Hi! Is it possible to include variables that might be cointegrated as exogenous variables in a VEC? For instance, I have cointegration between variable A and B, where A is my hypothesized explanatory variable. Now, variable C is also cointegrated with both of the other variables (C with A, C with B, and also with the VAR system of A, B and C). However, preferr to include C only as a exogenous variable to the A + B system. Is that valid? For instance, real house prices is cointegrated with both the profit share of functional incomes, and also with private debt/GDP. I wich to examine only one cointegrating relation, and impose real house prices as an exogenous variable in the VEC.
Strictly speaking in the VAR representation of the VEC the exogenous variable will always be only an additional regressor. The question is how to split the contributions of this regressor into the cointegration part and the adjustment part of the system. Or the other way round: if your exogenous variable is cointegrated with your endogenous variables, the system formed by them should be a stationary process. But if the endogenous system is already stationary itself in form of a cointegration equation, how can the same system appended by an additional variable still be stationary: well only if the exogenous process is stationary itself. But in this case there would be no value to include it into the cointegration system in the first place. If you have an additional variable, of which you believe that it would be also correlated with your set of endogenous variables, I would rather include it as an additional endogenous variable. Otherwise you might very well end up in a situation where you add non-stationary processes to an overall VEC system, which is supposed to be stationary already.
Ok, but what if my theory/hypothesis is finding a specific relationship between A and B, and I'm not interested in getting any more vectors with potential dependent variables? Should I just ignore the additional variable C? In one example I have five endogenous variables, with four cointegrating relations. Considering normalization/restrictions, it is very hard to interpret such a system - at least for me.
Btw, I'm playing around with the Rahbek/Mosconi(1998) approach, with critical values from Harbo(1998).
As I stated in an answer to a similar question there are two cointegrating relationships and one common trend in the variables A, B and C. If you estimate a vecm in the three variables you can test for weak exogeneity of the C variable. If it is exogenous you are still left with a two variable system. If it is not weakly exogenous you can not leave C out of the three variable system. There is some material on reducing a vecm system in the presence of weak exogeneity in Juselius, K (2006), The cointegrated Var model, OUP. David Hendry and his collaborators did a lot of research on weak exogeneity. If I remember right his book Dynamic Econometrics (1995), OUP, summarizes a lot of this theory.
There is an analysis of US Australian data in Johansen Soren (1995) which implements tests for weak exogeneity. Basically you identify the alpha and beta matrix, You then test that a row in the matrix corresponding to the exogenous variables is zero. The tests are probably not labeled as exogeneity tests but as zero tests on the alpha and beta. Such tests have more general use than simply as exogeneity tests.
These tests are implemented in CATS in RATS, gretl, the urca package in R, Eviews and probably in other software.
You probably need to do a bit of more background reading. A full answer would not be possible within the limitations imposed by a forum such as this.
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