We study the fluctuations of exchange rates and consumer prices in two small open economies, Sweden and Canada, using a structural Bayesian VAR. Four domestic and two global shocks are identified through zero and sign restrictions. For both economies, we find that the main driver of consumer price inflation is the global demand shock. A negative global demand shock is not only deflationary for the small open economy, but also depreciates its currency. Hence, the observed exchange rate pass-through following this shock is of opposite sign to what is usually expected. Finally, exogenous shocks to the exchange rate are less important drivers of exchange rate movements than in many other structural models.