Cointegrated VAR s. Eduardo Rossi University of Pavia. November Rossi Cointegrated VAR s Financial Econometrics / 56

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1 Cointegrated VAR s Eduardo Rossi University of Pavia November 2013 Rossi Cointegrated VAR s Financial Econometrics / 56

2 VAR y t = (y 1t,..., y nt ) is (n 1) vector. y t VAR(p): Φ(L)y t = ɛ t The process is stable if Φ(L) = I n Φ 1 L... Φ p L p ɛ t = (ɛ 1t,..., ɛ nt ) ɛ t independent VWN(0, Ω) det(i n Φ 1 z... Φ p z p ) 0 for z 1 On the assumption that the process has been initiated in the infinite past (t = 0, ±1, ±2,...) it generates stationary time series that have time-invariant means, variances, and covariances. Rossi Cointegrated VAR s Financial Econometrics / 56

3 VAR If det(i n Φ 1 z... Φ p z p ) = 0 for z = 1 i.e. it has a unit root, then some or all of the variables are integrated. We assume that the variables are at most I (1). Rossi Cointegrated VAR s Financial Econometrics / 56

4 VAR with Integrated variables VAR(p) process without deterministic term: given that Φ(L)y t = ɛ t Φ(L) Φ(L) 1 = Φ(L) adj Φ(L) I n = Φ(L) adj Φ(L) multiplying from the left by the adjoint Φ(L) adj of Φ(L) Φ(L) adj Φ(L)y t = Φ(L) adj ɛ t Φ(L) y t = Φ(L) adj ɛ t all components have the same AR operator Φ(L). Φ(L) adj ɛ t is a finite order MA process. Rossi Cointegrated VAR s Financial Econometrics / 56

5 VAR with Integrated variables If Φ(L) has d unit roots Φ(L) = α(l)(1 L) d = α(l) d where α(l) is a invertible operator. d y t is stable process. If a VAR(p) process is unstable because of a unit root, it can be made stable by differencing its components. Due to cancellations it may not be necessary to difference each component as many times as there are unit roots in Φ(L). Rossi Cointegrated VAR s Financial Econometrics / 56

6 VAR with Integrated variables Bivariate VAR(1) process: ([ ] [ ] ) [ ] [ ] y1t (1 L)y1t L = = ɛ y 2t (1 L)y t 2t Each component is stationary after differencing once, i.e. each component is I (1) [ ] Φ(L) = 1 L 0 = (1 L) L It is also possible that some components are stable and stationary as univariate processes whereas others need differencing. Rossi Cointegrated VAR s Financial Econometrics / 56

7 VAR with Integrated variables VAR(p) process with a nonzero intercept term Φ(L)y t = c + ɛ t Φ(L) adj Φ(L)y t = Φ(L) adj c + Φ(L) adj ɛ t Φ(L) y t = Φ(L) adj c + Φ(L) adj ɛ t and Φ(L) has one or more unit roots, then some of the components of y t may have deterministic trends in their mean values. It is also possible that none of the components of y t has a deterministic trend in mean. This occurs if Φ(L) adj c = 0 Rossi Cointegrated VAR s Financial Econometrics / 56

8 VAR with Integrated variables For instance Φ(L) = [ 1 L ηl 0 1 Φ(z) = 1 z ] Φ(z) has a unit root and Φ(L) adj c = Φ(L) adj = [ 1 ηl 0 1 L [ 1 ηl 0 1 L ] ] [ ] [ ] c1 c1 ηc = 2 c 2 c 2 c 2 which is zero if c 1 = ηc 2. In a VAR analysis an intercept cannot be excluded a priori if there are unit roots and none of the component series has a deterministic trend. Rossi Cointegrated VAR s Financial Econometrics / 56

9 Cointegrated VAR s If the variables have a common stochastic trend, it is possible there are linear combinations of them that are I (0): they are cointegrated. Systems of I (1) and I (0) variables are also considered. The concept of cointegration is extended by calling any linear combination that is I (0) a cointegration relation. Rossi Cointegrated VAR s Financial Econometrics / 56

