A variety of international macroeconomic models predict a relationship between the real exchange rate and consumption. The empirical evidence in favor of such a relationship is limited, the so-called Backus and Smith puzzle. In this paper, we extend the analysis to allow for nonlinear dynamics and volatility changes across exchange rate regimes. Our findings suggest that long-run relationships in line with standard international business cycle models do exist for many Organization for Economic Co-operation and Development (OECD) countries. Further, Monte Carlo experiments illustrate that the nonlinear models can generate the Backus and Smith and the exchange rate disconnect puzzles. In this paper, we also contribute to the nonlinear real exchange rate literature by establishing a theoretical relationship between volatility and persistence. In accordance with the theoretical results, our empirical findings suggest that the increase in volatility in the post-Bretton Woods era is associated with relatively fast mean reversion of the real rate toward its equilibrium value.