.2) and generate one random wak with T steps in each simuation, and associate a percentage of this to each sampe period. In order to avoid poor approximations for cases with reativey short sampe periods, representation (3.4) was used. The idea is to generate three random waks each with T steps and then scae them according to the reative engths of the sampe periods. The vaues of T were the integer part of 5/t for t = 1,...,1. The considered number of non-stationary reations was (p r) = 1,...,8. Finay 2 different vaues of a and b were chosen as iustrated in Figure 1, to be representative of a pairs (a,b) such that a < b and b <(1 a b). Note that there is a more dense samping when a =, corresponding to one singe break. This gives 16 cases which were repeated N = 1 times. The fit of (3.11) is exceent, even when the number of parameters is dramaticay reduced starting from the east significant, as iustrated in Tabe 1. Standard errors are about.2% for the mean and.9% for the variance, an order of magnitude actuay very cose to the Monte Caro samping variation in og(moment). Note that such sma errors in the moments are virtuay negigibe for a practica purposes when computing the quanties and tai probabiities of the Ɣ-distribution. This point is iustrated in Tabes 2 and 3, where quanties and tai probabiities are computed for the vaues mean = 8 and variance = 125 and sma variations thereof. These c Roya Economic Society 2

13 228 S. Johansen et a. Tabe 1. Goodness of fit measures for the response surface. Unrestricted Restricted Mode # Par. R σ # Par. R σ H c, og(mean) H c, og(variance) H, og(mean) H, og(variance) Tabe 2. 95th Percenties of the Ɣ-distribution. Variance Mean 125/ *1.9 8/ * Tabe 3. Right-hand tai probabiity of the Ɣ-distribution for the vaue Variance Mean 125/ *1.9 8/ * vaues of mean and variance are approximatey equa to the average of the vaues found in our simuations. It is important to remark that the residuas of (3.11) are approximatey homoscedastic in a cases, which means that there are no vaues of (p r), a, b and T in which the errors are much bigger as a percentage of the moment. This is what motivated the choice to mode the og of the moments rather than the moments themseves. We aso tried to mode the moments directy, without taking ogarithms, using weighted east squares with (p r) 2 as weights to account for heteroscedasticity, aong the ines of Doornik (1998). However, the fit appears to be sighty poorer with that specification. Simiar resuts can be achieved using PcGets, an agorithm for mode seection constructed by Krozig and Hendry (2). The estimated coefficients are reported in Tabe 4 where the coefficients referred to the variabe x 4 = T 1 are not reported, since they are irreevant for computing the asymptotic moments. In the case of q = 2 sampe periods instead of q = 3 the mean and variance can sti be computed by using (3.11) by etting b = and subtracting the mean and variance of a χ 2 (p r) variabe, see (3.5). In the case of just q = 1 sampe period et a = b = and subtract an additiona χ 2 (p r) variabe. Thus for q = 1, 2, 3 we have the approximation mean exp{ f mean (p r, a, b, )} (3 q)(p r) (3.12) variance exp{ f variance (p r, a, b, )} 2(3 q)(p r). (3.13) c Roya Economic Society 2

14 Cointegration anaysis 229 Tabe 4. Estimated response surface. H c H og(mean) og(variance) og(mean) og(variance) Constant ( p r) a b (p r) ( p r)a (p r)b.195 a ab 1.12 b (p r) (p r)a a ab a 2 b 1.33 b (p r) a(p r) b(p r) a 2 (p r) ab(p r) b 2 (p r) a 3 (p r) ab 2 (p r) b 3 (p r) (p r) b(p r) a 2 (p r) b 2 (p r) a 3 (p r) b 3 (p r) These formuas aow comparison with pubished approximations of the imit distribution for the case with no breaks. Tabe 5 compares the mean and variance based on formuas (3.12) and (3.13) with the expressions obtained by Doornik (1998) as we as comparing percenties based on these two sets of formua with those reported by Johansen (1996). Formuas (3.12) and (3.13) agree quite we with Doornik s approximation both with respect to moments and in particuar the rightmost percenties. Our approximation is based on surface responses with (p r) 8 whie c Roya Economic Society 2

