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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 :

yt = A1yt – 1 + º + Apyt p + Bxt + et

(38.1)

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,

p – 1

 

Dyt = Pyt – 1 + Â GiDyt i + Bxt + et

(38.2)

i = 1

686—Chapter 38. Cointegration Testing

where:

p

p

 

P = Â Ai I,

Gi = Â Aj

(38.3)

i =1

j =i +1

 

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

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” on page 692.

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” on page 379 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

x2 distribution and depends on the assumptions made with respect to deterministic trends.

Johansen Cointegration Test—687

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)

688—Chapter 38. Cointegration Testing

The terms associated with a^ 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 a space so that a^ is the null space of a such that a¢a^= 0 . 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 yt on a constant (and linear trend).

Exogenous Variables

The test dialog allows you to specify additional exogenous variables xt 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 yt . 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 q and month m , 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 Dyt on

Dyt –1 , Dyt –2 , and any other exogenous variables that you have specified. Note that in terms of the level series yt 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 OsterwaldLenum.

Johansen Cointegration Test—689

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).

690—Chapter 38. Cointegration Testing

Unrestricted Cointegration Rank Test (Trace)

 

Hypothesized

 

Trace

 

0.05

 

 

 

No. of CE(s)

Eigenvalue

Statistic

Critical Value

Prob.**

 

 

 

 

 

 

 

 

None

0.433165

49.14436

54.07904

0.1282

 

 

At most 1

0.177584

19.05691

35.19275

0.7836

 

 

At most 2

0.112791

8.694964

20.26184

0.7644

 

 

At most 3

0.043411

2.352233

9.164546

0.7071

 

 

 

 

 

 

 

 

 

 

 

 

Trace test indicates no cointegration at the 0.05 level

 

 

 

 

* denotes rejection of the hypothesis at the 0.05 level

 

 

 

 

**MacKinnon-Haug-Michelis (1999) p-values

 

 

 

 

 

Unrestricted Cointegration Rank Test (Maximum Eigenvalue)

 

 

 

 

 

 

 

 

 

 

 

Hypothesized

 

Max-Eigen

 

0.05

 

 

 

No. of CE(s)

Eigenvalue

Statistic

Critical Value

Prob.**

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

None *

0.433165

30.08745

28.58808

0.0319

 

 

At most 1

0.177584

10.36195

22.29962

0.8059

 

 

At most 2

0.112791

6.342731

15.89210

0.7486

 

 

At most 3

0.043411

2.352233

9.164546

0.7071

 

 

 

 

 

 

 

 

 

 

 

 

Max-eigenvalue test indicates 1 cointegrating eqn(s) at the 0.05 level

 

* denotes rejection of the hypothesis at the 0.05 level

 

 

 

 

**MacKinnon-Haug-Michelis (1999) p-values

 

 

 

 

To determine the number of cointegrating relations r

conditional on the assumptions made

about the trend, we can proceed sequentially from r

= 0 to r

= k – 1 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 r cointegrating relations against the alternative of k cointegrating relations, where k is the number of endogenous variables, for r = 0, 1, º, k – 1 . The alternative of k 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 r cointegrating relations is computed as:

LRtr(r

 

k) = T Âk

log(1 – li)

(38.4)

 

 

 

i=r + 1

 

 

where li is the i-th largest eigenvalue of the P matrix in (38.3) 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 r cointegrating relations against the alternative of r + 1 cointegrating relations. This test statistic is computed as:

LRmax(r r + 1) = T log(1 – lr + 1 )

(38.5)

= LRtr(r k) LRtr(r + 1 k)

Johansen Cointegration Test—691

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 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):

LRM

LRY

IBO

IDE

C

-21.97409

22.69811

-114.4173

92.64010

133.1615

14.65598

-20.05089

3.561148

100.2632

-62.59345

7.946552

-25.64080

4.277513

-44.87727

62.74888

1.024493

-1.929761

24.99712

-14.64825

-2.318655

 

 

 

 

 

 

Unrestricted Adjustment Coefficients (alpha):

 

 

 

 

 

 

 

 

 

 

 

 

D(LRM)

0.009691

-0.000329

0.004406

0.001980

D(LRY)

-0.005234

0.001348

0.006284

0.001082

D(IBO)

-0.001055

-0.000723

0.000438

-0.001536

D(IDE)

-0.001338

-0.002063

-0.000354

-4.65E -05

 

 

 

 

 

 

 

 

 

 

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.

692—Chapter 38. Cointegration Testing

Asymptotic standard errors are reported in parentheses for the parameters that are identified.

In our example, for one cointegrating equation we have:

1 Cointegrating Equation(s):

Log likelihood

669.1154

 

 

 

 

 

 

 

 

Normalized cointegrating coefficients (standard error in

 

parentheses)

 

 

 

 

LRM

LRY

IBO

IDE

C

1.000000

-1.032949

5.206919

-4.215880

-6.059932

 

(0.13897)

(0.55060)

(1.09082)

(0.86239)

Adjustment coefficients (standard error in parentheses) D(LRM) -0.212955

(0.06435) D(LRY) 0.115022 (0.06739)

D(IBO) 0.023177 (0.02547)

D(IDE) 0.029411 (0.01717)

Imposing Restrictions

Since the cointegrating vector b 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 b . Restrictions can be imposed on the cointegrating vector (elements of the b matrix) and/or on the adjustment coefficients (elements of the a 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” on page 460. 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:

Johansen Cointegration Test—693

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:

Hypothesized

Restricted

LR

Degrees of

 

No. of CE(s)

Log-likehood

Statistic

Freedom

Probability

 

 

 

 

 

 

 

 

 

 

1

668.6698

0.891088

1

0.345183

2

674.2964

NA

NA

NA

3

677.4677

NA

NA

NA

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

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