It is easily seen that in general a linear combination of y t and x t is I(1). However, in particular cases, it can be I(0), i.e. stationary.

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1 6. COINTEGRATION 1

2 1 Cointegration 1.1 Definitions I(1) variables. z t = (y t x t ) is I(1) (integrated of order 1) if it is not stationary but its first difference z t is stationary. It is easily seen that in general a linear combination of y t and x t is I(1). However, in particular cases, it can be I(0), i.e. stationary. Example 1: Common trends Consider for instance the case y t x t = T t + c t = βt t + e t where T t is a random walk process and c t, e t are co-stationary. y t ax t is I(1) for a 1/β but is stationary for for a = 1/β. Cointegration. y t and x t are cointegrated of order 1 if and only if z t = (y t x t ) is I(1) and there is a linear combination y t ax t which is stationary. The vector 2

3 α = (1 a) such that α z t is stationary is called the cointegrating vector. Examples are: money-prices, consumption-gdp labor productivity- real wages. Example 2. Consider the following system Y 1t Y 2t = γy 2t + u 1t = γy 2t 1 + u 2t where u 1t, u 2t are WN. It easy to see that both Y 1t, Y 2t are I(1) but the linear combination Y 1t γy 2t is I(0). Y 1t, Y 2t are cointegrated. Cointegration of n variables. The variables in Y t = (Y 1t,..., Y nt ) are cointegrated if they are jointly I(1) and there exist a non-zero vector α such that α Y t I(0). 3

4 1.2 Beveridge-Nelson decomposition Consider a difference stationary process (I L)Y t = C(L)ε t. Any matrix of polynomials in the lag operator C(L) can be written as C(L) = C(1) + (1 L)C (L) where C (L) = C(L) C(1) 1 L. Therefore we can write Therefore (1 L)Y t = C(1)ε t + (1 L)C (L)ε t (1 L)Y t = (1 L)Z t + (1 L)W t where Z t Z t 1 = C(1)ε t and W t = C (L)ε t and Y t = Z t + W t 4

5 1.3 Implications of cointegration Notice that from the above decomposition W t is stationary while Z t has a unit root. If the element of Y t are cointegrated then a j Y t has to be stationary. Let α = be the matrix formed by the h cointegrating vectors. By definition a 1 a 2. a h α Y t = α Z t + α W t where the term α W t is stationary while α Z t is either nonstationary or constant. Given that α Y t is stationary this imples that α C(1) = 0. This means that C(z) = 0 for z=1. That is the MA representation has a root at unity so that the model is not invertible and a finite order VAR is misspecified. 5

6 Example 2 cont d. Considering again the processes described in Example 2, by taking the differences we have Y 1t Y 2t = γu 2t + u 1t u 1t 1 = γu 2t Let ε 1t = u 2t and ε 2t = γu 2t + u 1t be the two forecast errors. Then Y t = C(L)ε t where ( ) 1 L γl C(L) = 0 1 The problem is that the matrix associated with the moving average operator for this process C(z) has a root at unity ( ) C(1) = 1 1 γ1 = implying that the MA representation is not invertible implying that no finite order VAR can describe Y t. 6

7 1.4 Common trend representation The Beveridge and Nelson representation for cointegrated variables is therefore very particular. Consider the case n = 2 and the cointegrating vector α = (1 a). By defining Z t = c 11 ε 1t + c 12 ε 2t we can write ( ) ( ) Y1t Zt = + (1 L)C (L)ε t Y 2t a Z ( ) ( ) t Y1t Z = + C (L)ε t Y 2t az t This representation is called the common trend representation, since the trends are collinear. We have seen that if two variables are cointegrated, they have a common trend representation. The converse is also true, as it is easily seen by observing that,if there is a common trend, there must be a linear combination such that the trends cancel out and only the stationary terms survive. In conclusion, we have the following result: x t and y t are cointegrated if and only if they have a common trend representation. 7

8 2 Error correction mechanism A well-known model in time series macroeconometrics is the so called Error- Correction Mechanism (ECM): A(L) Y t = γs t 1 + ε t (1) where ε t is stationary and s t = α Y t is the error and γ is the corrrection vector. The interpretation is easier when A(L) = I. In this case, we see that, if the entries of Y t are not in equilibrium, they move in the opposite direction (the minus sign in front of γ) with respect to the equilibrium deviation s t. It is easily seen that, if Y t is I(1) and (1) holds, then the variables in Y t are cointegrated, since s t = α Y t must be stationary. Granger s representation theorem shows that the converse is also true, i.e. if the variables in Y t are cointegrated they have an ECM representation (Engle and Granger, 1987). 8

