Suppose I have two I(1) time series X and Y, and I want to know whether X and Y are "related" (for some definition of "related").
The standard cointegration approach defines relationship as cointegration, and says that X and Y are cointegrated if some linear combination of X and Y is stationary. To test whether X and Y are cointegrated, you perform a regression on X and Y, and test for stationarity of the residual errors.
It seems to me like another approach might be to difference the I(1) time series X and Y, to get new I(0) time series X' and Y', and to use a standard linear regression relationship test on X' and Y' (i.e., perform a regression to get Y' = aX' + b, and use a t-test to see whether a is significantly non-zero). You could then define X and Y to be related if X' and Y' pass this test.
Is this second approach valid, or do you get spurious relationships? What's the difference between this approach and the cointegration approach, or what are the advantages of the cointegration definition?