I want to specify observation equation for interest rate when estimating DSGE model. In my model interest rate is defined in gross terms i.e. R_t =(1+r_t). In the data the interest rate is quoted in annualized terms (net interest rate) while in the model it is defined as quarterly gross interest rate. I divided original data by 100 and subtract the mean. I specified the following observation equation:

R_t_obs = 4*(R-1) + 4RR_hat.

Is my specification correct? Adolfson et al. for example report the following specification:

thank you for your answer. To clarify my question, please see attached file. I use the model developed by Adolfson et al. (2007) which is log-linearized (variables with hat - % deviations from SS).

Given my data (see attached file), I specified the following observation equation.
R_t_obs = 4*(R-1) + 4RR_hat.

where R is the seady state gross interest rate.

But on the other hand, Adolfson et al. (2007) in their technical appendix report the following specification:
R_t_obs = 4*(R-1)R + 4R*R_hat.

So, the difference is the term: 4*(R-1) vs. 4*(R-1)*R. Could you explain me the intuition behind this specification?

Sorry, but I am not exactly shure what they did there. You might have to ask the authors for their rationale. My hunch is that it has to do with the fact that the stated equation is the log-linearized observation equation where the approximation is around the gross interest rate. As the net interest rate is the gross interest rate minus 1, you have to adjust for the fact that, when taking logs, you are not exactly subtracting 1. Put differently, saying the log of the gross interest rate is equal to the net interest rate is not exactly correct (only if R=1).

In the more recent “Ramses II – Model Description” by Malin Adolfson, Stefan Laséen, Lawrence Christiano, Mathias Trabandt and Karl Walentin it is the former version that is used.