PROC VARMAX: Cointegration :: SAS/ETS(R) 9.22 User's Guide

The VARMAX Procedure

Cointegration

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

Definition 1.: (Engle and Granger 1987): If a series $\text{[math]}$ with no deterministic components can be represented by a stationary and invertible ARMA process after differencing $\text{[math]}$ times, the series is integrated of order $\text{[math]}$ , that is, $\text{[math]}$ .
Definition 2.: (Engle and Granger 1987): If all elements of the vector $\text{[math]}$ are $\text{[math]}$ and there exists a cointegrating vector $\text{[math]}$ such that $\text{[math]}$ for any $\text{[math]}$ , the vector process is said to be cointegrated $\text{[math]}$ .

A simple example of a cointegrated process is the following bivariate system:

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

with $\text{[math]}$ and $\text{[math]}$ being uncorrelated white noise processes. In the second equation, $\text{[math]}$ is a random walk, $\text{[math]}$ , $\text{[math]}$ . Differencing the first equation results in

$\text{[math]}$

Thus, both $\text{[math]}$ and $\text{[math]}$ are $\text{[math]}$ processes, but the linear combination $\text{[math]}$ is stationary. Hence $\text{[math]}$ is cointegrated with a cointegrating vector $\text{[math]}$ .

In general, if the vector process $\text{[math]}$ has $\text{[math]}$ components, then there can be more than one cointegrating vector $\text{[math]}$ . It is assumed that there are $\text{[math]}$ linearly independent cointegrating vectors with $\text{[math]}$ , which make the $\text{[math]}$ matrix $\text{[math]}$ . The rank of matrix $\text{[math]}$ is $\text{[math]}$ , which is called the cointegration rank of $\text{[math]}$ .

Common Trends

This section briefly discusses the implication of cointegration for the moving-average representation. Let $\text{[math]}$ be cointegrated $\text{[math]}$ , then $\text{[math]}$ has the Wold representation:

$\text{[math]}$

where $\text{[math]}$ is $\text{[math]}$ , $\text{[math]}$ with $\text{[math]}$ , and $\text{[math]}$ .

Assume that $\text{[math]}$ if $\text{[math]}$ and $\text{[math]}$ is a nonrandom initial value. Then the difference equation implies that

$\text{[math]}$

where $\text{[math]}$ and $\text{[math]}$ is absolutely summable.

Assume that the rank of $\text{[math]}$ is $\text{[math]}$ . When the process $\text{[math]}$ is cointegrated, there is a cointegrating $\text{[math]}$ matrix $\text{[math]}$ such that $\text{[math]}$ is stationary.

Premultiplying $\text{[math]}$ by $\text{[math]}$ results in

$\text{[math]}$

because $\text{[math]}$ and $\text{[math]}$ .

Stock and Watson (1988) showed that the cointegrated process $\text{[math]}$ has a common trends representation derived from the moving-average representation. Since the rank of $\text{[math]}$ is $\text{[math]}$ , there is a $\text{[math]}$ matrix $\text{[math]}$ with rank $\text{[math]}$ such that $\text{[math]}$ . Let $\text{[math]}$ be a $\text{[math]}$ matrix with rank $\text{[math]}$ such that $\text{[math]}$ ; then $\text{[math]}$ has rank $\text{[math]}$ . The $\text{[math]}$ has rank $\text{[math]}$ . By construction of $\text{[math]}$ ,

$\text{[math]}$

where $\text{[math]}$ . Since $\text{[math]}$ and $\text{[math]}$ , $\text{[math]}$ lies in the column space of $\text{[math]}$ and can be written

$\text{[math]}$

where $\text{[math]}$ is a $\text{[math]}$ -dimensional vector. The common trends representation is written as

$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

and

$\text{[math]}$

where $\text{[math]}$ , $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ .

Stock and Watson showed that the common trends representation expresses $\text{[math]}$ as a linear combination of $\text{[math]}$ random walks ( $\text{[math]}$ ) with drift $\text{[math]}$ plus $\text{[math]}$ components ( $\text{[math]}$ .

Test for the Common Trends

Stock and Watson (1988) proposed statistics for common trends testing. The null hypothesis is that the $\text{[math]}$ -dimensional time series $\text{[math]}$ has $\text{[math]}$ common stochastic trends, where $\text{[math]}$ and the alternative is that it has $\text{[math]}$ common trends, where $\text{[math]}$ . The test procedure of $\text{[math]}$ versus $\text{[math]}$ common stochastic trends is performed based on the first-order serial correlation matrix of $\text{[math]}$ . Let $\text{[math]}$ be a $\text{[math]}$ matrix orthogonal to the cointegrating matrix such that $\text{[math]}$ and $\text{[math]}$ . Let $\text{[math]}$ and $\text{[math]}$ . Then

$\text{[math]}$

Combining the expression of $\text{[math]}$ and $\text{[math]}$ ,

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

The Stock-Watson common trends test is performed based on the component $\text{[math]}$ by testing whether $\text{[math]}$ has rank $\text{[math]}$ against rank $\text{[math]}$ .

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 32.51, the first column is the null hypothesis that $\text{[math]}$ has $\text{[math]}$ common trends; the second column is the alternative hypothesis that $\text{[math]}$ has $\text{[math]}$ common trends; the third column contains the eigenvalues used for the test statistics; the fourth column contains the test statistics using AR( $\text{[math]}$ ) filtering of the data. The table shows the output of the case $\text{[math]}$ .

Figure 32.51 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.

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