How to choose the best filter?

Continuing the discussion from HP filter in linear models:

There is no uniformly best filter. You simply need to defend your choice. If your model explicitly models the trend, e.g. by having a unit root with drift or a linear trend, then the choice is obvious.

If you do not explicitly specify the trend, it becomes tricky, because your implicit concept may for it may be different the reader/reviewer’s. The article’s cited in the previous thread have one common theme: if the trend in the true DGP is different from the one removed by the filter used, results will be misleading.

Thank you, Professor.
Dear, Pfeifer,
Could you briefly explain to me the essence of the filtering method, which is described in the article “Estimation of DSGE models when the data are persistent”(2010), (Yuriy Gorodnichenko, Serena Ng ).
I can’t understand his approach.
Thank you,

You filter the data and the model variables in the same way and then do moment matching. Doing that with Bayesian estimation is pretty uncommon. That would most probably be something like

Thanks professor.