VARMAX Procedure

Engle and Granger (1987) define

and the following assumptions hold:

Then has the representation

where the Granger representation coefficient, C, is

where the full-rank matrix is orthogonal to and the full-rank matrix is orthogonal to . is an process, and depends on the initial values.

The Granger representation coefficient C can be defined only when the matrix is invertible.

One motivation for the VECM(p) form is to consider the relation as defining the underlying economic relations. Assume that agents react to the disequilibrium error through the adjustment coefficient to restore equilibrium. The cointegrating vector, , is sometimes called the long-run parameter.

Consider a vector error correction model that has a deterministic term, , which can contain a constant, a linear trend, and seasonal dummy variables. Exogenous variables can also be included in the model. The model has the form

where .

The alternative vector error correction representation considers the error correction term at lag and is written as

If the matrix has a full rank (), all components of are . On the other hand, are stationary in difference if . When the rank of the matrix is , there are linear combinations that are nonstationary and r stationary cointegrating relations. Note that the linearly independent vector is stationary and this transformation is not unique unless . There does not exist a unique cointegrating matrix because the coefficient matrix can also be decomposed as

where M is an nonsingular matrix.

The cointegration rank test determines the linearly independent columns of . Johansen and Juselius proposed the cointegration rank test by using the reduced rank regression (Johansen 1988, 1995b; Johansen and Juselius 1990).

The preceding log-likelihood function is maximized for

The estimators of the orthogonal complements of and are

and

Let denote the parameter vector . The covariance of parameter estimates is obtained as the inverse of the negative Hessian matrix . Because , the variance of and the covariance between and are calculated as follows:

For Case 2 (Case 4), because the coefficient vector () for the constant term (the linear trend term) is the product of and (), the variance of () and the covariance between () and are calculated as follows:

The following statements are examples of fitting the five different cases of the vector error correction models mentioned in the previous section.

For fitting Case 1,

model y1 y2 / p=2 noint;
cointeg rank=1 normalize=y1;

For fitting Case 2,

model y1 y2 / p=2;
cointeg rank=1 normalize=y1 ectrend;

For fitting Case 3,

model y1 y2 / p=2;
cointeg rank=1 normalize=y1;

For fitting Case 4,

model y1 y2 / p=2 trend=linear;
cointeg rank=1 normalize=y1 ectrend;

For fitting Case 5,

model y1 y2 / p=2 trend=linear;
cointeg rank=1 normalize=y1;

In the previous example, the output from the COINTTEST=(JOHANSEN) option shown in Figure 73 indicates that you can fit the model by using either Case 2 or Case 3 because the test of the restriction was not significant at the 0.05 level, but was significant at the 0.10 level. Following both models are fit to show the differences in the displayed output. Figure 76 is for Case 2, and Figure 77 is for Case 3.

For Case 2,

proc varmax data=simul2;
   model y1 y2 / p=2 print=(estimates);
   cointeg rank=1 normalize=y1 ectrend;
run;

Figure 76 can be reported as follows:

The keyword "EC" in the "Model Parameter Estimates" table means that the ECTREND option is used for fitting the model.

For fitting Case 3,

proc varmax data=simul2;
   model y1 y2 / p=2 print=(estimates);
   cointeg rank=1 normalize=y1;
run;

Figure 77 can be reported as follows:

Consider the example with the variables log real money, log real income, deposit interest rate, and bond interest rate. It seems a natural hypothesis that in the long-run relation, money and income have equal coefficients with opposite signs. This can be formulated as the hypothesis that the cointegrated relation contains only and through . For the analysis, you can express these restrictions in the parameterization of such that , where is a known matrix and is the parameter matrix to be estimated. For this example, is given by

Restriction

When the linear restriction is given, it implies that the same restrictions are imposed on all cointegrating vectors. You obtain the maximum likelihood estimator of by reduced rank regression of on corrected for , solving the following equation,

for the eigenvalues and eigenvectors , given in the preceding section. Then choose that corresponds to the r largest eigenvalues, and the is .

