Central puzzles in international macroeconomics are why fluctuations of the real exchange rate are so volatile with respect to other macroeconomic variables, and the contradiction of efficient risk-sharing. Several theoretical contributions have evaluated alternative forms of pricing under nominal rigidities along with different asset markets structures to explain real exchange dynamics. In this paper, we use a Bayesian approach to estimate a standard two-country New Open Economy Macroeconomics (NOEM) using data for the United States and the Euro Area, and perform model comparisons to study the importance of departing from the law of one price and complete markets assumptions. Our results can be summarized as follows. First, we find that the baseline model does a good job in explaining real exchange rate volatility, but at the cost of implying too high volatility in output and consumption. Second, the introduction of incomplete markets allows the model to better match the volatilities of all real variables. Third, introducing sticky prices in local currency pricing (LCP) improves the fit of the baseline model, but not by as much as by introducing incomplete markets. Finally, we show that monetary shocks have played a minor role in explaining the behaviour of the real exchange rate, while both demand and technology shocks have been important.