dear professor Pfeifer;
The smoothed variables are identical to observables without measurement error, but what exactly does “ smoothed variables” mean, are they identical to the variables from using simult with the posterior mode and all smoothed shocks ? if so ,then why many paper compare the business cycle from model to that from the data, there seems ono need to do so . looking forward to your reply .
The Kalman smoother linearly decomposes the historical observed variables into starting values for the unobserved states and the smoothed shocks. Thus, if you start at the smoothed initial states and simulate with the smoothed shocks, you will be getting the observed series. But that is only the historical decomposition. In short samples, the shock properties may not coincide with the model assumptions (e.g. typically uncorrelated). For that reason, moments from model simulations with random shocks can be different than the historical ones. This may also be a sign of model misspecification.
But the comparisons you are alluding to are often happening in non-estimated models.