Estimation : detrend actual data or log difference

Hello? I had a question below,

As for estimation, to my knowledge, we have to detrend variables by HP filter (for example).
so, detrended real data log(y) which is deviation from the trend could be consistent with y-hat (deviation from the steady-state) in DSGE models.

But in some literatures, typically, Adolfson et al. (2005, 2007), she used growth rate such as log(yt/yt-1), then estimate and forecast them (growth rate
of gdp, consumption, wage, etc) Is it coherent with interpretation of y^ in a DSGE ? does it matter?

Plz. explain these stuff. I’d greatly appreciate it.

both ways of proceeding are possible. Your model and the data are not stationary due to some trend. There are several ways to eliminate this trend. Usually people use some kind of filter like the HP-filter or the first-difference filter (or some band-pass-filter). If your data has a unit root (we usually think of technology as a unit root process), using first differences is usually seen as more appropriate, because the HP-filter does not explicitely deal with the unit root (although it has been shown to filter out most of it). This is the reason today most models like the one you cited or Smets/Wouters explicitely model the stochastic trend and use an observation equation.
In the end it is often a matter of convenience that depends on how you wrote down your model and with which way it is easier to match the model to the data.

Regarding your last question: Of course the way of filtering your data/eliminating the trend matters for the results. Ideally it should not, but the different filters filter out different components they identify as a trend. Note also that all filters usually introduce an artifact at the beginning/end of the data and this problem is usually more pronounce with an HP-filter.