206 The Basics of financial economeTrics
The Johansen-Juselius cointegration test^13 avoids these problems. Con-
sider the following multivariate model:
yt = Ayt− 1 + ut (10.8)
where yt = an n × 1 vector (y 1 t, y 2 t,... , ynt) of variables
ut = an n-dimensional error term at t
A = an n × n matrix of coefficients
If the variables display a time trend, we may wish to add the matrix
A 0 to equation (10.8). This matrix would reflect a deterministic time trend.
(The same applies to equation (10.9) presented below.) It does not change
the nature of our analysis.
The model (without the deterministic time trend) can then be repre-
sented as:
Δyt = B yt− 1 + ut (10.9)
where B = I − A, and I is the identity matrix of dimension n variables
The cointegration of the system is determined by the rank of B matrix. The
highest rank of B that can be obtained is n, the number of variables under
consideration. If B is zero, that means that there are no linear combinations
of yt that are stationary and so there are no cointegrating vectors.
If the rank of B is n, then each yit is an autoregressive process. This
means each yit is stationary and the relationship can be tested using a vector
autoregression model, which we cover in Chapter 9. For any rank between
1 and n − 1, the system is cointegrated and the rank of the matrix is the
number of cointegrating vectors.
The Johansen-Juselius cointegration test employs two statistics to test
for cointegration:
- λ trace test statistic
- maximum eigenvalue test
The λ trace test statistic verifies the null hypothesis that there are no
cointegration relations. The alternative hypothesis is that there is at least one
(^13) Soren Johansen and Katarina Juselius, “Maximum Likelihood Estimation and
Inference on Cointegration with Application to the Demand for Money,” Oxford
Bulletin of Economics and Statistics 52 (1990): 169–209.