Vector Error Correction Modeling

The VARMAX Procedure

Vector Error Correction Modeling

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

$y_t=\sum_{i=1}^p\Phi_i{y}_{t-i} + {\epsilon}_t$

$\Phi(B) y_t={\epsilon}_t$

where the initial values, y_-p+1, ... ,y₀, are fixed and ${\epsilon}_t \sim N(0,\Sigma)$ . Since the AR operator $\Phi(B)$ can be re-expressed as $\Phi(B)=\Phi^*(B)(1-B)+\Phi(1)B$ where $\Phi^*(B)=I_k-\sum_{i=1}^{p-1}\Phi^*_iB^i$ with $\Phi^*_i=- \sum_{j=i+1}^p \Phi_j$ , the vector error correction model is

$\Phi^*(B)(1-B)y_t={\alpha} {\beta}'y_{t-1}+{\epsilon}_t$

$\Delta y_t={\alpha} {\beta}'y_{t-1} + \sum_{i=1}^{p-1} \Phi^*_i \Delta y_{t-i} + {\epsilon}_t$

where ${\alpha}{\beta}'=-\Phi(1)=-I_k+\Phi_{1}+\Phi_{2}+ ... +\Phi_{p}$ .

One motivation for the VECM(p) form is to consider the relation ${\beta}'y_{t}=c$ as defining the underlying economic relations and assume that the agents react to the disequilibrium error ${\beta}'y_{t} - c$ through the adjustment coefficient ${\alpha}$ to restore equilibrium; that is, they satisfy the economic relations. The cointegrating vector, ${\beta}$ is sometimes called the long-run parameters.

You can consider a vector error correction model with a deterministic term. The deterministic term D_t can contain a constant, a linear trend, seasonal dummy variables, or nonstochastic regressors.

$\Delta y_{t}=\Pi y_{t-1} + \sum_{i=1}^{p-1} \Phi^*_i \Delta y_{t-i} + A D_t + {\epsilon}_t$

where $\Pi={\alpha} {\beta}'$ .

The alternative vector error correction representation considers the error correction term at lag t-p and is written as

$\Delta y_t=\sum_{i=1}^{p-1}\Phi^{\sharp}_i\Delta y_{t-i} +\Pi^{\sharp} y_{t-p} + AD_t +{\epsilon}_t$

If the matrix $\Pi$ has a full rank (r=k), all components of y_t are I(0). On the other hand, y_t are stationary in difference if $rank(\Pi)=0$ .When the rank of the matrix $\Pi$ is r < k, there are k-r linear combinations that are nonstationary and r stationary cointegrating relations. Note that the linearly independent vector $z_{t}={\beta}' y_{t}$ is stationary and this transformation is not unique unless r=1. There does not exist a unique cointegrating matrix ${\beta}$ since the coefficient matrix $\Pi$ can also be decomposed as

$\Pi={\alpha} MM^{-1}{\beta}'={\alpha}^{*}{\beta}^{*'}$

where M is an r×r nonsingular matrix.

Test for the Cointegration

The cointegration rank test determines the linearly independent columns of $\Pi$ . Johansen (1988, 1995a) and Johansen and Juselius (1990) proposed the cointegration rank test using the reduced rank regression.

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 will not be present in the VECM(p) form. On the other hand, if there are stochastic cointegrated relationships, deterministic terms appear in the VECM(p) form via the error correction term or as an independent term in the VECM(p) form.

Case 1: There is no separate drift in the VECM(p) form.
$\Delta y_t={\alpha} {\beta}'y_{t-1} + \sum_{i=1}^{p-1} \Phi^*_i \Delta y_{t-i} + {\epsilon}_t$
Case 2: There is no separate drift in the VECM(p) form, but a constant enters only via the error correction term.
$\Delta y_t={\alpha} ({\beta}', \beta_0)(y_{t-1}',1)' + \sum_{i=1}^{p-1} \Phi^*_i \Delta y_{t-i} + {\epsilon}_t$
Case 3: There is a separate drift and no separate linear trend in the VECM(p) form.
$\Delta y_t={\alpha} {\beta}'y_{t-1} + \sum_{i=1}^{p-1} \Phi^*_i \Delta y_{t-i} + {\delta}_0 + {\epsilon}_t$
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.
$\Delta y_t={\alpha} ({\beta}', \beta_1)(y_{t-1}',t)' + \sum_{i=1}^{p-1} \Phi^*_i \Delta y_{t-i} + {\delta}_0 + {\epsilon}_t$
Case 5: There is a separate linear trend in the VECM(p) form.
$\Delta y_t={\alpha} {\beta}'y_{t-1} + \sum_{i=1}^{p-1} \Phi^*_i \Delta y_{t-i} + {\delta}_0 + {\delta}_1t +{\epsilon}_t$

