I am estimating a DSGE model and I noticed that most of the smoothed variables have non-zero mean (non-negligible different from zero values), including shocks and innovations. The model is log-linaerized, no constant in any equation, all observables have zero mean. So in principle all smoothed variables should have zero mean.
I tried with a simple AR(1) model with no constant and one observable (217 observations) with zero mean to check this fact. It turns out that it is a small sample issue. In fact the mean of the smoothed innovation was 0.0011. Small enough to be considered zero, but still not quite. I then tried with a simulated series of 100 000 observations and I got exact zero mean on the innovation.
The important thing to notice though is that according to which way one specifies the shocks processes matters for the small sample issue. In fact one has two options
y_t = rho * y_t(-1) + sig_eps * eps_t with eps_t N(0,1)
y_t = rho * y_t(-1) + eps_t with eps_t N(0,sig_eps)
Despite the fact that those two formulations should be equivalent, the mean of the smoothed innovation with the second option was 0.0008 (vs. 0.0011 of the other case). So it seems that the second specification is more suited to tackle the small sample issue.
Any comments on that?
Also, once we know that there is this small sample issue, is it reasonable to take the mean out of the smoothed variables? The reason I am asking is that there are some variables that by definition have zero mean, like the output gap. If I get a smoothed series with a mean of say 0.5, this is not a negligible shift.