By disaggregating price indices, it becomes apparent that the real exchange rate consists of the real exchange rate for a single good and a weighted sum of relative prices between goods. When applying a battery of panel unit root tests to this sum and its components, it is found that both the sum and the relative prices are non-stationary. This implies that PPP is invalid even if the LOP holds for all goods. The findings contrast with the result from panel unit root tests that real exchange rates as a whole are stationary. Several suggestions for solving the conflict are discussed.