Non-zero mean smoothed variables


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.

HI Paolo,

if the observed variables are zero-mean, there is no result that warranty that the smoothed variables should be exactly zero-mean as well. Smoothed variables are derived from the observed variables and as such are submitted to sample variability. You would need to test your hypotheses taking into account the distribution of the smoothed variables. For example, I’m not convinced that 0.0008 is significantly different from 0.0011.
If you remove the mean from the smoothed variables you can’t say anymore that you are reporting the smoothed variables. It is not even sure that these transformed smoothed variables are the best estimated under the constraint that the mean should be zero.
When you get a model where the mean of smoothed output gap is 0.5, it may be interesting to compare with the results from a model that puts less structure on the data. It maybe that the “large” departure from zero is the sign of some feature of your model that conflicts with the data. But again remember that you are comparing random variables.

Thanks a lot Michel.