10 Cointegrated VAR s The VECM is a convenient model setup for cointegration analysis: y t = Πy t 1 + Γ 1 y t Γ p 1 y t p+1 + ɛ t Π = (I n Φ 1... Φ p ) = Φ(1) Γ i = (Φ i Φ p ) i = 1,..., p 1 Because y t does not contain I (1), the term Πy t 1 is the only one that includes I (1) variables, hence it must also be I (0). It contains the cointegrating relations. Γ i short-run parameters Πy t 1 long-run part Rossi Cointegrated VAR s Financial Econometrics / 56

11 Cointegrated VAR s From VAR(P) to VECM(p-1): substract y t 1 from both sides y t = Φ 1 y t Φ p y t p + ɛ t y t y t 1 = y t 1 + Φ 1 y t Φ p y t p + ɛ t y t = y t 1 + Φ 1 y t Φ p y t p +(Φ 2 + Φ Φ p )y t 1 (Φ 2 + Φ Φ p )y t 1 +(Φ 3 + Φ Φ p )y t 2 (Φ 3 + Φ Φ p )y t Φ p y t p+1 Φ p y t p+1 + ɛ t Rossi Cointegrated VAR s Financial Econometrics / 56

12 Cointegrated VAR s y t = (I Φ 1... Φ p )y t 1 + Φ 2 y t Φ p y t p (Φ 2 + Φ Φ p )y t 1 +(Φ 3 + Φ Φ p )y t 2 (Φ 3 + Φ Φ p )y t Φ p y t p+1 Φ p y t p+1 + ɛ t y t = Πy t 1 (Φ 2 + Φ Φ p )(y t 1 y t 2 ) (Φ 3 + Φ Φ p )(y t 2 y t 3 ) Φ p (y t p+1 y t p ) + ɛ t Rossi Cointegrated VAR s Financial Econometrics / 56

13 Cointegrated VAR s From VECM(p-1) to VAR(p): Φ 1 = Γ 1 + Π + I n Φ i = Γ i Γ i 1 i = 2,..., p 1 Φ p = Γ p 1 Rossi Cointegrated VAR s Financial Econometrics / 56

14 Cointegrated VAR s If the VAR(p) process has unit roots det(i n Φ 1 z... Φ p z p ) = 0 for z = 1 the matrix is singular. Suppose then we can write Π = (I Φ 1... Φ p ) r(π) = r Π = αβ α (n r) r(α) = r β (n r) r(β) = r Rossi Cointegrated VAR s Financial Econometrics / 56

15 Example VAR(2): suppose the process is unstable with y t = Φ 1 y t 1 + Φ 2 y t 2 + ɛ t I n Φ 1 z Φ 2 z 2 = (1 λ 1 z)... (1 λ 2n z) = 0 for z = 1 λ i are the reciprocals of the roots of the determinantal polynomial, one or more of them must be equal to 1. All other are assumed to lie outside the unit circle, that is if λ i = 1 then it must be that λ i < 1. Since I n Φ 1 Φ 2 = 0 the matrix suppose rk(π) = r < n Π = (I n Φ 1 Φ 2 ) Π = αβ Rossi Cointegrated VAR s Financial Econometrics / 56

16 Example where y t = (I n Φ 1 Φ 2 )y t 1 Φ 2 y t 1 + Φ 2 y t 2 + ɛ t y t = Πy t 1 + Γ 1 y t 1 + ɛ t Γ 1 = Φ 2 αβ y t 1 = y t (Γ 1 y t 1 + ɛ t ) the right-hand side involves stationary terms only then αβ y t 1 must also be stationary. β y t 1 represents a cointegrating relation. Rossi Cointegrated VAR s Financial Econometrics / 56

17 Cointegrated VAR s Simply taking the first differences of all variables eliminates the cointegration term which may contain relations of great importance. A VAR process with cointegrated variables does not admit a pure VAR representation in first differences. Rossi Cointegrated VAR s Financial Econometrics / 56