15 23 S. Johansen et a. Tabe 5. Approximate mean, variance and 95th percentie of the asymptotic distribution of the rank test in mode H c for a = b =, q = 1. Comparison of RS, the response surface anaysis, D, Doornik (1998), and J, Johansen (1996, Tabe 15.2). mean variance 95th percentie (p r) RS D RS D RS D J Doornik considered (p r) 15. It is seen that our formua is not suited for extrapoation beyond (p r) >1. The percenties tabuated by Johansen are based on a discretization of T = 4, whereas Doornik s and our formuas are based on response surfaces in T combined with the Ɣ- approximation. As expected there is agreement for ower dimensions whereas Johansen s figures are ess accurate for higher dimensions. Doornik gives a more detaied comparison of percenties found from the Ɣ-approximation and directy by simuation. 4. Restrictions on the sope parameters When the cointegrating rank is known it is usuay desirabe to test further restrictions on the parameters. In this section hypotheses on the sope of the deterministic trend are considered. Reca from Theorem 2.1 that the sope parameter is τ, j = Cµ j + (C I p )β(β β) 1 γ j in the jth sampe period. In particuar, the deterministic sope for the cointegrating reations, β X t, is therefore β τ, j = γ j. In brief, the resuts are 1. tests for inear restrictions on the sope for the cointegrating reation, β τ G =, are asymptoticay χ 2 -distributed. 2. tests for inear restrictions on the sope for the entire process, τ M =, are asymptoticay χ 2 -distributed. The two tests can be performed sequentiay, by first imposing restrictions on the sope for the cointegrating reation, β τ G =, and then, if accepted, imposing restrictions on the sope for the common deterministic trend, α τ M =, provided span(g) span(m). In this way it is possibe to impose more restrictions on the sope for the cointegrating reation, β τ, than on the sope in genera, τ. c Roya Economic Society 2

16 Cointegration anaysis 231 Note that it is not straightforward to do these two tests in the opposite order. In that case the test for sope of the common deterministic trend is burdened with nuisance parameters. This is reated to the issue that the non-stationary trends are not uniquey defined Sope of the cointegrating reation The sope for the inear trend in the cointegrating reation is given by the parameter γ. Linear restrictions on this parameter can be formuated as H γ (r) : γ = Gϕ, where G is a known (q g)-matrix of rank g, where g q, and the parameter ϕ is a (g r)-matrix. Under the hypothesis the sope for the cointegrating reations is therefore β τ E t = ϕ G E t. As an exampe suppose q = 2. By the choice G = (1, ) the inear trend is absent in the second period whereas if G = (1, 1) then the sope is not atered by the break. Note, that when there is no cointegration, r =, then γ vanishes, hence H γ () = H (). As before the ikeihood is maximized by canonica correation anaysis. The squared sampe canonica correations of the residuas, 1 > ˆλ γ 1 > > ˆλ γ p >, are given by CanCor { ( ) } Xt 1 X t, tg F E t, t and the ikeihood ratio test for the hypothesis H γ (r) in H (r) is LR{ H γ (r) H (r)} =T r og{(1 ˆλ γ i )/(1 ˆλ i )}, see Johansen (1996, Theorem 7.2). The asymptotic distribution is as foows. i=1 Theorem 4.1. Suppose H γ (r) and Assumption 1 are satisfied. Then the ikeihood ratio test statistic for H γ (r) in H (r) is asymptoticay χ 2 {r(q g)}-distributed. The restriction on the inear term of the cointegrating vector, γ = Gϕ, coud be combined with, for instance, a inear restriction on the cointegrating vector itsef, β = Hψ, where H is a known, fu-rank, (p h)-matrix and the parameter ψ is of dimension (h r). The ikeihood ratio test statistic for this hypothesis is aso asymptoticay χ 2. The degrees of freedom is {r(p h)} if the aternative is H γ (r) and {r(p h + q g)} if H (r) is the aternative Sope of the process Restrictions on the sope τ for the process X t are now studied. This sope is a inear combination of the sope for the cointegrating reation, β τ = γ, and that of the common deterministic trends, α τ = α µ. Restrictions on γ were studied above and we now turn to restricting α µ. c Roya Economic Society 2

