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Estimation of Vector Error Correction Model

The preceding log-likelihood function is maximized for

The estimators of the orthogonal complements of and are

and

The ML estimators have the following asymptotic properties:

where

and

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 ecm=(rank=1 normalize=y1) noint;

For fitting Case 2,

   model y1 y2 / p=2 ecm=(rank=1 normalize=y1 ectrend);

For fitting Case 3,

   model y1 y2 / p=2 ecm=(rank=1 normalize=y1);

For fitting Case 4,

   model y1 y2 / p=2 ecm=(rank=1 normalize=y1 ectrend)
                 trend=linear;

For fitting Case 5,

   model y1 y2 / p=2 ecm=(rank=1 normalize=y1) trend=linear;

From Figure 32.53 that uses the COINTTEST=(JOHANSEN) option, you can fit the model by using either Case 2 or Case 3 because the test was not significant at the 0.05 level, but was significant at the 0.10 level. Here both models are fitted to show the difference in output display. Figure 32.56 is for Case 2, and Figure 32.57 is for Case 3.

For Case 2,

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

Test for the Linear Restriction on the Parameters

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 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 ecm=(rank=1 normalize=y1);
   cointeg rank=1 h=(1,-2);
run;

Figure 32.58 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.

Test for the Weak Exogeneity and Restrictions of Alpha

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 is a matrix with .

From the previous residual regression equation

you can obtain

where and is orthogonal to 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 largest eigenvalues. The likelihood ratio test for is

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 ecm=(rank=1 normalize=y1);
   cointeg rank=1 exogeneity;
run;

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

Figure 32.59 shows that each variable is not the weak exogeneity of other variable.

Forecasting of the VECM

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

Assume that . The linear process can be written as

Therefore, for any ,

The -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() 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 -step-ahead forecast is computed as

Cointegration with Exogenous Variables

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 ecm=(rank=1);
   run;

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

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

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