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I see some state space models specify their innovation process as log innovations and some squaring the term. For example, the examples in the R package DLM favours the use of log innovations when estimating the parameters of a state space model - i.e. wrapping the parameter to be estimated in an exp(). Where some other examples I have seen simply square the term - which makes sense since we are estimating a variance. The argument given in the DLM examples is to stop the parameter estimate to go negative - which squaring obviously also achieves.

What are the benefits and costs of choosing one over the other? What should guide such choice?

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