17 232 S. Johansen et a. Linear restrictions on α µ wi be expressed in terms of a known (q m)-matrix M where m q. The formuation H µ (r) : µ = ζ M + αδ M, where ζ,δ are of dimension (p m) and {(q m) r}, respectivey, eaves α µ unrestricted, whereas α µ is restricted so α µm =. If the sope of the cointegrating reation is restricted correspondingy as β τ M =, or more generay as β τ G = for some G satisfying span(g) span(m), we have that the sope of the entire process satisfies τ M =. As opposed to the concept of cointegrating reations the non-stationary trends are inear combinations of the eves of the time series which do not cointegrate. They coud be chosen in various ways, for instance as β X t or α X t. Under the above restriction, H µ, the sope of the former of these is β τ = β Cζ M + β C β(β β) 1 γ, showing that restrictions on γ are necessary to interpret H µ in terms of the non-stationary trends. In the foowing we therefore discuss ikeihood ratio tests for H γµ (r) : γ = Gϕ,µ = ζ M + αδ M in H γ (r) and H (r). Note, that in the unrestricted mode, with up to p cointegrating reations, the hypothesis H γµ (p) entais no restrictions as compared with H γ ( p). The squared sampe canonica correations, 1 > ˆλ γµ 1 > > ˆλ γµ p >, are now based on X t 1 CanCor X t, tg E t M E F t µ, (4.1) t where the notation F t µ indicates that the regressor E t is repaced by M E t. The ikeihood ratio test statistics for H γµ (r) are LR{ H γµ (r) p γ H (r)}=t og{(1 ˆλ γ i )/(1 ˆλ γµ i )}, i=r+1 LR{ H γµ (r) H (r)}=lr{ H γµ (r) γ H (r)}+lr{ H γ (r) H (r)}, see Johansen (1996, Theorem 6.2), where it is expained that it is convenient to express the first of these statistics in terms of the sma eigenvaues, using the fact that H γµ (p) = H γ (p). For the asymptotic anaysis the restriction span(g) span(m) is crucia for avoiding nuisance parameters. Theorem 4.2. Suppose H γµ (r) and Assumption 1 are satisfied. If in addition span(g) span(m) then L R{ H γµ (r) H γ (r)} and L R{ H γµ (r) H (r)} are asymptoticay χ 2 -distributed with {(p r)(q m)} and {(p r)(q m) + r(q g)} degrees of freedom, respectivey. If span(g) span(m), the asymptotic distributions of the test statistics invove nuisance parameters. 5. Empirica Iustration This section iustrates the suggested statistica anaysis, appied to a five-dimensiona data set with variabes reevant for anaysing the Uncovered Interest Parity (UIP) hypothesis between Germany c Roya Economic Society 2

