{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T13:22:19Z","timestamp":1776172939720,"version":"3.50.1"},"reference-count":54,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,13]],"date-time":"2023-08-13T00:00:00Z","timestamp":1691884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000038","name":"Natural Science and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN-2017-05720"],"award-info":[{"award-number":["RGPIN-2017-05720"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Science and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["RGPIN-2023-05655"],"award-info":[{"award-number":["RGPIN-2023-05655"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Science and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["71931004"],"award-info":[{"award-number":["71931004"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Science and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["92046005"],"award-info":[{"award-number":["92046005"]}],"id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"name":"State Key Program of National Natural Science Foundation of China","award":["RGPIN-2017-05720"],"award-info":[{"award-number":["RGPIN-2017-05720"]}]},{"name":"State Key Program of National Natural Science Foundation of China","award":["RGPIN-2023-05655"],"award-info":[{"award-number":["RGPIN-2023-05655"]}]},{"name":"State Key Program of National Natural Science Foundation of China","award":["71931004"],"award-info":[{"award-number":["71931004"]}]},{"name":"State Key Program of National Natural Science Foundation of China","award":["92046005"],"award-info":[{"award-number":["92046005"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["RGPIN-2017-05720"],"award-info":[{"award-number":["RGPIN-2017-05720"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["RGPIN-2023-05655"],"award-info":[{"award-number":["RGPIN-2023-05655"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["71931004"],"award-info":[{"award-number":["71931004"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["92046005"],"award-info":[{"award-number":["92046005"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The value at risk based on expectile (EVaR) is a very useful method to measure financial risk, especially in measuring extreme financial risk. The double-threshold autoregressive conditional heteroscedastic (DTARCH) model is a valuable tool in assessing the volatility of a financial asset\u2019s return. A significant characteristic of DTARCH models is that their conditional mean and conditional variance functions are both piecewise linear, involving double thresholds. This paper proposes the weighted composite expectile regression (WCER) estimation of the DTARCH model based on expectile regression theory. Therefore, we can use EVaR to predict extreme financial risk, especially when the conditional mean and the conditional variance of asset returns are nonlinear. Unlike the existing papers on DTARCH models, we do not assume that the threshold and delay parameters are known. Using simulation studies, it has been demonstrated that the proposed WCER estimation exhibits adequate and promising performance in finite samples. Finally, the proposed approach is used to analyze the daily Hang Seng Index (HSI) and the Standard &amp; Poor\u2019s 500 Index (SPI).<\/jats:p>","DOI":"10.3390\/e25081204","type":"journal-article","created":{"date-parts":[[2023,8,14]],"date-time":"2023-08-14T10:13:57Z","timestamp":1692008037000},"page":"1204","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The Financial Risk Measurement EVaR Based on DTARCH Models"],"prefix":"10.3390","volume":"25","author":[{"given":"Xiaoqian","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhenni","family":"Tan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8858-1289","authenticated-orcid":false,"given":"Yuehua","family":"Wu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yong","family":"Zhou","sequence":"additional","affiliation":[{"name":"Key Laboratory of Advanced Theory and Application in Statistics and Data Science, MOE, and Academy of Statistics and Interdisciplinary Sciences and School of Statistics, East China Normal University, Shanghai 200062, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1002\/(SICI)1099-1255(199605)11:3<253::AID-JAE393>3.0.CO;2-8","article-title":"On a double-threshold autoregressive heteroscedastic time series model","volume":"11","author":"Li","year":"1996","journal-title":"J. Appl. Econom."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1111\/j.1368-423X.2005.00157.x","article-title":"Robust modelling of DTARCH models","volume":"8","author":"Jiang","year":"2005","journal-title":"Econom. J."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1111\/ectj.12023","article-title":"Weighted composite quantile regression estimation of DTARCH models","volume":"17","author":"Jiang","year":"2014","journal-title":"Econom. J."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2571","DOI":"10.1007\/s11425-016-9321-x","article-title":"Likelihood ratio-type tests in weighted composite quantile regression of DTARCH models","volume":"62","author":"Liu","year":"2019","journal-title":"Sci. China Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.jeconom.2008.12.002","article-title":"Assessing value at risk with CARE, the conditional autoregressive expectile models","volume":"150","author":"Kuan","year":"2009","journal-title":"J. Econom."