This section discusses the implication of cointegration for the autoregressive representation. Assume that the cointegrated series can be represented by a vector error correction model according to the Granger representation theorem (Engle and Granger, 1987). Consider the vector autoregressive process with Gaussian errors defined by
or
where the initial values, , are fixed and . Since the AR operator can be re-expressed as , where with , the vector error correction model is
or
where .
One motivation for the VECM(p) form is to consider the relation as defining the underlying economic relations and assume that the agents react to the disequilibrium error through the adjustment coefficient to restore equilibrium; that is, they satisfy the economic relations. The cointegrating vector, is sometimes called the long-run parameters.
You can consider a vector error correction model with a deterministic term. The deterministic term can contain a constant, a linear trend, and seasonal dummy variables. Exogenous variables can also be included in the model.
where .
The alternative vector error correction representation considers the error correction term at lag and is written as
If the matrix has a full-rank (), all components of are . On the other hand, are stationary in difference if . When the rank of the matrix is , there are linear combinations that are nonstationary and r stationary cointegrating relations. Note that the linearly independent vector is stationary and this transformation is not unique unless . There does not exist a unique cointegrating matrix since the coefficient matrix can also be decomposed as
where is an nonsingular matrix.
The cointegration rank test determines the linearly independent columns of . Johansen and Juselius proposed the cointegration rank test by using the reduced rank regression (Johansen, 1988, 1995b; Johansen and Juselius, 1990).
Different Specifications of Deterministic Trends
When you construct the VECM(p) form from the VAR(p) model, the deterministic terms in the VECM(p) form can differ from those in the VAR(p) model. When there are deterministic cointegrated relationships among variables, deterministic terms in the VAR(p) model are not present in the VECM(p) form. On the other hand, if there are stochastic cointegrated relationships in the VAR(p) model, deterministic terms appear in the VECM(p) form via the error correction term or as an independent term in the VECM(p) form. There are five different specifications of deterministic trends in the VECM(p) form.
Case 1: There is no separate drift in the VECM(p) form.
Case 2: There is no separate drift in the VECM(p) form, but a constant enters only via the error correction term.
Case 3: There is a separate drift and no separate linear trend in the VECM(p) form.
Case 4: There is a separate drift and no separate linear trend in the VECM(p) form, but a linear trend enters only via the error correction term.
Case 5: There is a separate linear trend in the VECM(p) form.
First, focus on Cases 1, 3, and 5 to test the null hypothesis that there are at most r cointegrating vectors. Let
where can be empty for Case 1, 1 for Case 3, and for Case 5.
In Case 2, and are defined as
In Case 4, and are defined as
Let be the matrix of parameters consisting of , …, , , and , …, , where parameters corresponds to regressors . Then the VECM(p) form is rewritten in these variables as
The log-likelihood function is given by
The residuals, and , are obtained by regressing and on , respectively. The regression equation of residuals is
The crossproducts matrices are computed
Then the maximum likelihood estimator for is obtained from the eigenvectors that correspond to the r largest eigenvalues of the following equation:
The eigenvalues of the preceding equation are squared canonical correlations between and , and the eigenvectors that correspond to the r largest eigenvalues are the r linear combinations of , which have the largest squared partial correlations with the stationary process after correcting for lags and deterministic terms. Such an analysis calls for a reduced rank regression of on corrected for , as discussed by Anderson (1951). Johansen (1988) suggests two test statistics to test the null hypothesis that there are at most r cointegrating vectors
Trace Test
The trace statistic for testing the null hypothesis that there are at most r cointegrating vectors is as follows:
The asymptotic distribution of this statistic is given by
where is the trace of a matrix , is the dimensional Brownian motion, and is the Brownian motion itself, or the demeaned or detrended Brownian motion according to the different specifications of deterministic trends in the vector error correction model.
Maximum Eigenvalue Test
The maximum eigenvalue statistic for testing the null hypothesis that there are at most r cointegrating vectors is as follows:
The asymptotic distribution of this statistic is given by
where is the maximum eigenvalue of a matrix . Osterwald-Lenum (1992) provided detailed tables of the critical values of these statistics.
