1) Linear approximation is for every system of equations. It is just that the approximation error has been shown to be unacceptably large for many economic models. The nonlinearities in the dynamics matter.

2) Further, many models have multiple stationary equilibria or have instabilities such as unbalanced growth or a more fundamental instability such as unsustainable fiscal policy. Standard methods cannot even obtain a solution in these cases.

Linearization about the current state (or CSL, current state linearization) is an iteratively updating linear approximation about the current state. This method has the computational speed of linear approximation, while still accounting for the nonlinear dynamics of the underlying problem. There are some potential issues with this method.

A) We have not proven a theoretical guarantee that CSL will converge to the truth asymptotically. That said, CSL performs very well in all the applications we check in the paper.

B) When you are linearizing around a point other than the steady state, the method of undermined coefficients has a strange problem. It is not clear which eigenvalues to use in order to find the P matrix. Linearizing around the steady-state, you just choose the stable roots. However, away from the steady-state, it is likely that the linear approximation is not stable. We therefore have an issue about which roots to use. But again, in the paper, all the examples we checked used the stable roots and performed very well in terms of Euler errors and in terms of deviations from the true solution, both in impulse response functions and across simulations.

We think this approach is very valuable for simulating models that have the instabilities that are so important to many current research questions.

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