This section briefly introduces the concepts of cointegration (Johansen 1995b).
(Engle and Granger 1987): If a series with no deterministic components can be represented by a stationary and invertible ARMA process after differencing times, the series is integrated of order , that is, .
(Engle and Granger 1987): If all elements of the vector are and there exists a cointegrating vector such that for any , the vector process is said to be cointegrated .
A simple example of a cointegrated process is the following bivariate system:






with and being uncorrelated white noise processes. In the second equation, is a random walk, , . Differencing the first equation results in

Thus, both and are processes, but the linear combination is stationary. Hence is cointegrated with a cointegrating vector .
In general, if the vector process has components, then there can be more than one cointegrating vector . It is assumed that there are linearly independent cointegrating vectors with , which make the matrix . The rank of matrix is , which is called the cointegration rank of .
This section briefly discusses the implication of cointegration for the movingaverage representation. Let be cointegrated , then has the Wold representation:

where is , with , and .
Assume that if and is a nonrandom initial value. Then the difference equation implies that

where and is absolutely summable.
Assume that the rank of is . When the process is cointegrated, there is a cointegrating matrix such that is stationary.
Premultiplying by results in

because and .
Stock and Watson (1988) showed that the cointegrated process has a common trends representation derived from the movingaverage representation. Since the rank of is , there is a matrix with rank such that . Let be a matrix with rank such that ; then has rank . The has rank . By construction of ,

where . Since and , lies in the column space of and can be written

where is a dimensional vector. The common trends representation is written as









and

where , , , and .
Stock and Watson showed that the common trends representation expresses as a linear combination of random walks () with drift plus components (.
Stock and Watson (1988) proposed statistics for common trends testing. The null hypothesis is that the dimensional time series has common stochastic trends, where and the alternative is that it has common trends, where . The test procedure of versus common stochastic trends is performed based on the firstorder serial correlation matrix of . Let be a matrix orthogonal to the cointegrating matrix such that and . Let and . Then

Combining the expression of and ,






The StockWatson common trends test is performed based on the component by testing whether has rank against rank .
The following statements perform the StockWatson test for common trends:
proc iml; sig = 100*i(2); phi = {0.2 0.1, 0.5 0.2, 0.8 0.7, 0.4 0.6}; call varmasim(y,phi) sigma=sig n=100 initial=0 seed=45876; cn = {'y1' 'y2'}; create simul2 from y[colname=cn]; append from y; quit; data simul2; set simul2; date = intnx( 'year', '01jan1900'd, _n_1 ); format date year4. ; run; proc varmax data=simul2; model y1 y2 / p=2 cointtest=(sw); run;
In Figure 36.51, the first column is the null hypothesis that has common trends; the second column is the alternative hypothesis that has common trends; the third column contains the eigenvalues used for the test statistics; the fourth column contains the test statistics using AR() filtering of the data. The table shows the output of the case .
Figure 36.51: Common Trends Test (COINTTEST=(SW) Option)
Common Trend Test  

H0: Rank=m 
H1: Rank=s 
Eigenvalue  Filter  5% Critical Value  Lag 
1  0  1.000906  0.09  14.10  2 
2  0  0.996763  0.32  8.80  
1  0.648908  35.11  23.00 
The test statistic for testing for 2 versus 1 common trends is more negative (–35.1) than the critical value (–23.0). Therefore, the test rejects the null hypothesis, which means that the series has a single common trend.