The following statements use the JOHANSEN option to compute the Johansen cointegration rank trace test of integrated order 1:
proc varmax data=simul2; model y1 y2 / p=2 cointtest=(johansen=(normalize=y1)); run;
Figure 35.52 shows the output based on the model specified in the MODEL statement, an intercept term is assumed. In the "Cointegration Rank Test Using Trace" table, the column Drift in ECM indicates that there is no separate drift in the error correction model, and the column Drift in Process indicates that the process has a constant drift before differencing. The "Cointegration Rank Test Using Trace" table shows the trace statistics and p-values based on Case 3, and the "Cointegration Rank Test Using Trace under Restriction" table shows the trace statistics and p-values based on Case 2. For a specified significance level, such as 5%, the output indicates that the null hypothesis that the series are not cointegrated (H0: Rank = 0) can be rejected, because the p-values for both Case 2 and Case 3 are less than 0.05. The output also shows that the null hypothesis that the series are cointegrated with rank 1 (H0: Rank = 1) cannot be rejected for either Case 2 or Case 3, because the p-values for these tests are both greater than 0.05.
Figure 35.52: Cointegration Rank Test (COINTTEST=(JOHANSEN=) Option)
Cointegration Rank Test Using Trace | ||||||
---|---|---|---|---|---|---|
H0: Rank=r |
H1: Rank>r |
Eigenvalue | Trace | Pr > Trace | Drift in ECM | Drift in Process |
0 | 0 | 0.4644 | 61.7522 | <.0001 | Constant | Linear |
1 | 1 | 0.0056 | 0.5552 | 0.4559 |
Cointegration Rank Test Using Trace Under Restriction | ||||||
---|---|---|---|---|---|---|
H0: Rank=r |
H1: Rank>r |
Eigenvalue | Trace | Pr > Trace | Drift in ECM | Drift in Process |
0 | 0 | 0.5209 | 76.3788 | <.0001 | Constant | Constant |
1 | 1 | 0.0426 | 4.2680 | 0.3741 |
Figure 35.53 shows which result, either Case 2 (the hypothesis H0) or Case 3 (the hypothesis H1), is appropriate depending on the significance level. Since the cointegration rank is chosen to be 1 by the result in Figure 35.52, look at the last row that corresponds to rank=1. Since the p-value is 0.054, the Case 2 cannot be rejected at the significance level 5%, but it can be rejected at the significance level 10%. For modeling of the two Case 2 and Case 3, see Figure 35.56 and Figure 35.57.
Figure 35.54 shows the estimates of long-run parameter (Beta) and adjustment coefficients (Alpha) based on Case 3.
Using the NORMALIZE= option, the first low of the "Beta" table has 1. Considering that the cointegration rank is 1, the long-run relationship of the series is
Figure 35.55 shows the estimates of long-run parameter (Beta) and adjustment coefficients (Alpha) based on Case 2.
Considering that the cointegration rank is 1, the long-run relationship of the series is
The preceding log-likelihood function is maximized for
The estimators of the orthogonal complements of and are
and
The ML estimators have the following asymptotic properties:
where
and
The following statements are examples of fitting the five different cases of the vector error correction models mentioned in the previous section.
For fitting Case 1,
model y1 y2 / p=2 ecm=(rank=1 normalize=y1) noint;
For fitting Case 2,
model y1 y2 / p=2 ecm=(rank=1 normalize=y1 ectrend);
For fitting Case 3,
model y1 y2 / p=2 ecm=(rank=1 normalize=y1);
model y1 y2 / p=2 ecm=(rank=1 normalize=y1 ectrend) trend=linear;
For fitting Case 5,
model y1 y2 / p=2 ecm=(rank=1 normalize=y1) trend=linear;
From Figure 35.53 that uses the COINTTEST=(JOHANSEN) option, you can fit the model by using either Case 2 or Case 3 because the test was not significant at the 0.05 level, but was significant at the 0.10 level. Here both models are fitted to show the difference in output display. Figure 35.56 is for Case 2, and Figure 35.57 is for Case 3.
