In this post I will try to address your UCM questions so far. First some general comments:
Modeling and forecasting a time series is not easy without some understanding of the series being modeled. Very often several models can be proposed that appear to to fit the historical data reasonably well (his is true of both ARIMA models and UCMs). Model diagnostics (such as residual analysis) is useful but still requires context to decide which of the discovered features of the model are real and which might not be so. Cross-validation type methods, which are very effective in addressing overfitting in the ordinary regression modeling are not as effective in the time series setting. The policy about the handling of the outliers discovered during the exploratory stage is also not quite clear cut and (again) requires context info. In light of these, my personal preference is to try simple models that fit the data reasonably well and not to try to overfit the historical region. Outliers are left unhandled unless they distort the main features (such as trend) of the series. Without additional context, the model given at the end of this post seems adequate to me. Of course, whether the discovered cycle (of period 13 years) is "real" or not cannot be answered without domain info. Now answers to your specific questions:
1. Negative R-square: The R-square in usual ordinary regression is based on "regression residuals" (Y - X beta-hat). The UCM R-square is based on "one-step-ahead" residuals. One-step-ahead residual at a particular time is based on data prior to that time point. Therefore, UCM R-square is not guaranteed to be non-negative (this is mentioned in the UCM doc). Moreover, when the UCM model contains dummy regressors, very often only a few non-missing one-step-ahead residuals are available for residual analysis. This is because non-missing residuals are availble only after adequate number of observations are processed to initialize the diffuse components (which include regressors) in the model. All of your models suffer from this condition of inadequate number of non-missing residuals for residual analysis.
2. You can use the OUTFOR= option in the FORECAST statement to output series forecasts, residuals (their standard errors) and many other things. UCM provides rich graphical support for residual analysis (as you have noticed). If you want to compute some of the statistics you mention by hand, you can use the OUTFOR data set and use PROC IML or PROC UNIVARIATE.
My suggested program:
proc ucm data=metals;
model ZI;
irregular;
level variance=0 noest checkbreak;
slope;
cycle plot=smooth;
estimate plot=panel;
forecast plot=decomp;
run;
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