This section briefly introduces the concepts of cointegration (Johansen, 1995a).
(Engle and Granger, 1987): If a series with no deterministic components can be represented by a stationary and invertible ARMA process after differencing d times, the series is integrated of order d, 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 k components, then there can be more than one cointegrating vector . It is assumed that there are r linearly independent cointegrating vectors with , which make the matrix . The rank of matrix is r, which is called the cointegration rank of .
This section briefly discusses the implication of cointegration for the moving-average 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 moving-average representation. Since the rank of is , there is a matrix with rank r such that . Let be a matrix with rank m such that ; then has rank m. The has rank k. By construction of H,
where . Since and , lies in the column space of and can be written
where is a k-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 m random walks () with drift plus components (.
Stock and Watson (1988) proposed statistics for common trends testing. The null hypothesis is that the k-dimensional time series has m common stochastic trends, where and the alternative is that it has s common trends, where . The test procedure of m versus s common stochastic trends is performed based on the first-order 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 Stock-Watson common trends test is performed based on the component by testing whether has rank m against rank s.
The following statements perform the Stock-Watson 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 35.51, 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(p) filtering of the data. The table shows the output of the case .
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.