Abstract

This article introduces a novel double dynamic trinomial GARCH model with the simultaneous modeling of the time-varying mean and variance to analyze bounded full-integer-valued time series. In this paper, we construct a trinomial autoregressive process to analyze zero-mean process with time-varying variance process, and then propose a novel double dynamic trinomial autoregressive process to analyze non-zero and time-varying mean process, besides the time-varying variance process. Furthermore, we discuss their stationarity and ergodicity, give the conditional maximum likelihood estimate, and derive their asymptotic properties.

We also provide details on one-step-ahead forecasting.

The performances of these estimators are compared via simulation study. Finally, we apply the proposed model to two datasets to demonstrate its favorable performance.

Key words and phrases: trinomial difference distribution, bounded full-integer-valued time series, GARCH model, time-varying variance, time-varying mean, statistical inference

Information

Preprint No.SS-2025-0448
Manuscript IDSS-2025-0448
Complete AuthorsHuaping Chen, QianQian Zhao, Liugen Xue, Fukang Zhu
Corresponding AuthorsFukang Zhu
Emailszfk8010@163.com

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Acknowledgments

The authors thank the Editor-in-Chief and the two anonymous referees for the valuable comments and suggestions that result in a substantial improvement of this paper. This work is supported by National Nature Science Foundation of China (Nos. 12671320, 12471252, 12271206), Natural Science Foundation of Jilin Province (No. 20250102003JC) and Science and Technology Research Program of Henan Province (No. 262102320106).

Supplementary Materials

The online supplementary materials provide detailed results for the first and second derivatives of lt(θ) with respect to θ of Section 3.1 (S1), the proofs of the Propositions, Lemma 1, Remark 2 and Theorems (S2), detailed results for the simulation study of Section 4 (S3) and some auxiliary results (S4).

Supplementary materials are available for download.