18 Cointegration rank p 1 y t = Πy t 1 + Γ i y t i + ɛ t i=1 rk(π) = n the system is stationary (y t I (0)), standard asymptotic theory applies of estimation for impulse response analysis. rk(π) = 0, the system is I (1) and NOT cointegrated. The system is driven by n common stochastic trends (unit roots). rk(π) = 0 Π = 0 the VECM reduces to a stationary VAR in first differences. Standard asymptotic theory applies for estimation of Γ i. 0 < rk(π) < n the system is integrated of order one and cointegrated. The system is driven by n r common stochastic trends. Rossi Cointegrated VAR s Financial Econometrics / 56

19 Granger Representation Theorem Another useful representation of a cointegrated system is given by the Granger Representation Theorem (Johansen s version (1995, Th.4.2)). For m n we denote by M, with rk(m ) = m n, an orthogonal complement of the (m n) matrix M with rk(m) = n. and if n = 0 M : m (m n) M M = 0 M = I m Assumption: the process y t is a n-dimensional cointegrated process with cointegration rank r, 0 r < n. Rossi Cointegrated VAR s Financial Econometrics / 56

20 Granger Representation Theorem Suppose p 1 y t = Πy t 1 + Γ i y t i + ɛ t t = 1, 2,... i=1 where y t = 0 for t 0, ɛ t VWN for t = 1, 2,... and ɛ t = 0 for t 0. Define p 1 C(z) = (1 z)i n αβ z Γ i (1 z)z i with the following conditions for the parameters: 1 C(z) = 0 z > 1 or z = 1 i=1 2 The number of unit roots z = 1 is exactly n r 3 α and β are (n r) matrices with rk(α) = rk(β) = r Rossi Cointegrated VAR s Financial Econometrics / 56

21 Granger Representation Theorem Then y t has the representation t y t = Ξ ɛ i + Ξ (L)ɛ t + y0 i=1 where ( ) ] p 1 1 Ξ = β [α I N Γ i β α i=1 α : n (n r) β : n (n r) Ξ (L)ɛ t = Ξ j ɛ t j j=0 is an I (0) process and y0 contains initial values. Rossi Cointegrated VAR s Financial Econometrics / 56

22 Granger Representation Theorem This proposition decomposes the process y t into I(1) and I(0) components which have to be treated accordingly. It makes precise under what conditions the process is driven by (n r) I(1) components and r I(0) components. This is a multivariate version of the Beveridge-Nelson decomposition: Ξ t i=1 ɛ t, n random walks t i=1 ɛ t multiplied by a matrix of rank n r. There are n r stochastic trends driving the system. y t is I(1) if it has the representation with Ξ 0. For Ξ to have the form given in the theorem, the (n r) (n r) matrix [ α ( ) ] p 1 I N Γ i β must be invertible. Only under that condition, rk(ξ) = n r. The latter condition ensures that y t is driven by n r random walk components. i=1 Rossi Cointegrated VAR s Financial Econometrics / 56

23 Granger Representation Theorem An immediate consequence of the representation is that β y t is stationary, since β Ξ = 0. β Ξ (L)ɛ t is a representation of the disequilibrium error β y t. For large t the random walk dominates the stochastic component of y t and the long-run variance Ξ ΩΞ. This matrix is singular. Rossi Cointegrated VAR s Financial Econometrics / 56

24 Granger Representation Theorem One can interpret the matrix Ξ as indicating how the common trends t α ɛ t i=1 contribute to the various variables through the matrix β. Another interpretation is that a random shock to the first equation at time t = 1 is represented by the coefficients of Ξ (L) which die out over time. A long-run effect is given by u 1(Ξɛ t ). Rossi Cointegrated VAR s Financial Econometrics / 56

