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Cointegration Test—739

The starting values are those for the unconstrained parameters after substituting out the constraints. Fixed sets all free parameters to the value specified in the edit box. User Specified uses the values in the coefficient vector as specified in text form as starting values. For restrictions specified in pattern form, user specified starting values are taken from the first m elements of the default C coefficient vector, where m is the number of free parameters. Draw from... options randomly draw the starting values for the free parameters from the specified distributions.

Estimation Output

Once convergence is achieved, EViews displays the estimation output in the VAR window. The point estimates, standard errors, and z-statistics of the estimated free parameters are reported together with the maximized value of the log likelihood. The estimated standard errors are based on the inverse of the estimated information matrix (negative expected value of the Hessian) evaluated at the final estimates.

For overidentified models, we also report the LR test for over-identification. The LR test statistic is computed as:

LR = 2( lu − lr) = T( tr( P) − log	P	− k)	(24.21)

where P = A′B−TB−1AΣ . Under the null hypothesis that the restrictions are valid, the LR statistic is asymptotically distributed χ2( q − k) where q is the number of identifying restrictions.

If you switch the view of the VAR window, you can come back to the previous results (without reestimating) by selecting View/SVAR Output from the VAR window. In addition, some of the SVAR estimation results can be retrieved as data members of the VAR; see “Var Data Members” on page 191 of the Command and Programming Reference for a list of available VAR data members.

Cointegration Test

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.

The purpose of the cointegration test is to determine whether a group of non-stationary series are cointegrated or not. As explained below, the presence of a cointegrating relation forms the basis of the VEC specification. EViews implements VAR-based cointegration tests using the methodology developed in Johansen (1991, 1995a).

740—Chapter 24. Vector Autoregression and Error Correction Models

Consider a VAR of order p :
yt = A1yt − 1 + … + Apyt − p + Bxt + t	(24.22)

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

p − 1
∆yt = Πyt − 1 + Σ Γi∆yt − i + Bxt + t		(24.23)
i = 1
where:
p	p
Π = Σ Ai − I,	Γi = − Σ Aj	(24.24)
i =1	j =i +1

Granger’s representation theorem asserts that if the coefficient matrix Π has reduced rank r < k , then there exist k × r matrices α and β each with rank r such that

Π = αβ′ and β′yt is I(0). r is the number of cointegrating relations (the cointegrating rank) and each column of β is the cointegrating vector. As explained below, the elements of α are known as the adjustment parameters in the VEC model. Johansen’s method is to estimate the Π matrix from an unrestricted VAR and to test whether we can reject the restrictions implied by the reduced rank of Π .

How to Perform a Cointegration Test

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

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 Test” on

page 329 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

Cointegration Test—741

equations may have intercepts and deterministic trends. The asymptotic distribution of the LR test statistic for cointegration does not have the usual χ2 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.

Technical Discussion

EViews considers the following five deterministic trend cases considered by Johansen (1995a, pp. 80–84):

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

H2( r) : Πyt − 1 + Bxt = αβ′yt − 1

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

H1*( r) : Πyt − 1 + Bxt = α( β′yt − 1 + ρ0)

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

H1( r) : Πyt − 1 + Bxt = α( β′yt − 1 + ρ0) + α γ0

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

∆yt −1

742—Chapter 24. Vector Autoregression and Error Correction Models

H*( r) : Πyt − 1 + Bxt = α( β′yt − 1 + ρ0 + ρ1t) + α γ0

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

H( r) : Πyt − 1 + Bxt = α( β′yt − 1 + ρ0 + ρ1t) + α ( γ0 + γ1t)

The terms associated with α 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 (1995a) identifies the part that belongs inside the error correction term by orthogonally projecting the exogenous terms onto the α space so that α is the null space of α such that α′α = 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 β′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 (1995a, 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 ∆yt on

, ∆yt −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.

Cointegration Test—743

Interpreting Results of a Cointegration Test

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

Date: 01/16/04 Time: 11:40

Sample (adjusted): 1974:3 1987:3 Included observations: 53 after adjusting endpoints

Trend assumption: No deterministic trend (restricted constant) Series: LRM LRY IBO IDE

Lags interval (in first differences): 1 to 1

Unrestricted Cointegration Rank Test (Trace)

Hypothesized		Trace	0.05
No. of CE(s)	Eigenvalue	Statistic	Critical Value	Prob.**


None	0.469677	52.71087	54.0790	0.0659
At most 1	0.174241	19.09464	35.1928	0.7814
At most 2	0.118083	8.947661	20.2618	0.7411
At most 3	0.042249	2.287849	9.1645	0.7200

* denotes rejection of the hypothesis at the 0.05 level Trace test indicates no cointegration at the 0.05 level **MacKinnon-Haug-Michelis (1999) p-values

As indicated in the header of the output, the test assumes no trend in the series with a restricted intercept in the cointegration relation (second trend specification in the dialog), 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 first part 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 Π matrix in (24.24), the third column is the test statistic, and the last two columns are the 5% and 1% critical values. The (nonstandard) critical values are taken from Osterwald-Lenum (1992), which differ slightly from those reported in Johansen and Juselius (1990).

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 table block.

