LIKELIHOOD INFERENCE FOR A FRACTIONALLY COINTEGRATED VECTOR AUTOREGRESSIVE MODEL

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1 Econometrica, Vol. 8, No. 6 November, 212), LIKELIHOOD INFERENCE FOR A FRACTIONALLY COINTEGRATED VECTOR AUTOREGRESSIVE MODEL SØREN JOHANSEN University of Copenhagen, 1353 København K, Denmark and CREATES MORTEN ØRREGAARD NIELSEN Queen s University, Kingston, Ontario K7L 3N6, Canada and CREATES The copyright to this Article is held by the Econometric Society. It may be downloaded, printed and reproduced only for educational or research purposes, including use in course packs. No downloading or copying may be done for any commercial purpose without the explicit permission of the Econometric Society. For such commercial purposes contact the Office of the Econometric Society contact information may be found at the website or in the back cover of Econometrica). This statement must be included on all copies of this Article that are made available electronically or in any other format.

2 Econometrica, Vol. 8, No. 6 November, 212), LIKELIHOOD INFERENCE FOR A FRACTIONALLY COINTEGRATED VECTOR AUTOREGRESSIVE MODEL BY SØREN JOHANSEN AND MORTEN ØRREGAARD NIELSEN 1 We consider model based inference in a fractionally cointegrated or cofractional) vector autoregressive model, based on the Gaussian likelihood conditional on initial values. We give conditions on the parameters such that the process X t is fractional of order d and cofractional of order d b; that is, there exist vectors β for which β X t is fractional of order d b and no other fractionality order is possible. For b = 1, the model nests the Id 1) vector autoregressive model. We define the statistical model by <b d, but conduct inference when the true values satisfy d b < 1/2andb 1/2, for which β X t is asymptotically) a stationary process. Our main technical contribution is the proof of consistency of the maximum likelihood estimators. To this end, we prove weak convergence of the conditional likelihood as a continuous stochastic process in the parameters when errors are independent and identically distributed with suitable moment conditions and initial values are bounded. Because the limit is deterministic, this implies uniform convergence in probability of the conditional likelihood function. If the true value b > 1/2, we prove that the limit distribution of T b ˆβ β ) is mixed Gaussian, while for the remaining parameters it is Gaussian. The limit distribution of the likelihood ratio test for cointegration rank is a functional of fractional Brownian motion of type II. If b < 1/2, all limit distributions are Gaussian or chi-squared. We derive similar results for the model with d = b, allowing for a constant term. KEYWORDS: Cofractional processes, cointegration rank, fractional cointegration, likelihood inference, vector autoregressive model. 1. INTRODUCTION AND MOTIVATION THE COINTEGRATED VECTOR AUTOREGRESSIVE VAR) model for a p- dimensional nonstationary time series X t is 1) X t = α β X t 1 + ρ ) + k Γ i X t i + ε t i=1 t = 1T where X t i = X t i X t i 1. This model has been widely used to analyze longrun economic relations given by the stationary combinations β X t and to build empirical dynamic models in macroeconomics and finance; see for instance Juselius 26). 1 We are very grateful to Jim Stock and five referees for many useful and constructive comments that were above and beyond the call of duty and led to significant improvements to the paper. We are also grateful to Uwe Hassler, James MacKinnon, Ilya Molchanov, and seminar participants at various universities and conferences for comments, and to the Danish Social Sciences Research Council FSE Grant ), the Social Sciences and Humanities Research Council of Canada SSHRC Grant ), and the Center for Research in Econometric Analysis of Time Series CREATES, funded by the Danish National Research Foundation) for financial support. A previous version of this paper was circulated under the title Likelihood Inference for a Vector Autoregressive Model Which Allows for Fractional and Cofractional Processes. 212 The Econometric Society DOI: /ECTA9299

3 2668 S. JOHANSEN AND M. Ø. NIELSEN Fractional processes are a useful tool for describing time series with slowly decaying autocorrelation functions and have played a prominent role in econometrics see, e.g., Henry and Zaffaroni 23) andgil-alana and Hualde 29) for reviews and examples). It appears to be important to allow fractional orders of integration fractionality) in time series models. In this paper, we analyze VAR models for fractional processes. The models allow X t to be fractional of order d and β X t to be fractional of order d b, so as to extend the usefulness of model 1) to fractional processes. We also consider a model with d = b, allowing for a constant term. The model can be derived in two steps. First, in 1) we replace the usual lag operator L = 1 and the difference operator by the fractional lag and difference operators L b = 1 b and b = 1 L) b defined by the binomial expansion b Z t = 1)n b n= n) Zt n. Second, we apply the resulting model to Z t = d b X t. This defines the fractional VAR model, VAR db k) see Johansen 28)), 2) H r : d X t = d b L b αβ X t + k Γ i d L i X b t + ε t t = 1T i=1 where ε t is p-dimensional independent and identically distributed) i.i.d. Ω), Ω is positive definite, and α and β are p r r p. Theparameter space of H r is given by the otherwise unrestricted parameters λ = dbαβγ 1 Γ k Ω). In the special case r = p, the p p matrix Π = αβ is unrestricted, and if r =, the parameters α and β are not present. Finally, if k = r =, the model is d X t = ε t, so the parameters are d Ω). Note that, for b = 1 d 1 X t satisfies a VAR model with k + 1 lags or, equivalently, X t satisfies a vector autoregressive fractionally integrated moving average, VARFIMAk + 1d 1),model. If we model data Y t by Y t = μ + X t,wherex t is given by 2), then a Y t = a X t + μ) = a X t because a 1 = fora> so that Y t satisfies the same equations. For the same reason, when d>b, the model 2) is invariant to a restricted constant term ρ when included in a way similar to that in 1). Thus 2) is a model for the stochastic properties of the data and when they have been determined, one can, for example, estimate the mean of the stationary linear combinations by the average. Therefore, we also consider the model with d = b and a constant term, 3) H r d = b): d X t = αl d β X t + ρ ) k + Γ i d L i X d t + ε t t = 1T i=1 with a similar interpretation of β X t except now β X t + ρ is a mean zero process of fractional order zero. Note that L d ρ = ρ because d 1 =

