A new approach to using sign restrictions for identifying market models

Sign restriction identification provides us with well-behaved market models. However, they are set-identified: there are many structural models that are consistent with both the observed data and imposed sign restrictions. This issue has raised many critics against using sign restrictions. Operationally, structural models are usually recovered using randomly drawn rotation matrices.  We show that, for a given reduced form estimation, there is a unique locus of slopes (elasticities) describing all the possibles models satisfying sign restrictions. Such a locus is invariant to the type of rotation performed (clockwise or counterclockwise) or the variables’ order used for the Cholesky decomposition. This representation also highlights that rotations do not generate probability distributions of structural parameters, but a deterministic set of structural models. Consequently, confidence bands associated with the rotation process, or the structural model selection, are pointless. In this setting, we also put forward a new criterion for model selection. This criterion, named the Rotation Minimization, chooses the model that has more stable slopes in the neighborhood of the optimal rotating angle. Those slopes also have a straightforward interpretation and generate variance decompositions that are easily recovered without engaging in any stochastic simulation.