I am doing time series data analysis by state space methods. With my data the stochastic local level model totally outperformed the deterministic one. But the deterministic level and slope model gives better results than with stochastic level and stochastic/deterministic slope. Is this something usual? All methods in R require initial values, and I read somewhere that fitting an ARIMA model first and taking values from there as initial values for state space analysis is one way; possible? or any other proposition? I should confess here that I'm totally new to state space analysis.
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To answer your first question. Yes, all is possible. It is not usual or unusual. You should let the data tell you what the correct model is. Try augmenting the model further with seasonals, cycles, and explanatory regressors if possible.
You should not only be comparing the Akaike Information Criterion (AIC) to compare models, but also checking to see that residuals (irregular term) are normal, homoskedastic, and independent (Ljung-Box test). If you can find a model that has all of these desirable properties. This should be your preferred model (it is likely that a model with all these properties will have the best AIC).
Although the initial values will affect which maximum point of the log-likelihood function is found, if your model is well specified, it shouldn't vary too much and there should be an obvious candidate for the best model with the best initial values. I do a lot of this type of analysis in Matlab and I found the best way to find initial values is just to play around for a bit. It can be tedious but it works out well in the end.