### 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 .

#### 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 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 (.

#### Test for the Common Trends

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 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)

The VARMAX Procedure

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.