4 INFERENCE FOR A FRACTIONALLY COINTEGRATED VAR 2669 We show that when <r<px t is fractional of order d and cofractional of order d b; that is, β X t is fractional of order d b. Moreover, if d b<1/2, then β X t in model 2) is asymptotically a mean zero stationary process. The model has the attractive feature of straightforward interpretations of β as the cointegrating parameters in the long-run relations β X t =, which are stable in the sense that they are fractional of a lower order, and of α describing adjustment toward the long-run equilibria and through the orthogonal complement) the common stochastic trends, which are fractional of order d The lag structure of models 2) and3) admits simple criteria for fractionality and cofractionality of X t or fractional cointegration; henceforth we use these terms synonymously). At the same time, the model is relatively easy to estimate because for fixed d b), the model is estimated by reduced rank regression, which reduces the numerical problem to an optimization of a function of just two variables. Finally, an appealing feature of the model is that it gives the possibility of the usual misspecification tests based on estimated residuals, although of course the theory for these would need to be developed in the current setting. The purpose of this paper is to conduct quasi) Gaussian maximum likelihood inference in models 2) and3) to show that the maximum likelihood estimator MLE) exists uniquely and is consistent, and to find the asymptotic distributions of MLEs and some likelihood ratio test statistics. We analyze the conditional likelihood function for X 1 X T ) given initial values X n n= 1 under the assumption that ε t is i.i.d. N p Ω). For the calculations of the likelihood function and the MLE, we need a X t for a>. Because we do not know all initial values, we assume that we have observations of X t t = N + 1Tand define initial values X n = X n n= N 1 and X n = n N, and base the calculations on these. Thus we set aside N observations for initial values. For the asymptotic analysis, we represent X t by its past values and we make suitable assumptions about their behavior. Apart from that, we assume only that ε t is i.i.d. Ω)with suitable moments. We treat d b) as parameters to be estimated jointly with the other parameters. Another possibility is to impose the restriction d = d for some prespecified d, for example, d = 1, or to impose b = b where b = 1 yields the VARFIMAk+1d 1 ) or Id 1) VAR model. We note here that the models with d = d and/or b = b are submodels in H r, and results for these models can be derived by the methods developed for the general model H r.thesame holds for the restriction d = b in model H r d = b) see 3)), even though a simple modification is needed due to the constant term. The univariate version of model 2) with a unit root was analyzed by Johansen and Nielsen 21) henceforth JN 21)), and we refer to that paper for some technical results. The inspiration for model 2) comes from Granger 1986), who noted the special role of the fractional lag operator L b = 1 b and suggested the model 4) A L) d X t = d b L b αβ X t 1 + dl)ε t ;

5 267 S. JOHANSEN AND M. Ø. NIELSEN see also Davidson 22). One way to derive the main term of this model is to assume that we have linear combinations γ β) of rank p for which d γ X t and d b β X t are I). Simple algebra shows that d X t = d b L b αβ X t + u t, where α is a function of γ and u t is I); see Johansen 28, p. 652) for details. The main technical contribution in this paper is the proof of existence and consistency of the MLE, which allows standard likelihood theory to be applied. This involves an analysis of the influence of initial values as well as proving tightness and uniform convergence in d b) of product moments of processes that can be close to critical processes of the form 1/2 ε t. In our asymptotic distribution results, we distinguish between weak cointegration when the true value b < 1/2) and strong cointegration b > 1/2), using the terminology of Hualde and Robinson 21). Specifically, we prove that for i.i.d. errors with sufficient finite moments, the estimated cointegration vectors are locally asymptotically mixed normal LAMN) when b > 1/2 and asymptotically Gaussian when b < 1/2, so that in either case, standard chisquared) asymptotic inference can be conducted on the cointegrating relations. Thus, for Gaussian errors, we get asymptotically optimal inference, but the results hold more generally. Note that the parameter value b = 1/2 is a singular point in the sense that inference is different for b < 1/2 andb > 1/2. Close to b = 1/2, we need many observations for the asymptotic results to be useful, and a similar situation occurs when the true value of either α or β is close to a matrix with lower rank; see Elliott 1998). Although such LAMN results are well known from the standard nonfractional) cointegration model e.g., Johansen 1988, 1991), Phillips and Hansen 199), Phillips 1991), and Saikkonen 1991) among others), they are novel for fractional models. Only recently, asymptotically optimal inference procedures have been developed for fractional processes e.g., Jeganathan 1999), Robinson and Hualde 23), Lasak 28, 21), Avarucci and Velasco 29), and Hualde and Robinson 21)). Specifically, in a vector autoregressive context, but in a model with d = 1 and a different lag structure from ours, Lasak 21) analyzed a test for no cointegration, and she analyzed Lasak 28)) maximum likelihood estimation and inference, in both cases assuming strong cointegration. In the same model as Lasak, but assuming weak cointegration, Avarucci and Velasco 29) extended the univariate test of Lobato and Velasco 27) to analyze a Wald test for cointegration rank; see also Marmol and Velasco 24). However, the present paper seems to be the first to develop LAMN results for the MLE in a fractional cointegration model in a vector error correction framework and with two fractional parameters d and b). The rest of the paper is laid out as follows. In the next section we describe the solution of the fractionally cointegrated vector autoregressive model and its properties. In Section 3, we derive the likelihood function and estimators, and show consistency. In Section 4, we find the asymptotic distribution of estimators and, in Section 5, that of the likelihood ratio test for cointegration rank. Section 6 concludes and technical material is presented in the Appendices.

6 INFERENCE FOR A FRACTIONALLY COINTEGRATED VAR 2671 A word on notation. We let C p K) denote the space of continuous p-vectorvalued functions on a compact set K R q, that is, continuous functions f : K R p,andletd p K) denote the corresponding space of cadlag functions. When p = 1, the superscript is omitted. For a symmetric matrix A, we write A > to mean that it is positive definite. The Euclidean norm of a matrix, vector, or scalar A is denoted A =tra A)) 1/2 and the determinant of a square matrix is denoted deta). Throughout, c denotes a generic positive constant that may take different values in different places. 2. SOLUTION OF THE COFRACTIONAL VECTOR AUTOREGRESSIVE MODEL We discuss the fractional difference operator d, a truncated version d +,and calculation of d X t. We show how equation 2) can be solved for X t as a function of initial values, parameters, and errors ε i i= 1t and give properties of the solution in Theorem 2. We then give assumptions for the asymptotic analysis and discuss identification of parameters, and finally we briefly discuss initial values The Fractional Difference Operator The fractional coefficients, π n a) are defined by the expansion ) a 1 z) a = 1) n z n n n= aa + 1) a + n 1) = z n = π n a)z n n! n= and satisfy π n a) = n<, and π n a) cn a 1 n 1; see Lemma A.5. The fractional difference operator applied to a process Z t t = 11T, is defined by a Z t = π n a)z t n n= provided the right-hand side exists. Note that a 1 a 2 = a 1 a 2 and the useful relation a 1 πt a 2 ) = π t a 1 + a 2 ), using that π t a) = fort<. We collect a few simple results in a lemma, where D m a Z t denotes the mth derivative with respect to a. LEMMA 1: Let Z t = ξ n= nε t n, where ξ n are s p, ε t are p-dimensional i.i.d. Ω), and ξ n= n < i) If the initial values Z n, n, are bounded, then D m a Z t exists for a and is almost surely continuous in a for a> n=

