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We investigate identification issues in DSGE models and their consequences for parameter estimation and model evaluation when the objective function measures the distance between estimated and model-based impulse responses. Observational equivalence, partial and weak identification problems are widespread and typically produced by an ill-behaved mapping between the structural parameters and the coefficients of the solution. Different objective functions affect identification and small samples interact with parameters identification. Diagnostics to detect identification deficiencies are provided and applied to a widely used model.