The test statistic for is given by

If the series has no deterministic trend, the constant term should be restricted by as in Case 2. Then is given by

The following statements test that 2 :

proc varmax data=simul2;
   model y1 y2 / p=2;
   cointeg rank=1 h=(1,-2) normalize=y1;
run;

Figure 78 shows the results of testing . The input matrix is . The adjustment coefficient is reestimated under the restriction, and the test indicates that you cannot reject the null hypothesis.

Consider a vector error correction model:

Divide the process into with dimension and and the into

Similarly, the parameters can be decomposed as follows:

Then the VECM(p) form can be rewritten by using the decomposed parameters and processes:

The conditional model for given is

and the marginal model of is

where .

The test of weak exogeneity of for the parameters determines whether . Weak exogeneity means that there is no information about in the marginal model or that the variables do not react to a disequilibrium.

Restriction

Consider the null hypothesis , where J is a matrix with .

From the previous residual regression equation

you can obtain

where and is orthogonal to J such that .

Define

and let . Then can be written as

Using the marginal distribution of and the conditional distribution of , the new residuals are computed as

where

In terms of and , the MLE of is computed by using the reduced rank regression. Let

Under the null hypothesis , the MLE is computed by solving the equation

Then , where the eigenvectors correspond to the r largest eigenvalues and are normalized such that ; . The likelihood ratio test for is

For more information, see Theorem 6.1 in Johansen and Juselius (1990).

The test of weak exogeneity of is a special case of the test , considering . Consider the previous example with four variables ( ). If , you formulate the weak exogeneity of () for as and the weak exogeneity of for () as .

The following statements test the weak exogeneity of other variables, assuming :

proc varmax data=simul2;
   model y1 y2 / p=2;
   cointeg rank=1 exogeneity normalize=y1;
run;

Figure 79 shows that each variable is not the weak exogeneity of other variable.

The previous sections discuss some special forms of tests on and , namely the long-run relations that are expressed in the form , the weak exogeneity test, and the null hypotheses on in the form . In fact, with the help of the RESRICT and BOUND statements, you can estimate the models that have linear restrictions on any parameters to be estimated, which means that you can implement the likelihood ratio (LR) test for any linear relationship between the parameters.

The restricted error correction model must be estimated through numerical optimization. You might need to use the NLOPTIONS statement to try different options for the optimizer and the INITIAL statement to try different starting points. This is essentially important because the and are usually not identifiable.

You can also use the TEST statement to apply the Wald test for any linear relationships between parameters that are not long-run. Even more, you can test the constraints on and in Case 2 or in Case 4 when the constant term or linear trend is restricted to the error correction term.

For more information and examples, see the section Analysis of Restricted Cointegrated Systems.

Consider the cointegrated moving-average representation of the differenced process of

Assume that . The linear process can be written as

Therefore, for any ,

The l-step-ahead forecast is derived from the preceding equation:

Note that

since and . The long-run forecast of the cointegrated system shows that the cointegrated relationship holds, although there might exist some deviations from the equilibrium status in the short-run. The covariance matrix of the predict error is

When the linear process is represented as a VECM(p) model, you can obtain

The transition equation is defined as

where is a state vector and the transition matrix is

where 0 is a zero matrix. The observation equation can be written

where .

The l-step-ahead forecast is computed as

The error correction model with exogenous variables can be written as follows:

The following statements demonstrate how to fit VECMX(), where and from the P=2 and XLAG=1 options:

proc varmax data=simul3;
   model y1 y2 = x1 / p=2 xlag=1;
   cointeg rank=1;
run;

The following statements demonstrate how to BVECMX(2,1):

proc varmax data=simul3;
   model y1 y2 = x1 / p=2 xlag=1
                      prior=(lambda=0.9 theta=0.1);
   cointeg rank=1;
run;