7 2672 S. JOHANSEN AND M. Ø. NIELSEN We next consider fractional differences of Z t without fixing initial values. ii) If a, then D m a Z t is a stationary process with absolutely summable coefficients and is almost surely continuous in a>. iii) If a> 1/2, then D m a Z t is a stationary process with square summable coefficients. PROOF: The existence is a simple consequence of the evaluation D m π n a) c1 + log n) m n a 1 for n 1 see Lemma A.5), which implies that D m π n a) is absolutely summable and continuous in a for a > and square summable for a > 1/2. For case ii), the continuity follows because D m a Z t D m ãz t c a ã n=1 1 + log n)m+1 n η 1 1 Z t n for mina ã) η 1 >. This random variable has a finite mean and is hence finite except on a null set that depends on η 1 but not a or ã. It follows that D m a Z t D mãz t as fora ã Q.E.D. For a<1/2 an example of these results is the stationary linear process a ε t = 1 L) a ε t = π n a)ε t n n= For a 1/2 the infinite sum does not exist, but we can define a nonstationary process by the operator a + defined on doubly infinite sequences, as t 1 ε + t = π n a)ε t n a n= t = 1T Thus, for a 1/2, we do not use a directly but apply instead a +,which is defined for all processes. Other authors have used the notation a ε t 1 {t 1}, where 1 {A} denotes the indicator function for the event A, and called this a type II process; see, for instance, Marinucci and Robinson 2). The idea of conditioning on initial values is used in the analysis of autoregressive models for nonstationary processes, and we modify the definition of a fractional process to take initial values into account. DEFINITION 1: Let ε t be i.i.d. Ω)in p dimensions and consider s p matrices ξ n for which ξ n= n < and define Cz) = ξ n= nz n z < 1. Then the linear process CL)ε t = ξ n= nε t n is fractional of order if C1). A process X t is fractional of order d> denoted X t Fd)) if d X t is fractional of order zero, and X t is cofractional with cofractionality vector β if β X t

8 INFERENCE FOR A FRACTIONALLY COINTEGRATED VAR 2673 is fractional of order d b forsomeb>. The same definitions hold for any d R and b> for the truncated linear process 5) C + L)ε t + ω t = 1 {t 1} t 1 n= ξ n ε t n + ω t where ω t is a deterministic term. The main result in Theorem 2 in Section 2.3 is the representation of the solution of equation 2) in terms of certain stationary processes, which we introduce next. DEFINITION 2: We define the class Z b as the set of multivariate linear stationary processes Z t that can be represented as Z t = ξε t + b ξ ε n t n n= where b>, ε t is i.i.d. Ω), and the coefficient matrices satisfy n= ξ n <. We also define the corresponding truncated process t 1 Z + t = ξε t + b + ξ ε n t n n= Definition 2 is a fractional version of the usual Beveridge Nelson decomposition, where ξ n= nε t n = ξ n= n)ε t + n= ξ ε n t n Z 1. For the asymptotic analysis, we apply the result that when a>1/2 and E ε t q < for some q>1/a 1/2), then for Z t Z b b>, we have 6) T a+1/2 a + Z+ [Tu] W a 1 u) u = Ɣa) 1 u s) a 1 dw s) on D p [ 1] ) a>1/2 where Ɣa) is the gamma function and W denotes p-dimensional Brownian motion BM) generated by ε t. The process W a 1 is the corresponding fractional Brownian motion fbm) of type II, and is used for convergence in distribution as a process on a function space C p or D p ); see Billingsley 1968) or Kallenberg 22). The proof of 6) is given in JN 21, Lemma D.2) for Z t Z b b>; see also Taqqu 1975)forZ t = ε t.

9 2674 S. JOHANSEN AND M. Ø. NIELSEN We also have under the same conditions on ε t and for Z t Z b b>, that 7) T a T t=1 a + L az + t ε t D 1 W a 1 dw a>1/2 where D denotes convergence in distribution on R p p. This result is proved in JN 21, p. 65) for univariate processes building on the result of Jakubowski, Mémin, and Pagès 1989) for the case Z t = ε t and L a = L 1. The same proof can be applied for processes in Z b Solution of Fractional Autoregressive Equations The properties of the solution of 2) are given by the properties of the polynomial 8) Ψy)= 1 y)i p αβ y k Γ i 1 y)y i i=1 = αβ y + 1 y) k Ψ i 1 y) i i= where the coefficients satisfy k i= Ψ i = I p, Ψ = I p k i=1 Γ i, and Ψ k = 1) k+1 Γ k Equation 2) can be written as ΠL)X t = d b ΨL b )X t = ε t,so that 9) Πz) = 1 z) d b Ψ 1 1 z) b) ; that is, d b X t satisfies a VAR in the lag operator L b rather than the standard lag operator L = L 1. This structure means that the solution of 2) and the criteria for fractionality of order d and cofractionality of order d b can be found by analyzing the polynomial Ψy), just as for the cointegrated VAR model. We want to solve X t as a function of initial values X n n = 1 and random shocks ε 1 ε t. A solution can be found using the two operators see Johansen 28)) Π + L)X t = 1 {t 1} t 1 i= Π i X t i and Π L)X t = Π i X t i i=t for which ΠL)X t = Π + L)X t + Π L)X t Here the operator Π + L) is defined for any sequence as a finite sum. Because Π) = I p Π + L) is invertible on sequences that are zero for t and the coefficients of the inverse are found by expanding Πz) 1 around zero. The expression Π L)X t is defined

