## The Annals of Statistics

### On model selection from a finite family of possibly misspecified time series models

#### Abstract

Consider finite parametric time series models. “I have $n$ observations and $k$ models, which model should I choose on the basis of the data alone” is a frequently asked question in many practical situations. This poses the key problem of selecting a model from a collection of candidate models, none of which is necessarily the true data generating process (DGP). Although existing literature on model selection is vast, there is a serious lacuna in that the above problem does not seem to have received much attention. In fact, existing model selection criteria have avoided addressing the above problem directly, either by assuming that the true DGP is included among the candidate models and aiming at choosing this DGP, or by assuming that the true DGP can be asymptotically approximated by an increasing sequence of candidate models and aiming at choosing the candidate having the best predictive capability in some asymptotic sense. In this article, we propose a misspecification-resistant information criterion (MRIC) to address the key problem directly. We first prove the asymptotic efficiency of MRIC whether the true DGP is among the candidates or not, within the fixed-dimensional framework. We then extend this result to the high-dimensional case in which the number of candidate variables is much larger than the sample size. In particular, we show that MRIC can be used in conjunction with a high-dimensional model selection method to select the (asymptotically) best predictive model across several high-dimensional misspecified time series models.

#### Article information

Source
Ann. Statist., Volume 47, Number 2 (2019), 1061-1087.

Dates
Revised: March 2018
First available in Project Euclid: 11 January 2019

https://projecteuclid.org/euclid.aos/1547197248

Digital Object Identifier
doi:10.1214/18-AOS1706

Mathematical Reviews number (MathSciNet)
MR3909960

Zentralblatt MATH identifier
07033161

Subjects
Primary: 63M30
Secondary: 62F07: Ranking and selection 62F12: Asymptotic properties of estimators

#### Citation

Hsu, Hsiang-Ling; Ing, Ching-Kang; Tong, Howell. On model selection from a finite family of possibly misspecified time series models. Ann. Statist. 47 (2019), no. 2, 1061--1087. doi:10.1214/18-AOS1706. https://projecteuclid.org/euclid.aos/1547197248

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#### Supplemental materials

• Supplement to “On model selection from a finite family of possibly misspecified time series models”. The supplementary material contains the proofs of all theorems, an extension of MRIC to a class of nonlinear models and simulation studies and real data analysis to illustrate the performance of the proposed methods in both low- and high-dimensional cases.