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The marginal utility of wealth in incomplete markets small open economy models follows a unit root process. I study the nonlinear properties of devices often used to remove the unit root and I find that they generate different dynamics when matching emerging markets. Models with endogenous discount factors reinforce consumption response to shocks and increase the countercyclicality of the trade balance to output ratio. Conversely, models with debt frictions ameliorate the responses of consumption and trade balance. Hence, to generate dynamics similar to those in emerging economies, the debt frictions need to be small, inducing a near unit root behavior in their Euler equations. This difference across models is hidden when matching developed economies because of consumption smoothing and the mild countercyclicality of the trade balance.