A fundamental prerequisite of VECM estimation is a priori knowledge of the number of cointegrating relations . Accordingly, we consider formal methods for determining the cointegrating rank using Johansen cointegration tests.

We begin with a VECM with endogenous variables and no deterministic terms,

Recall that Granger’s representation theorem asserts that if the coefficient matrix has rank for , then there exist matrices and each with rank such that .

where denotes the maximized value of the Gaussian likelihood function for cointegration rank , and are the eigenvalues associated with the relevant symmetric matrix given in Equation (45.23).

The asymptotic distributions of are nonstandard and are given by:

where and are respectively the trace and maximum eigenvalue of the matrix

where is the -dimensional standard Wiener process. and are commonly referred to as the trace eigenvalue test and maximum eigenvalue test, respectively.

The basic approach to cointegration rank determination is to run a battery of these hypothesis tests starting with the null hypothesis . If a test rejects, increase the null by one and repeat the test. The estimated cointegration rank is then the first (minimum) for which the test fails to reject the null.

When deterministic terms or exogenous variables are present in the specification, the test statistics and the test strategy outlined above are unchanged, although the eigenvalues of the relevant matrix given in Equation (45.23) will differ since they are based on the extended regressors and instead of and . ( “Estimating VECM with Deterministics”).

For example, if you type “1 2” in the edit field, the test VAR regresses on , , and any other exogenous variables that you have specified. Note that in terms of the level series the largest lag is 3. To run a cointegration test with one lag in the level series, type “0 0” in the edit field.

Keep in mind that the exogenous variables entered in the test dialog refer to short-run variables added to the VEC difference specification Equation (45.41). This treatment is in contrast to that of the built-in deterministic trends which are specified in terms of the levels equation. For example, one of the most commonly added exogenous variables are seasonal dummy variables. Note, however, that if you include 0–1 seasonal dummy variables in the test specification, this will affect both the mean and the trend of the level series . To handle this problem, Johansen (1995, page 84) suggests using centered (orthogonalized) seasonal dummy variables, which shift the mean without contributing to the trend. Centered seasonal dummy variables for quarterly and monthly series can be generated by the commands:

for quarter and month , respectively.

Your series may have nonzero means and deterministic trends as well as stochastic trends. Similarly, the cointegrating equations may have intercepts and deterministic trends. The asymptotic distribution of the LR test statistic for cointegration does not have the usual distribution and depends on the assumptions made with respect to deterministic trends. To carry out the test, you need to make an assumption regarding the trend underlying your data.

The asymptotic distribution of the LR test statistic for cointegration does not have the usual distribution and depends on the assumptions made with respect to deterministic trends.

Critical values are available for up to series.

Following the summary is a Rank Tests table showing Johansen cointegration tests based on the trace and maximum eigenvalue statistics. The trace statistic reported in the first block tests the null hypothesis of at most cointegrating relations against the alternative of cointegrating relations, where is the number of endogenous variables. The maximum eigenvalue statistics tests the null hypothesis of cointegrating relations against the alternative of cointegrating relations.

In interpreting the results for these tests, we begin with the null hypothesis . If a test rejects, increase the null by one and repeat the test. The estimated cointegration rank is then the first (minimum) for which the test fails to reject the null. In this case, we reject all three null hypotheses at conventional significance levels, and conclude that there are 3 cointegrating equations.

The next node contains the Unrestricted Coefficients table which provides estimates of the cointegrating vector and the adjustment parameters . As is well known, the cointegrating vector is not identified unless we provide an arbitrary normalization. This table reports estimates of and under the normalization where is defined in Equation (45.22). Note that the transpose of is reported in the table so that the first row is the first cointegrating vector, the second row is the second cointegrating vector, and so on.

The final node contains the Normalized Coefficients table which reports estimates from a different normalization for each possible number of cointegrating relations. This alternative normalization expresses the first variables as functions of the remaining variables in the system. Asymptotic standard errors are reported in parentheses for the parameters that are identified.