First, focus on cases 1, 3, and 5 to test the null hypothesis that there are at most r cointegrating vectors. Let

$Z_{0t}&=&\Delta{y}_t \ Z_{1t}&=&y_{t-1} \ Z_{2t}&=&[\Delta{y}_{t-1}', ... ,\Delt... ...' \ Z_{1} &=& [Z_{11}, ... , Z_{1T}]' \ Z_{2} &=& [Z_{21}, ... , Z_{2T}]'$

where D_t can be empty for Case 1; 1 for Case 3; (1,t) for Case 5. Let $\Psi$ be the matrix of parameters consisting of $\Phi^{*}_1, ... , \Phi^{*}_{p-1}$ and A, where parameters A corresponds to regressors D_t. Then the VECM(p) form is rewritten in these variables as

$Z_{0t}={\alpha} {\beta}' Z_{1t} + \Psi Z_{2t} + {\epsilon}_t$

The log-likelihood function is given by

$\ell &=& - \frac{kT}2 \log 2\pi -\frac{T}2 \log |\Sigma| \ & & - \frac{1}2 \sum_... ..._{1t} -\Psi Z_{2t})'\Sigma^{-1} (Z_{0t} -{\alpha} {\beta}' Z_{1t} -\Psi Z_{2t})$

The residuals, R_0t and R_1t, are obtained by regressing Z_0t and Z_1t on Z_2t, respectively. The regression equation in residuals is

$R_{0t}={\alpha} {\beta}' R_{1t} + \hat { {\epsilon}}_t$

The crossproducts matrices are computed

$S_{ij}=\frac{1}T\sum_{t=1}^TR_{it}R_{jt}',i,j=0,1$

Then the maximum likelihood estimator for ${\beta}$ is obtained from the eigenvectors corresponding to the r largest eigenvalues of the following equation:

$|\lambda S_{11} - S_{10}S_{00}^{-1}S_{01}|=0$

The eigenvalues of the preceding equation are squared canonical correlations between R_0t and R_1t, and the eigenvectors corresponding to the r largest eigenvalues are the r linear combinations of y_t-1, which have the largest squared partial correlations with the stationary process $\Delta{y}_t$ after correcting for lags and deterministic terms. Such an analysis calls for a reduced rank regression of $\Delta{y}_t$ on y_t-1 corrected for $(\Delta{y}_{t-1}, ... ,\Delta{y}_{t-p+1},D_t)$ , as discussed by Anderson (1951). Johansen (1988) suggested two test statistics to test the null hypothesis that there are at most r cointegrating vectors

$H_0: \lambda_i=0 {\rm for} i=r+1, ... ,k$

Trace Test

$\lambda_{trace}=-T\sum_{i=r+1}^k\log(1-\lambda_i)$

The asymptotic distribution of this statistic is given by

$tr\{ \int_0^1 (dW){\tilde W}' (\int_0^1 {\tilde W}{\tilde W}'dr)^{-1}\int_0^1 {\tilde W}(dW)' \}$

where tr(A) is the trace of a matrix A, W is the k-r dimensional Brownian motion, and $\tilde W$ 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

$\lambda_{max}=-T\log(1-\lambda_{r+1})$

The asymptotic distribution of this statistic is given by

$max\{ \int_0^1 (dW){\tilde W}' (\int_0^1 {\tilde W}{\tilde W}'dr)^{-1}\int_0^1 {\tilde W}(dW)' \}$

where max(A) is the maximum eigenvalue of a matrix A. Osterwald-Lenum (1992) provided the detailed tables of critical values of these statistics.

In case 2, consider that Z_1t = (y_t',1)' and $Z_{2t}=(\Delta{y}_{t-1}', ... ,\Delta{y}_{t-p+1}')'$ .In case 4, Z_1t = (y_t',t)' and $Z_{2t}=(\Delta{y}_{t-1}', ... ,\Delta{y}_{t-p+1}',1)'$ .