25 Deterministic components When two (or more) variables share the same stochastic and deterministic trends it is possible to find a linear combination that cancels both the trends. The resulting cointegration is not trending, even if the variables by themselves are. This case can be accounted for by including a trend in the cointegration space. A linear combination of variables removes the stochastic trend(s), but not the deterministic trend, so we again need to allow for a linear trend in the cointegration space. Rossi Cointegrated VAR s Financial Econometrics / 56

26 Deterministic components Biased (misleading) parameter estimates if the deterministic components are incorrectly formulated, partly because the asymptotic distributions of the cointegrating tests are not invariant to the specification of these components. Parameter inference, policy simulations, and forecasting are much more sensitive to the specifications of the deterministic than the stochastic components of the VAR model. Rossi Cointegrated VAR s Financial Econometrics / 56

27 Deterministic components y t VAR(1), VECM: y t = αβ y t 1 + µ 0 + µ 1 t + ɛ t µ 0 = αµ + γ (n 1) µ 1 = αρ + τ (n 1) µ (r 1) γ (n 1) ρ (r 1) τ (n 1) decomposition into two vectors: one is related to the mean of the cointegrating relations (αµ) and the other to growth rates in y t. y t = αβ y t 1 + αµ + γ + αρt + τ t + ɛ t Rossi Cointegrated VAR s Financial Econometrics / 56

28 Deterministic components y t = α [ β : µ : ρ ] y t 1 1 t + (γ + τ t) + ɛ t = αβ y t 1 + (γ + τ t) + ɛ t β = [ β : µ : ρ ] we can choose µ and ρ such that (n (r + 2)) β y t = v t with E[v t ] = 0 Rossi Cointegrated VAR s Financial Econometrics / 56

29 Deterministic components so that E[ y t ] = γ + τ t τ = 0, γ 0 constant growth in y t τ 0 linear trends in growth and so quadratic trends in y t µ 0 and µ 1 play a dual role in the cointegrated model: α describes a linear trend αt and an intercept αµ in the steady-state relations. γ describes quadratic and linear trends in the data. In empirical work we have some idea whether there are linear deterministic trends in some or all of the variables. It is more difficult to know if they cancel in the cointegrating relations or not. Rossi Cointegrated VAR s Financial Econometrics / 56

30 Five Cases 1. No restrictions on µ 0 and µ 1 E[ y t ] = (γ + τ t) + αβ E(y t 1) 2. τ = 0, (µ 1 = αρ), γ, µ, ρ unrestricted y t = αβ y t 1 + γ + ɛ t E[ y t ] = γ + αβ E(y t 1) = γ Linear trend restricted to lie in the cointegration space but constant is unrestricted in the model. Linear trend in y t but no quadratic trend. These linear trends in the variables do not cancel in the cointegrating relations. The model contains trend stationary relations which can either describe a single trend-stationary variable or an equilibrium relation (y 1t b 1 t) I (0) (β 1y t b 2 t) I (0) Rossi Cointegrated VAR s Financial Econometrics / 56

31 Five Cases 3. µ 1 = 0 (ρ = 0 and τ = 0). Since the constant µ 0 is unconstrained there are still linear trends in y t, but no deterministic trends in any cointegration relations. Non-zero intercept in the cointegration relations y t = αβ y t 1 + γ + ɛ t β = [β : µ ] y t 1 = [y t 1, 1] 4. µ 1 = 0, γ = 0, µ 0. Constant term: αµ constant term restricted to lie in the cointegration space. y t = αβ y t 1 + ɛ t β = [β : µ ] y t 1 = [y t 1, 1] There are no linear deterministic trends in the data E[ y t ] = 0 The only deterministic components are µ, intercepts, in any cointegrating relation, implying that some equilibrium means are different from zero. Rossi Cointegrated VAR s Financial Econometrics / 56

32 Five Cases 5. µ 0 = 0 and µ 1 = 0, the model excludes all deterministic components in the data E[ y t ] = 0 E[β y t ] = 0 no growth and zero intercepts in every cointegrating relation. Since an intercept is generally needed to account for the initial level of measurements, y 0, only in the exceptional case where the measurement starts from zero, can the restriction µ 0 = 0 be justified. Rossi Cointegrated VAR s Financial Econometrics / 56