744—Chapter 24. Vector Autoregression and Error Correction Models

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, …, 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:

k
LRtr( r k ) = − T Σ log ( 1	− λi)	(24.25)

i=r + 1

where λi is the i-th largest eigenvalue of the Π matrix in (24.24) 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 − λr + 1)

(24.26)

= LRtr( r k) − LRtr( r + 1 k)

for r = 0, 1, …, 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 integrated, but the cointegration test will indicate that the Π 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 β and the adjustment parameters α . As is well known, the cointegrating vector β is not identified unless we impose some arbitrary normalization. The first block reports estimates of β and α based on the normalization β′ S11β = I , where S11 is defined in Johansen (1995a).

Cointegration Test—745

Note that the transpose of β 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 r = 0, 1, …, k − 1 . This alternative normalization expresses the first r variables as functions of the remaining k − r variables in the system. Asymptotic standard errors are reported in parentheses for the parameters that are identified.

Imposing Restrictions

Since the cointegrating vector β is not identified, you may wish to impose your own identifying restrictions. Restrictions can be imposed on the cointegrating vector (elements of the β matrix) and/or on the adjustment coefficients (elements of the α matrix). To impose restrictions in a cointegration test, select View/Cointegration Test... and specify the options in the Trend 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:

Restrictions on the Cointegrating Vector

To impose restrictions on the cointegrating vector β , you must refer to the (i,j)-th element of the transpose of the β matrix by B(i,j). The i-th cointegrating relation has the representation:

746—Chapter 24. Vector Autoregression and Error Correction Models

B(i,1)*y1 + B(i,2)*y2 + ... + B(i,k)*yk

where y1, y2, ... are the (lagged) endogenous variable. Then, if you want to impose the restriction that the coefficient on y1 for the second cointegrating equation is 1, you would type the following in the edit box:

B(2,1) = 1

You can impose multiple restrictions by separating each restriction with a comma on the same line or typing each restriction on a separate line. For example, if you want to impose the restriction that the coefficients on y1 for the first and second cointegrating equations are 1, you would type:

B(1,1) = 1

B(2,1) = 1

Currently all restrictions must be linear (or more precisely affine) in the elements of the β matrix. So for example

B(1,1) * B(2,1) = 1

will return a syntax error.

Restrictions on the Adjustment Coefficients

To impose restrictions on the adjustment coefficients, you must refer to the (i,j)-th elements of the α matrix by A(i,j). The error correction terms in the i-th VEC equation will have the representation:

A(i,1)*CointEq1 + A(i,2)*CointEq2 + ... + A(i,r)*CointEqr

Restrictions on the adjustment coefficients are currently limited to linear homogeneous restrictions so that you must be able to write your restriction as Rvec( α) = 0 , where R is a known qk × r matrix. This condition implies, for example, that the restriction,

A(1,1) = A(2,1)

is valid but:

A(1,1) = 1

will return a restriction syntax error.

One restriction of particular interest is whether the i-th row of the α matrix is all zero. If this is the case, then the i-th endogenous variable is said to be weakly exogenous with respect to the β parameters. See Johansen (1992b) for the definition and implications of weak exogeneity. For example, if we assume that there is only one cointegrating relation in the VEC, to test whether the second endogenous variable is weakly exogenous with respect to β you would enter:

Cointegration Test—747

A(2,1) = 0

To impose multiple restrictions, you may either separate each restriction with a comma on the same line or type each restriction on a separate line. For example, to test whether the second endogenous variable is weakly exogenous with respect to β in a VEC with two cointegrating relations, you can type:

A(2,1) = 0

A(2,2) = 0

You may also impose restrictions on both β and α . However, the restrictions on β and α must be independent. So for example,

A(1,1) = 0

B(1,1) = 1

is a valid restriction but:

A(1,1) = B(1,1)

will return a restriction syntax error.

Identifying Restrictions and Binding Restrictions

EViews will check to see whether the restrictions you provided identify all cointegrating vectors for each possible rank. The identification condition is checked numerically by the rank of the appropriate Jacobian matrix; see Boswijk (1995) for the technical details. Asymptotic standard errors for the estimated cointegrating parameters will be reported only if the restrictions identify the cointegrating vectors.

If the restrictions are binding, EViews will report the LR statistic to test the binding restrictions. The LR statistic is reported if the degrees of freedom of the asymptotic χ2 distribution is positive. Note that the restrictions can be binding even if they are not identifying, (e.g. when you impose restrictions on the adjustment coefficients but not on the cointegrating vector).

Options for Restricted Estimation

Estimation of the restricted cointegrating vectors β and adjustment coefficients α generally involves an iterative process. The VEC Restrictions tab provides iteration control for the maximum number of iterations and the convergence criterion. EViews estimates the restricted β and α using the switching algorithm as described in Boswijk (1995). Each step of the algorithm is guaranteed to increase the likelihood and the algorithm should eventually converge (though convergence may be to a local rather than a global optimum). You may need to increase the number of iterations in case you are having difficulty achieving convergence at the default settings.

748—Chapter 24. Vector Autoregression and Error Correction Models

Results of Restricted Cointegration Test

If you impose restrictions in the Cointegration Test view, the output will first display the test results without the restrictions 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 α31 = 0 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 β and α 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 r = 2, 3, …, k − 1 (if the VAR has only k = 2 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 β .

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