What data should I use for historical shock decomposition in a calibrated model?

Estimation ML/Bayesian estimation
1

As the title suggested, what data should I use for historical shock decomposition in a calibrated model?

For instance, if I use actual GDP data in my datafile, should the observation equation be y_obs=y?

Or if I use the deviation of GDP from its trend after HP filtering, should the observation equation be y_obs= (y-steady_state(y))/steady_state(y)?

Thank you for your time.

2

There is no general answer. See Pfeifer(2013): “A Guide to Specifying Observation Equations for the Estimation of DSGE Models”
Note that you should not use HP filtered data as it is inconsistent with the recursive/backward-looking nature of the DSGE model’s solution.