VARMAX Procedure
Cointegration
This section briefly introduces the concepts of cointegration (Johansen 1995a).
- Definition 1.
(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, 
. - Definition 2.
(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 .
Common Trends
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 
,
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 (
.
Test for the Common Trends
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 71, 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 
.
Figure 71: 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.