33 Reduced Rank Regression If the cointegrating rank is known Y = [ y 1,..., y T ] (n T ) Y 1 = [y 0,..., y T 1 ] (n T ) ɛ = [ɛ 1,..., ɛ T ] (n T ) Γ = [Γ 1,..., Γ p 1 ] (n n(p 1)) X = [X 0,..., X T 1 ] (n(p 1) T ) X T 1 = T sample values, p presample values. y t 1. y t p+1 (n(p 1) 1) Rossi Cointegrated VAR s Financial Econometrics / 56

34 Reduced Rank Regression VECM without deterministic trends: Y = ΠY 1 + ΓX + ɛ Given a specific Π the equationwise OLS estimator of Γ is Γ = ( Y ΠY 1 ) X (XX ) 1 substituting YM = ΠY 1 M + ɛ M = I T X (XX ) 1 X For a given integer r, 0 < r < n an estimate of Π with r( Π) = r can be obtained by canonical correlation analysis. Rossi Cointegrated VAR s Financial Econometrics / 56

35 Reduced Rank Regression If the process y t is Gaussian, or ɛ t N(0, Ω) the VECM can be estimated by ML taking also the rank restrictions (Π = αβ ) into account (Johansen (1995)). The log-likelihood function L(θ) = Tn 2 log 2π T log Ω tr [ ( Y ΠY 1 ΓX)Ω 1 ( Y ΠY 1 ΓX) ] For the estimation we assume that rank(π) = r which implies that the matrix Π = αβ Rossi Cointegrated VAR s Financial Econometrics / 56

36 Reduced Rank Regression The estimator of β may be defined by S 00 = T 1 YM Y (n n) S 01 = T 1 YMY 1 (n n) S 11 = T 1 Y 1 MY 1 (n n) M = I T X (XX ) 1 X and solving the generalized eigenvalue problem Ordered eigenvalues: det ( λs 11 S 01S 1 00 S 01) = 0 λ 1 λ 2... λ n Rossi Cointegrated VAR s Financial Econometrics / 56

37 Reduced Rank Regression The estimator of β may be obtained from the the generalized eigenvectors. The corresponding orthonormal eigenvectors V = [b 1,..., b n ] The reduced rank estimator of is thus obtained by choosing Π = αβ β = [b 1,..., b r ] α = YMY 1 β ( β Y 1 MY 1 β) 1 = S01 β( β S11 β) 1 Rossi Cointegrated VAR s Financial Econometrics / 56

38 Reduced Rank Regression Thus α can be viewed as the OLS estimator from the model YM = α β Y 1 M + ɛ The corresponding estimator Π = α β the feasible estimator of Γ: Γ = ( Y ΠY 1 ) X (XX ) 1 under Gaussian assumptions these estimators are ML estimators conditional on the presample values. Consistent and asymptotically normal: T vec ([ Γ1,..., Γ p 1 ] ) [Γ 1,..., Γ p 1 ] ) T vec ( Π Π d d N(0, Σ Γ) N(0, Σ Π) Rossi Cointegrated VAR s Financial Econometrics / 56

39 Normalization of β The ML estimators α and β are not unique, since, for any nonsingular matrix Q α β = αq 1 Q β The parameter estimator β is made unique by the normalization of the eigenvectors, and α is adjusted accordingly. These are not econometric restrictions. Only the cointegration space but not the cointegrating parameters are estimated consistently. To estimate α and β consistently it is necessary to impose identifying restrictions. Rossi Cointegrated VAR s Financial Econometrics / 56

40 Normalization of β An example of identifying restrictions is [ ] I β = r β (n r) for r = 1 this amounts to normalizing the coefficient of the first variable to be 1. This normalization requires care in choosing the order of the variables. There may be a cointegrating relation only between a subset of variables in a given system. Therefore, normalizing an arbitrary coefficient may result in dividing by an estimate corresponding to a parameter that is actually zero because the associated variable does not belong in the cointegrating relation. Rossi Cointegrated VAR s Financial Econometrics / 56

