We show that empirical results concerning the behavior of floating exchange rates differ between otherwise identical cointegrated and non-cointegrated VAR models. In particular, virtually all ten-year movements in nominal exchange rates are due to fundamental supply and demand shocks when long run equilibrium relationships between the levels of the variables are included in the empirical specification. Another major difference between the models with the opposite implication for the shock creation versus shock absorption debate is that non-fundamental exchange rate shocks have much larger effects on output and inflation in the cointegrated models. Finally, impulse response functions in the first difference specification die out within a year whereas adjustment to long run equilibrium continues for up to ten years in the cointegrated models. Hence a correct specification of the long-run equilibrium dynamics of exchange rates is essential for capturing also short-run behavior of exchange rates.