Your textbook so far considered variables for cointegration that are integrated of the same order

For example, the log of consumption and personal disposable income might both be I(1) variables, and the error correction term would be I(0), if consumption and personal disposable income were cointegrated.
(a) Do you think that it makes sense to test for cointegration between two variables if they are integrated of different orders? Explain.
(b) Would your answer change if you have three variables, two of which are I(1) while the third is I(0)? Can you think of an example in this case?
What will be an ideal response?

Answer:
(a) To test for cointegration requires that the two variables have the same stochastic trend. If one variable is I(1) while the other is I(0), then obviously they do not have the same stochastic trend and therefore cannot be cointegrated.
(b) In this case there would possibly be cointegration between the two I(1) variables, but not between all three variables. This does not imply that the third variable could not enter into the relationship. Think, for example, about a money demand relationship between the (log of) real money balances, income, and the nominal interest rate. It may well be that in some samples the nominal interest rate is I(0), while real money balances and income are I(1). Finding real money balances and income to be cointegrated does not imply that the nominal interest rate does not enter the money demand function. There is simply no need for the interest rate to enter the cointegrating relation because it is I(0). The cointegrating relation only involves zero-frequency relationships between the first differences of real money balances and income, and the zero-frequency component of the first difference of the interest rate is non-existent.