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I've got a random process of positive observations. I've built a state space model (following an ARMA(1,3) structure), found the parameters that fit the process observations through log-likelihood maximization, and after simulating the obtained model I've seen that negative values occur. I really don't have any idea on how to correct these results. Is it that some type of lower bound must be formulated at the stage of parameter estimation?

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If you require strictly positive values, then an untransformed ARMA may be inappropriate. One option is to transform the observations using, for example, the logarithm which would guarantee your response variable never goes negative. But this assumes that the model is at least approximately linear in log space. This can give very poor performance in predictive settings. Another option is to utilize a nonnegative distribution of errors, for example Gamma. Here is a comprehensive description of such a state space model, replete with the details of likelihood inference.

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