ECON 4160, 2016. Lecture 11 and 12 Co-integration Ragnar Nymoen Department of Economics 9 November 2017 1 / 43
Introduction I So far we have considered: Stationary VAR ( no unit roots ) Standard inference Non-stationary VAR ( all unit-root ) The spurious regression example with two independent random walks is an example of a non-stationary VAR Have discovered that the Dickey-Fuller distribution can be used to test the null hypothesis of unit-root for a single time series We next consider cointegration, the case of some, but not only unit-roots in the VAR. 2 / 43
Introduction II In such systems, there exist one or more linear combinations of I (1) variables that are I (0) they are called cointegration relationships. Cointegration is the flip of the coin of spurious regression: If we have two dependent I (1) variables, they are cointegrated. We can guess that the right distribution to use for testing cointegration is going to be of the Dickey-Fuller type: Intuition: If we reject Y t I (1) against Y t I (0) using critical values from DF-distributions, we have shown that Y t is cointegrated with itself. To test whether Y t is cointegrated with another variable, X t I (1), as similar distribution will be needed. 3 / 43
Introduction III In these two lectures we sketch the theory of cointegration more fully: Granger s representation theorem. This shows the implications of at least one unit root in the VAR, and the rest < 1. Leads to the concept of the cointegrated VAR, and to ECM model equations as an implication of cointegration Methods for testing the null-hypothesis of no cointegration Testing residuals form static cointegrating regression for unit-root Using the conditional ECM model equation to test for unit-root (Ericsson and MacKinnon article in particular). 4 / 43
Introduction IV These two methods are single equation methods, and still the most used tests in practice. A multi-equation method uses the VAR directly. It is often called testing for cointegration rank (aka the Johansen method). Note about HN: They present an example of the VAR method first, in Chapter 17.3, and the single equation test in Chapter 17.4 Important take-away: After concluding the test for absence of cointegration, we are back in the stationary case: Can estimate cointegrated VARs, and simultaneous equations models in the way we have learned for stationary systems. Or, if there are no evidence of cointegration: formulate VARs and models that only include differences of the I (1) series. 5 / 43
Introduction V References: HN: Ch. 17. BN(2104): Kap. 11. The posted paper by Ericsson and MacKinnon (to Lecture 9). 6 / 43
The VAR with one unstable and one stable root I Consider the bi-variate first order VAR y t = Φy t 1 + ε t (1) where y t = (Y t, X t ), Φ is a 2 2 matrix with coeffi cients and ε t is a vector with Gaussian disturbances. The characteristic equation for Φ: Φ λi = 0, Our interest is the case with one unit-root and one stationary root: λ 1 = 1, and λ 2 = λ, λ < 1. (2) implying that both X t and Y t are I (1). 7 / 43
The VAR with one unstable and one stable root II Φ has full rank, equal to 2. It can be diagonalized in terms of its eigenvalues and the corresponding eigenvectors: [ 1 0 Φ = P 0 λ ] P 1 (3) P has the eigenvectors as columns: [ a b P = c d ] (4) Set P = 1 without loss of generality, then: [ P 1 d b = c a ]. 8 / 43
Cointegrated VAR ECM implication I Multiply from the right on both sides of (1) by P 1. The VAR is then re-written as: [ ] ] [ Wt Wt 1 EC t = [ 1 0 0 λ EC t 1 where η t = P 1 ε t,and EC t and W t are defined as: ] + η t, (5) W t = dy t bx t, (6) EC t = cy t + ax t. (7) EC t I (0), is a stationary variable, the Equilibrium Correction variable. Why? 9 / 43
Cointegrated VAR ECM implication II W t I (1), is a common (stochastic) trend, i.e.y t and X t contain the same trend We say that there is cointegration between X t and Y t, since EC t is a stationary variable, and it is a linear combination of X t and Y t. c and a are the cointegrating parameters in this example. 10 / 43