10 INFERENCE FOR A FRACTIONALLY COINTEGRATED VAR 2675 if we assume that the initial values of X t are bounded. Then the equations in model 2) can be expressed as ε t = ΠL)X t = Π + L)X t + Π L)X t,andby applying Π + L) 1 on both sides, we find, for t = 1 2, that 1) X t = Π + L) 1 ε t Π + L) 1 Π L)X t = Π + L) 1 ε t + μ t The first term is the stochastic component generated by ε 1 ε t ; the second term is a deterministic component generated by initial values. An example of the solution 1) is the well known result that y t = vy t 1 + ε t has the solution y t = t 1 i= vi ε t i + v t y for any v and t = 1T.Whend<1/2, we use a representation of the solution that explicitly contains the stationarity of X t.in the simple example y t = vy t 1 + ε t with v < 1, this corresponds to using the solution y t = i= vi ε t i for t = 1T Properties of the Solution: Representation Theorem The solution 1) ofequation2) is valid without any assumptions on the parameters. We next give results that guarantee that X t is fractional of order d and cofractional from d to d b; that is, d X t and d b β X t are fractional of order zero. These results are given in terms of an explicit condition on the roots of the polynomial detψ y)) and the set C b which is the image of the unit disk under the mapping y = 1 1 z) b ; see Johansen 28, p. 66). Note that C 1 is the unit disk and that C b is increasing in b. The following result is Granger s representation theorem for the cofractional VAR models 2) and3); see also Johansen 28, Theorem8,29, Theorem 3). It is related to previous representation theorems of Engle and Granger 1987) and Johansen 1988, 1991) for the cointegrated VAR model. Below we use the notation β for a p p r) matrix of full rank for which β β = and we note the orthogonal decomposition, which defines β and β : 11) I p = β β β ) 1 β + β β β ) 1 β = β β + β β THEOREM 2: Let Πz) = 1 z) d b Ψ1 1 z) b ) be given by 8) and 9) for any <b d, and let y = 1 1 z) b. We assume that α and β have rank r p and that detψ y)) = implies that either y = 1 or y/ C maxb1) and we define Γ = I p k Γ i=1 i. i) Then it holds that 12) 1 z) d Πz) 1 = C + 1 z) b C + 1 z) 2b H 1 1 z) b) = C + 1 z) b H 1 1 z) b) if and only if detα Γβ ), where H y) is regular in a neighborhood of C maxb1) 13) C = β α Γβ ) 1 α and β C α = I r

11 2676 S. JOHANSEN AND M. Ø. NIELSEN For F z) = H 1 1 z) b ) = n= τ n zn and Fz) = H1 1 z) b ) = n= τ nz n z < 1 we have 14) τ n < and n= n= τ n < ii) For d 1/2, we represent the solution of 2) as 15) X t = C d + ε t + d b) + Y + t + μ t t = 1T where μ t = Π + L) 1 Π L)X t depends on initial values of X t and Y t = τ n= nε t n Z b is fractional of order zero with EY h= ty t h ) <. In this case, β X t is asymptotically stationary with mean zero. The solution of 3) with d = b and a constant term is represented as 16) X t = C d + ε t + Y + t + μ t + C αρ t = 1T and β X t + ρ is asymptotically stationary with mean zero. iii) For d<1/2, we represent the solutions of 2) and 3) as 17) 18) X t = C d ε t + d b) Y t t = 1T X t = C d ε t + d b) Y t + C αρ t = 1T iv) In all cases, there is no γ for which γ X t Fc) for some c<d b. PROOF: i) The proofs of 12) and 13) are given in Johansen 28, Theorem 8, 29, Theorem 3). The condition detα Γβ ) is necessary and sufficient for the representation of X t as an Fd) variable, because if detα Γβ ) =, then we get terms of the form 1 z) d+ib) i 2, corresponding to models for Ii) variables, i 2, in the cointegrated VAR context; see Johansen 28, Theorem 9). To prove 14), it is enough to prove it for τ because τ n n = τ n τ n 1n 1 We note that because H y) = n= τ n yn is regular in a neighborhood of C b, we can extend H 1 1 z) b ) by continuity to z =1 and define the transfer function φ e iλ) = H 1 1 e iλ) b) i = 1 We then apply the proof in JN 21, Lemma 1), which shows that because φe iλ )/ λ is square integrable when b>1/2 we have n= τ n n)2 < and hence n= τ <. n For b 1/2, we need another proof. The assumption y/ C 1 implies that H y) = k= h k yk is regular for y < 1 + δ for some δ> so that h k decrease exponentially. From the expansion 1 1 z) b = b m=1 mz m with

12 INFERENCE FOR A FRACTIONALLY COINTEGRATED VAR 2677 b m = π m b) we find that if b 1/2, then b m and b m=1 m = 1. Therefore, H 1 1 z) b) ) k = b m z m n= n n= k= = h k k= m=1 h k n= k= so that τ satisfies n τ h k= k k m=1 m 1 + +m k =n i=1 m 1 + +m k =n i=1 ) k b mi ) k h b m = h < k= k k b mi )z n = τ n zn ii) For d 1/2, we define Y t = τ n= nε t n = C ε t + b n= τ ε n t n Z b, which is fractional of order zero because C see13)) and has τ n= n <, which implies EY h= ty t h ) <. Then15) follows from 1); see also Johansen 28, Theorem 8). For ρ = andd = b, we find the solution X, t say, from 15). Then ΠL)X t + C αρ ) = ε t +ΠL)C αρ = ε t αβ C αρ = ε t + αρ so that X t = X t + C αρ is a solution of 3). In this case, we therefore find β X t +ρ = β X t +β C αρ +ρ = β X t, which is asymptotically stationary with mean zero and fractional of order zero. iii) For d<1/2 C d ε t + d b) Y t is stationary and represents a solution of 2)and3)forρ = We then add C αρ for ρ. iv) We find from i) and ii) that if, for some c<d b, γ X t is fractional of order c, then t 1 γ X t = γ C d ε + t + γ C d+b + ε t + d+2b + γ τ ε n t n + γ μ t Fc) implies that γ C = andγ C =. Hence γ = βξ and, therefore, γ C α = ξ β C α = ξ = so that γ =. Q.E.D. Thus for model 2) with<r<px t is fractional of order d, and because β C =, X t is cofractional since β X t = d b) + β Y + t + β μ t for d 1/2 and β X t = d b) β Y t for d<1/2 are fractional of order d b and no linear combination gives other orders of fractionality. If r =, we have α = β = ρ = α = β = I p,andc = Γ 1 is assumed to have full rank, and thus X t is fractional of order d and not cofractional. Finally, n= n=

