Is it possible to get "spurious cointegration"? If two time series $X$ and $Y$ exhibit a cointegration relationship while theory would suggest that this absolutely makes no sense, can we speak of "spurious cointegration" (like we sometimes speak of "spurious correlation", in a different context)?
One could then argue that the empirical results contradict theory for the sample period but that it does not mean the two time series are necessarily related in the long run ($\neq$ "real" cointegration).
Is this reasoning valid? 
 A: The "spurious regression" phenomenon, has to do not with a contradiction between "theory" and "statistically measured relation", but with the appearance of a statistically significant relation when in reality no such statistical relationship exists, due to the mathematical properties of the processes involved and the technical weaknesses of the statistical procedure used in light of these properties. What "causal/association/behavioral theory" has to say about the two variables is not a criterion to characterize a statistically measured relationship as spurious. 
Analogously, a "spurious co-integration" would be the case where  


*

*We obtain a cointegrating vector that appears to make the linear combination of the two variables stationary (based on some statistical tests) but 

*The linear combination is not stationary, "in reality"
Such a case could in principle happen due to limitations in the available data. Namely with a small sample, our stationarity tests may tell us that we have stationarity, but in reality, non-stationarity may exist but be "slowly emerging", and so would need more data to be detected.
Again, what theory has to say about the two variables does not enter the picture at that level.
Of course, if one feels that one has strong theoretical arguments regarding the non-association of the two variables, one can always argue "this cannot be happening, bring more data and eventually statistics will also show that no co-integration exists, validating the theory", but here, the reverse argument is also valid: "data appears to disprove your theory, try to figure out why and change/enhance the theory".