9 2.1 Granger representation theorem Granger s Theorem The entries of the I (1) vector Y t are cointegrated if and only if they have an ECM representation. Let us show the only if part of the result. Premultiply the Wold representation (2) by A(L), the adjugate matrix of C(L), to get A(L) Y t = detc(l)ε t = (1 L)d(L)ε t (for simplicity we assume a zero constant term). Integrating both sides we get A(L)Y t = g + d(l)ε t where g is a vector of constants. Hence there is a VAR representation in levels, with a stationary residual and if if d(l) is invertible, there is a representation with a white noise residual. Now let us ignore g for simplicity and set w t = d(l)ε t, A(L) = (A(L) A(1)L)/(1 L)). We have A(L) Y t = A(1)Y t 1 + w t Now, A(1) is singular, since it is the adjugate of C(1), which is singular. Hence the columns are collinear and we can write A(1) = γα - 9

10 In addition α must be a cointegrating vector, since the first term on the RHS must be stationary. Hence we have the ECM representation where s t = α Y t 1. A(L) Y t = γs t 1 + w t 10

11 2.2 Solutions We have seen that if Y 1t and Y 2t are cointegrated, then the Wold MA matrix of Y t has a unit root. This implies that the MA representation is not invertible and a VAR representation for Y t does not exist. As a consequence, if we specify a VAR for the first differences of cointegrated variables we have estimation problems. Hence, before estimating a VAR for Y t we should test for cointegration and reject cointegration. On the other hand, if we find that the variables are cointegrated, we can either estimate an ECM or a VAR for Y t in levels. Indeed, in the proof of Granger s representation theorem we have seen that a VAR in levels does exist. Moreover it can be shown that the OLS estimator is super-consistent. 11

12 2.3 Engle and Granger two-step testing procedure The simplest way to test for cointegration was suggested in Engle and Granger (1987): 1. estimate by OLS the regression equation x t = b + ay t + e t 2. take the estimated residuals ê t and test for stationarity by using the ADF test. An alternative procedure, which is the most used one, particularly in the case n > 2, is Johansen s procedure. Start from OLS estimation of and test for the rank of the matrix Π. N(L) Y t = θ + ΠY t 1 + w t 12

13 2.4 Johansen s trace test More precisely, there are two tests, both based on the eigenvalues λ 1 > λ 1 >... > λ n > 0, related to the matrix Π. The former is the so called trace test, based on the sequence of statistics n η r = T log(1 λ i ),, r = 0, 1,..., n 1 i=r+1 The null here is that the smallest n r eigenvalues are all zero, which implies rank r for Π, i.e. cointegration rank r (r independent cointegrating vectors). The alternative is that there are more than r cointegrating vectors. The procedure starts from H 0 : r = 0 (no cointegration) against H 1 : r > 0 (at least one cointegrating vector). If H 0 is not significant, the null of no cointegration is not rejected. If, by contrast, η 0 is significant, we test H 0 : r = 1 against H 1 : r > 1,..., H 0 : r = n 1 against H 1 : r = n (joint stationarity). If the last significant statistic 13

14 is η r, the number of cointegrating vector selected is r Johansen s maximum eigenvalue test The second test is the so called maximum eigenvalue test, based on the statistic ζ r = T log(1 λ r+1 ), r = 0, 1,..., n 1 The null hypotheses, the alternative and the whole procedure are the same as before. If ζ r is the last significant statistic, the number of cointegrating vector selected is r + 1. Both tests have non-standard distributions which have been tabulated by Monte Carlo simulations. 14

15 3 VAR in levels An alternative is to use always variables in levels and completely forget about cointegration. As seen earlier the VAR in leevels always exists. Sims, Stock and Watson (1991) has shown that OLS is consistent also with nonstationary variables. Standard testing procedures however cannot applied since statistics have non standard ditributions. Nonetheless impulse response functions analysis and forecast error variance decomposition can be performed using bootstraped confidence bands. 15