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We consider conditional sum of squares estimation of a parametric model composed of a fractional time series with an additive general deterministic component. Both the memory parameter, which characterizes the behaviour of the stochastic part of the model, and the parameter which drives the particular shape of the deterministic component, are considered not only unknown, but also lying in arbitrarily large (but finite) intervals. Thus our model captures different forms of nonstationarity and noninvertibility. As in related settings, the proof of consistency (which is a prerequisite for proving asymptotic normality) can be difficult owing to non-uniform convergence of the objective function over a large admissible parameter space, but our framework is substantially more involved due to the competition between the stochastic and deterministic elements of the model. We establish consistency and asymptotic normality under quite general circumstances, finding that results differ substantially depending on the relative strength of the deterministic and stochastic components.
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