18 Cointegration anaysis 233 Tabe 6. Maximum ag anaysis (p-vaue for the Godfrey test). k Akaike Hannan Quinn Schwartz Godfrey χar5 2 (125) and Itay. The economic mode is very simpe, and shoud be regarded as an iustration rather than a contribution to the ongoing economic debate. The anaysis has been done using MALCOLM 2.4 (Mosconi1998), where a the techniques iustrated in this paper are impemented in a user friendy menu driven environment. Let us consider the vector Y t = ( p I t, pd t, e t+1, i I t, i D t ) where pt I and pt D are first differences of og Consumer Price Index and represent infation rates in Itay and Germany. The variabe e t+1 is the first differences of og nomina exchange rate between Itaian Lira and German Mark (LIT/DM) and represents the rationa expectation to future exchange rates. Finay, it I and it D are Itaian and German nomina interest rates on ong-term treasury bonds, given as annua rates divided by 4, to make them dimensionay matching with the other variabes. As for the sources, prices are from EUROSTAT (except , where prices are from UN Monthy Buetin of Statistics); note that, after October 199, German prices refer to unified Germany. Exchange Rates are from the Bank of Itay (average quartery exchange rates). Interest Rates are from IMF, Internationa Financia Statistics. The data are avaiabe from the Econometrics Journa website. The data, which are shown in Figure 2, are quartery, ranging from to (T = 91). To mode these data, based on prior knowedge of reevant historica events, we introduce two breaks. The ast observation of the first period is , whie the ast observation of the second period is (T 1 = 27, T 2 = 77; v 1 =.297, v 2 =.846; a =.154, b =.297). The first break coincides with the creation of the EMS, but it is aso supposed to catch the oi shock and the modification of the US monetary poicy. The second break corresponds to the exit of Itay from the EMS, but aso to the unification of Germany. The pot ceary shows the presence of trends: the trend in infation and interest rates in both countries in the second period is apparent, but one might suspect trends in some of the variabes aso in the first and third period. This suggests modeing the data using mode H. In fact, the presence of trends in the variabes may be expained within mode H c ony by random waks, whereas mode H aows for interpreting trends either as reated to random waks or trend stationarity. Within mode H, the nature of the trend may be decided according to appropriate tests once the cointegration rank is determined in a setting which is robust with respect to trend stationarity. The anaysis to determine the maximum ag k is reported in Tabe 6. The information criteria suggest different vaues of k, in which case it is common practice to prefer the Hannan Quinn criterion. Therefore, k = 2 has been seected, since it is aso the first ag to give approximatey white noise residuas, according to the Godfrey test. Jarque Bera normaity tests, reported in Tabe 7, show some probems with skewness in the c Roya Economic Society 2

19 234 S. Johansen et a..64 Infation in Itay (dashed) and Germany Interest Rates in Itay (dashed) and Germany og difference of LIT/DM Exchange Rate Figure 2. The data. Tabe 7. Jarque Bera Normaity tests (p-vaues). Equation Skewness Kurtosis Sk + Kur pt I pt D e t it I it D System c Roya Economic Society 2

20 Cointegration anaysis 235 Tabe 8. Rank tests. Hypothesis Test p-vaue r = r r r r Tabe 9. Characteristic roots of the modes. Root r = 5 r = i i.11.71i i i i.45.27i e t equation and kurtosis in the it D equation, so that, at the system eve, normaity is rejected. Due to the iustrative aim of this anaysis we did not try to anayse these probems any further. Note, however, that a residua-based misspecification tests, ike Godfrey and Jarque Bera, shoud be modified in the present setting to take into account that the first k residuas of each period are set to zero by the presence of the dummies D j,t i, whose purpose is to condition upon the first k observations of each period. This might party expain the probems with kurtosis. Coming to cointegration anaysis, UIP impies that i I t {i D t + E t ( e t+1 )}= (5.1) so that for an Itaian investor the return from investing in Itay equas the expected return from investing in Germany. The interpretation of the reation in the context of a vector autoregressive mode is it I (it D + e t+1 ) = zero mean stationary. (5.2) Therefore, the cointegration rank r is expected to be at east equa to one, but of course it may be higher, since we do not have theoretica reasons to excude more stationarity in the data. The tests for cointegration rank are reported in Tabe 8. The anaysis supports r = 3, which is consistent with our prior expectation. Therefore, we estimate the mode with r = 3. Tabe 9 reports the five argest characteristic roots for both the unrestricted and the restricted modes, which seem to be consistent with the I(1) assumption rank (α β ) = (p r). Before trying to set up identifying restrictions on the cointegration space, et us iustrate some interesting tests on the deterministic components. The sopes of the deterministic trends for the cointegrating reations are given by the eements of the (3 3)-matrix γ, whose ith coumn represents the trend coefficients of the ith stationary reation in the three different periods. A suggested routine anaysis consists in testing for the excusion of the inear trend in the stationary c Roya Economic Society 2