The ECM representation of the cointegrated VAR I Re-parameterize the VAR (1) as y t = Πy t 1 + ε t (8) where we use the notation (in compliance with the literature): Π = (Φ I) (9) Next, define two (2 1) parameter vectors α and β in such a way that the product αβ gives Π: Π=αβ (10) In our example (compare BN 2014 Kap 11): [ ] (1 λ)b [ ] Π= c a (1 λ)d }{{}}{{} β α 11 / 43
The ECM representation of the cointegrated VAR II and then (8) can be expressed as: [ Yt X t ] = αβ [ Yt 1 X t 1 ] + ε t, (11) α is the matrix with equilibrium correction coeffi cients ( aka adjustment coeffi cients, or loadings), α = [ (1 λ)b (1 λ)d β is the matrix with the cointegration coeffi cients ] (12) 12 / 43
The ECM representation of the cointegrated VAR III In this formulation we see that [ ] c β = a rank(π) = 0, reduced rank and no cointegration. Both eigenvalues are zero. rank(π) = 1, reduced rank and cointegration. One eigenvalue is different from zero. rank(π) = 2, full rank, both eigenvalues are different from zero and the VAR in (1) is stationary. (13) Cointegration and Granger causality Since λ < 1 is equivalent with cointegration, we see from (12) that cointegration implies Granger-causality in at least one direction: (1 λ)b = 0 and/or (1 λ)b = 0. 13 / 43
The ECM representation of the cointegrated VAR IV Cointegration and weak exogeneity Assume d = 0, from (12). This implies [ ] [ ] Yt b = (1 λ) [cy X t 0 t 1 ax t 1 ] + ε t [ ] [ ] Yt (1 λ)b[cyt 1 ax = t 1 ] + ε y,t X t ε x,t The marginal model contains no information about the cointegration parameters (c, a). Y t is weakly exogenous for the cointegration parameters β = (c, a). So how can we test for WE of X t with respect to β? 14 / 43
Generalization of ECM VAR(p) > ECM general case I If y t is n 1 with I (1) variables. The VAR is: y t = ( p i=0 where ε t is multivariate Gaussian. The generalization of the ECM form (8): Φ i+1 L i )y t 1 + ε t (14) y t = Φ (L) y t 1 + Πy t 1 + ε t (15) where Φ (L) contains known expressions of Φ i+1, and Π = ( p i=0 Φ i+1 I n ). (16) 15 / 43
Generalization of ECM VAR(p) > ECM general case II In the case of no cointegration, Π is the null matrix: Π = 0 (17) but, it is: Π = αβ (18) in the case of r cointegrating vectors. β n r contains the CI-vectors as columns, while α n r shows the strength of equilibrium correction in each of the equations for Y 1t, Y 2t,..., Y nt. In general rank(β) = r and rank(π) = r < n. 16 / 43
Generalization of ECM VAR(p) > ECM general case III If β is known, the system y t = Φ (L) y t 1 + α[β y] t 1 + ε t (19) contains only I (0) variables and conventional asymptotic inference applies. Moreover: If β is regarded as known, after first estimating β, conventional asymptotic inference also applies. (19) is then a stationary VAR, called the VAR-ECM or the cointegrated VAR. This system can be estimated and modelled as we have seen for the stationary case. 17 / 43
The role of the intercept and trend Restricted and unrestricted constant term I Usually we include separate Constants in each row of the VAR. We call them unrestricted constant terms. In the unit-root case the implication is that each Y jt contains a deterministic trend (think of a Random Walk with drift) But if the constants are restricted to be in the EC t 1 variables, there are no drifts and therefore no trend in the levels variables. When we test for cointegration, based on the VAR, there are different critical values for the two cases. 18 / 43
Conditional cointegeration and exogeous I(1) variables in the VAR Conditional cointegrated ECM I Consider the bivariate case: y t = (Y t, X t ) above. Assume that α 21 = 0, then X t is weakly exogenous for β. With Gaussian disturbances ε t = N(0, Ω), where Ω has elements ω ij,we can derive the conditional model for Y 1t : Y t = ω 21 ω 1 22 } {{ } β 0 [ Yt 1 X t + α 11β X t 1 a single equation ECM. If we write it as ] + ε 1t ω 21 ω 1 22 ε 2t }{{} ɛ t (20) Y 1t = β 0 X t + α 11 β 11 Y 1t 1 + α 11 β 12 X t 1 + ɛ t we see that Π=α 11 β 11 = 0, i.e., the Π matrix has full rank. 19 / 43
