Vector Error Correction (VEC) Models—749

Vector Error Correction (VEC) Models

A vector error correction (VEC) model is a restricted VAR designed for use with nonstationary series that are known to be cointegrated. The VEC has cointegration relations built into the specification so that it restricts the long-run behavior of the endogenous variables to converge to their cointegrating relationships while allowing for short-run adjustment dynamics. The cointegration term is known as the error correction term since the deviation from long-run equilibrium is corrected gradually through a series of partial short-run adjustments.

To take the simplest possible example, consider a two variable system with one cointegrating equation and no lagged difference terms. The cointegrating equation is:

In this simple model, the only right-hand side variable is the error correction term. In long run equilibrium, this term is zero. However, if y1 and y2 deviate from the long run equilibrium, the error correction term will be nonzero and each variable adjusts to partially restore the equilibrium relation. The coefficient αi measures the speed of adjustment of the i-th endogenous variable towards the equilibrium.

How to Estimate a VEC

As the VEC specification only applies to cointegrated series, you should first run the Johansen cointegration test as described above and determine the number of cointegrating relations. You will need to provide this information as part of the VEC specification.

To set up a VEC, click the Estimate button in the VAR toolbar and choose the Vector Error Correction specification from the VAR/VEC Specification tab. In the VAR/VEC Specification tab, you should provide the same information as for an unrestricted VAR, except that:

•The constant or linear trend term should not be included in the Exogenous Series edit box. The constant and trend specification for VECs should be specified in the Cointegration tab (see below).

•The lag interval specification refers to lags of the first difference terms in the VEC. For example, the lag specification “1 1” will include lagged first difference terms on the right-hand side of the VEC. Rewritten in levels, this VEC is a restricted VAR with two lags. To estimate a VEC with no lagged first difference terms, specify the lag as “0 0”.

750—Chapter 24. Vector Autoregression and Error Correction Models

•The constant and trend specification for VECs should be specified in the Cointegration tab. You must choose from one of the five trend specifications as explained in “Deterministic Trend Specification” on page 740. You must also specify the number of cointegrating relations in the appropriate edit field. This number should be a positive integer less than the number of endogenous variables in the VEC.

•If you want to impose restrictions on the cointegrating relations and/or the adjustment coefficients, use the Restrictions tab. “Restrictions on the Cointegrating Vector” on page 745 describes this restriction in greater detail. Note that this tab is grayed out unless you have clicked the Vector Error Correction specification in the

VAR/VEC Specification tab.

Once you have filled the dialog, simply click OK to estimate the VEC. Estimation of a VEC model is carried out in two steps. In the first step, we estimate the cointegrating relations from the Johansen procedure as used in the cointegration test. We then construct the error correction terms from the estimated cointegrating relations and estimate a VAR in first differences including the error correction terms as regressors.

VEC Estimation Output

The VEC estimation output consists of two parts. The first part reports the results from the first step Johansen procedure. If you did not impose restrictions, EViews will use a default normalization that identifies all cointegrating relations. This default normalization expresses the first r variables in the VEC as functions of the remaining k − r variables, where r is the number of cointegrating relations and k is the number of endogenous variables. Asymptotic standard errors (corrected for degrees of freedom) are reported for parameters that are identified under the restrictions. If you provided your own restrictions, standard errors will not be reported unless the restrictions identify all cointegrating vectors.

The second part of the output reports results from the second step VAR in first differences, including the error correction terms estimated from the first step. The error correction terms are denoted CointEq1, CointEq2, and so on in the output. This part of the output has the same format as the output from unrestricted VARs as explained in “VAR Estimation Output” on page 723, with one difference. At the bottom of the VEC output table, you will see two log likelihood values reported for the system. The first value, labeled Log Likelihood (d.f. adjusted), is computed using the determinant of the residual covariance matrix (reported as Determinant Residual Covariance), using small sample degrees of freedom correction as in (24.3). This is the log likelihood value reported for unrestricted VARs. The Log Likelihood value is computed using the residual covariance matrix without correcting for degrees of freedom. This log likelihood value is comparable to the one reported in the cointegration test output.

752—Chapter 24. Vector Autoregression and Error Correction Models

vectors β′ , and C holds the short-run parameters (the coefficients on the lagged first difference terms).

•The first index of A is the equation number of the VEC, while the second index is the number of the cointegrating equation. For example, A(2,1) is the adjustment coefficient of the first cointegrating equation in the second equation of the VEC.

•The first index of B is the number of the cointegrating equation, while the second index is the variable number in the cointegrating equation. For example, B(2,1) is the coefficient of the first variable in the second cointegrating equation. Note that this indexing scheme corresponds to the transpose of β .

•The first index of C is the equation number of the VEC, while the second index is the variable number of the first differenced regressor of the VEC. For example, C(2, 1) is the coefficient of the first differenced regressor in the second equation of the VEC.

You can access each element of these coefficients by referring to the name of the VEC followed by a dot and coefficient element:

var01.a(2,1)

var01.b(2,1)

var01.c(2,1)

To see the correspondence between each element of A, B, and C and the estimated coefficients, select View/Representations from the VAR toolbar.

A Note on Version Compatibility

The following changes made in Version 4 may yield VAR results that do not match those reported from previous versions of EViews:

•The estimated residual covariance matrix is now computed using the finite sample adjustment so the sum-of-squares is divided by T − p where p is the number of

estimated coefficients in each VAR equation. Previous versions of EViews divided the sum-of-squares by T .

•The standard errors for the cointegrating vector are now computed using the more general formula in Boswijk (1995), which also covers the restricted case.