Testing for cointegration

Let the (K × 1) vector yt be I(1) with 0 < r < K cointegration vectors if there

exists an (r × K) matrix B0 such that B0y_t∼ I(0). Testing for cointegration may be thought of as testing for the existence of long-run equilibrium among the elements of yt. Cointegration tests cover two situations: (1) There is at most one cointegration

vector and (2) there are possible r, where 0 < r < K cointegration vectors. Among many procedures, two approaches will be described here. The Engle and Granger [42] test related to the first situation and the Johansen [63] methodology for the second situation.

Engle and Granger’s two-step procedure

Engle and Granger [42], show that if there is a cointegration vector a simple two-step residual-based testing procedure can be employed to test for cointegration. In this case, a long-run equilibrium relationship components of yt can be estimated by

running y1t= βy2t+ ut, where y2t= (y2t, ..., ykt)0 is an ((K − 1) × 1) vector.

To test the null hypothesis that the yt is not cointegrated, we should test for

a unit root in the residuals described in subsection 2.1.4, ut ∼ I(1) vs ut ∼ I(0).

The hypothesis are H0 : ut = β0yt ∼ I(1) (no-cointegration) and H1 : ut = β0yt ∼

I(0) (cointegration). If the unit root hypothesis is rejected, the hypothesis of no-

cointegration is also rejected, thus, the cointegration tests are similar to the unit root tests, the differences are founded in their critical values.

Mackinnon [79], provides tables based on simulation accurately enough for all practical purposes. The potential problems with Engle - Granger procedure is that the cointegrating vector will not involve y1tcomponent. In this case the cointegrating

3 . 4 . V E C T O R E R RO R C O R R E C T I O N M O D E L

Estimation of the static model y1t = βy2t+ ut is equivalent to omitting the

short-term component from the VEC model. If this results in autocorrelation in the residuals, although the results will still hold asymptotically, it might create a severe bias in finite samples. Because of this, it makes sense to estimate the full dynamic model. Since all variables in the VEC are I(0), the model can be consistently estimated using the OLS method, this approach leads to a better performance as it does not push the short-term dynamic into residuals.

Johansen methodology

The Johansen [63] procedure is an alternative approach to test for cointegration which allows to avoid some drawbacks for straightforward test the r unit roots and avoids normalization problem existing in the Engle-Granger approach and test the number of cointegrating relations directly. Because the Johansen procedure is based on ML estimator, that is why, we need the time series to be multivariate normal.

The Johansen test procedure builds on the vector autoregression of order p, here in a dynamic form or VEC model, equation 3.28 updated.

∆y_t= Φ0+ Πyt−1+ p−1

X

i=1

Γi∆yt−i+ ut (3.29)

For yt nonstationary I(1) time series components, in order to get a stationary

error term ut, Πyt−1should also be stationary. If the VAR(p) process has unit roots

then the coefficient matrix Π has reduced rank, rank(Π) = r < K. In fact, testing for cointegration is equivalent to checking the rank of the matrix Π.

1) If Π has a full rank then all time series in yt are stationary;

2) If the rank of Π = 0, then there are no cointegrating relationships;

3) If 0 <rank(Π) = r < K, this implies that yt is I(1) with r linearly independent

cointegrating vectors and m = K − r non-stationary vectors.

Since Π has rank r there exist K × r matrices with rank(α) = rank(β) = r such that Π = αβ0 and β0yt is stationary, r is the number of cointegrating vector.

The VEC model becomes: ∆y_t= αβ0yt−1+ Γ1∆yt−1+ ... + Γp−1∆yt−p+1+ ut,

with β0yt−1∼ I(0).

Algorithm for Johansen procedure

b) Determine rank(Π) equal to number of cointegrating vectors; the ML estimate for β equals the matrix of eigenvectors corresponding to the r largest eigenvalue of K × K residual matrix;

c) If necessary, impose normalization and identifying restrictions on the cointegrat- ing vectors;

d) Given the normalized cointegrating vectors estimate the resulting cointegrated VEC model by ML.

In general, it is known that for a given r, the ML estimator of β defines the combination of yt−1 that yields the r largest canonical correlations of ∆yt with

yt−1 after correcting for lagged differences and deterministic variables when present.

From these idea Johansen [63], proposes two different likelihood ratio tests of the significance of these canonical correlations and there by the reduced rank of the Π matrix: the trace test and maximum eigenvalue test.

a) Trace statistic test. Since the rank of the long-run impact matrix Π gives the number of cointegrating relationships in yt, the Johansen likelihood ratio (LR)

statistic for determining the rank of Π are based on the estimated eigenvalues ˆλ1>

ˆ

λ2> ... > ˆλk of the matrix Π. The null hypothesis of r cointegrating vectors against

the alternative hypothesis of K cointegrating vectors, are H0(r0) : r ≤ r0 vs H1(r0) :

r > r0. The LR statistics called trace statistic is LRtrace(r0) = −TPki=r0+1log(1− ˆλ).

The trace statistic checks whether the smallest K −r0eigenvalues are statistically

different from zero. If rank(Π) = r0 then ˆλr0+1, ..., ˆλk should all be close to zero

and LRtrace(r0) should be small. If rank(Π) > r0 then some of ˆλr0+1, ..., ˆλk will

be nonzero, but less than 1 and LRtrace(r0) should be large. The asymptotic null

distribution of LRtrace(r0) in not chi-square but instead is a multivariate version of

the Dickey-Fuller unit root distribution which depends on the dimension K − r0 and

the specification of the deterministic terms.

b) Maximum eigenvalue statistic. The maximum eigenvalue test, tests the null hypothesis of r cointegrating vectors against the alternative hypothesis of r + 1 cointegrating vectors, that is H0(r0) : r = r0 vs H1(r0) : r = r0+ 1. The LR statistic,

called maximum eigenvalue statistic is given by LRmax(r0) = −T log(1 − ˆλr0+1).

The asymptotic null distribution of LRmax(r0) is a function of Brownian motion,

which depends on the dimension K − r0 and the specification of the deterministic

terms. Critical values for LRtrace(r0) and LRmax(r0) statistics are in Osterwald and

Lenum [94] for K − r0= 1, ..., 10.

The algorithm for sequential procedure to determining r cointegrating vectors are:

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a) Test H0: r0= 0 vs H1: r0> 0. If we N RH0 then, there are no cointegrating

vectors among K variables in yt. If we RH0 then, there is at least 1 cointegrating

vector and proceeds to test.

b) Test H0: r0= 1 vs H1: r0> 1. If we N RH0 then, there is only one cointegrat-

ing vector. If we RH0 then, here is at least two cointegrating vectors. The sequence

is continued until we N RH0.