13 2678 S. JOHANSEN AND M. Ø. NIELSEN if r = p, then αβ has full rank and C = so that X t = d b) + Y + t + μ t the d 1/2 representation) is fractional of order d b. Note, however, that the coefficients of Y + t and μ t depend on both d and b so that d b) is identified; see Theorem 3. The stochastic properties of X t aregivenintheorem2 in terms of the process U t = Cε t + b Y t Z b see Definition 2) and it follows from Theorem 2 that also Y t Z b Assumptions for the Data Generating Process We here formulate assumptions on the true parameter λ = d b α β Γ 1 Γ k Ω ) needed for identification and for the asymptotic properties of the estimators and the likelihood function for model H r. For the model H r d = b) with d = b and a constant term i.e. 3)), we replace b with ρ in the definition of λ. We define the parameter set 19) N ={db:<b d d 1 } for some d 1 >, which can be arbitrarily large. ASSUMPTION 1: For k and r p, the process X t t = 1T, is generated by model H r in 2) or model H r d = b) in 3) with the parameter value λ. ASSUMPTION 2: The errors ε t are i.i.d. Ω ) with Ω > and E ε t 8 <. ASSUMPTION 3: The initial values X n n are uniformly bounded, and X n = X n for n<n and X n = for n N. ASSUMPTION 4: The true parameter value λ satisfies d b ) N, d b < 1/2, b 1/2, and the identification conditions Γ k if k>), α and β are p r of rank r, α β I p, and detα Γ β ). Thus, if r<p, then detψ y)) = has p r unit roots and the remaining roots are outside C maxb 1). If k = r =, only <d 1/2 is assumed. Importantly, in Assumption 2,the errorsarenot assumedtobegaussianfor the asymptotic analysis, but are only assumed to be i.i.d. with eight moments; we later specify the existence of further moments needed for the asymptotic properties of the maximum likelihood estimator. Assumption 3 about initial values is needed for nonstationary processes so that d X t is defined for any d ; see Lemma 1. In Assumption 4 about the true values, we include the condition that d b < 1/2 which appears to be perhaps the most empirically relevant range of values for d b see, e.g., Henry and Zaffaroni 23), Gil-Alana and Hualde 29), and the references in the Introduction),

14 INFERENCE FOR A FRACTIONALLY COINTEGRATED VAR 2679 because in this case, β X t is asymptotically) stationary with mean zero. Assumption 4 also includes the condition for cofractionality when r>, which ensures that X t is fractional of order d and β X t is fractional of order d b. The identification conditions in Assumption 4 guarantee that the lag length is well defined, that the parameters are identified see Section 2.5), and that the asymptotic distribution of the maximum likelihood estimator is nonsingular see Lemma 7) Identification of Parameters In a statistical model with parameter λ, we say that the parameter value λ is identified if, for all λ for which P λ = P λ, it holds that λ = λ. We say that the model is generically identified if the set of unidentified parameter values has Lebesgue measure zero. In model 2), the parameters α and β enter, when r>, only through their product αβ, so they are not individually identified. This is usually solved by normalizing β. We use the decomposition 11) and define β = β β β) 1 α = αβ β,and ρ = ρ β β) 1, so that αβ = α β. We assume in the following discussion that this normalization has been performed and we use the notation α β. Note that β β = I r.wedefineλ = dbαβγ 1 Γ k Ω) suitably modified if r = p, r =, or k = see the discussion after 2)), and apply the notation Π λ L). THEOREM 3: For any k and r p, we let λ denote all parameters of model H r with k lags; see 2). We assume see Assumption 4) that for λ and λ, it holds that Γ k if k>), α and β are p r of rank r, αβ I p, and detα Γβ ). Then P λ = P λ implies λ = λ so that λ is identified. It follows that model H r in 2) is generically identified. A similar result holds for model 3). PROOF: IfP λ = P λ, the mean and variance of X t given the past are the same with respect to P λ and P λ, so that Ω = Ω,and,forallz, Π λ z) = 1 z) d b Ψ ) λ 1 1 z) b 2) = 1 z) d b Ψ λ 1 1 z) b ) = Πλ z) If k>andr>, then Ψ λ 1 1 z) b ) is a polynomial in 1 z) b see 8)) with highest order term Ψ k 1 z) k+1)b and lowest order term αβ. Hence 2) implies that 1 z) d b Ψ k 1 z) k+1)b = 1 z) d b Ψk 1 z) k+1)b and 1 z) d b αβ = 1 z) d b α β This evidently implies that d b) = d b ) and, therefore, αβ = α β, and that Ψ λy) and Ψ λ y) have the same coefficients; that is, λ = λ.ifk>andr =, then α = β =, α = β = I p, and Ψ = I p k Γ i=1 i = Γ = α Γβ, and the same conclusion holds. In case k = andr>, where the model is d X t = d b L b αβ X t + ε t, the conditions αβ andαβ I p for λ and λ imply that λ is identified. Finally, if k = r =, the model is d X t = ε t and λ = d Ω ) is identified.

15 268 S. JOHANSEN AND M. Ø. NIELSEN Since the set of values of λ that do not satisfy the given conditions has Lebesgue measure zero, it follows that model 2) is generically identified. Q.E.D. Identification was discussed in JN 21, Section 2.3, Lemma 3, and Corollary 4) in the univariate case, and an example of an indeterminacy between d, b, andk was given. Theorem 3 shows that once the lag length has been determined, the model is generically identified Initial Values For a X t a > to be well defined, we assume that the initial values X n n are uniformly bounded. The theory in this paper will be developed for observations X 1 X T generated by 2) or3) with fixed bounded initial values; that is, conditional on X n n, as developed in JN 21), and we choose the representations given in Theorem 2. The likelihood function depends on a X t for different values of a, andbecausewedonotobservetheinfinitelymanypastvaluesofx t, we choose initial values X n for the calculations and define a X t = a + X t + a X t The first term is a function of the observations X 1 X T but the second is a function of initial values. A possible choice is X n = n, but we derive the theory for the choice X n = X n n<n, and set X n = forn N Thus we set aside N observations for initial values, as is usually done in the analysis of an ARk) model. We prove consistency under the assumption that X n is uniformly bounded for n and derive the asymptotic distributions under the further assumption that X n = forn T υ for a small υ. 2 In this way, we allow the number of initial values in the representation of X t to increase with T thereby approximating the situation where the representation has infinitely many initial values. ThechoiceofN entails a small sample bias/efficiency trade-off, with fewer initial values introducing bias, but also leaving more observations for parameter estimation. Simulations suggest that about a handful of initial values is usually sufficient, which is also what is used in the univariate) empirical application in Hualde and Robinson 211, Section 5) who assumed that both X n and X n are zero in their theoretical analysis, but in their empirical application, they actually condition on nonzero initial values. Such simulations and analytical results will be reported elsewhere. For d 1/2, we use the representations 15)and16) in terms of μ t,which depends on the correct initial values, and approximate it as discussed above. For d < 1/2, we use the representations 17) and18) ofx t as a stationary process around its mean. The initial values term μ t plays no role in that case, because the initial values have been given their invariant distribution. 2 An alternative assumption is n=1 n 1/2 X n < ; see Lemma A.8.

