Economists are often confronted with the choice between a completely specified, yet approximate model and an incomplete model that only imposes a set of credible behavioral conditions. We offer a reconciliation of these approaches and demonstrate its usefulness for estimation and economic inference. The approximate model, which can be structural or statistical, is distorted such that to satisfy the behavioral conditions. We provide the asymptotic theory and Monte Carlo evidence, and illustrate that counterfactual experiments are possible. We apply the methodology to the model of long run risks in aggregate consumption (Bansal and Yaron, 2004), where the auxiliary model is generated using the Campbell and Shiller (1988) approximation. Using US data, we investigate the empirical importance of the neglected non-linearity. We find that distorting the model to satisfy the original equilibrium condition is strongly preferred by the data and substantially improves the identification of the structural parameters. More importantly, it completely
overturns key qualitative predictions of the linear model. Risk premia are endogenously time varying and stochastic volatility had a minor explanatory power during the Great Recession.