We consider the 
-dimensionaland integrated of one VAR(p) process of 
given by
or
where
,
and 
is an 
vector of an independently and identically distributed disturbance term with a finite variance
, where 
denotes an 
-dimensional identity matrix and
.
Further we assume the cointegration of Model (1) (e.g., Johansen, 1995) as:
AssumptionWe suppose
, where 
and 
are 
matrices of the full-column rank
.
Note that Model (2) may be written in an ECM as
under Assumption 2.1, where
.
Model (3) obviously consists of all stationary variables of 
and 
under Assumption 2.1. We are now interested in the dynamic interaction between these variables 
and
. In this regard, we may apply the results of Pesaran and Shin (1996) and Hansen (2005). However Sims’ (1980) VAR approach is sometimes convenient for practical reasons. For instance, the optimal forecasting of the k-th-period-ahead cointegration error 
is executed mechanically in a VAR system.
Thus, we now show that Model (3) may be represented as a stationary VAR model of a part of variables 
and 
when there is an ECM.
of the rank
, where 
is
. Then a confirmable non-singular square matrix can be defined as
.
transforms the VAR variable 
into a variable 
of 
and cointegration error
.
.
is the explanatory variable of 
in a cointegration relation as
.We then transform the above VAR model (1) of 
into a VAR model of the variable
. In particular, we multiply the 
on the left of Model (1) and modify the VAR coefficients to get the following VAR model:
for 
and
.
has a Gaussian distribution. However, we cannot determine the correct model by simply analyzing observed data.
and
.
in Equation (5) as follows:
is
, 
is 
and
.
in the third equality of (6) by using 
and
, where 
under Assumption 2.1. For this, note that
and 
in 
and
. Thus, we show how we may transform Model (5) of 
into a VAR model of a purely stationary variable
.
and 
.
:
(17)
from Theorem 3.2, and 
does not appear in Equation (13). If these restrictions (17) are not exploited and the coefficients 
is estimated with errors, then the efficiency of dynamic analyses by using these estimated coefficients may be reduced. For instance, if we provide forecasts using a VAR model, then forecasting mean squared error may be increased not by using restrictions (17). Campbell and Shiller (1987, Equation (5)) used the system (13) without referring to the VAR model (1) of level variable
, the rank deficiency of matrix 
in Assumption 2.1, and the restriction of Theorem 3.2 as 
of the coefficient
.
is defined as 
from (13) if 
is invertible, where L denotes a time-lag operator and 
This is conformable to the result in Hansen (2005, Corollary1).
is also possible as
conformably and
:
and 
conformably, where
. Thus 
does not appear in Equation (21). Thus, the ECM (18) is written as a stationary VAR (2) model, as in (21). ■
for 
is known as long as the cointegration vector 
(and thus
) is known. Thus we may consistently estimate coefficients 
in Model (13) as 
with variables
, using ordinary least square (OLS) method. However remind that the variable 
should be excluded from this regression.
.
is not known and estimated as 
with super-consistency as 
[c.f., Johansen (1995)] where n is a sample number. Then 
should be replaced by
where
. Thus we may consistently estimate coefficients 
in Model (13) as 
with variables 
by using the OLS, where the variable 
should be excluded from the regression. Finally, we may conduct standard dynamic analyses by using the VAR model
may be considered as known 
because of its rapid convergence. See Hamilton (1994) for a nice review on this issue.
, 
and 
denote the log of the price level in the US (
), the log of the Japanese price level (
), and the nominal yen/dollar exchange rate (
), respectively. The LOOP states that 
and that the same goods should be sold at the same prices both at home and in foreign countries, where prices are converted into local prices at prevailing exchange rates. Here if the exchange rate 
is manipu lated as 
through state intervention, then it is possible that 
in the short run, where 
is the relative price of the foreign good. Then the cheaper price of the home country good may induce a trade surplus.
in (13), where 
and
.
by using the Model (13). An ECM (3) does not have an explicit VAR form, and thus, it is not convenient for conducting such impulse response analysis. For this, we employ monthly data for the period from January 1998 to December 2008. We excluded the recent global financial crisis because of a possible structural break.
, 
and
. The results of the KPSS test do not reject the null hypothesis of stationarity at the 1% level for the cointegration error 
in the yen/dollar exchange rate. However, the null hypothesis of a unit root cannot not be rejected for the other variables. In a VAR model of
, we obtained order 2 by using the Akaike information criterion, and the results of the Johansen test reject the null hypothesis of no cointegration at the 1% level. The possible number of ordering for the identification of structural shocks is 6 for the trivariate model, and we investigate all these cases. However, changing the order has no influence on the results. Thus, we report only the result for the identification order 
. The impulses from the US price and the yen/dollar exchange rate have significant effects on the Yen/Dollar real exchange rate. See