The VARMAX Procedure

I(2) Model

Test for I(2)

The VARX(,) model can be written in the error correction form:

$\displaystyle \Delta \mb {y} _{t} = \balpha \bbeta ’ \mb {y} _{t-1} + \sum _{i=1}^{p-1} \Phi ^*_ i \Delta \mb {y} _{t-i} + A D_ t + \sum _{i=0}^{s}\Theta ^*_ i\mb {x} _{t-i} + \bepsilon _ t$

Let $\Phi ^* = I_ k - \sum _{i=1}^{p-1} \Phi ^*_ i$ .

If $\balpha$ and $\bbeta$ have full-rank , and $rank(\balpha ’_{\bot } \Phi ^* \bbeta _{\bot }) =k-r$ , then $\mb {y} _{t}$ is an process.

If the condition $rank(\balpha ’_{\bot } \Phi ^* \bbeta _{\bot }) =k-r$ fails and $\balpha ’_{\bot } \Phi ^* \bbeta _{\bot }$ has reduced-rank $\balpha ’_{\bot } \Phi ^* \bbeta _{\bot }=\bxi \bm {\eta }’$ where $\bxi$ and $\bm {\eta }$ are $(k-r)\times s$ matrices with $s\leq k-r$ , then $\balpha _{\bot }$ and $\bbeta _{\bot }$ are defined as $k\times (k-r)$ matrices of full rank such that $\balpha ’\balpha _{\bot }=0$ and $\bbeta ’\bbeta _{\bot }=0$ .

If $\bxi$ and $\bm {\eta }$ have full-rank , then the process $\mb {y} _ t$ is , which has the implication of model for the moving-average representation.

$\displaystyle \mb {y} _ t = B_0 + B_1 t + C_2\sum _{j=1}^ t\sum _{i=1}^ j\bepsilon _ i + C_1\sum _{i=1}^ t\bepsilon _ i + C_0(B)\bepsilon _ t$

The matrices , , and are determined by the cointegration properties of the process, and and are determined by the initial values. For details, see Johansen (1995a).

The implication of the model for the autoregressive representation is given by

$\displaystyle \Delta ^2 \mb {y} _{t} = \Pi \mb {y} _{t-1} -\Phi ^* \Delta \mb {y} _{t-1} + \sum _{i=1}^{p-2} \Psi _ i \Delta ^2 \mb {y} _{t-i} + A D_ t + \sum _{i=0}^{s}\Theta ^*_ i\mb {x} _{t-i} +\bepsilon _ t$

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 cointegrated model is given by the following parameter restrictions:

$\displaystyle H_{r,s}\colon \Pi =\balpha \bbeta ’\; \mbox{and}\; \balpha _{\bot }’\Phi ^* \bbeta _{\bot } = \bxi \bm {\eta }’$

where $\bxi$ and $\bm {\eta }$ are $(k-r)\times s$ matrices with $0\leq s \leq k-r$ . Let represent the model where $\balpha$ and $\bbeta$ have full-rank , let $H_{r,s}^0$ represent the model where $\bxi$ and $\bm {\eta }$ have full-rank , and let $H_{r,s}$ represent the model where $\bxi$ and $\bm {\eta }$ have rank $\leq s$ . The following table shows the relation between the models and the models.

Table 36.2: Relation between the and Models


$r \backslash k-r-s$	k		k-1		$\cdots$
0	$H_{00}$	$\subset$	$H_{01}$	$\subset$	$\cdots$	$\subset$	$H_{0,k-1}$	$\subset$	$H_{0k}$		$H_{0}^0$
1			$H_{10}$	$\subset$	$\cdots$	$\subset$	$H_{1,k-2}$	$\subset$	$H_{1,k-1}$		$H_{1}^0$
$\vdots$							$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$
							$H_{k-1,0}$	$\subset$	$H_{k-1,1}$		$H_{k-1}^0$

Johansen (1995a) proposed the two-step procedure to analyze the model. In the first step, the values of $(r, \balpha , \bbeta )$ are estimated using the reduced rank regression analysis, performing the regression analysis $\Delta ^2\mb {y} _{t}$ , $\Delta \mb {y} _{t-1}$ , and $\mb {y} _{t-1}$ on $\Delta ^2\mb {y} _{t-1},\ldots ,\Delta ^2\mb {y} _{t-p+2},$ and . This gives residuals $R_{0t}$ , $R_{1t}$ , and $R_{2t}$ , and residual product moment matrices

$M_{ij} = \frac{1}{T} \sum _{t=1}^ TR_{it}R_{jt}’ ~ ~ \mr {for~ ~ } i,j=0,1,2$

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

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

where $M_{ij\mb {.} 1} = M_{ij} - M_{i1}M_{11}^{-1}M_{1j}$ for .

