Johansen Cointegration Test

EViews supports VAR-based cointegration tests using the methodology developed in Johansen (1991, 1995) performed using a Group object or an estimated Var object.

Consider a VAR of order :

(48.1) |

where is a ‑vector of non-stationary I(1) variables, is a -vector of deterministic variables, and is a vector of innovations. We may rewrite this VAR as,

(48.2) |

where:

(48.3) |

Granger’s representation theorem asserts that if the coefficient matrix has reduced rank , then there exist matrices and each with rank such that and is I(0). is the number of cointegrating relations (the cointegrating rank) and each column of is the cointegrating vector. As explained below, the elements of are known as the adjustment parameters in the VEC model. Johansen’s method is to estimate the matrix from an unrestricted VAR and to test whether we can reject the restrictions implied by the reduced rank of .

How to Perform a Johansen Cointegration Test

To carry out the Johansen cointegration test, select View/Cointegration Test/Johansen System Cointegration Test... from a group window or View/Cointegration Test... from a Var object window. The Cointegration Test Specification page prompts you for information about the test.

The dialog will differ slightly depending on whether you are using a group or an estimated Var object to perform your test. We show here the group dialog; the Var dialog has an additional page as described in “Imposing Restrictions”.

Note that since this is a test for cointegration, this test is only valid when you are working with series that are known to be nonstationary. You may wish first to apply unit root tests to each series in the VAR. See “Unit Root Testing” for details on carrying out unit root tests in EViews.

Deterministic Trend Specification

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. Therefore, in order to carry out the test, you need to make an assumption regarding the trend underlying your data.

For each row case in the dialog, the COINTEQ column lists the deterministic variables that appear inside the cointegrating relations (error correction term), while the OUTSIDE column lists the deterministic variables that appear in the VEC equation outside the cointegrating relations. Cases 2 and 4 do not have the same set of deterministic terms in the two columns. For these two cases, some of the deterministic term is restricted to belong only in the cointegrating relation. For cases 3 and 5, the deterministic terms are common in the two columns and the decomposition of the deterministic effects inside and outside the cointegrating space is not uniquely identified; see the technical discussion below.

In practice, cases 1 and 5 are rarely used. You should use case 1 only if you know that all series have zero mean. Case 5 may provide a good fit in-sample but will produce implausible forecasts out-of-sample. As a rough guide, use case 2 if none of the series appear to have a trend. For trending series, use case 3 if you believe all trends are stochastic; if you believe some of the series are trend stationary, use case 4.

If you are not certain which trend assumption to use, you may choose the Summary of all 5 trend assumptions option (case 6) to help you determine the choice of the trend assumption. This option indicates the number of cointegrating relations under each of the 5 trend assumptions, and you will be able to assess the sensitivity of the results to the trend assumption.

We may summarize the five deterministic trend cases considered by Johansen (1995, p. 80–84) as:

1. The level data have no deterministic trends and the cointegrating equations do not have intercepts:

2. The level data have no deterministic trends and the cointegrating equations have intercepts:

3. The level data have linear trends but the cointegrating equations have only intercepts:

4. The level data and the cointegrating equations have linear trends:

5. The level data have quadratic trends and the cointegrating equations have linear trends:

The terms associated with are the deterministic terms “outside” the cointegrating relations. When a deterministic term appears both inside and outside the cointegrating relation, the decomposition is not uniquely identified. Johansen (1995) identifies the part that belongs inside the error correction term by orthogonally projecting the exogenous terms onto the space so that is the null space of such that . EViews uses a different identification method so that the error correction term has a sample mean of zero. More specifically, we identify the part inside the error correction term by regressing the cointegrating relations on a constant (and linear trend).

Exogenous Variables

The test dialog allows you to specify additional exogenous variables to include in the test VAR. The constant and linear trend should not be listed in the edit box since they are specified using the five Trend Specification options. If you choose to include exogenous variables, be aware that the critical values reported by EViews do not account for these variables.

The most commonly added deterministic terms are seasonal dummy variables. Note, however, that if you include standard 0–1 seasonal dummy variables in the test VAR, 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:

series d_q = @seas(q) - 1/4

series d_m = @seas(m) - 1/12

for quarter and month , respectively.

Lag Intervals

You should specify the lags of the test VAR as pairs of intervals. Note that the lags are specified as lags of the first differenced terms used in the auxiliary regression, not in terms of the levels. 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.

Critical Values

By default, EViews will compute the critical values for the test using MacKinnon-Haug-Michelis (1999) p-values. You may elect instead to report the Osterwald-Lenum (1992) at the 5% and 1% levels by changing the radio button selection from MHM to Osterwald-Lenum.

Interpreting Results of a Johansen Cointegration Test

As an example, the header portion of the cointegration test output for the four-variable system used by Johansen and Juselius (1990) for the Danish data is shown below.

