We use a Bayesian approach to estimate a standard two-country New Open Economy Macroeconomics model using data for the United States and the euro area, and we 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 overestimating 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 improves the fit of the baseline model but does not improve the fit as much as introducing incomplete markets. Finally, we show that monetary shocks have played a minor role in explaining the behavior of the real exchange rate, while both demand and technology shocks have been important.