The following statements use the JOHANSEN option to compute the Johansen cointegration rank test of integrated order 1:

   proc varmax data=simul2;
      model y1 y2 / p=2 cointtest=(johansen=(normalize=y1));
   run;

The VARMAX Procedure

Cointegration Rank Test
H_0: Rank=r	H_1: Rank>r	Eigenvalue	Trace	Critical Value	DriftInECM	DriftInProcess
0	0	0.4644	61.75	15.34	Constant	Linear
1	1	0.0056	0.56	3.84

Cointegration Rank Test under the Restriction
H_0: Rank=r	H_1: Rank>r	Eigenvalue	Trace	Critical Value	DriftInECM	DriftInProcess
0	0	0.5209	76.38	19.99	Constant	Constant
1	1	0.0426	4.27	9.13

Figure 4.40: Cointegration Rank Test (JOHANSEN option)

Suppose that the model has an intercept term. The first table in Figure 4.32 shows the trace statistics based on case 3; the second table, case 2. The output indicates that the series are cointegrated with rank 1.

The VARMAX Procedure

Test of the Restriction when Rank=r
Hypo	DriftInECM	DriftInProcess
H_0	Constant	Linear
H_1	Constant	Constant

Test of the Restriction when Rank=r
Rank	Eigenvalue On Restrict	Eigenvalue	Chi-Square	DF	Prob>ChiSq
0	0.5209	0.4644	14.63	2	0.0007
1	0.0426	0.0056	3.71	1	0.0540

Figure 4.41: Cointegration Rank Test Continued

Figure 4.33 shows which result, case 3 (the hypothesis H_0) and case 2 (the hypothesis H_1), is appropriate. Since the cointegration rank is chosen to be 1 by the result in Figure 4.32, look at the last row corresponding to rank=1.

The VARMAX Procedure

Long-Run Parameter BETA Estimates
Variable	Dummy 1	Dummy 2
y1	1.00000	1.00000
y2	-2.04869	-0.02854

Adjustment Coefficient ALPHA Estimates
Variable	Dummy 1	Dummy 2
y1	-0.46421	-0.00502
y2	0.17535	-0.01275

Figure 4.42: Cointegration Rank Test Continued

Figure 4.34 shows the estimates of long-run parameter and adjustment coefficients based on case 3. Considering that the cointegration rank is 1, the long-run relationship of the series is

y_1t = 2.049 y_2t

The VARMAX Procedure

Long-Run Coefficient BETA based on the Restricted Trend
Variable	Dummy 1	Dummy 2	Dummy 3
y1	1.00000	1.00000	1.00000
y2	-2.04366	-2.75773	-0.73198
1	6.75919	101.37051	-42.50051

Adjustment Coefficient ALPHA based on the Restricted Trend
Variable	Dummy 1	Dummy 2	Dummy 3
y1	-0.48015	0.01091	5.96583E-14
y2	0.12538	0.03722	-2.834E-13

Figure 4.43: Cointegration Rank Test Continued

Figure 4.35 shows the estimates of long-run parameter and adjustment coefficients based on case 2. Considering that the cointegration rank is 1, the long-run relationship of the series is

y_1t = 2.044 y_2t - 6.760

Estimation of Vector Error Correction Model

Now consider cases 1, 3, and 5 for the vector error correction model. The preceding log-likelihood function is maximized for

$\hat {{\beta}} &=& S_{11}^{-1/2} [v_1, ... ,v_r] \\hat {{\alpha}} &=& S_{01}\hat... ...{2} \hat \Psi' - Z_{1} \hat\Pi')' (Z_{0} - Z_{2} \hat \Psi' - Z_{1} \hat\Pi')/T$

The estimators of the orthogonal complements of ${\alpha}$ and ${\beta}$ are

$\hat {{\beta}}_{\bot}=S_{11} [v_{r+1}, ... ,v_{k}]$

and

$\hat {{\alpha}}_{\bot}=S_{00}^{-1} S_{01} [v_{r+1}, ... ,v_{k}]$

The ML estimators have the following asymptotic properties:

