The VARMAX Procedure

I(2) Model

The VARX($p$,$s$) 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 $r$, and $ rank(\balpha ’_{\bot } \Phi ^* \bbeta _{\bot }) =k-r$, then $\mb {y} _{t}$ is an $I(1)$ 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 $s$, then the process $\mb {y} _ t$ is $I(2)$, which has the implication of $I(2)$ 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 $C_1$, $C_2$, and $C_0(B)$ are determined by the cointegration properties of the process, and $B_0$ and $B_1$ 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

$\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 $I(2)$ 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 $H_ r^0$ represent the $I(1)$ model where $\balpha $ and $\bbeta $ have full-rank $r$, let $H_{r,s}^0$ represent the $I(2)$ model where $\bxi $ and $\bm {\eta }$ have full-rank $s$, and let $H_{r,s}$ represent the $I(2)$ model where $\bxi $ and $\bm {\eta }$ have rank $\leq s$. The following table shows the relation between the $I(1)$ models and the $I(2)$ models.

Table 36.2: Relation between the $I(1)$ and $I(2)$ Models






$r \backslash k-r-s$





$\cdots $


$ 1$




$\subset $


$\subset $

$\cdots $

$\subset $


$\subset $







$\subset $

$\cdots $

$\subset $


$\subset $




$\vdots $


$\vdots $

$\vdots $

$\vdots $

$\vdots $

$\vdots $




$\subset $




Johansen (1995a) proposed the two-step procedure to analyze the $I(2)$ 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 $D_ t$. This gives residuals $R_{0t}$, $R_{1t}$, and $R_{2t}$, and residual product moment matrices

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

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 $D_ t$, 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 $i,j=0,2$.

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  $


$\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));

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 $r=1$. 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