21 236 S. Johansen et a. Tabe 1. Test statistics on severa hypotheses, a tested against H (r). Hypothesis χ 2 (n) degrees of freedom p-vaue H γ,1 (3) (3) H γ,2 H γ,3 (3) H γµ,1 (3) H γµ,2 (3) H I (3) components in each period. In our exampe, this is done using the matrices 1 1 G 1 = 1, G 2 =, G 3 = and the restrictions H γ,i (3) : γ = G iϕ. Note that these hypotheses may be aso written as H γ,i (3) : βγ = ( β γ ) = ( I p G i )( β ϕ which is easiy impemented in standard cointegration software. The resuts are given in Tabe 1, which shows that a trend stationary component cannot be removed from any of the three periods. However, the test takes on borderine vaues for H γ γ,1 (3) and H,2 (3), whereas the rejection is much stronger for H γ,3 (3). Let us now iustrate some joint hypotheses about the common trends and the stationary components, namey H γµ,i (3) : { γ = M i ϕ,µ = ζ M i + αδ M i with 1 M 1 = 1 M 2 =. 1 1 Referring to the discussion in Section 4.2, note that, in this case, M = G, so that the condition span(g) span(m) of Theorem 4.2 is fufied. When H γµ,1 (3) hods, then τ M 1 = (i.e. τ = ψ M 1 ), so the sopes of both the stationary and the non-stationary components are zero in the first period. The interpretation of the other restriction is simiar. We perform these tests for iustrative purposes ony, since part of H γµ,i (3), i.e. H γ,i (3), invoving the stationary components, has aready been tested and rejected athough not very strongy. As shown in Tabe 1, the joint test aso takes on borderine vaues, rejecting H γµ γµ,1 (3) and accepting H,2 (3). Stricty speaking, this means that, according to the joint test, inear trends may be excuded from both the stationary and non-stationary components in the second period. In order to (over)-identify the cointegration space, we suggest the foowing stationary inear combinations ) c Roya Economic Society 2

22 Cointegration anaysis 237 z 1t = it I (it D + e t+1 ) z 2t = (it D pt D ) z 3t = (it I pt I ) (i t D pt D ). The equations represent the UIP hypothesis, the German rea interest rate, and the rea interest rate differentia, respectivey. Note that, if these inear combinations are stationary, then aso z 4t = z 1t z 3t = ( p I t p D t ) e t+1 z 5t = z 2t + z 3t = (i I t p I t ) are stationary, and coud be used to find an aternative and equivaent basis of the cointegration space. Identifying restrictions may be written as ( ) H I (3) : β γ β = = (B γ 1 b 1, B 2 b 2, B 3 b 3 ). In order to test the oca trend stationarity of z 1t,z 2t and z 3t, together with some pausibe restrictions on the deterministic part, we set up the foowing identifying restrictions: B 1 = 1 1, B 2 = 1, B 3 = which excude the inear trend from z 1t in the second period, from z 2t in the first and second periods, and from z 3t in a periods. The degrees of freedom for testing H I (r) against H(r) are given by r j=1 (p + q r dim B j + 1), see Johansen (1996, Theorem 7.5). As shown in Tabe 1, H I (3) cannot be rejected. Figure 3 represents z 1t, z 2t and z 3t, together with their deterministic components estimated under H I (3). This shows that in the period (79.1, 92.2), in which Itay beonged to the EMS, z 1t is approximatey zero on average, which is evidence in favour of the UIP in that period. Conversey, the mean of z 1t is negative and quite arge in the first period, athough trending towards zero. This means that the interest rate in Itay was much ower than predicted by UIP pus rationa expectations. An interpretation coud be that the extreme devauation of Itaian Lira in the 197s was unexpected, or in other words (5.2) shoud be repaced in the first period by i I t (i D t + e t+1 ) = zero mean stationary + ρ t, where ρ t represents a systematic bias in expectations. An aternative interpretation of the ow Itaian interest rate coud be reated to the presence of a negative risk premium on Itay: if Itay is perceived as ess risky than Germany, then it I shoud be ower than (it D + e t+1 ). However, this interpretation seems impausibe in the 197s. c Roya Economic Society 2