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I am testing for the existence of purchasing power parity (PPP) relationships using cointegration tests, among others, and I am a bit tempted to test whether or not my cointegrating vectors are equal to the canonical vector, that is $r=(1,-1,1)$. However, I am a bit unsure whether or not this makes sense. Many papers do that, but I am not convinced. Absolute purchasing power parity in logs is $s=p-p^*$, but this equivalence only holds in theory if you use actual price levels, not price indices: there is no theory of purchasing power parity that says one times the log of a domestic price index minus one times the log of a foreign price index should be equal to the log of the spot exchange rate. That means no theory prescribes the canonical vector as an equilibrium relationship when dealing with price indices. What do you think would be the right procedure to follow while testing on price indices? Would you test for the canonical vector, or is it superfluous?

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