16 INFERENCE FOR A FRACTIONALLY COINTEGRATED VAR LIKELIHOOD FUNCTION AND MAXIMUM LIKELIHOOD ESTIMATORS The log likelihood function log L T λ) is continuous in λ and we show that for the probability measure P determined by λ, T 1 log L T λ) converges weakly as a continuous function on a compact set. Because the limit is deterministic, we get uniform convergence in the parameter λ, and we use that to prove existence and uniqueness of the MLE. We first discuss the calculation of the MLE, and then find the likelihood and profile likelihood functions and their limits. We apply this to prove consistency of the MLE Calculation of MLE, Profile Likelihood Function, and Its Limit In 8), we eliminate Ψ k = I p k 1 Ψ i= i and define d+ib X t = d+ib X t, the regressors d+ib + X t + 21) X 1t = d b d) X t X kt = d+kb X t X it = d+ib d+kb) X t for i = k 1 and the residuals 22) k 1 ε t λ) = Π + L)X t + Π L) X t = X kt αβ X 1t + Ψ i X it i= where λ = dbαβψ Ω) is freely varying and Ψ = Ψ Ψ k 1 ).The Gaussian likelihood function is now T 23) 2T 1 log L T λ) = log detω) + tr Ω 1 T 1 ε t λ)ε t λ) ) For the model with d = b, wedefinex 1t = 1 d )X t C α ρ ) and θ = ρ ρ + β C α ρ so that 1 d )β X t + ρ ) = β X 1t + θ and ρ t=1 24) k 1 ε t λ) = X kt αβ X 1t αθ + ρ Ψ i X it i= Note that for ρ β) = ρ β ),wefindθ ρ = because β C α = I r ;see13). For fixed ψ = d b), the MLE based on 23) is found by reduced rank regression of X kt on X 1t corrected for {X it } k 1 i= ;seeanderson 1951) orjohansen 1995). Note that this is equivalent to reduced rank regression of d X t on d b L b X t corrected for { d L i X b t} k i=1. The calculations are organized as follows. For fixed ψ in model H r, we define, in analogy with the notation for the I1) modelseejohansen1995, pp )), the residuals R t ψ) = X kt X t X k 1t )

17 2682 S. JOHANSEN AND M. Ø. NIELSEN and R 1t ψ) = X 1t X t X k 1t ) from regressions of X kt and X 1t on X t X k 1t, respectively. We then define the product moments S ij ψ) = T 1 T R t=1 itψ)r jt ψ) and the eigenvalue problem 25) = det ωs 11 ψ) S 1 ψ)s ψ) 1 S 1 ψ) ) which gives eigenvalues 1 > ˆω 1 ψ) > > ˆω p ψ) > and the maximized profile likelihood function expressed as 26) l Tr ψ) = 2T 1 log L max H r ) = log det S ψ) ) r + log 1 ˆω i ψ) ) i=1 Finally the MLE and maximized likelihood can be calculated by minimizing l Tr ψ) asafunctionofψ = d b) by a numerical optimization procedure. For model 3), we assume b = d and include αρ in the definition of ε t λ) see 24)), and apply reduced rank regression of X kt on X 1t 1) corrected for {X it } k 1 to define the concentrated likelihood function l i= Trψ). Belowwe focus on 2) and only include comments on 3) when the results or arguments are different. A computer package for conducting statistical inference using the procedure described in this paper is available; see Nielsen and Morin 212). Using non- or semiparametric estimates of d and b, followed by reduced rank regression estimation of the remaining parameters, would entail an efficiency loss for the asymptotically normal estimators, that is, ˆα Γ ˆ 1 Γ ˆ k ) when b > 1/2, and for all the estimators when b < 1/2, because ˆd and ˆb are asymptotically correlated with them, but would entail no efficiency loss for ˆβ when b > 1/2. In addition, we have found that using d = b = 1asstarting values in the numerical iterations is a good choice, so there seems to be no advantage from initializing the search with preliminary estimates. The calculation of the fractional differences in {X it } k i= 1 in each step of the numerical optimization algorithm can be time consuming for very large samples, but the actual optimization of l Tr ψ) seems to be unproblematic. Note that for r = p l Tp ψ) is found by regression of X kt on {X it } k 1 27) l Tp ψ) = log det SSR T ψ) ) = log det T 1 ) T R t R t t=1 i= 1 : where R t = X kt {X it } k 1 ) denotes the regression residual and SSR i= 1 T ψ) = T 1 T R t=1 tr t denotes the normalized) sum of squared residuals.

18 INFERENCE FOR A FRACTIONALLY COINTEGRATED VAR 2683 The stochastic properties of X t are given in Theorem 2 in terms of the stationary process U t = C ε t + b Yt. We note that for any ψ = d b) for which d + ib d > 1/2 the process d+ib d Ut is stationary. On the other hand, d+jb d β U t = d+jb d +b β Y t is stationary for all j = 1kbecause d + jb d + b d + b > 1/2. Thus, corresponding to X it see 21)), we define 28) U 1t = d b d L b U t U it = d+ib d+kb) d U t U kt = d+kb d U t if they are stationary and define the class of stationary processes for a given ψ as F stat ψ) = { β U jt for all j and U it for d + ib d > 1/2 } For d < 1/2 d+ ib d d > 1/2, so in that case F stat ψ) contains U it for all i We next want to define the probability limit, l p ψ) of the profile likelihood function l Tp ψ) in 27). The limit of log detssr T ψ)) is infinite if X kt is nonstationary and is finite if X kt is asymptotically) stationary; see Theorem 4.We therefore define the subsets of N, N div κ) = N {db: d + kb d 1/2 + κ} N conv κ) = N {db: d + kb d 1/2 + κ} N conv ) = N {db: d + kb d > 1/2} κ κ> and note that N = N div κ) N conv κ) for all κ. The family of sets N div κ) decreases as κ ) to the set N div ) which is exactly the set where X kt is nonstationary and log detssr T ψ)) diverges. Similarly, N conv κ) is a family of sets increasing as κ ) to N conv ), which is the set where X kt is stationary and log detssr T ψ)) converges pointwise in ψ in probability. We therefore define the limit likelihood function as { if ψ Ndiv ), 29) l p ψ) = log det Var U kt F stat ψ) )) if ψ N conv ), where we use the notation for any random vectors W and V with finite variance: VarW V)= VarW ) CovW V ) VarV ) 1 CovV W ) 3.2. Convergence of the Profile Likelihood Function and Consistency of the MLE For η>, we define the family of compact sets Kη) ={db: η b d d 1 }