${\sqrt T} {\rm vec}([\hat \Pi,\hat \Psi] - [\Pi, \Psi]) \stackrel{d}{arrow} N(0, \Sigma_{co})$

where

$\Sigma_{co}=\Sigma \otimes ( [ {\beta} & 0 \ 0 & I_k ] \Omega^{-1} [ {\beta}' & 0 \ 0 & I_k ] )$

and

$\Omega={\rm plim} \frac{1}T [ {\beta}'Z_{1}'Z_{1}{\beta} & {\beta}'Z_{1}'Z_{2} \ Z_{2}'Z_{1}{\beta} & Z_{2}'Z_{2} \ ]$

Test for the Linear Restriction of ${\beta}$

Consider the example with the variables m_t, log real money, y_t, log real income, i^d_t, deposit interest rate, and i^b_t, 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 m_t and y_t through m_t - y_t. For the analysis, you can express these restrictions in the parameterization of H such that ${\beta}=H\psi$ , where H is a known k×s matrix and $\psi$ is the $sx r ( r\leq s \lt k)$ parameter matrix to be estimated. For this example, H is given by

$H=[ 1 & 0 & 0 \ -1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \ ]$

Restriction $H_0: {\beta}=H\phi$

When the linear restriction ${\beta}=H\phi$ is given, it implies that the same restrictions are imposed on all cointegrating vectors. You obtain the maximum likelihood estimator of ${\beta}$ by reduced rank regression of $\Delta{y}_t$ on Hy_t-1 corrected for $(\Delta{y}_{t-1}, ... ,\Delta{y}_{t-p+1},D_t)$ , solving the following equation

$|\rho H'S_{11}H - H'S_{10}S^{-1}_{00}S_{01}H|=0$

for the eigenvalues $1\gt\rho_1\gt ... \gt\rho_s\gt$ and eigenvectors (v₁, ... ,v_s), S_ij are given in the preceding section. Then choose $\hat \phi=(v_1, ... ,v_r)$ corresponding to the r largest eigenvalues, and the $\hat {{\beta}}$ is $H\hat \phi$ .

The test statistic for $H_0: {\beta}=H\phi$ is given by

$T\sum_{i=1}^r \log\{(1-\rho_i)/(1-\lambda_i)\} \stackrel{d}{arrow} \chi^2_{r(k-s)}$

If the data have no deterministic trend, the constant term should be restricted by ${\alpha}_{\bot}'{\delta}_0=0$ like Case 2. Then H is given by

$H=[ 1 & 0 & 0 & 0\ -1 & 0 & 0 & 0\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0\ 0 & 0 & 0 & 1\ ]$

The following statements test that $\beta_1 + 2 \beta2=0$ :

   proc varmax data=simul2;
      model y1 y2 / p=2 ecm=(rank=1 normalize=y1);
      cointeg rank=1 h=(1,-2); 
   run;

The VARMAX Procedure

Long-Run Coefficient BETA with respect to Hypothesis on BETA
Variable	Dummy 1
y1	1.00000
y2	-2.00000

Adjustment Coefficient ALPHA with respect to Hypothesis on BETA
Variable	Dummy 1
y1	-0.47404
y2	0.17534

Test for Restricted Long-Run Coefficient BETA
Index	EigenvalueOnRestrict	Eigenvalue	Chi-Square	DF	Prob>ChiSq
1	0.4616	0.4644	0.51	1	0.4738

Figure 4.44: Testing of Linear Restriction ${\beta}$ (H= option)

Figure 4.36 shows the results of testing $H_0:\beta_1 +2\beta2=0$ .The input H matrix is H=(1, -2)'. The adjustment coefficient is reestimated under the restriction, and the test indicates that you cannot reject the null hypothesis.

Test for the Weak Exogeneity and Restrictions of ${\alpha}$

Consider a vector error correction model:

$\Delta y_t={\alpha} {\beta}'y_{t-1} + \sum_{i=1}^{p-1} \Phi^*_i \Delta y_{t-i} + AD_t + {\epsilon}_t$

Divide the process y_t into (y_1t',y_2t')' with dimension k₁ and k₂ and the $\Sigma$ into

$\Sigma=[ \Sigma_{11} & \Sigma_{12} \ \Sigma_{21} & \Sigma_{22} ]$

Similarly, the parameters can be decomposed as follows:

