On the identifiability of Euler equation estimates under saddlepath stability

In all but the most trivial settings Euler equation estimation in saddlepath stable systems is faced with a fundamental identification problem: the Euler equation allows for an unstable root (of its characteristic equation), while the data used to estimate the Euler equation also obey the transversality condition, which rules out unstable roots. Thus, even if the model is true, then the data are completely uninformative with respect to the unstable root of the Euler equation. But ignorance of the unstable root implies that the parameters are not identified if the relationship between parameters and roots is one to one. We illustrate the issue using a linearized Euler equation and present an application with OECD consumption data