{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,12]],"date-time":"2026-04-12T15:38:40Z","timestamp":1776008320274,"version":"3.50.1"},"reference-count":25,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,8]],"date-time":"2024-05-08T00:00:00Z","timestamp":1715126400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper makes a significant contribution by focusing on estimating the coefficients of a sample of non-linear time series, a subject well-established in the statistical literature, using bilinear time series. Specifically, this study delves into a subset of bilinear models where Generalized Autoregressive Conditional Heteroscedastic (GARCH) models serve as the white noise component. The methodology involves applying the Klimko\u2013Nilsen theorem, which plays a crucial role in extracting the asymptotic behavior of the estimators. In this context, the Generalized Autoregressive Conditional Heteroscedastic model of order (1,1) noted that the GARCH (1,1) model is defined as the white noise for the coefficients of the example models. Notably, this GARCH model satisfies the condition of having time-varying coefficients. This study meticulously outlines the essential stationarity conditions required for these models. The estimation of coefficients is accomplished by applying the least squares method. One of the key contributions lies in utilizing the fundamental theorem of Klimko and Nilsen, to prove the asymptotic behavior of the estimators, particularly how they vary with changes in the sample size. This paper illuminates the impact of estimators and their approximations based on varying sample sizes. Extending our study to include the estimation of bilinear models alongside GARCH and GARCH symmetric coefficients adds depth to our analysis and provides valuable insights into modeling financial time series data. Furthermore, this study sheds light on the influence of the GARCH white noise trace on the estimation of model coefficients. The results establish a clear connection between the model characteristics and the nature of the white noise, contributing to a more profound understanding of the relationship between these elements.<\/jats:p>","DOI":"10.3390\/sym16050581","type":"journal-article","created":{"date-parts":[[2024,5,8]],"date-time":"2024-05-08T12:27:11Z","timestamp":1715171231000},"page":"581","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Comparative Analysis of Bilinear Time Series Models with Time-Varying and Symmetric GARCH Coefficients: Estimation and Simulation"],"prefix":"10.3390","volume":"16","author":[{"given":"Ma\u2019mon","family":"Abu Hammad","sequence":"first","affiliation":[{"name":"Department of Mathematics, Al-Zaytoonah University of Jordan, Amman 11733, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rami","family":"Alkhateeb","sequence":"additional","affiliation":[{"name":"Department of Basic Sciences, Al-Ahliyya Amman University, Amman 19328, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nabil","family":"Laiche","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Seville University, 41004 Sevilla, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Adel","family":"Ouannas","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Oum El Bouaghi, Oum El Bouaghi 04000, Algeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3313-506X","authenticated-orcid":false,"given":"Shameseddin","family":"Alshorm","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Al-Zaytoonah University of Jordan, Amman 11733, Jordan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,8]]},"reference":[{"key":"ref_1","first-page":"7","article-title":"An introduction to bilinear time series models","volume":"8","author":"Granger","year":"1978","journal-title":"Angew. 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