19 2684 S. JOHANSEN AND M. Ø. NIELSEN which has the property that Kη) N increases to N as η. We now show that for all A>andallγ> there exists a κ > andt > so that with probability greater than 1 γ, the profile likelihood l Tp ψ) is uniformly greater than A on Kη) N div κ ) for T T. Thus the minimum of l Tp ψ) cannot be attained on Kη) N div κ ). On the rest of Kη), however, we show that l Tp ψ) converges uniformly in probability as T to the deterministic limit l p ψ), which has a strict minimum, log detω ),atψ. We prove this by showing weak convergence, on a compact set, of the likelihood as a continuous process in the parameters. Because the limit is deterministic, weak convergence implies uniform convergence in probability; see Lemma A.4. THEOREM 4: The function l p ψ) has a strict minimum at ψ = ψ that is, 3) l p ψ) l p ψ ) = log detω ) ψ N and equality holds if and only if ψ = ψ Let Assumptions 1 4 hold, so that, in particular, E ε t 8 < and assume that d b ) Kη). For r = p, it holds that 31) l Tr ψ ) P log detω ) and, furthermore, there are two alternative cases: i) Suppose E ε t q < for some q>1/ minη/31/2 d + b )/2). Then the likelihood function for H p satisfies that, for any A> and γ>, there exists a κ > and a T > such that ) 32) P inf l Tpψ) A 1 γ ψ N div κ ) Kη) for all T T. It also holds that 33) l Tp ψ) l p ψ) on C N conv κ ) Kη) ) as T ii) Suppose η b = d d 1 and E ε t q < for q>3/η. Then, for the model H p d = b) with a constant, the results 32) and 33) hold on the respective sets intersected with {b = d}. The proof is given in Appendix B. Note that, in general, the larger is the compact set Kη), the more moments are needed. When consideration is restricted to the model H r d = b) and a parameter set defined by η>3/8 i.e., in particular, if consideration is restricted to the case of strong cointegration, where b > 1/2), then the moment condition reduces to E ε t 8 < from Assumption 2). We now derive the important consequence of Theorem 4.

20 INFERENCE FOR A FRACTIONALLY COINTEGRATED VAR 2685 THEOREM 5: Let the assumptions of Theorem 4 be satisfied and let ˆλ denote the MLE in model H r, respectively, model H r d = b). Corresponding to Theorem 4i) and ii), we have the following cases: i) With probability converging to 1, ˆλ in model H r r = p exists uniquely for ψ Kη) η >, and is consistent. ii) For model H r d = b) with a constant, existence, uniqueness, and consistency of ˆλ hold for d {d :<η d d 1 }. PROOF: To prove existence and consistency of the MLE, we define the open neighborhood N ψ ɛ)={ψ : ψ ψ <ɛ}, and want to find a set A T with PA T ) 1 2γ so that ˆψ exists on A T and P ˆψ A T N ψ ɛ) ) 1 3γ We first analyze model H p see 2)), where α and β are p p. Forany γ>, 32) shows that we can find κ = κ γ) and T = T γ), and define A 1T = {inf ψ Ndiv κ ) Kη) l Tp ψ) 2 + l p ψ )} so that PA 1T ) 1 γ for all T T. We find from 33) that l Tp ψ) = log detssr T ψ)) l p ψ) on the compact set N = N conv κ ) Kη) so that l p ψ) is continuous on N.Because l p ψ) is continuous and greater than l p ψ ) if ψ ψ see 3)), and N \ N ψ ɛ)is compact and does not contain ψ we have min ψ N \N ψ ɛ) l p ψ) l p ψ ) + 3c for some c >. By the uniform convergence of l Tp ψ) to l p ψ) on N see 33)), we can find T 1 = T 1 γ) and define { A 2T = min ltp ψ) l p ψ) } c ψ N \N ψ ɛ) such that PA 2T ) 1 γ for all T T 1. We now turn to the model H r r = p. On the set A 2T,wehaveforany r p min l Trψ) min l Tpψ) min l pψ) c ψ N \N ψ ɛ) ψ N \N ψ ɛ) ψ N \N ψ ɛ) which is bounded below by l p ψ ) + 3c c = l r ψ ) + 2c, recalling l r ψ ) = log detω ) = l p ψ ) see 3)). On the set A 1T,wehavel Tr ψ) l Tp ψ) 2 + l p ψ ) anditfollowsthatona T = A 1T A 2T with PA T ) 1 2γ min l Trψ) l r ψ ) + 2min1c ) ψ Kη)\N ψ ɛ) On the other hand, at the point ψ = ψ,wehavel Tr ψ ) P l r ψ ) = log detω ) see 31)), so that for all T T 2 = T 2 γ) P ltr ψ ) l r ψ ) minc 1) ) 1 γ

21 2686 S. JOHANSEN AND M. Ø. NIELSEN which implies that on A T, the minimum of l Tr ψ) is attained inside N ψ ɛ). Thus the MLE ˆψ r of ψ in model H r exists on A T and is contained in the set N ψ ɛ) which proves consistency; see also van der Vaart 1998, Theorem 5.7). The estimators ˆαψ) ˆβψ) ˆΨ ψ) and ˆΩψ) see Section 3.1) are continuous functions of ψ and are therefore also consistent. The second derivative of 2T 1 log L T λ) is positive definite in the limit almost surely at λ = λ ; see Lemma 9. It is therefore also positive definite in a neighborhood N λ ɛ) for ɛ small. It follows from Theorem 6 and Lemma 9 that also the second derivative of 2T 1 log L T λ) is positive definite inside N λ ɛ) with probability converging to 1, but then 2T 1 log L T λ) is convex and the minimum is unique. Q.E.D. The result in Theorem 5 on existence and consistency of the MLE involves analyzing the likelihood function on the set of admissible values <b d.the likelihood depends on product moments of d+ib X t for all such d b), evenif the true values are fixed at some b and d. Since the main term in X t is d + ε t see 15)), analysis of the likelihood function leads to analysis of d+ib d + ε t, which may be asymptotically stationary, be nonstationary, or be critical in the sense that it may be close to the process 1/2 + ε t. The possibility that d+ib X t can be critical or close to critical, even if X t is not, implies that we have to split up the parameter space around values where d+ib X t is close to critical and give separate proofs of uniform convergence of the likelihood function in each subset of the parameter space. This is true, in general, for any fractional model, where the main term in X t is typically of the form d + ε t, and analysis of the likelihood function requires analysis of d X t and, therefore, of a term like d d + ε t that may be close to critical. To the best of our knowledge, all previous consistency results in the literature for parametric fractional models either have been of a local nature or have covered only the set where d X t is asymptotically stationary, due to the difficulties in proving uniform convergence of the likelihood function when d X t is close to critical and, hence, on the whole parameter set; see the discussion in Hualde and Robinson 211, pp ). 3 The consistency results in our Theorem 5 apply to admissible parameter sets that are so large that they include values of d b) where d+ib X t is asymptotically stationary, nonstationary, or critical. The inclusion of the near critical processes in the proof is made possible by a truncation argument, allowing us to show that when v [ 1/2 κ 1 1/2 + κ] for κ sufficiently small, then the 3 In independent and concurrent work, Hualde and Robinson 211) proved consistency for a large set of admissible values in a fractional model with one fractional parameter and initial values equal to zero, that is, both X n = and X n = forn. Also, their consistency proof applies only to the univariate case see their discussion on pp ).

