We consider truncated (or conditional) sum-of-squares estimation of a parametric fractional time series model with an additive deterministic structure. The latter consists of both a drift term and a generalized power law trend. The memory parameter of the stochastic component and the power parameter of the deterministic trend component are both considered unknown real numbers to be estimated and belonging to arbitrarily large compact sets. Thus, our model captures different forms of nonstationarity and noninvertibility as well as a very flexible deterministic specification. 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 and due to the competition between stochastic and deterministic components. As expected, parameter estimates related to the deterministic component are shown to be consistent and asymptotically normal only for parts of the parameter space depending on the relative strength of the stochastic and deterministic components. In contrast, we establish consistency and asymptotic normality of parameter estimates related to the stochastic component for the entire parameter space. Furthermore, the asymptotic distribution of the latter estimates is unaffected by the presence of the deterministic component, even when this is not consistently estimable. We also include Monte Carlo simulations to illustrate our results.
J. Hualde’s research is supported by the Spanish Ministerio de Ciencia e Innovación through project PGC2018-093542-B-I00.
We are grateful to the editor, Domenico Marinucci, two anonymous referees, Søren Johansen, Peter M. Robinson, and seminar participants at various universities and conferences for useful comments.
"Truncated sum-of-squares estimation of fractional time series models with generalized power law trend." Electron. J. Statist. 16 (1) 2884 - 2946, 2022. https://doi.org/10.1214/22-EJS2009