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This is a matter of taste and theory. If you believe your model, then the interest rate is stationary, even if in the short sample you are using you cannot reject the null of a unit root. In that case, from a model perspective the problem is with econometric testing in the data. However, if you are unhappy with using the level, you can still use first differences for the interest rates as long as you are using a correct observation equation, i.e. you are matching the first differences in the data to first differences in the model. A different, but related issue is the the presence of deterministic trend. Such a trend may be due to e.g. an unmodeled drift in the inflation target. In that case, detrending the data suitably before feeding it into the model may be required to make the data consistent with the model.
The point about moment matching I do not understand. The Kalman filter is about unobserved variables, not observed ones. How would you match the moments of an unobserved state to the moments in unobserved data? -
The one-sided HP filter is based on a state space filtering problem. The cyclical component asymptotically has mean 0, but in short samples it can have a mean different from 0. What you report is no reason to worry. On average, your data was 0.1 percent below steady state, which is negligible. Of course you are right that shocks have to account for the mean not being 0 over the sample. So if you strongly dislike the small non-zero mean, you can demean the result. I doubt that it will make a big difference.