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Title: The asymptotic excess risk of possibly non-stationary time-series Authors:  Shu-Hui Yu - Institute of Statistics (Taiwan) [presenting]
Abstract: 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, the conventional AIC-type and BIC-type information criteria, which are arguably most suitable for univariate time series in which the lags are ordered, are assessed. Throughout we consider a univariate AR process in which the AR order and the order of integratedness are finite but unknown. We prove that the BIC-type information criterion, which penalty goes to infinity, attains zero asymptotic excess risk. In contrast, the AIC-type information criterion, which penalty goes to a finite number strictly greater than 1, renders a strictly positive asymptotic excess risk. Further, the asymptotic excess risk increases with the admissible number of lags, a result that gives a warning about certain high-dimensional analyses when the true data generating process is of low-dimension. In sum, we extend the existing results in threefold: (i) a general I(d) process; (ii) same realization prediction; and (iii) an information criterion more general than AIC. Some simulation study shows these asymptotic results are valid for fairly small sample sizes.