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I am reading through "Monetary Policy Shocks: What Have we Learned and to What End?" and am hoping someone here can offer a bit of clarity about a claim made within about how measurement error in data used by policy makers can lead to a violation of the recursiveness assumption:

For reference, equation (2.1): $S_t = f(\Omega_t) +\sigma_s \epsilon_t^s$ where $f()$ is a feedback rule that maps the information set, gamma, into action taken (systematic component of movements in the policy instrument) by the policy maker and $\sigma_s \epsilon_t^s$ represents an exogenous monetary policy shock.

The authors give this definition for the recursiveness assumption:

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Now, they lay out a simple enough equation for the monetary policy instrument that incorporates a feedback rule and allows for some type of exogenous monetary policy shock:

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Now, what I want to sort out is why the different states for the parameters within, given the assumption of classical measurement error, do/do not cause a violation of the recursiveness assumption:

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