Abstract
A famous puzzle holds that real exchange rates carry unit roots and that their deviations take three to five years to die, far too long for sticky prices to explain. We show the puzzle is not about persistence but about the equilibrium: the fair value itself drifts smoothly, and imposed as a constant it is read as a unit root. A microeconomic buyer-seller difference equation and a macroeconomic Taylor loss-minimisation problem both deliver this moving fair value, and both map into one estimable smooth-transition (arctangent) trend, so the economic model and the econometric test share a single object. Around it, sixteen of twenty OECD real exchange rates are stationary, six of ten independent currency areas after euro consolidation; nonlinear ESTAR adjustment adds no rejection; a formal rule separates permanent from temporary shifts in parity; and the median half-life of deviations falls from years to five months.

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