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On asymptotic risk of selecting models for possibly nonstationary time-series

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Version 2 2020-07-02, 14:02
Version 1 2020-06-19, 13:12
journal contribution
posted on 2020-07-02, 14:02 authored by Shu-Hui Yu, Chor-yiu (CY) Sin

Model selection criteria are often assessed by the so-called asymptotic risk. Asymptotic risk is defined either with the mean-squared error of estimated parameters; or with the mean-squared error of prediction. The literature focuses on i.i.d. or stationary time-series data though. Using the latter definition of asymptotic risk, this paper assesses the conventional AIC-type and BIC-type information criteria, which are arguably most suitable for univariate time series in which the lags are naturally ordered. Throughout we consider a univariate AR process in which the AR order and the order of integratedness are finite but unknown. We prove the BIC-type information criterion, whose penalty goes to infinity, attains zero asymptotic excess risk. In contrast, the AIC-type information criterion, whose penalty goes to a finite number, renders a strictly positive asymptotic excess risk. Further, the asymptotic excess risk increases with the admissible number of lags. The last result gives a warning on possible over-fitting of certain high-dimensional analyses, should the underlying data generating process be strongly sparse, that is, the true dimension be finite. In sum, we extend the existing asymptotic risk results in threefold: (i) a general I(d) process; (ii) same-realization prediction; and (iii) an information criterion more general than AIC. A simulation study and a small-scale empirical application compare the excess risk of AIC with those of AIC3, HQIC, BIC, Lasso as well as adaptive Lasso.

Funding

Yu and Sin’s researches are partially supported by the Ministry of Science and Technology of Taiwan under grants MOST 108-2118-M-390-005 and MOST 108-2410-H-007-014, respectively.

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