For Case 2,
proc varmax data=simul2; model y1 y2 / p=2 ecm=(rank=1 normalize=y1 ectrend) print=(estimates); run;
Figure 35.56: Parameter Estimation with the ECTREND Option
Parameter Alpha * Beta' Estimates | |||
---|---|---|---|
Variable | y1 | y2 | 1 |
y1 | -0.48015 | 0.98126 | -3.24543 |
y2 | 0.12538 | -0.25624 | 0.84748 |
AR Coefficients of Differenced Lag | |||
---|---|---|---|
DIF Lag | Variable | y1 | y2 |
1 | y1 | -0.72759 | -0.77463 |
y2 | 0.38982 | -0.55173 |
Model Parameter Estimates | ||||||
---|---|---|---|---|---|---|
Equation | Parameter | Estimate | Standard Error |
t Value | Pr > |t| | Variable |
D_y1 | CONST1 | -3.24543 | 0.33022 | 1, EC | ||
AR1_1_1 | -0.48015 | 0.04886 | y1(t-1) | |||
AR1_1_2 | 0.98126 | 0.09984 | y2(t-1) | |||
AR2_1_1 | -0.72759 | 0.04623 | -15.74 | 0.0001 | D_y1(t-1) | |
AR2_1_2 | -0.77463 | 0.04978 | -15.56 | 0.0001 | D_y2(t-1) | |
D_y2 | CONST2 | 0.84748 | 0.35394 | 1, EC | ||
AR1_2_1 | 0.12538 | 0.05236 | y1(t-1) | |||
AR1_2_2 | -0.25624 | 0.10702 | y2(t-1) | |||
AR2_2_1 | 0.38982 | 0.04955 | 7.87 | 0.0001 | D_y1(t-1) | |
AR2_2_2 | -0.55173 | 0.05336 | -10.34 | 0.0001 | D_y2(t-1) |
Figure 35.56 can be reported as follows:
The keyword "EC" in the "Model Parameter Estimates" table means that the ECTREND option is used for fitting the model.
For fitting Case 3,
proc varmax data=simul2; model y1 y2 / p=2 ecm=(rank=1 normalize=y1) print=(estimates); run;
Figure 35.57: Parameter Estimation without the ECTREND Option
Parameter Alpha * Beta' Estimates | ||
---|---|---|
Variable | y1 | y2 |
y1 | -0.46421 | 0.95103 |
y2 | 0.17535 | -0.35923 |
AR Coefficients of Differenced Lag | |||
---|---|---|---|
DIF Lag | Variable | y1 | y2 |
1 | y1 | -0.74052 | -0.76305 |
y2 | 0.34820 | -0.51194 |
Model Parameter Estimates | ||||||
---|---|---|---|---|---|---|
Equation | Parameter | Estimate | Standard Error |
t Value | Pr > |t| | Variable |
D_y1 | CONST1 | -2.60825 | 1.32398 | -1.97 | 0.0518 | 1 |
AR1_1_1 | -0.46421 | 0.05474 | y1(t-1) | |||
AR1_1_2 | 0.95103 | 0.11215 | y2(t-1) | |||
AR2_1_1 | -0.74052 | 0.05060 | -14.63 | 0.0001 | D_y1(t-1) | |
AR2_1_2 | -0.76305 | 0.05352 | -14.26 | 0.0001 | D_y2(t-1) | |
D_y2 | CONST2 | 3.43005 | 1.39587 | 2.46 | 0.0159 | 1 |
AR1_2_1 | 0.17535 | 0.05771 | y1(t-1) | |||
AR1_2_2 | -0.35923 | 0.11824 | y2(t-1) | |||
AR2_2_1 | 0.34820 | 0.05335 | 6.53 | 0.0001 | D_y1(t-1) | |
AR2_2_2 | -0.51194 | 0.05643 | -9.07 | 0.0001 | D_y2(t-1) |
Figure 35.57 can be reported as follows:
Consider the example with the variables log real money, log real income, deposit interest rate, and bond interest rate. It seems a natural hypothesis that in the long-run relation, money and income have equal coefficients with opposite signs. This can be formulated as the hypothesis that the cointegrated relation contains only and through . For the analysis, you can express these restrictions in the parameterization of H such that , where H is a known matrix and is the parameter matrix to be estimated. For this example, H is given by
Restriction
When the linear restriction is given, it implies that the same restrictions are imposed on all cointegrating vectors. You obtain the maximum likelihood estimator of by reduced rank regression of on corrected for , solving the following equation
for the eigenvalues and eigenvectors , given in the preceding section. Then choose that corresponds to the r largest eigenvalues, and the is .
