Chapter 38. Cointegration Testing

The finding that many macro time series may contain a unit root has spurred the development of the theory of non-stationary time series analysis. Engle and Granger (1987) pointed out that a linear combination of two or more non-stationary series may be stationary. If such a stationary linear combination exists, the non-stationary time series are said to be cointegrated. The stationary linear combination is called the cointegrating equation and may be interpreted as a long-run equilibrium relationship among the variables.

This chapter describes several tools for testing for the presence of cointegrating relationships among non-stationary variables in non-panel and panel settings.

The first two parts of this chapter focus on cointegration tests employing the Johansen (1991, 1995) system framework or Engle-Granger (1987) or Phillips-Ouliaris (1990) residual based test statistics. The final section describes cointegration tests in panel settings where you may compute the Pedroni (1999), Pedroni (2004), and Kao (1999) tests as well as a Fisher-type test using an underlying Johansen methodology (Maddala and Wu, 1999).

The Johansen tests may be performed using a Group object or an estimated Var object. The residual tests may be computed using a Group object or an Equation object estimated using nonstationary regression methods. The panel tests may be conducted using a Pool object or a Group object in a panel workfile setting. Note that additional cointegration tests are offered as part of the diagnostics for an equation estimated using nonstationary methods. See “Testing for Cointegration” on page 234.

If cointegration is detected, Vector Error Correction (VEC) or nonstationary regression methods may be used to estimate the cointegrating equation. See “Vector Error Correction (VEC) Models” on page 478 and Chapter 25. “Cointegrating Regression,” beginning on page 219 for details.

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 p :

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

i = 1

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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 yt have no deterministic trends and the cointegrating equations do not have intercepts:

H2(r): Pyt – 1 + Bxt = ab¢yt – 1

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

H1*(r): Pyt – 1 + Bxt = a(b¢yt – 1 + r0)

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

H1(r): Pyt – 1 + Bxt = a(b¢yt – 1 + r0) + a^g0

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

H*(r): Pyt – 1 + Bxt = a(b¢yt – 1 + r0 + r1t) + a^g0

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

H(r): Pyt – 1 + Bxt = a(b¢yt – 1 + r0 + r1t) + a^(g0 + g1t)

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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.

Date: 09/21/09 Time: 11:12

Sample (adjusted): 1974Q3 1987Q3

Included observations: 53 after adjustments

Trend assumption: No deterministic trend (restricted constant)

Series: LRM LRY IBO IDE

Exogenous series: D1 D2 D3

Warning: Critical values assume no exogenous series

Lags interval (in first differences): 1 to 1

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 P matrix in (38.3), 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-Miche- lis (1999) so they differ slightly from those reported in Johansen and Juselius (1990).

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for r = 0, 1, º, k – 1 .

There are a few other details to keep in mind:

•Critical values are available for up to k = 10 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 yt .

•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 inte-

grated, but the cointegration test will indicate that the P matrix has full rank

(r = k ). 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 b and the adjustment parameters a . As is well known, the cointegrating vector b is not identified unless we impose some arbitrary normalization. The first block reports estimates of b and a based on the normalization b¢S11b = I , where S11 is defined in Johansen (1995). Note that the transpose of b 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.

Unrestricted Cointegrating Coefficients (normalized by b'*S11*b=I):

The remaining blocks report estimates from a different normalization for each possible number of cointegrating relations r = 0, 1, º, k – 1 . This alternative normalization expresses the first r variables as functions of the remaining k – r variables in the system.

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A full description of how to enter your restrictions is provided in “Imposing Restrictions” on page 481.

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.

Restrictions:

a(3,1)=0

Tests of cointegration restrictions:

NA indicates restriction not binding.

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 a31 = 0 is binding only under the assumption that