Different forms of Shocks

Hi everyone,

I have a basic question regarding the types of shockes we can use in a DSGE model. For example in one type, there is an AR(1) process without any intercept or other terms:


In another type, we include the steady state of the variable into the shock formula:


Sometimes, the steady state of a variable is 1 and the second type changes the first one but what if the steady state is different than 1? Does using different forms depend on whether the model is written in a nonlinear form?

The first formulation hard-codes the steady state to be 1. The second one allows to set the steady state to any arbitrary value. Both versions would not appear in a linear model due to the use of log.

Thanks a lot. I always appreciate your help.
If the second one is more general, what is the reason of using the first formolation for many researchers? I am wondering that they can easily use the second one and set the steady state value to 1 to get the first one.

For many processes, the natural steady state is 1. Why go for something more complicated, if the easy one suffices?

Thanks again for your help.
Correct. It makes sense.
Just a quick question, could you provide me with some processes which their natural steady states are 1. I’ve seen that for the technolody shock, what about other examples?

Pretty much any shocks that enters multiplicatively. Examples are TFP shocks or monetary policy shocks.

Thank you so much!