The test statistic for is given by
If the series has no deterministic trend, the constant term should be restricted by as in Case 2. Then H is given by
The following statements test that 2 :
proc varmax data=simul2; model y1 y2 / p=2 ecm=(rank=1 normalize=y1); cointeg rank=1 h=(1,-2); run;
Figure 35.58 shows the results of testing . The input H matrix is . The adjustment coefficient is reestimated under the restriction, and the test indicates that you cannot reject the null hypothesis.
Consider a vector error correction model:
Divide the process into with dimension and and the into
Similarly, the parameters can be decomposed as follows:
Then the VECM(p) form can be rewritten by using the decomposed parameters and processes:
The conditional model for given is
and the marginal model of is
where .
The test of weak exogeneity of for the parameters determines whether . Weak exogeneity means that there is no information about in the marginal model or that the variables do not react to a disequilibrium.
Restriction
Consider the null hypothesis , where is a matrix with .
From the previous residual regression equation
you can obtain
where and is orthogonal to such that .
Define
and let . Then can be written as
Using the marginal distribution of and the conditional distribution of , the new residuals are computed as
where
In terms of and , the MLE of is computed by using the reduced rank regression. Let
Under the null hypothesis , the MLE is computed by solving the equation
Then , where the eigenvectors correspond to the r largest eigenvalues and are normalized such that ; . The likelihood ratio test for is
See Theorem 6.1 in Johansen and Juselius (1990) for more details.
The test of weak exogeneity of is a special case of the test , considering . Consider the previous example with four variables ( ). If , you formulate the weak exogeneity of () for as and the weak exogeneity of for () as .
The following statements test the weak exogeneity of other variables, assuming :
proc varmax data=simul2; model y1 y2 / p=2 ecm=(rank=1 normalize=y1); cointeg rank=1 exogeneity; run;
proc varmax data=simul2; model y1 y2 / p=2 ecm=(rank=1 normalize=y1); cointeg rank=1 j=exogeneity; run;
Figure 35.59 shows that each variable is not the weak exogeneity of other variable.
Consider the cointegrated moving-average representation of the differenced process of
Assume that . The linear process can be written as
Therefore, for any ,
The l-step-ahead forecast is derived from the preceding equation:
Note that
since and . The long-run forecast of the cointegrated system shows that the cointegrated relationship holds, although there might exist some deviations from the equilibrium status in the short-run. The covariance matrix of the predict error is
When the linear process is represented as a VECM(p) model, you can obtain
The transition equation is defined as
where is a state vector and the transition matrix is
where 0 is a zero matrix. The observation equation can be written
where .
The l-step-ahead forecast is computed as
The error correction model with exogenous variables can be written as follows:
The following statements demonstrate how to fit VECMX(), where and from the P=2 and XLAG=1 options:
proc varmax data=simul3; model y1 y2 = x1 / p=2 xlag=1 ecm=(rank=1); run;
The following statements demonstrate how to BVECMX(2,1):
proc varmax data=simul3; model y1 y2 = x1 / p=2 xlag=1 ecm=(rank=1) prior=(lambda=0.9 theta=0.1); run;