The RATS Software Forum

Hi Tom,

I have a question with regard to unit root or cointegration test.

When testing for unit roots or cointegration in a VAR with exogenous stochastic variables, is it necessary to take the presence of these variables into account? Most of the econometric textbooks haven't mentioned these cases....And do you know how to implement the resulting testing procedure in RATS?

Many thanks!

MC

Unit root is a property of a series, not of a regression; similarly, cointegration is a property of a set of series. If, for instance, you run the regression

y(t)-y(t-1)=alpha+beta*y(t-1)+gamma*z(t)+e(t)

where z is some exogenous stochastic variable, you can't use beta=0 as a test for a unit root in y. If y and z are cointegrated, beta could be anything. If z is I(1) but not cointegrated with y, you have a spurious regression situation - the distribution of all the coefficients are non-standard. If z is I(0), then it can be washed into the lag structure on an ADF test or into the correlated errors in a PP test.

Hi Tom,

So if z(t) is I(0), then is it true that there will be no need to specify it in @dfunit or @ppunit? (@dfunit or @ppunit only allow constant or trend term in the option det.)

How about the case of cointegration? I know from Mosconi and Rahbek (1999) that the presence of other stationary regressors (i.e. z(t)) will have an impact on the asymptotic distribution of the likelihood ratio test on reduced rank (on beta in your example).....does Rats take that into account?

Thank you so much!

MC

CATS includes adjustments of the cointegration test critical values for exogenous variables. That requires running simulations, since the adjustment is different for each. I think most people just let I(0) shift variables wash into the I(0) remainder in the cointegrating relationship.

Dear Tom,

Thanks for your reply! but I am not sure about "letting I(0) shift variables wash into the I(0) remainder in the cointegrating relationship" mean?

Consider the following:

y(t) - y(t-1) = alpha*beta*y(t-1) + z(t) + e(t) ---------------------------(1)

where z(t) is the exogenous variable and e(t) is the error term.....

So you think I should put z(t) into the cointegration relationship (i.e. alpha*beta*{y(t-1)+z(t)} or just leave it there as a conditioning variable in equation (1)?

On the other hand, after the estimation of the above VECM, can I decompose the error term using standard SVAR technique? Will the presence of exogenous variables pose any issue?

Many thanks,

MC

A set of variables y is cointegrated if beta'y(t) is I(0). Not 0, not white noise. I(0). A very broad category of processes. The overall process can be written as:

y(t)-y(t-1)=alpha beta' y(t) + v(t)

where v(t) is another I(0) process. In practice, the "residual" v(t) is a VAR of unknown form. It's captured by adding a sufficient number of lags of y(t)-y(t-1) to the VECM. Suppose that you think that beta'y(t)=gamma'Z(t)+w(t) where Z(t) is an "exogenous" I(0) process and w(t) is another I(0) process, presumably with a smaller variance. So now you would be estimating

y(t)-y(t-1)=alpha (beta'y(t)-gamma'z(t)) + (v(t)+alpha gamma'z(t))

The residual in this formulation (v(t)+alpha gamma'z(t)) is still a VAR of unknown form, which is still captured by adding a sufficient number of lags of y(t)-y(t-1) to the VECM. In other words, you really haven't changed things all that much, other than to complicate the test statistics.

When exogenous variables are included, it will almost always be variables which are not I(0); instead, they're shift variables and other types of deterministic dummies. Addition of those will also require adjusting the test statistics (CATS includes this as an option), but if those are appropriate, there is no real alternative.