I have a question about trying to estimate impluse-response functions
for stationary I(0) variables when the other variables in the system are
I(1) and cointegrated.

My basic problem is that I have four variables, Y1 to Y4, where Y1 is
I(0) and Y2,Y3 and Y4 are all I(1) and cointegrated. The primary
variable of interest is Y1.

I can run a VAR but this will suffer from cointegration bias, since the
ECMs of the I(1) variables are not modelled. I can also run a VEC, but
the variable of interest is Y1 and this will only be included as an
exogenous or deterministic variable and I won't be able to calculate the
impulse response function of this variable.

A colleague has suggested a work-around, based on the 2sls principle. My
gut feeling is that this is very wrong, but I can't put my finger on
where or why. I was hoping someone might have an answer.

The work-around is based on the idea of estimating the ECMs in a VEC
model and then plugging the fitted values into a VAR model. This will
supposedlyenable the cointegration between the I(1) variables to be
explicitly incorporated into the VAR and yet enable the I(0) variable to
be on the LHS of the system.

The Procedure:

What I do is run the I(1) variables by themselves in a VEC. The Johansen
ML test indicates that r=3, which is a problem since we need r<=n-1.
However, this is not the major concern at this stage.
The VEC is run on the I(1) variables alone - this results in only the
ECMs being on the RHS of the system, and the fitted values of the I(1)
variables will thus only incorporate the error correction component of
the DGP.
The fitted values are then saved and a VAR is then run on all the four
variables with the lagged fitted values from the VEC and other
deterministic variables (seasonal dummies etc.) included as exogenous
variables. This two-step procedure allows the ECM effects to be included
in the VAR framework while allowing the I(0) variable to be endogenous.

My question is whether this is legitimate, and/or does someone else have
a better method?

Following on from this is another question, of a more practical nature
and is a result of the actual dataset.

The VEC results indicate that the ECMs explain very little of the DGP
for the I(1) variables - this is to be expected as only the ECMs are
regressed on the RHS of the VEC system. The R-squareds are very low
(0.35 - 0.42). This is okay.
The problem is when the fitted values from the VEC are plugged into the
VAR. The t-tests for the fitted values indicate that the ECMs are not
significant. Dropping the fitted values from the VAR and re-estimating
the VAR results in the R-squareds of the VAR falling from, for example,
0.91-0.99 to 0.90-0.985. This indicates to me that the ECMs are not
significant determinates of the DGP and that the VAR without the fitted
values does not suffer substantial cointegration bias from the lack of
incorporation of the ECMs. The implication is that I can run the VAR
without worring about cointegration bias and then get the
impluse-response functions that I require.

There seems to be a dichotomy between what the Johansen ML test for
cointegration is suggesting (that there are 3 significant cointegrating
vectors) and the practical application shown above. My worry is that the
result obtained above is an artifact of the (dodgy or not) two step
procedure.

I was wondering if anyone had any thoughts on the matter.

Thanks

Tim Purcell
Department of Economics
University of Queensland
St. Lucia 4072
Australia

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