In modern complex data environments, time series across fields such as economics, agriculture, and energy exhibit increasingly intricate and dynamic characteristics. They include not only periodic fluctuations driven by market mechanisms and natural laws, but are also subject to the nonlinear influence of multi-dimensional covariates.

Particularly in applications like agricultural prices, financial cycles, and climate indices, the cycle length is often not fixed or known, and the covariates can manifest as functional, time-varying, or exponential effects.

If traditional periodic decomposition methods are still used, treating the cycle as known or ignoring the complex covariate structure, it often leads to problems such as inaccurate cycle identification, confounding of covariate effects, and insufficient model interpretability.

Therefore, simultaneously considering the unknown periodic structure and the complex covariate mechanism in time series modeling, and developing a comprehensive model capable of flexibly characterizing these two types of dynamics, holds significant practical importance for enhancing the interpretability, reliability, and predictive accuracy of time series analysis.

To address these issues, Associate Professor (Researcher) Liu Hua from the School of Economics and Finance at Xi'an Jiaotong University (XJTU), along with Assistant Professor Wang Shouxia, Professor Huang Tao, and Professor You Jinhong from the School of Statistics and Data Science at Shanghai University of Finance and Economics, proposed a novel time series modeling framework capable of simultaneously identifying the unknown periodic structure and characterizing the effect of complex covariates.

They validated the effectiveness of the model and its estimation method through rigorous theoretical proofs and extensive simulation experiments. This framework not only can automatically infer the unknown cycle length from data, breaking away from the dependence of traditional methods on a pre-set period, but also handles multi-dimensional complex data including functional covariates, nonlinear single-index structures, and time-varying effects, demonstrating stronger expressive power in dealing with multimodal, nonlinear, and dynamic changes.

Furthermore, this method is applicable to nonstationary and even functional time series, broadening the application boundaries of classic time series decomposition models in large-scale and structurally complex scenarios. By incorporating periodic fluctuation and covariate effects into a unified framework, this research can simultaneously identify the source of the cycle, the evolution of cycle strength, and the dynamic role of covariates, achieving a deep structural explanation that is difficult to attain with traditional models.

This research proposes a brand-new time series modeling method to resolve issues such as inaccurate cycle identification, difficulty in separating covariate effects, and insufficient model interpretability in classic time series decomposition models within large-scale and structurally complex scenarios.

The above research findings were recently published in the Journal of Econometrics, an international top-tier journal for econometrics, under the title Functional Semiparametric Modeling for Nonstationary and Periodic Time Series Data.