The VARMAX Procedure |

Cointegration |

This section briefly introduces the concepts of cointegration (Johansen 1995b).

**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 times, the series is integrated of order , 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 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 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 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 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 against rank .

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 30.51, 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 .

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.

Copyright © 2008 by SAS Institute Inc., Cary, NC, USA. All rights reserved.