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The VARMAX Procedure


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




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
      cn = {'y1' 'y2'};
      create simul2 from y[colname=cn];
      append from y;
   data simul2;
      set simul2;
      date = intnx( 'year', '01jan1900'd, _n_-1 );
      format date year4. ;
   proc varmax data=simul2;
      model y1 y2 / p=2 cointtest=(sw);

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 .

Figure 30.51 Common Trends Test (COINTTEST=(SW) Option)
The VARMAX Procedure

Common Trend Test
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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