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

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.