Truncated sum of squares estimation of fractional time series models with deterministic trends

We consider truncated (or conditional) sum of squares estimation of a parametric model composed of a fractional time series and an additive generalized polynomial trend. Both the memory parameter, which characterizes the behaviour of the stochastic component of the model, and the exponent parameter, which drives the shape of the deterministic component, are considered not only unknown real numbers, 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) is challenging due to non-uniform convergence of the objective function over a large admissible parameter space, but, in addition, our framework is substantially more involved due to the competition between stochastic and deterministic components. We establish consistency and asymptotic normality under quite general circumstances, finding that results differ crucially depending on the relative strength of the deterministic and stochastic components.