Why would you want to do this? The equation you wrote down can easily be estimated using a OLS regression.

However, the literature has shown that the coefficients of Taylor rules are typically not identified, because of endogeneity (see e.g. Cochrane’s recent JPE paper). Thus, you would need to either use GMM or joint estimation of an equilibrium system. This would imply that you need to specify equations for y and pi which are NOT exogenous, but jointly determined.

First, I believe that almost nothing is exogenous stricto sensu.
Second, OLS would allow to estimate “(1-rho)*gamma”, not directly gamma, right? I would use NLS instead.

However, there two main reasons why I want to do this:

Not a lot of observations + idea of parameters value/signs + want to constraint rho between 0 and 1 => Bayesian econometrics seems useful in that case.

It is for quick estimation, not for a research paper (more a preliminary/basic analysis) => Do not want to estimate a full model.

If I consider to estimate an equilibrium system:
What would be the smaller equilibrium system for a small open economy (I have exchange rate in the Taylor rule)?
Or alternatively, could I estimate a small closed economy model and consider that exchange rate is exogenous (following an AR(1) for example)?

Thank you for the reference, it seems to be a must read!
L.

Cochrane is a must read. You will see that your preliminary approach most likely will result in wrong results.

You don’t need NLS. The equation is linear in R(-1), pi and y. The coefficient in front of lagged R identifies rho. Using this information, you can back out gamma from the (1-rho)*gamma coefficient in front of pi. This what e.g. Clarida/Gali/Gertler (2000) do (but they use GMM). If you have access to Eviews, performing a similar GMM estimation to them is really straightforward.

The smallest model would be something like the toy models from Gali’s 2008 textbook.