Problem with estimating the model by Sims, Garin and Lester (2018)

Hello to everyone! I am trying to estimate a small-scale NK model (Garin3estimation zip file), inspired by Garìn, Sims and Lester (2018). From my estimation it seems impossible to find the posterior mode with the normal optimizer routine (mode_compute = 4). I am using not detrended data, but I removed the empirical mean from the data with the option “prefilter=1”. I attach the Mode_check plots together with the .mod file and the data I used. I think that the model could be not well identified, because there is not interest smoothing in the Monetary Policy, and I don’t know how to solve this issue. Neverthless, the most surpising thing is that if I consider only half of the dataset ( the data goes from 1966 to 2018) the estimation works. Are there any reasons why this happen? Can you please suggest me some readings to understand this phenomenon?
Furthermore, I am able to complete the estimation with a different optimization method (MCMC), putting mode_compute=6 option. I attach the .mod file and the results (Garin3estimation2); in this case I have used two Measurement errors instead of AR(1) shock processes, as suggested by Pfeifer, 2018 (“A Guide to Specifying Observation Equations for the Estimation of DSGE Models”, chp.6), to handle possible stochastic singularity problems. Are these results reliable? should I prefer one estimation to another?
Thank you very much for the help that you will be able to give me; I am truly sorry if these questions are too trivial, I am still a (206.7 KB) (45.8 KB)

You are not handling parameter dependency correctly. Use identification to see the problems.