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February 2011 Feature Matching in Time Series Modeling
Yingcun Xia, Howell Tong
Statist. Sci. 26(1): 21-46 (February 2011). DOI: 10.1214/10-STS345

Abstract

Using a time series model to mimic an observed time series has a long history. However, with regard to this objective, conventional estimation methods for discrete-time dynamical models are frequently found to be wanting. In fact, they are characteristically misguided in at least two respects: (i) assuming that there is a true model; (ii) evaluating the efficacy of the estimation as if the postulated model is true. There are numerous examples of models, when fitted by conventional methods, that fail to capture some of the most basic global features of the data, such as cycles with good matching periods, singularities of spectral density functions (especially at the origin) and others. We argue that the shortcomings need not always be due to the model formulation but the inadequacy of the conventional fitting methods. After all, all models are wrong, but some are useful if they are fitted properly. The practical issue becomes one of how to best fit the model to data.

Thus, in the absence of a true model, we prefer an alternative approach to conventional model fitting that typically involves one-step-ahead prediction errors. Our primary aim is to match the joint probability distribution of the observable time series, including long-term features of the dynamics that underpin the data, such as cycles, long memory and others, rather than short-term prediction. For want of a better name, we call this specific aim feature matching.

The challenges of model misspecification, measurement errors and the scarcity of data are forever present in real time series modeling. In this paper, by synthesizing earlier attempts into an extended-likelihood, we develop a systematic approach to empirical time series analysis to address these challenges and to aim at achieving better feature matching. Rigorous proofs are included but relegated to the Appendix. Numerical results, based on both simulations and real data, suggest that the proposed catch-all approach has several advantages over the conventional methods, especially when the time series is short or with strong cyclical fluctuations. We conclude with listing directions that require further development.

Citation

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Yingcun Xia. Howell Tong. "Feature Matching in Time Series Modeling." Statist. Sci. 26 (1) 21 - 46, February 2011. https://doi.org/10.1214/10-STS345

Information

Published: February 2011
First available in Project Euclid: 9 June 2011

zbMATH: 1219.62142
MathSciNet: MR2849904
Digital Object Identifier: 10.1214/10-STS345

Keywords: acf , Bayesian statistics , black-box models , blowflies , Box’s dictum , Calibration , catch-all approach , data mining , ecological populations , epidemiology , feature consistency , Feature matching , Least squares estimation , maximum likelihood , measles , Measurement errors , misspecified models , model averaging , multi-step-ahead prediction , nonlinear time series , observation errors , optimal parameter , periodicity , population models , sea levels , short time series , SIR epidemiological model , skeleton , substantive models , sunspots , threshold autoregressive models , Whittle’s likelihood , XT-likelihood

Rights: Copyright © 2011 Institute of Mathematical Statistics

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Vol.26 • No. 1 • February 2011
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