Conditional cointegeration and exogeous I(1) variables in the VAR Conditional cointegrated ECM II The common I (1)-trend is now the observable exogenous variable X t. Generalization: The VAR contains n 1 endogenous I (1) variables and n 2 non-modelled I (1) variables. (Often called an open system or VAR-EX). Cointegration is then consistent with: 0 < rank(π) n 1 20 / 43
Estimating a single cointegrating vector The cointegrating regression I When rank(π) = 1, the cointegration vector is unique (subject only to normalization). Without loss of generality we set n = 1 and write y t = (Y t, X ) as in a usual regression. The cointegration parameter β can be estimated by OLS on Y t = α + βx t + u t (21) where u t I (0) by assumption of cointegration. ( ˆβ β) = T t=1 X t u t T. (22) t=1 Xt 2 21 / 43
Estimating a single cointegrating vector The cointegrating regression II Since X t I (1), we are in a the same situation as with the first order AR case with autoregressive parameter equal to one (Lecture 10) In direct analogy, we need to multiply ( ˆβ β) by T obtain a non-degenerate asymptotic distribution: in order to T ( ˆβ β) = 1 T T t=1 X t u t, (23) 1 T 2 T t=1 Xt 2 = ( ˆβ β) converges to zero at rate T, instead of T as in the stationary case. This result is called the Engle-Granger super-consistency theorem. 22 / 43
Estimating a single cointegrating vector The cointegrating regression III Remember: This is based on r = 1 so the cointegration vector is unique if it exists. 23 / 43
Estimating a single cointegrating vector The distribution of the Engle-Granger estimator I It has been shown that the E-G estimator is not normally distributed asymptotically (it is so called mixed-normal). The same applies to the t value based on ˆβ: It does not have an asymptotic normal distribution (under the assumption of cointegration). This means that it is impractical to use the static regression for doing valid inference about β. (Test whether β = 1 for example). The drawback is even more severe in DGPs with higher order dynamics, due of residual autocorrelation as a results of dynamic misspecification. 24 / 43
Estimating a single cointegrating vector ECM estimator I The ECM represents a logical way of avoiding the bias due to dynamic mis-specification of the E-G cointegration regression. Because, if cointegration is valid, the ECM is implied (from the representation theorem). With n = 2, p = 1 and weak exogeneity of X t with respect to the cointegration parameter, we have seen that the cointegrated VAR can be re-written as a conditional model and a marginal model: Y t = β 0 X t + }{{} π Y t 1 + γ X }{{} t 1 + ɛ t (24) α 11 β 11 α 11 β 12 X t = ε 2t (25) 25 / 43
Estimating a single cointegrating vector ECM estimator II where β 0 is the regression coeffi cient, and ɛ t and ε 2t are uncorrelated normal variables. Normalization on Y t 1 by setting β 11 = 1, and defining β 12 = β, for comparison with the E-G estimator, gives Y t = β 0 X t α 11 (Y t 1 βx t 1 ) + ɛ t The ECM estimator ˆβ ECM, is obtained from OLS on (24) ˆβ ECM = ˆγ ˆπ (26) ˆβ ECM is consistent if both ˆγ and ˆπ are consistent. OLS chooses the ˆγ and ˆπ that give the best predictor (Y t 1 ˆβ ECM X t 1 of Y t. As T grows towards infinity, the true parameters γ, π and β will therefore be found. 26 / 43
Estimating a single cointegrating vector ECM estimator III In fact ˆβ ECM is super-consistent (as the E-G estimator is) ˆβ ECM has better small sample properties than the E-G estimator, since it is based on a well specified econometric model (avoids the second-order bias problem). Inference: The distributions of γ and π (under cointegration) can be shown to be so called mixed normal for large T. This means that variances are stochastic variables rather than fixed parameters. However, the OLS based t-values of γ and π are asymptotically N(0, 1). 27 / 43
