I have estimated optimal monetary policy rule for economy A. I want to check how the current rule used by the central bank deviates from the optimal. So I guess I have to do the following.

If I believe the central bank’s current rule is, for example, i_t = \rho_1 i_{t1} + \rho_2 \pi_{t1} + \rho_3 y_{t1} + \beta_2 \pi_{t} + \beta_3 y_{t}. Then I should also use this reaction function to approximate the optimal rule, right? So that I can compare the current rule to the approximated optimal rule.

Where should that belief come from? Should i_t = \rho_1 i_{t1} + \rho_2 \pi_{t1} + \rho_3 y_{t1} + \beta_2 \pi_{t} + \beta_3 y_{t} be the reaction function that best fits historical data (I have not seen that argument yet in structural models)? Should it be based on other papers (I see this sometimes, e.g., we evaluate rules based on the specification in Clarida et al. (1998), Gali (2015), etc.)? Maybe we cannot know the true datagenerating process for the policy rate, but why so many specifications in (not only empirical models) but also structural models?
I am not sure I understand the exercise. Fully optimal monetary policy will not consist of a simple rule. Empirically, you can estimate a rule that approximates actual behavior. You can then compare it to the Ramsey planner’s behavior.
With respect to misspecification, have a look at
Yes. I can do that using the IRFs. But Ramsey’s optimal rule is not easy to implement, yeah? So I want to approximate it with some implementable rule and compare that implementable rule with the rule that approximates the actual behavior of the central bank.
For example, empirically, the actual rule can be, say, i_t = 0.58y_t + 1.6\pi_t. I can compare this to the optimal rule using IRFs. No problem. But if I want an implementable ‘optimal’ rule in the form of the actual rule (describing the behavior of the CB), then I need to use the specification i_t = ay_t + b\pi_t to approximate the optimal rule, right? Let’s say the approximated optimal implementable rule is i_t = 0.45y_t + 3.6\pi_t; then I can make statements like the central bank would need to raise the interest rate 3.6to1 to minimize its loss function.
So here, I want to compare two implementable rules, one that approximates the Ramsey Planner’s behavior and one that empirically approximates the central bank’s actual behavior. So I thought, maybe the specification I want to use to approximate the Ramsey planner behavior should be based on the one I use to approximate the actual behavior of the central bank.
In my mind, why we have so many osr (optimal simple rule) specifications is because they are based on empirical rules that approximates actual central bank behavior, of which we there many. No? I know sometimes, authors defend their osr specification by saying they borrowed it from another paper. But generally, how would you justify an osr specification? Like it can be based on the empirical specification that best fits the data? Thanks for the paper
The simple rules are meant to approximate the Ramsey solution. The open question is always how good that approximation is. Most of the literature seems to show that rather simple rules already provide a good fit. In that case, more complicated rules may not add that much of value and you may opt for a parsimonious specification. But that rather general statement may not be true in all cases.
A second step is about implementable rules, i.e. rules considering variables that the central bank can actually observe. For example, it’s easier to observe output than capital.
When you put both together (and potentially combine them with actual information on the central bank’s mandate), you can end up with rather simple rules like the one you describe.
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