${\alpha}=[ {\alpha_1} \ {\alpha_2} ] \Phi^*_i=[ \Phi^*_{1i} \ \Phi^*_{2i} ] A=[ A_{1} \ A_{2} ]$

Then the VECM(p) form can be rewritten using the decomposed parameters and processes:

$[ \Delta y_{1t} \ \Delta y_{2t} ]=[ {\alpha_1} \ {\alpha_2} ] {\beta}'y_{t... ...elta y_{t-i} + [ A_{1} \ A_{2} ] D_t + [ {\epsilon}_{1t} \ {\epsilon}_{2t} ]$

The conditional model for y_1t given y_2t is

$\Delta{y}_{1t} &=& \omega\Delta{y}_{2t} + (\alpha_1-\omega\alpha_2){\beta}'y_... ...-i} \ && + (A_1 - \omega A_2) D_t + {\epsilon}_{1t} - \omega {\epsilon}_{2t}$

and the marginal model of y_2t,

$\Delta{y}_{2t}=\alpha_2{\beta}'y_{t-1} + \sum_{i=1}^{p-1} \Phi^{*}_{2i}\Delta{y}_{t-i} + A_2 D_t + {\epsilon}_{2t}$

where $\omega=\Sigma_{12}\Sigma_{22}^{-1}$ .

The test of weak exogeneity of y_2t for the parameters $(\alpha_1, {\beta})$ determines whether $\alpha_2=0$ .Weak exogeneity means that there is no information about ${\beta}$ in the marginal model or that the variables y_2t do not react to a disequilibrium.

Restriction $H_0\colon {\alpha}=J\psi$

Consider the null hypothesis $H_0\colon {\alpha}=J\psi$ , where J is a k×m matrix with $r\leq m \lt k$ .

From the previous residual regression equation

$R_{0t}={\alpha}{\beta}'R_{1t} + \hat{{\epsilon}}_t=J\psi{\beta}'R_{1t} + \hat{{\epsilon}}_t$

you can obtain

$\bar{J}'R_{0t} &=& \psi{\beta}'R_{1t} +\bar{J}'\hat{{\epsilon}}_t \ J_{\bot}'R_{0t}&=& J_{\bot}'\hat{{\epsilon}}_t$

where $\bar{J}=J(J'J)^{-1}$ and $J_{\bot}$ is orthogonal to J such that $J_{\bot}'J=0$ .

Define

$\Omega_{JJ_{\bot}}=\bar{J}'\Omega J_{\bot} {\rm and} \Omega_{J_{\bot}J_{\bot}}=J_{\bot}'\Omega J_{\bot}$

and let $\omega=\Omega_{JJ_{\bot}}\Omega_{J_{\bot}J_{\bot}}^{-1}$ .Then $\bar{J}'R_{0t}$ can be written

$\bar{J}'R_{0t}=\psi{\beta}'R_{1t} + \omega J_{\bot}'R_{0t} + \bar{J}'\hat{{\epsilon}}_t - \omega J_{\bot}' \hat{{\epsilon}}_t$

Using the marginal distribution of $J_{\bot}'R_{0t}$ and the conditional distribution of $\bar{J}'R_{0t}$ , the new residuals are computed as

$\tilde{R}_{Jt} &=& \bar{J}'R_{0t} - S_{JJ_{\bot}} S_{J_{\bot}J_{\bot}}^{-1}J_... ...e{R}_{1t} &=& R_{1t} - S_{1J_{\bot}} S_{J_{\bot}J_{\bot}}^{-1}J_{\bot}'R_{0t}$

where

$S_{JJ_{\bot}}=\bar{J}'S_{00}J_{\bot}, S_{J_{\bot}J_{\bot}}=J_{\bot}'S_{00}J_{\bot}, {\rm and } S_{J_{\bot}1}=J_{\bot}'S_{01}$

In terms of $\tilde{R}_{Jt}$ and $\tilde{R}_{1t}$ , the MLE of ${\beta}$ is computed by using the reduced rank regression. Let

$S_{ij{.}J_{\bot}}=\frac{1}T\sum_{t=1}^T\tilde{R}_{it} \tilde{R}_{jt}', {\rm for}i,j=1,J$

Under the null hypothesis $H_0: {\alpha}=J\psi$ , the MLE $\tilde{{\beta}}$ is computed by solving the equation