41 Model selection Determining the Autoregressive order Specifying the cointegrating rank Determining the lag order: 1 Sequential testing procedures 2 Model selection criteria (information criteria) Rossi Cointegrated VAR s Financial Econometrics / 56

42 Model selection Fit VAR(m) models with orders m = 0, 1,..., p max and choose an estimator that minimizes the preferred criterion. The general form of the criteria: Cr(m) = log Ω(m) + c T ϕ(m) Ω(m) = T 1 the residual covariance matrix estimator for a model of order m, c T is a sequence that depends on T, ϕ(m) is a function that penalizes large m. The term log Ω(m) measures the fit of the model with order m. This decreases when m increases. T t=1 ɛ t ɛ t Rossi Cointegrated VAR s Financial Econometrics / 56

43 Model selection AIC(m) = log Ω(m) + 2 T mn2 2 log log T HQ(m) = log Ω(m) + T SC(m) = log Ω(m) + log T T mn2 The AIC asymptotically overestimates the order with positive probability. HQ and SC estimate the order consistently under quite general conditions if the actual DGP has a finite VAR order and the maximum order is larger than the true order. These results not only hold for I (0) processes but also for I (1) processes with cointegrated variables (Paulsen (1984)). For T 16, the orders selected by the three criteria p(sc) p(hq) p(aic) mn 2 Rossi Cointegrated VAR s Financial Econometrics / 56

44 Specifying the cointegrating rank VECM without deterministic part: y t = Πy t 1 + Γ 1 y t Γ p 1 y t p+1 + ɛ t The cointegrating rank r has to be chosen in addition to the lag-order. Suppose we wish to test The LR test statistic H 0 : rank(π) = r 0 against H 1 : r 0 < rank(π) r 1 LR(r 0, r 1 ) = 2[L(r 1 ) L(r 0 )] [ = T = T r 1 i=1 r 1 i=r 0+1 log(1 λ i ) + log (1 λ i ) r 0 i=1 log (1 λ i ) ] Rossi Cointegrated VAR s Financial Econometrics / 56

45 Specifying the cointegrating rank The asymptotic distribution of the LR under the H 0 fr givern r 0 and r 1 is nonstandard. It is not a χ 2 -distribution. It depends on the number of common trends n r 0 under H 0 and on the alternative hypothesis. Two different pairs of hypothesis have received attention H 0 : rank(π) = r 0 against H 1 : r 0 < rank(π) N the LR statistics LR(r 0, N) is called the trace test statistic for testing the cointegrating rank and H 0 : rank(π) = r 0 against H 1 : r with LR(r 0, r 0 + 1) is called the maximum eigenvalue statistic. Rossi Cointegrated VAR s Financial Econometrics / 56

46 Specifying the cointegrating rank Sequential procedures based on LR-type tests. The sequence of hypotheses: H 0 (0) : rk(π) = 0 vs H 1 (0) : rk(π) > 0 H 0 (1) : rk(π) = 1 vs H 1 (1) : rk(π) > 1... H 0 (n 1) : rk(π) = n 1 vs H 1 (n 1) : rk(π) = n The testing procedures terminates when the null hypothesis cannot be rejected for the first time. Rossi Cointegrated VAR s Financial Econometrics / 56

47 Specifying the cointegrating rank If H 0 (0) cannot be rejected a VAR in first differences is considered. If H 0 (n 1) cannot be rejected a levels VAR should be considered. Under Gaussian assumptions the LR statistic under H 0 (r 0 ) is nonstandard. It depends on the difference n r 0. The deterministic trend terms and shift dummy variables in the DGP have an impact on the distribution of the LR test statistic under the null. LR-type tests have been derived under different assumptions regarding the deterministic trend. On the assumption that the lag order is specified correctly, the limiting null distributions do not depend on the short-term dynamics. Rossi Cointegrated VAR s Financial Econometrics / 56