Estimating a single cointegrating vector ECM estimator IV ˆβ ECM is also mixed normal, but t β ECM = ˆβ ECM / Var( ˆβ ECM ) N(0, 1) (27) T and Var( ˆβ ECM ) can be found by using the delta-method. The generalization to n 1 explanatory variables, intercept and dummies is also unproblematic. Remember:The effi ciency of the ECM estimator depends on the assumed weak exogeneity of X t. 28 / 43
Testing r=0 against r=1 Engle-Granger test The easiest approach is to use an ADF regression to the test the null-hypothesis of a unit-root in the residuals û t from the cointegrating regression (21). The motivation for the û t j terms is as before: to whiten the residuals of the ADF regression The DF critical values are shifted to the left as deterministic terms, and/or more I (1) variables in the regression are added. 29 / 43
Testing r=0 against r=1 The ECM test As we have seen, r = 0 corresponds to π = 0 in the ECM model in (24): Y t = β 0 X t + πy t 1 + γx t 1 + ɛ t It also comes as no surprise that the t-value t π have Dickey-Fuller like distributions under H 0 : π = 0. See Ericsson and MacKinnon (2002) for critical values. Type-1 error probability: Not much difference between E-G and ECM test. In general. Type-2 error probabilities are smaller for ECM (The two becomes equivalent when the ECM model equation obeys Common factor restrictions.) 30 / 43
The Johansen method Testing cointegrating rank, general case I For the vector y t consisting of n 1 variables, we have the Gaussian VAR(p) above: y t = Φ (L) y t 1 + Πy t 1 + ε t (28) We are interested in both the cointegrating case and the case with no cointegration 0 < rank(π) < n (29) rank(π) = 0. (30) 31 / 43
The Johansen method Testing cointegrating rank, general case II rank(π) is given by the number of non-zero eigenvalues of Π. But can we find the number of eigenvalues that are significantly different from zero? Fortunately, this problem has a solution. An eigenvalue of Π is a special kind of squared correlation coeffi cient known as a canonical correlation in multivariate statistics. This method has become known as the Johansen approach. It is likelihood based, see HN 17.3.2. Unlike the ECM method: The Johansen method does not require weak exogeneity of X with respect to cointegration parameters. 32 / 43
The Johansen method Intuition I For concreteness, consider n = 3 so r can be 0,1 or 2 r = 0 corresponds to Π = 0 From the representation theorem; with two unit-roots [ ] 1 0 Π=Φ I = P P 1 I = 0. 0 1 r = 1 corresponds to α 3 1 = 0 for a single cointegration vector β 1 3. β 1 3 y t 1 must then be I (0) and it must be a significant predictor of y t. The strength of the relationship can be estimated by the highest squared correlation coeffi cient, call it ˆρ 2 1,between y t 33 / 43
The Johansen method Intuition II and all the possible linear combinations of the variables in y t 1. If ˆρ 2 1 > 0 is statistically significant, we reject that r = 0. ˆρ 2 1 is the same as the highest eigenvalue of ˆΠ, and ˆβ 1 3 is the corresponding eigenvector. If r = 0 is rejected we can,continue, and test r = 1 against r = 2. If the second largest correlation coeffi cient ˆρ 2 2 is also significantly different from zero, we conclude that the number of cointegrating vectors is two. ˆβ 2 3 is the corresponding eigenvector It can be shown that, for the Gaussian VAR, ˆβ 1 3 and ˆβ 2 3 are ML estimates. 34 / 43
The Johansen method Trace-test and max-eigenvalue test I We order the canonical correlations from largest to smallest and construct the so called trace test: Trace-test = T 3 ln(1 ˆρ 2 i ), r = 0, 1, 2 (31) i=r +1 If ˆρ 2 1 is close to zero, then clearly Trace-test will be close to zero, and we we will not reject the H 0 of r = 0 against r 1. and so on for H 0 of r = 1 against r 2 Of course: to make this a formal testing procedure, we need the critical values from the distribution of the Trace-test for the sequence of null-hypotheses. 35 / 43