As indicated in the header of the output, the test assumes no trend in the series with a restricted intercept in the cointegration relation (We computed the test using assumption 2 in the dialog, Intercept (no trend) in CE - no intercept in VAR), includes three orthogonalized seasonal dummy variables D1–D3, and uses one lag in differences (two lags in levels) which is specified as “1 1” in the edit field.

Number of Cointegrating Relations

The next portion of the table reports results for testing the number of cointegrating relations. Two types of test statistics are reported. The first block reports the so-called trace statistics and the second block (not shown above) reports the maximum eigenvalue statistics. For each block, the first column is the number of cointegrating relations under the null hypothesis, the second column is the ordered eigenvalues of the matrix in (48.3), the third column is the test statistic, and the last two columns are the 5% and 1% critical values. The (nonstandard distribution) critical values are taken from MacKinnon-Haug-Michelis (1999) so they differ slightly from those reported in Johansen and Juselius (1990).

To determine the number of cointegrating relations conditional on the assumptions made about the trend, we can proceed sequentially from to until we fail to reject. The result of this sequential testing procedure is reported at the bottom of each block.

The trace statistic reported in the first block tests the null hypothesis of cointegrating relations against the alternative of cointegrating relations, where is the number of endogenous variables, for . The alternative of cointegrating relations corresponds to the case where none of the series has a unit root and a stationary VAR may be specified in terms of the levels of all of the series. The trace statistic for the null hypothesis of cointegrating relations is computed as:

(48.4) |

where is the i-th largest eigenvalue of the matrix in (48.3) which is reported in the second column of the output table.

The second block of the output reports the maximum eigenvalue statistic which tests the null hypothesis of cointegrating relations against the alternative of cointegrating relations. This test statistic is computed as:

(48.5) |

for .

There are a few other details to keep in mind:

• Critical values are available for up to series. Also note that the critical values depend on the trend assumptions and may not be appropriate for models that contain other deterministic regressors. For example, a shift dummy variable in the test VAR implies a broken linear trend in the level series .

• The trace statistic and the maximum eigenvalue statistic may yield conflicting results. For such cases, we recommend that you examine the estimated cointegrating vector and base your choice on the interpretability of the cointegrating relations; see Johansen and Juselius (1990) for an example.

• In some cases, the individual unit root tests will show that some of the series are integrated, but the cointegration test will indicate that the matrix has full rank (). This apparent contradiction may be the result of low power of the cointegration tests, stemming perhaps from a small sample size or serving as an indication of specification error.

Cointegrating Relations

The second part of the output provides estimates of the cointegrating relations and the adjustment parameters . As is well known, the cointegrating vector is not identified unless we impose some arbitrary normalization. The first block reports estimates of and based on the normalization , where is defined in Johansen (1995). Note that the transpose of is reported under Unrestricted Cointegrating Coefficients so that the first row is the first cointegrating vector, the second row is the second cointegrating vector, and so on.

The remaining blocks report 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.

In our example, for one cointegrating equation we have:

Imposing Restrictions

Since the cointegrating vector is not fully identified, you may wish to impose your own identifying restrictions. If you are performing your Johansen cointegration test using an estimated Var object, EViews offers you the opportunity to impose restrictions on . Restrictions can be imposed on the cointegrating vector (elements of the matrix) and/or on the adjustment coefficients (elements of the matrix)

To perform the cointegration test from a Var object, you will first need to estimate a VAR with your variables as described in “Estimating a VAR in EViews”. Next, select View/Cointegration Test... from the Var menu and specify the options in the Cointegration Test Specification tab as explained above. Then bring up the VEC Restrictions tab. You will enter your restrictions in the edit box that appears when you check the Impose Restrictions box:

A full description of how to enter your restrictions is provided in “Imposing VEC Restrictions”.

Results of Restricted Cointegration Test

If you impose restrictions in the Cointegration Test view, the top portion of the output will display the unrestricted test results as described above. The second part of the output begins by displaying the results of the LR test for binding restrictions.

If the restrictions are not binding for a particular rank, the corresponding rows will be filled with NAs. If the restrictions are binding but the algorithm did not converge, the corresponding row will be filled with an asterisk “*”. (You should redo the test by increasing the number of iterations or relaxing the convergence criterion.) For the example output displayed above, we see that the single restriction is binding only under the assumption that there is one cointegrating relation. Conditional on there being only one cointegrating relation, the LR test does not reject the imposed restriction at conventional levels.

The output also reports the estimated and imposing the restrictions. Since the cointegration test does not specify the number of cointegrating relations, results for all ranks that are consistent with the specified restrictions will be displayed. For example, suppose the restriction is:

B(2,1) = 1

Since this is a restriction on the second cointegrating vector, EViews will display results for ranks (if the VAR has only variables, EViews will return an error message pointing out that the “implied rank from restrictions must be of reduced order”).

For each rank, the output reports whether convergence was achieved and the number of iterations. The output also reports whether the restrictions identify all cointegrating parameters under the assumed rank. If the cointegrating vectors are identified, asymptotic standard errors will be reported together with the parameters .