I(2) Model

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The VARMAX Procedure

I(2) Model

The VAR(p) model can be written as the error correction form.

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

Let $\Phi^*=I_k - \sum_{i=1}^{p-1} \Phi^*_i$ .If ${\alpha}$ and ${\beta}$ have full rank r, and if

$rank({\alpha}'_{\bot} \Phi^* {\beta}_{\bot})=k-r$

then y_t is an I(1) process. If the condition $rank({\alpha}'_{\bot} \Phi^* {\beta}_{\bot})=k-r$ fails and ${\alpha}'_{\bot} \Phi^* {\beta}_{\bot}$ has reduced rank ${\alpha}'_{\bot} \Phi^* {\beta}_{\bot}={\xi} {\eta}'$ where ${\xi}$ and ${\eta}$ are (k-r)×s matrices with $s\leq k-r$ , ${\alpha}_{\bot}$ and ${\beta}_{\bot}$ are defined as k×(k-r) matrices of full rank such that ${\alpha}'{\alpha}_{\bot}=0$ and ${\beta}'{\beta}_{\bot}=0$ .If ${\xi}$ and ${\eta}$ have full rank s, then the process y_t is I(2), which has the implication of I(2) model for the moving-average representation.

$y_t=B_0 + B_1 t + C_2\sum_{j=1}^t\sum_{i=1}^j{\epsilon}_i + C_1\sum_{i=1}^t{\epsilon}_i + C_0(B){\epsilon}_t$

The matrices C₁, C₂, and C₀(B) are determined by the cointegration properties of the process, and B₀ and B₁ are determined by the initial values. For details, see Johansen (1995a).

The implication of the I(2) model for the autoregressive representation is given by

$\Delta^2 y_{t}=\Pi y_{t-1} -\Phi^* \Delta y_{t-1}+ \sum_{i=1}^{p-2} \Psi_i \Delta^2 y_{t-i} + A D_t + {\epsilon}_t$

where $\Psi_i=-\sum_{j=i+1}^{p-1} \Phi^*_i$ and $\Phi^*=I_k - \sum_{i=1}^{p-1} \Phi^*_i$ .

Test for I(2)

The I(2) cointegrated model is given by the following parameter restrictions:

$H_{r,s}: \Pi={\alpha}{\beta}'\;{and}\; {\alpha}_{\bot}'\Phi^* {\beta}_{\bot}={\xi} {\eta}'$

where ${\xi}$ and ${\eta}$ are (k-r)×s matrices with $0\leq s \leq k-r$ .Let H_r⁰ represent the I(1) model where ${\alpha}$ and ${\beta}$ have full rank r, let H_r,s⁰ represent the I(2) model where ${\xi}$ and ${\eta}$ have full rank s, and let H_r,s represent the I(2) model where ${\xi}$ and ${\eta}$ have rank $\leq s$ .The following table shows the relation between the I(1) models and the I(2) models.

Table 4.1: Relation between the I(1) and I(2) Models

					I(2)					I(1)
r \ k-r-s	k		k-1		...		1
0	H₀₀	$\subset$	H₀₁	$\subset$	...	$\subset$	H_0,k-1	$\subset$	H_0k	=	H₀⁰
1			H₁₀	$\subset$	...	$\subset$	H_1,k-2	$\subset$	H_1,k-1	=	H₁⁰
$\vdots$							$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$
k-1							H_k-1,0	$\subset$	H_k-1,1	=	H_k-1⁰

Johansen (1995a) proposed the two-step procedure to analyze the I(2) model. In the first step, the values of $(r, {\alpha}, {\beta})$ are estimated using the reduced rank regression analysis, performing the regression analysis $\Delta^2{y}_{t}$ , $\Delta y_{t-1}$ , and y_t-1 on $\Delta^2{y}_{t-1}, ... ,\Delta^2{y}_{t-p+2},D_t$ .This gives residuals R_0t, R_1t, and R_2t and residual product moment matrices

$M_{ij}=\frac{1}T \sum_{t=1}^TR_{it}R_{jt}' {\rm for} i,j=0,1,2$

Perform the reduced rank regression analysis $\Delta^2{y}_{t}$ on y_t-1 corrected for $\Delta y_{t-1}$ , $\Delta^2{y}_{t-1}, ... ,\Delta^2{y}_{t-p+2},D_t$ and solve the eigenvalue problem of the equation

$|\lambda M_{22{.}1} - M_{20{.}1}M_{00{.}1}^{-1}M_{02{.}1}|=0$

where M_ij.1 = M_ij - M_i1M₁₁^-1M_1j for i,j=0,2.

In the second step, if $(r, {\alpha}, {\beta})$ are known, the values of $(s, {\xi}, {\eta})$ are determined using the reduced rank regression analysis, regressing $\hat {{\alpha}}_{\bot}'\Delta^2{y}_{t}$ on $\hat {{\beta}}_{\bot}'\Delta{y}_{t-1}$ corrected for $\Delta^2{y}_{t-1}, ... ,\Delta^2{y}_{t-p+2},D_t$ and $\hat {{\beta}}'\Delta{y}_{t-1}$ .

The reduced rank regression analysis reduces to the solution of an eigenvalue problem for the equation

$|\rho M_{{\beta}_{\bot}{\beta}_{\bot}{.}{\beta}} - M_{{\beta}_{\bot}{\alpha}_{... ...\alpha}_{\bot}{.}{\beta}}^{-1} M_{{\alpha}_{\bot}{\beta}_{\bot}{.}{\beta}}|=0$

where

$M_{{\beta}_{\bot}{\beta}_{\bot}{.}{\beta}} &=& {\beta}_{\bot}'(M_{11} - M_... ... M_{01}{\beta}({\beta}'M_{11}{\beta})^{-1}{\beta}'M_{10})\bar{{\alpha}}_{\bot}$

where $\bar{{\alpha}}={\alpha}({\alpha}'{\alpha})^{-1}$ .

The solution gives eigenvalues $1\gt\rho_1\gt ... \gt\rho_s\gt$ and eigenvectors (v₁, ... , v_s). Then, the ML estimators are

$\hat{{\eta}} &=& (v_1, ... , v_s) \ \hat{{\xi}} &=& M_{{\alpha}_{\bot}{\beta}_{\bot}{.}{\beta}}\hat{\eta}$

The likelihood ratio test for the reduced rank model H_r,s with rank $\leq s$ in the model H_r,k-r = H_r⁰ is given by

$Q_{r,s}=-T\sum_{i=s+1}^{k-r}\log(1-\rho_i), s=0, ... ,k-r-1$

The following statements are to test the rank test for the cointegrated order 2:

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

The VARMAX Procedure

Trace of Cointegration Rank Test for I(2)
r\k-r-s	2	1	Trace_I1	CriticalValue_I1
0	720.40735	308.69199	61.75	15.34
1		211.84512	0.56	3.84
Critical	15.34000	3.84000
Value_I2

Figure 4.46: Cointegrated I(2) Test (IORDER= option)

The last two columns in Figure 4.38 explain the cointegration rank test with integrated order 1. The results indicate that there is the cointegrated relationship with the cointegration rank 1 with respect to a 0.05 significance level. Now, look at the row in the case of r=1. Compare the value to the critical value for the cointegrated order 2. There is no evidence that the series are integrated order 2 with a 0.05 significance level.

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