The Johansen method Trace-test and max-eigenvalue test II The distributions are non-standard, but at least the main cases are tabulated in PcGive. A closely related test is called the max-eigenvalue test,(but the trace test is today judged most reliable) If there is a single cointegrating vector, and there are n 1 weakly-exogenous variables, the Johansen method reduces to the testing and estimation based on a single ECM equation (with the remarks we have made above). 36 / 43
The Johansen method Constant and other deterministic trends I It matters a great deal whether the constant is restricted to be in the cointegrating space or not. The advise for data with visible drift in levels: include an deterministic trend as restricted together with an unrestricted constant. After rank determination, can test significance of the restricted trend with standard inference Shift in levels Include restricted step dummy and a free impulse dummy. Exogenous I(1) variables, see table and program by MacKinnon, Haug and Michelis (1999). 37 / 43
Identification of multiple cointegration vectors. I As we have seen, if n = 2, cointegration implies rank(π) = 1. There is one cointegration vector (β 11,β 12 ) which is uniquely identified after normalization. For example with β 11 = 1 the ECM variable becomes ECM 1t = Y 1t + β 12 Y 2t I (0). When n > 2, we can have rank(π) > 1, and in these cases the cointegrating vectors are not identified in Johansen s method. 38 / 43
Identification of multiple cointegration vectors. II Assume that is known (in practice, consistently estimated), and β is a n r cointegrating matrix: Π = αβ However for a r r non-singular matrix Θ: Π = αθθ 1 β = α Θ β Θ showing that β Θ is also a matrix with cointegrating vectors. This identification problem is equivalent to the identification problem in simultaneous equation models! 39 / 43
Identification of multiple cointegration vectors. III Assume rank(π) = 2 for a n = 3 VAR Y 1t + β 12 Y 2t + β 13 Y 3t = ECM 1t β 21 Y 1t Y 2t + β 13 Y 3t = ECM 2t By simply viewing these as a pair of simultaneous equations, we see that they are not identified on the order-condition. Exact identification requires for example 1 linear restrictions on each of the equations. For example β 13 = 0 and β 21 + β 13 = 0 will result in exact identification Identification Theory!!! If we include non-modelled (exogenous) variables in the cointegration relationship, the identification problem is again directly analogous to the simultaneous equation model. 40 / 43
I(0) variables in the VAR? A misunderstanding that sometimes occurs is that there can be no stationary variables in he cointegrating relationships. Consider for example: Y 1t + β 12 Y 2t + β 13 Y 3t + β 14 Y 4t = ecm 1t (32) β 21 Y 1t Y 2t + β 23 Y 3t + β 24 Y 4t = ecm 2t (33) If Y 1 is the log of real-wages, Y 2 productivity, Y 3 relative import prices, and Y 4 the rate of unemployment, then the first relationship may be a bargaining based wage and the second a mark-up equation. Y 4t I (0), most sensibly, but we want to estimate and test the theory β 14 = 0. Hence: specify the VAR with Y 4t included. 41 / 43
From I(1) to I(0) When the rank has been determined, we are back in the stationary-case. The distribution of the identified cointegration coeffi cients are mixed normal so that conventional asymptotic inference can be performed on this ˆβ. The determination of rank allows us to move from the I (1) VAR, to the cointegrated VAR that contains only I (0) variables Another name for this I (0) model is the vector equilibrium correction model, VECM. The VECM can be analyzed further, using the tools of the stationary VAR! Hence, cointegration analysis is an important step in the analysis, but just one step. 42 / 43
Some important additional references Johansen, S. (1995), Likelihood-Based Inference in Cointegrated Vector Auto-Regressive Models, Oxford University Press Juselius, K (2004) The Cointegrated VAR Model, Methodology and Applications, Oxford University Press MacKinnon, J., A. A. Haug and L. Michelis (1999) Numerical Distributions Functions of Likelihood Ratio Tests for Cointegration, with programs. 43 / 43