Abstract This paper extends the real interest differential (RID) model of Frankel [Am. Econ. Rev. 69 (1979) 610] by introducing Markov regime switches for three exchange rates, over the years 1973–2000. Evidence of a non-linear relationship between exchange rates and underlying fundamentals is provided. It turns out that one of the estimated regimes represents exactly the RID case. The key fundamental which determines regimes turns out to be the interest rate. The established relationship is shown to be stable in several respects: regimes are highly persistent, provide a much better description of the data than alternatives and are robust towards several modifications.