Priors - Monetary policy Taylor rule

Hi everyone

I was modeling the taylor rule for a small open economy that includes the depreciation of the nominal exchange rate, as follows:

1 + i_t = (1 + i_{t-1})^{\varphi_{i}}[(\frac{1 + \pi_t}{1 + \pi_{ss}})^{\varphi_{\pi}}(\frac{1+\Delta rer_t}{1 + \Delta rer_{ss}})^{\varphi_{rer}}(\frac{GDP_t}{GDP_ss})^{\varphi_{GDP}}]^{(1-\varphi_{i})}\times exp(\varepsilon_t)

where \varepsilon_{t} follows an AR(1) process. The country for which I am estimating the model is Mexico. So far I have not found literature on DSGE models that include exchange rate depreciation for this country, however, I found an empirical paper that does, even testing for various models removing the exchange rate, including lags, etc. and the one that presents the best results is the following:

1 + i_t = \beta_0 + \beta_1(1+i_{t-1}) + \beta_2 \widehat{GDP}_t + \beta_3 \pi_t + \beta_4 \Delta rer_t + e_t ,
\widehat{GDP}_t = GDP_t - (GDP_t^{HP/Trend}),
\pi_t = 100\times log(CPI_t/CPI_{t-1})

They find that: \beta_1 = 0.871, \beta_2 = 0.470, \beta_3 = 0.153, \beta_4 = -0.630. It is worth mentioning that in the empirical paper they do not take into account the autocorrelation of the error term. My questions are:

1) Should I stick to the empirical findings to define the parameters a prior?

That is to say:
\varphi_i \sim B(0.871, 0.1^2)
\varphi_{\pi} \sim N(1.186, 0.5^2)
\varphi_{rer} \sim N(-4.884, 0.5^2)
\varphi_{GDP} \sim N(3.643, 0.5^2)

or is it better to rely on the literature of models who have used this modeling? (even if it’s for another country)

That is to say:
\varphi_i \sim B(0.5, 0.2^2)
\varphi_{\pi} \sim N(1.25, 0.25^2)
\varphi_{rer} \sim N(0.5, 0.1^2)
\varphi_{GDP} \sim N(0.25, 0.1^2)

Or should I do the empirical estimation with updated data covering the period I want to analyze, considering the autocorrelation of the error and define my priors based on this estimate?

Thanks in advance.

I am not sure I understand the last part. But generally, your prior must be independent of the data you are using for the actual estimation. So option 1 would only work if you are considering a different sample.