{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T05:52:38Z","timestamp":1773553958519,"version":"3.50.1"},"reference-count":27,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T00:00:00Z","timestamp":1773360000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12071176"],"award-info":[{"award-number":["12071176"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12271231"],"award-info":[{"award-number":["12271231"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>GARCH models play a fundamental role in modeling time-varying volatility in financial return series. In practice, financial returns are also well known to exhibit heavy-tailed distributions, which naturally motivates the use of quasi-maximum exponential likelihood estimation (QMELE) for accurately capturing tail behavior and risk measures such as Value-at-Risk. At the same time, the increasing availability of intraday high-frequency data has led to the development of high-frequency augmented GARCH models, which incorporate intraday information into conventional low-frequency volatility frameworks. By exploiting transaction-level data recorded at very fine time scales, these models are able to capture intraday volatility dynamics and market microstructure effects that are not reflected in standard low-frequency observations. Against this background, this paper studies conditional quantile estimation for high-frequency augmented GARCH models. We develop QMELE-based estimators for both model parameters and conditional quantiles, and construct an adjusted test statistic for assessing model adequacy. The asymptotic properties of the proposed estimators and test statistic are established, and their finite-sample performance is examined through extensive simulation studies. Empirical applications to three major stock indices demonstrate that augmenting GARCH models with high-frequency information leads to substantial improvements in conditional quantile estimation compared with traditional low-frequency approaches.<\/jats:p>","DOI":"10.3390\/e28030326","type":"journal-article","created":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T15:06:22Z","timestamp":1773414382000},"page":"326","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Quasi-Maximum Exponential Likelihood Estimation of Conditional Quantiles for GARCH Models Based on High-Frequency Augmented Data"],"prefix":"10.3390","volume":"28","author":[{"given":"Zhenming","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Mathematics, Jilin University, Changchun 130012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shishun","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Mathematics, Jilin University, Changchun 130012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8615-1496","authenticated-orcid":false,"given":"Jianhua","family":"Cheng","sequence":"additional","affiliation":[{"name":"School of Mathematics, Jilin University, Changchun 130012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anze","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Mathematics, Jilin University, Changchun 130012, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1016\/0304-4076(86)90063-1","article-title":"Generalized autoregressive conditional heteroscedasticity","volume":"31","author":"Bollerslev","year":"1986","journal-title":"J. Econom."},{"key":"ref_2","first-page":"162","article-title":"GARCH parameter estimation using high-frequency data","volume":"9","author":"Visser","year":"2011","journal-title":"J. Financ. Econom."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"967","DOI":"10.1093\/biomet\/90.4.967","article-title":"Least absolute deviations estimation for ARCH and GARCH models","volume":"90","author":"Peng","year":"2003","journal-title":"Biometrika"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1093\/biomet\/asn014","article-title":"Least absolute deviation estimation for fractionally integrated autoregressive moving average time series models with conditional heteroscedasticity","volume":"95","author":"Li","year":"2008","journal-title":"Biometrika"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2131","DOI":"10.1214\/11-AOS895","article-title":"Global self-weighted and local quasi-maximum exponential likelihood estimators for ARMA-GARCH\/IGARCH models","volume":"39","author":"Zhu","year":"2011","journal-title":"Ann. Stat."},{"key":"ref_6","first-page":"251","article-title":"Quasi-maximum exponential likelihood estimators for a double AR (p) model","volume":"23","author":"Zhu","year":"2013","journal-title":"Stat. Sin."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1000","DOI":"10.1080\/03610926.2013.851225","article-title":"Quasi-maximum exponential likelihood estimation for a non stationary GARCH(1,1) model","volume":"45","author":"Pan","year":"2016","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1007\/s10255-015-0488-y","article-title":"Robust M-estimate of GJR model with high frequency data","volume":"31","author":"Huang","year":"2015","journal-title":"Acta Math. Appl. Sin."},{"key":"ref_9","first-page":"115","article-title":"Composite quantile regression for GARCH models using high-frequency data","volume":"7","author":"Wang","year":"2018","journal-title":"Econom. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"33","DOI":"10.2307\/1913643","article-title":"Regression Quantiles","volume":"46","author":"Koenker","year":"1978","journal-title":"Econometrica"},{"key":"ref_11","unstructured":"Taylor, S. (1986). Modelling Financial Time Series, Wiley."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"793","DOI":"10.1017\/S0266466600007167","article-title":"Conditional quantile estimation and inference for ARCH models","volume":"12","author":"Koenker","year":"1996","journal-title":"Econom. Theory"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1696","DOI":"10.1198\/jasa.2009.tm09170","article-title":"Conditional quantile estimation for generalized autoregressive conditional heteroscedasticity models","volume":"104","author":"Xiao","year":"2009","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1111\/j.1467-9469.2011.00759.x","article-title":"Quantile regression estimator for GARCH models","volume":"40","author":"Lee","year":"2013","journal-title":"Scand. J. Stat."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"975","DOI":"10.1111\/rssb.12277","article-title":"Hybrid quantile regression estimation for time series models with conditional heteroscedasticity","volume":"80","author":"Zheng","year":"2018","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1016\/0304-4076(96)84507-6","article-title":"An interior point algorithm for nonlinear quantile regression","volume":"71","author":"Koenker","year":"1996","journal-title":"J. Econom."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/S0927-5398(97)00004-2","article-title":"Intraday periodicity and volatility persistence in financial markets","volume":"4","author":"Andersen","year":"1997","journal-title":"J. Empir. Financ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"717","DOI":"10.1007\/s10255-017-0694-x","article-title":"M-estimation for periodic GARCH model with high-frequency data","volume":"33","author":"Fan","year":"2017","journal-title":"Acta Math. Appl. Sin."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4235","DOI":"10.1080\/03610918.2024.2374900","article-title":"The nonparametric GARCH model estimation using intraday high-frequency data","volume":"54","author":"Chai","year":"2025","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"7390","DOI":"10.1080\/03610918.2020.1836215","article-title":"On the test of the volatility proxy model","volume":"51","author":"Deng","year":"2022","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1394","DOI":"10.1198\/016214505000000169","article-title":"A tale of two time scales: Determining integrated volatility with noisy high-frequency data","volume":"100","author":"Zhang","year":"2005","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1007\/s00362-025-01704-y","article-title":"Conditional quantile estimation for GARCH model based on mixed-frequency data","volume":"66","author":"Zhang","year":"2025","journal-title":"Stat. Pap."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"605","DOI":"10.3150\/bj\/1093265632","article-title":"Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes","volume":"10","author":"Francq","year":"2004","journal-title":"Bernoulli"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1093\/biomet\/88.2.381","article-title":"A simple resampling method by perturbing the minimand","volume":"88","author":"Jin","year":"2001","journal-title":"Biometrika"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1093\/biomet\/92.3.691","article-title":"Diagnostic checking for time series models with conditional heteroscedasticity estimated by the least absolute deviation approach","volume":"92","author":"Li","year":"2005","journal-title":"Biometrika"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1080\/01621459.2014.892007","article-title":"Quantile correlations and quantile autoregressive modeling","volume":"110","author":"Li","year":"2015","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1214\/aos\/1028144858","article-title":"Limiting distributions for l1 regression estimators under general conditions","volume":"26","author":"Knight","year":"1998","journal-title":"Ann. Stat."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/3\/326\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T05:14:29Z","timestamp":1773551669000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/3\/326"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,13]]},"references-count":27,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2026,3]]}},"alternative-id":["e28030326"],"URL":"https:\/\/doi.org\/10.3390\/e28030326","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,13]]}}}