I am estimating two non-stationary models in Dynare. Both models described below decompose the time series y into trend and cycle. The only difference is that in Model 1. drift is a random walk process while in Model 2. it is a stationary process. The thing is that Dynare is able to estimate the first model, by using the diffuse_filter option, without specifiying steady state in a mod-file (or in a separate m-file) while in the second case it is not, i.e. one has to specify the steady state of the model in which the steady-state of unit root variables can be set to any arbitrary value. The thing is that in Model 1. I directly get the estimates of the trend while in the case of Model 2. I have to construct it by substracting the cycle from an observed series. I canâ€™t figure out why this is the case so if anyone has some clarification I would really appreciate it.

Model 1.

y = trend(-1) + cycle(-1)

trend = trend(-1) + drift(-1) + e_t

drift = drift(-1) + e_d

cycle = rho*cycle(-1) + e_c

Model 2.

y = trend(-1) + cycle(-1)

trend = trend(-1) + drift(-1) + e_t

drift = (1-phi)*drift_ss + phi*drift(-1) + e_d

cycle = rho*cycle(-1) + e_c