Cointegrated VAR s Eduardo Rossi University of Pavia November 2014 Rossi Cointegrated VAR s Fin. Econometrics - 2014 1 / 31
B-N decomposition Give a scalar polynomial α(z) = α 0 + α 1 z +... + α p z p α(z) = α(1) α(1) + α(z) = α(1) + α (z)(1 z) Given that α(z) α(1) = α 0 + α 1 z +... + α p z p (α 0 + α 1 +... + α p ) = α 1 (1 z) α 2 (1 z 2 )... α p (1 z p ) now since (1 z p ) = (1 +... + z p 1 )(1 z) α(z) α(1) = α 1 (1 z) α 2 (1 z)(1 + z)... α p (1 +... + z p 1 )(1 z) = (α 1 +... + α p )(1 z)... α p (1 z p ) = [(α 1 +... + α p ) + (α 2 +... + α p )z +... + α p z p 1 ](1 z) = α (z)(1 z) where α (z) = [(α 1 +... + α p ) + (α 2 +... + α p )z +... + α p z p 1 ] Rossi Cointegrated VAR s Fin. Econometrics - 2014 2 / 31
B-N decomposition Now, α(z) = α(1) + α (z)(1 z) = α(1)z α(1)z + α(1) + α (z)(1 z) = α(1)z + [α(1) + α (z)](1 z) = α(1)z + α (z)(1 z) with α (z) = α(1) + α (z) = α 0 [(α 2 +... + α p )z +... + α p z p 1 ] Rossi Cointegrated VAR s Fin. Econometrics - 2014 3 / 31
Cointegrated VAR s From VAR to VECM Φ(L)y t = ɛ t [Φ(1)L + Φ (L)(1 L)]y t = ɛ t Φ (L) y t = Φ(1)y t 1 + ɛ t with Φ (L) = I N [(Φ 2 +... + Φ p )L +... + Φ p L p 1 ] Rossi Cointegrated VAR s Fin. Econometrics - 2014 4 / 31
Cointegrated VAR s The VECM is a convenient model setup for cointegration analysis: y t = Πy t 1 + Γ 1 y t 1 +... + Γ p 1 y t p+1 + ɛ t Π = (I n Φ 1... Φ p ) = Φ(1) Γ i = (Φ i+1 +... + 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 Fin. Econometrics - 2014 5 / 31
Cointegrated VAR s From VAR(P) to VECM(p-1): substract y t 1 from both sides y t = Φ 1 y t 1 +... + Φ p y t p + ɛ t y t y t 1 = y t 1 + Φ 1 y t 1 +... + Φ p y t p + ɛ t y t = y t 1 + Φ 1 y t 1 +... + Φ p y t p +(Φ 2 + Φ 3 +... + Φ p )y t 1 (Φ 2 + Φ 3 +... + Φ p )y t 1 +(Φ 3 + Φ 4 +... + Φ p )y t 2 (Φ 3 + Φ 4 +... + Φ p )y t 2 +... + +Φ p y t p+1 Φ p y t p+1 + ɛ t Rossi Cointegrated VAR s Fin. Econometrics - 2014 6 / 31
Cointegrated VAR s y t = (I Φ 1... Φ p )y t 1 + Φ 2 y t 2 +... + Φ p y t p (Φ 2 + Φ 3 +... + Φ p )y t 1 +(Φ 3 + Φ 4 +... + Φ p )y t 2 (Φ 3 + Φ 4 +... + Φ p )y t 2 +... + +Φ p y t p+1 Φ p y t p+1 + ɛ t y t = Πy t 1 (Φ 2 + Φ 3 +... + Φ p )(y t 1 y t 2 ) (Φ 3 + Φ 4 +... + Φ p )(y t 2 y t 3 ) +... + Φ p (y t p+1 y t p ) + ɛ t Rossi Cointegrated VAR s Fin. Econometrics - 2014 7 / 31
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 Fin. Econometrics - 2014 8 / 31
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. M : m (m n) and if n = 0 M M = 0 M = I m Rossi Cointegrated VAR s Fin. Econometrics - 2014 9 / 31
Granger Representation Theorem Suppose p 1 y t = Π(y t 1 + ΥD t ) + Γ i y t i + Ψd t + ɛ t t = 1, 2,... i=1 p 1 [(1 L)I n HL Γ i (1 L)L i ]y t = ɛ t i=1 where y t = 0 for t 0, ɛ t V W N for t = 1, 2,... and ɛ t = 0 for t 0. The terms D t and d t are deterministic terms, like constant, trend, seasonal or intervention dummies. Define p 1 A(z) = (1 z)i n Πz Γ i (1 z)z i i=1 Rossi Cointegrated VAR s Fin. Econometrics - 2014 10 / 31
Granger Representation Theorem If the process {y t } has a unit root then C(1) = Π is singular, i.e. rk(π) = r < n then there exist (p r) matrices α and β of rank r such that Π = αβ. A necessary and sufficient condition that y t E( y t ) and β y t E(β y t ) can be given initial distributions such that they become I(0) is that det [ α ( ) ] p 1 I N Γ i β 0 α : n (n r) β : n (n r) i=1 In this case the solution of VECM has the MA representation y t = C t (ɛ i + Ψd i ) + C i (ɛ t i + Ψd t i + αυd t i ) + A i=1 i=0 where A depends on initial values, so that β A = 0. It follows that y t is a cointegrated I(1) process with cointegrating vectors β. Rossi Cointegrated VAR s Fin. Econometrics - 2014 11 / 31
Granger Representation Theorem The function C (z) satisfies (1 z)a(z) 1 = C(z) = C i z i = C + (1 z)c (z) converges for z 1 + δ for some δ > 0. The matrix C is defined by ( ) ] p 1 1 C = β [α I N Γ i β α. i=0 i=1 Rossi Cointegrated VAR s Fin. Econometrics - 2014 12 / 31
Granger Representation Theorem An immediate consequence of the GRT is that β y t = β C t (ɛ i + Ψd i ) + β C i (ɛ t i + Ψd t i + αυd t i ) + β A i=1 is stationary, since while i=0 β y t β C = 0 β A = 0 β C (L)(ɛ t + Ψd t + αυd 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 is singular. Rossi Cointegrated VAR s Fin. Econometrics - 2014 13 / 31
Granger Representation Theorem This result implies that y t and β y t are stationary around their mean, so that y t is a cointegrated I(1) process with r cointegrating vectors β and n r common stochastic trends. We can also write p 1 y t E[ y t ] = α(β y t 1 E[β y t 1 ]) + Γ i ( y t i E[ y t i ]) + ɛ t 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. t i=1 (ɛ i + Ψd i ) is multiplied by a matrix (C) of rank n r. There are n r stochastic trends driving the system. i=1 Rossi Cointegrated VAR s Fin. Econometrics - 2014 14 / 31
