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Title: A two-steps specification test for functional time series Authors:  Alejandra Lopez-Perez - Universidade de Santiago de Compostela (Spain) [presenting]
Javier Alvarez-Liebana - University of Oviedo (Spain)
Wenceslao Gonzalez-Manteiga - University of Santiago de Compostela (Spain)
Manuel Febrero-Bande - University of Santiago de Compostela (Spain)
Abstract: The functional data analysis framework allows the representation of continuous-time stochastic processes as sequences of random variables in function spaces. Focusing on Hilbert spaces, the autoregressive Hilbertian process (ARH) plays a central role in modeling time-series dynamics. We propose a two-steps test for the null composite hypothesis of the autoregressive Hilbertian model for a given order z, ARH(z), against a general alternative. The approach is twofold: we check if the functional sample and its lagged functional values are related via a Functional Linear Model with Functional Response (FLMFR) and, through the construction of linear representations, we propose a specification test for stochastic diffusion models. The later is a two-stage methodology where we first check the null hypothesis of the FLMFR and, secondly, under linearity, a functional F-test is performed. As an example, the Ornstein-Uhlenbeck process is characterized as ARH(1) to illustrate the finite sample performance of the proposed test. The new methodology is also applied to real datasets.