22 INFERENCE FOR A FRACTIONALLY COINTEGRATED VAR 2687 appropriately normalized product moment of critical processes v + ε t is tight in v and is uniformly large for T sufficiently large; see 17) in Lemma A ASYMPTOTIC DISTRIBUTION OF MAXIMUM LIKELIHOOD ESTIMATORS In this section, we exploit consistency of the MLE and expand the likelihood in a neighborhood of the true parameter to find the asymptotic distribution of the conditional MLE A Local Reparametrization and the Profile Likelihood Function for dbαψ Ω The likelihood function for model 2) in a neighborhood of the true value is expressed in terms of ε t λ); see22)and23). We have identified β by β β = I r see Section 2.5) andweuse11) towrite β = β + β β β) = β + β ϑ,say.whenb > 1/2, we let N ψ ɛ) = {ψ : ψ ψ ɛ}. Thenford b) N ψ ɛ) and ɛ<1/2 sufficiently small, we have that δ 1 = d b d = d b d + b ) b b + 2ɛ < 1/2 and d + ib d ɛ for i. Hence, β X 1t is the only nonstationary process in ε t λ) see 22)), and this is only possible for b > 1/2 The information for ϑ is proportional to T t=1 β X 1t)β X 1t) = O P T 2δ 1 ),andwe therefore introduce the normalized parameter θ = β β β )T δ 1+1/2) = ϑt δ 1+1/2) or β = β + β θt δ 1+1/2, so the information for θ is proportional to T.Wehaveβ X 1t = β X 1t + T δ 1+1/2 θ β X 1t; see21). Let V t = X β 1t {X it }k 1 i= X kt ) and define as in 22), for φ = dbαψ ), 34) ε t λ) = ε t φ θ) = αt δ 1+1/2 θ β X 1t + α Ψ I p )V t For the model H r d = b) in 3) withd > 1/2, we change the definitions in this section and use θ β = T d 1/2 β β β ), θ ρ = ρ + β C α ρ,and ε t λ) = ε t φ θ) = α ) β T d+1/2 θ X β θ 1t ρ 1 ) + α Ψ I p )V t When b > 1/2, the product moments needed to calculate the conditional likelihood function 2T 1 log L T φ θ) see 23)) are ) AT ψ) C T ψ) 35) C T ψ) B T ψ) T T = T 1 δ 1 +1/2 β X ) 1t T δ 1 +1/2 β X ) 1t t=1 V t V t

23 2688 S. JOHANSEN AND M. Ø. NIELSEN We sometimes suppress the dependence on ψ in A T ψ) B T ψ) and C T ψ). We indicate the values for ψ = ψ by A T B T C and T X 1t. Finally we define 36) T C = T 1/2 εt T 1/2 b β X 1t ε t t=1 When b < 1/2, all processes are asymptotically) stationary and we replace δ 1 + 1/2 by zero in the definitions of A T B T C T and C εt The conditional likelihood 2T 1 log L T λ) can now be expressed as 37) log detω) + tr Ω 1 αθ A T θα + α Ψ I p )B T α Ψ I p ) 2αθ C T α Ψ I p ) )) For fixed dbαψ Ω), we estimate θ by regression and find 38) ˆθψ α Ψ Ω)= A 1C T T α Ψ I p ) Ω 1 α α Ω 1 α ) 1 and the profile likelihood function 2T 1 log L profilet ψ α Ψ Ω)is then 39) log detω) + tr Ω 1 α Ψ I p )B T α Ψ I p ) ) tr α Ψ I p )C T A 1C T T α Ψ I p ) Ω 1 α α Ω 1 α ) 1 α Ω 1) For d b) N ψ ɛ)ɛ < 1/2 and i = 1k, U it, β U 1t, and their derivatives with respect to d b) are stationary because d + ib d d d ɛ > 1/2. Only β X 1t is nonstationary and only when b > 1/2. When normalized by T δ 1+1/2 it will converge to fbm provided E ε t q < for some q>1/b 1/2) see 6)), so that on D p r [ 1]), 4) T δ 1+1/2 β X 1[Tu] β C W d d+b 1u) = F ψ u) We show that the deterministic term in the process can be neglected asymptotically and that the stationary processes {β U 1tU jt } k j= 1 can replace the regressors {β X 1tX jt } k. This means that the limit of B j= 1 T can be calculated as B = Var U 1t β U t U kt) For b < 1/2 all regressors X it are stationary in a neighborhood of the true value. The various quantities A T B T C T and C εt are defined as above without the factor T δ 1+1/2, but their asymptotic properties are now different. The estimator of θ and the profile likelihood function are given by 38)and39). The next theorem summarizes the asymptotic results for the product moments and their derivatives with respect to ψ,denotedd m,whenψ Nψ ɛ).

24 INFERENCE FOR A FRACTIONALLY COINTEGRATED VAR 2689 THEOREM 6: Let Assumptions 1 4 be satisfied and let N ψ ɛ)={ψ : ψ ψ ɛ} N. i) Suppose 1/2 <b <d and ε t q < for some q>b 1/2) 1, and let ɛ be chosen so small that q>b d + d 1/2) 1 for all ψ N ψ ɛ). Then, for m and with n = p r) 2 + r + kp + p) 2 + p r)r + kp + p) the process D m A T ψ) D m B T ψ) D m C T ψ)) is tight on N ψ ɛ) and we have on C n N ψ ɛ))that see 4)), 41) AT ψ) D m B T ψ) D m C T ψ) ) 1 ) F ψ u)f ψ u) dud m Bψ) which holds jointly with 42) C εt D 1 F dw ) F u) = F ψ u) ii) Suppose <b < 1/2 and b <d. Then, for m, the process D m A T ψ) D m B T ψ) D m C T ψ)) is tight on N ψ ɛ)and we find on C n N ψ ɛ)) that AT ψ) D m B T ψ) D m C T ψ) ) Aψ) D m Bψ) D m Cψ) ) which is deterministic, and the convergence holds jointly with 43) C εt D N p r) p Ω A ) iii) For model H r d = b) with a constant, the same results hold with the relevant restriction imposed and the relevant modifications to the definitions, for example, F ψ u) is replaced by F u) 1). PROOF: i)ford > 1/2, it follows from Theorem 2 that for U + t = C ε t + b + Y + t 44) d+ib X t = d+ib d + U + t + d+ib μ + t + d+ib X t and hence the regressors satisfy see 21)) t = 1T 45) X 1t = ) d b d + d d + U + t + d b ) + d + μt + d b ) X d t = U + + D 1t 1tψ) X kt = d+kb d + U + t + d+kb + μ t + d+kb X t = U + + D kt ktψ)