In the second step, if $(r, \balpha , \bbeta )$ are known, the values of $(s, \bxi , \bm {\eta })$ are determined using the reduced rank regression analysis, regressing $\hat{\balpha }_{\bot }’\Delta ^2\mb {y} _{t}$ on $\hat{\bbeta }_{\bot }’\Delta \mb {y} _{t-1}$ corrected for $\Delta ^2\mb {y} _{t-1},\ldots ,\Delta ^2\mb {y} _{t-p+2},D_ t$ , and $\hat{\bbeta }’\Delta \mb {y} _{t-1}$ .

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

$\displaystyle |\rho M_{\bbeta _{\bot }\bbeta _{\bot }\mb {.} \bbeta } - M_{\bbeta _{\bot }\balpha _{\bot }\mb {.} \bbeta } M_{\balpha _{\bot }\balpha _{\bot }\mb {.} \bbeta }^{-1} M_{\balpha _{\bot }\bbeta _{\bot }\mb {.} \bbeta }| = 0$

where

$\displaystyle M_{\bbeta _{\bot }\bbeta _{\bot }\mb {.} \bbeta }$	$\displaystyle =$	$\displaystyle \bbeta _{\bot }’(M_{11} - M_{11}\bbeta (\bbeta ’M_{11}\bbeta )^{-1}\bbeta ’M_{11})\bbeta _{\bot }$
$\displaystyle M_{\bbeta _{\bot }\balpha _{\bot }\mb {.} \bbeta }’$	$\displaystyle =$	$\displaystyle M_{\balpha _{\bot }\bbeta _{\bot }\mb {.} \bbeta } ~ =~ \bar{\balpha }_{\bot }’(M_{01} - M_{01}\bbeta (\bbeta ’M_{11}\bbeta )^{-1}\bbeta ’M_{11})\bbeta _{\bot }$
$\displaystyle M_{\balpha _{\bot }\balpha _{\bot }\mb {.} \bbeta }$	$\displaystyle =$	$\displaystyle \bar{\balpha }_{\bot }’(M_{00} - M_{01}\bbeta (\bbeta ’M_{11}\bbeta )^{-1}\bbeta ’M_{10})\bar{\balpha }_{\bot }$

where $\bar{\balpha }=\balpha (\balpha ’\balpha )^{-1}$ .

The solution gives eigenvalues $1>\rho _1>\cdots >\rho _ s>0$ and eigenvectors $(v_1,\ldots , v_ s)$ . Then, the ML estimators are

$\displaystyle \hat{\bm {\eta }}$	$\displaystyle =$	$\displaystyle (v_1,\ldots , v_ s)$
$\displaystyle \hat{\bxi }$	$\displaystyle =$	$\displaystyle M_{\balpha _{\bot }\bbeta _{\bot }\mb {.} \bbeta }\hat{\eta }$

The likelihood ratio test for the reduced rank model $H_{r,s}$ with rank $\leq s$ in the model $H_{r,k-r} = H_ r^0$ is given by

$\displaystyle Q_{r,s} = -T\sum _{i=s+1}^{k-r}\log (1-\rho _ i), ~ ~ s=0,\ldots ,k-r-1$

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

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

The last two columns in Figure 36.60 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 the 0.05 significance level because the test statistic of 0.5552 is smaller than the critical value of 3.84. Now, look at the row associated with . Compare the test statistic value, 211.84512, to the critical value, 3.84, for the cointegrated order 2. There is no evidence that the series are integrated order 2 at the 0.05 significance level.

Figure 36.60: Cointegrated I(2) Test (IORDER= Option)

The VARMAX Procedure

Cointegration Rank Test for I(2)
r\k-r-s	2	1	Trace of I(1)	5% CV of I(1)
0	720.40735	308.69199	61.7522	15.34
1		211.84512	0.5552	3.84
5% CV I(2)	15.34000	3.84000