$|\rho S_{11{.}J_{\bot}} - S_{1J{.}J_{\bot}}S_{JJ{.}J_{\bot}}^{-1} S_{J1{.}J_{\bot}}|=0$

Then $\tilde{{\beta}}=(v_1, ... , v_r)$ , where the eigenvectors correspond to the r largest eigenvalues. The likelihood ratio test for $H_0: {\alpha}=J\psi$ is

$T\sum_{i=1}^r\log\{(1-\rho_i)/(1-\lambda_i)\} \stackrel{d}{arrow} \chi^2_{r(k-m)}$

The test of weak exogeneity of y_2t is the special case of the test ${\alpha}=J\psi$ , considering J=(I_k₁,0)'. Consider the previous example with four variables ( m_t, y_t, i_t^b, i_t^d ). If r=1, you formulate the weak exogeneity of (y_t,i_t^b,i_t^d) for m_t as J=[1, 0, 0, 0]' and the weak exogeneity of i_t^d for (m_t, y_t, i_t^b) as J = [I₃,0]'.

The following statements test the weak exogeneity of other variables:

   proc varmax data=simul2;
      model y1 y2 / p=2 ecm=(rank=1 normalize=y1);
      cointeg rank=1 exogeneity; 
   run;

The VARMAX Procedure

Tests of Weak Exogeneity of Each of Variables
Variable	Chi-Square	DF	Prob>ChiSq
y1	53.46	1	<.0001
y2	8.76	1	0.0031

Figure 4.45: Testing of Weak Exogeneity (EXOGENEITY option)

Figure 4.37 shows that each variable is not the weak exogeneity of other variable.

Forecasting of the VECM

Consider the cointegrated moving-average representation of the differenced process of y_t

$\Delta{y}_t={\delta} + \Psi(B){\epsilon}_t$

Assume that y₀ = 0. The linear process y_t can be written

$y_t={\delta}t + \sum_{i=1}^t\sum_{j=0}^{t-i}\Psi_j{\epsilon}_i$

Therefore, for any l > 0,

$y_{t+l}={\delta}(t+l) + \sum_{i=1}^t\sum_{j=0}^{t+l-i}\Psi_j{\epsilon}_i + \sum_{i=1}^l\sum_{j=0}^{l-i}\Psi_j{\epsilon}_{t+i}$

The l-step-ahead forecast is derived from the preceding equation:

$y_{t+l| t}={\delta}(t+l) + \sum_{i=1}^t\sum_{j=0}^{t+l-i}\Psi_j{\epsilon}_i$

Note that

$\lim_{larrow \infty} {\beta}'y_{t+l| t}=0$

since $\lim_{larrow \infty}\sum_{j=0}^{t+l-i}\Psi_j=\Psi(1)$ and ${\beta}'\Psi(1)=0$ . The long-run forecast of the cointegrated system shows that the cointegrated relationship holds, though there might exist some deviations from the equilibrium status in the short-run. The covariance matrix of the predict error $e_{t+l| t}=y_{t+l}-y_{t+l| t}$ is

$\Sigma(l)=\sum_{i=1}^l[(\sum_{j=0}^{l-i}\Psi_j)\Sigma (\sum_{j=0}^{l-i}\Psi_j')]$

When the linear process is represented as a VECM(p) model, you can obtain

$\Delta{y}_t=\Pi{y}_{t-1} + \sum_{j=1}^{p-1} \Phi^{*}_j\Delta{y}_{t-j} + {\delta} + {\epsilon}_t$

The transition equation is defined as

$z_{t}=F z_{t-1} + e_{t}$

where $z_t=(y_{t-1}',\Delta{y}_{t}', \Delta{y}_{t-1}', ... ,\Delta{y}_{t-p+2}')'$ is a state vector and the transition matrix is

$F=[ I_k & I_k & 0 & ... & 0 \ \Pi &(\Pi+\Phi^*_1)&\Phi^*_2 & ... &\Phi^*_{... ... & 0 \ \vdots & \vdots & \vdots & \ddots & \vdots \ 0 & 0 & ... & I_k & 0 \ ]$

where <U>0U> is a k-dimensional zero vector. The observation equation can be written

$y_t=\delta t + H z_t$

where H = [I_k,I_k,0, ... ,0].

The l-step-ahead forecast is computed as

$y_{t+l| t}=\delta (t+l) + H F^l z_t$

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