What is the relationship between the trend term of HP filtering for output variable Y and the steady-state value of output variable Y in the DSGE model? Can we roughly understand the two as one thing?

Thank you!

Conceptually, the trend in the data corresponds to the steady state in the model. For conceptualizing the trend in the data, you can use a filter like the HP filter. However, using two-sided filters in the context of estimation is strongly discouraged because the filter is non-causal, i.e. the two-sidedness does not match the backward-looking recursive structure of the model solution. For that reason, you should rather use the one-sided HP filter.

Thanks for the answer!

In addition, in the DSGE model, the welfare loss corresponding to the steady state is 0, so does this mean that the steady state must be the optimal state for the economic operation of the model?

Without further explanations on what exactly you are doing and comparing, it is impossible to tell.

I am trying to analyze the transmission path of negative productivity shocks. For example, “negative productivity shocks lead to an increase in risk premium, a decrease in investment scale, and a contraction in output levels…The economy deviates from steady state and is no longer in the optimal operating path”.

I want to know if this statement is correct? Thank you!

That is impossible to tell without knowing a lot more. The big question is if the response to a shock is optimal, i.e. the first best a planner would choose. The answer usually depends on the frictions in the model.

Thank you for your answer!