Granger Representation Theorem Note that d t cumulates in the process with a coefficient CΨ, but that D t does not because CαΥ = 0 A leading special case is the model with and D t = t d t = 1 which ensures that any linear combination of the components of y t is allowed to have a linear trend. Rossi Cointegrated VAR s Fin. Econometrics - 2014 15 / 31
Granger Representation Theorem One can interpret the matrix C as indicating how the common trends α i=1 t (ɛ i + Υd i ) 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 C (L) which die out over time. A long-run effect is given by u 1(Cɛ t ). Rossi Cointegrated VAR s Fin. Econometrics - 2014 16 / 31
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 Fin. Econometrics - 2014 17 / 31
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 j < 1, j i. 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 Fin. Econometrics - 2014 18 / 31
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 Fin. Econometrics - 2014 19 / 31
Cointegration rank p 1 y t = Πy t 1 + Γ i y t i + ɛ t rk(π) = n the system is stationary (y t I(0)), standard asymptotic theory of estimation applies to impulse response analysis. i=1 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 Fin. Econometrics - 2014 20 / 31
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 Fin. Econometrics - 2014 21 / 31
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 Fin. Econometrics - 2014 22 / 31
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 Fin. Econometrics - 2014 23 / 31
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 Fin. Econometrics - 2014 24 / 31
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 Fin. Econometrics - 2014 25 / 31
Specifying the cointegrating rank VECM without deterministic part: y t = Πy t 1 + Γ 1 y t 1 +... + Γ p 1 y t p+1 + ɛ t The two Johansen tests for cointegration are used to establish the rank of β, or in other words the number of cointegrating vectors. The cointegrating rank r has to be chosen in addition to the lag-order. Suppose we wish to test H 0 : rank(π) = r 0 against H 1 : r 0 < rank(π) r 1 under H 0 : rank(π) = r 0 there are n r 0 unit roots. The LR test statistic 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 Fin. Econometrics - 2014 26 / 31
Specifying the cointegrating rank Suppose that the eigenvalues λ i, i = 1,..., r are sorted from largest to smallest. When H 0 : rank(π) = n 1 this corresponds to the null hypothesis that the smallest eigenvalue is zero. The asymptotic distribution of the LR under the H 0 for given 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: Trace test statistic LR(r 0, n) for joint hypotheses: H 0 : rank(π) = r 0 against H 1 : r 0 < rank(π) n Maximum eigenvalue statistic LR(r 0, r 0 + 1) on individual eigenvalues H 0 : rank(π) = r 0 against H 1 : rank(π) = r 0 + 1 Rossi Cointegrated VAR s Fin. Econometrics - 2014 27 / 31
Specifying the cointegrating rank 1 If LR(r 0, n) > C n r we reject the null of n r 0 unit roots and conclude that there are fewer unit roots than assumed. 2 If LR(r 0, n) C n r we accept the hypothesis of at least n r 0 unit roots in the model, but conclude that there may be more. 3 Hence, the trace test does not give us the exact number of unit roots n r (or cointegration relations r). It only tells us whether n r < n r 0 (r r 0 ) when LR(r 0, n) > C n r, or alternatively n r n r 0 (r < r 0 ) when LR(r 0, n) C n r. Therefore, to estimate the value of r we have to perform a sequence of tests. The question is whether this sequence should be from top to bottom, i.e. {r = 0, n unit roots}, {r = 1, n 1 unit roots},..., {r = n, 0 unit roots} or the other way around. Rossi Cointegrated VAR s Fin. Econometrics - 2014 28 / 31
Specifying the cointegrating rank The reason why the top bottom procedure is asymptotically more correct is because the probability of incorrectly accepting r < r 0 is asymptotically zero, whereas the probability of incorrectly accepting r < r 0 in the bottom top procedure is generally greater than the chosen p-value. Rossi Cointegrated VAR s Fin. Econometrics - 2014 29 / 31
Specifying the cointegrating rank Sequential procedures based on LR-type tests. The sequence of hypotheses: H 0 (0) : rank(π) = 0 vs H 1 (0) : rank(π) > 0 H 0 (1) : rank(π) = 1 vs H 1 (1) : rank(π) > 1... H 0 (n 1) : rank(π) = n 1 vs H 1 (n 1) : rank(π) = n The testing procedures terminates when the null hypothesis cannot be rejected for the first time. Rossi Cointegrated VAR s Fin. Econometrics - 2014 30 / 31
Specifying the cointegrating rank If H 0 (0) cannot be rejected a VAR in first differences is considered. If H 0 (n 1) is 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 Fin. Econometrics - 2014 31 / 31