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"819","DOI":"10.2307\/1911031","article-title":"Asymmetric least squares estimation and testing","volume":"55","author":"Newey","year":"1987","journal-title":"Econometrica"},{"key":"ref_7","first-page":"93","article-title":"Regression percentiles using asymmetric squared error loss","volume":"1","author":"Efron","year":"1991","journal-title":"Stat. Sin."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/0167-7152(94)90031-0","article-title":"Expectiles and m-quantiles are quantiles","volume":"20","author":"Jones","year":"1994","journal-title":"Stat. Probab."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1080\/10485259608832675","article-title":"Asymmetric least squares regression estimation: A nonparametric approach","volume":"6","author":"Yao","year":"1996","journal-title":"J. Nonparametric Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1111\/sjos.12518","article-title":"Expectile-based measures of skewness","volume":"49","author":"Eberl","year":"2022","journal-title":"Scand. J. Stat."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.jeconom.2009.01.001","article-title":"Quantiles, expectiles and splines","volume":"152","author":"Harvey","year":"2009","journal-title":"J. Econom."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1016\/j.csda.2010.11.015","article-title":"Geoadditive expectile regression","volume":"56","author":"Sobotka","year":"2012","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"304","DOI":"10.1016\/j.spl.2018.02.006","article-title":"Expectile regression for analyzing heteroscedasticity in high dimension","volume":"137","author":"Zhao","year":"2018","journal-title":"Stat. Probab. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1731","DOI":"10.1080\/03610926.2017.1324989","article-title":"Variable selection in expectile regression","volume":"47","author":"Zhao","year":"2018","journal-title":"Commun. Stat.-Theory Methods"},{"key":"ref_15","first-page":"671","article-title":"Aggregated expectile regression by exponential weighting","volume":"29","author":"Gu","year":"2019","journal-title":"Stat. Sin."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"531","DOI":"10.3150\/19-BEJ1137","article-title":"Tail expectile process and risk assessment","volume":"26","author":"Daouia","year":"2020","journal-title":"Bernoulli"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"104758","DOI":"10.1016\/j.jmva.2021.104758","article-title":"Semi-parametric estimation of multivariate extreme expectiles","volume":"184","author":"Beck","year":"2021","journal-title":"J. Multivar. Anal."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/j.jeconom.2020.02.003","article-title":"ExpectHill estimation, extreme risk and heavy tails","volume":"221","author":"Daouia","year":"2021","journal-title":"J. Econom."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3358","DOI":"10.1214\/21-AOS2087","article-title":"Extreme conditional expectile estimation in heavy-tailed heteroscedastic regression models","volume":"49","author":"Girard","year":"2021","journal-title":"Ann. Stat."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Pan, Y., Liu, Z., and Song, G. (2021). Weighted expectile regression with covariates missing at random. Commun. -Stat.-Simul. Comput., 1\u201320. just-accepted.","DOI":"10.1080\/03610918.2021.1975133"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1080\/10485252.2022.2027412","article-title":"Nonparametric estimation of expectile regression in functional dependent data","volume":"34","author":"Almanjahie","year":"2022","journal-title":"J. Nonparametric Stat."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.jspi.2021.08.003","article-title":"Optimal model averaging estimator for expectile regressions","volume":"217","author":"Bai","year":"2022","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1111\/sjos.12502","article-title":"Nonparametric extreme conditional expectile estimation","volume":"49","author":"Girard","year":"2022","journal-title":"Scand. J. Stat."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"109682","DOI":"10.1016\/j.spl.2022.109682","article-title":"Local linear estimate of the functional expectile regression","volume":"192","author":"Litimein","year":"2023","journal-title":"Stat. Probab. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"576","DOI":"10.1080\/07350015.2014.917979","article-title":"A varying-coefficient expectile model for estimating value at risk","volume":"32","author":"Xie","year":"2014","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_26","first-page":"1089","article-title":"Weighted composite expectile regression estimate of autoregressive models with application","volume":"36","author":"Liu","year":"2016","journal-title":"Syst. Eng.-Theory Pract."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1080\/1351847X.2015.1052150","article-title":"Risk management with expectiles","volume":"23","author":"Bellini","year":"2017","journal-title":"Eur. J. Financ."},{"key":"ref_28","first-page":"183","article-title":"Assessing tail risk using expectile regressions with partially varying coefficients","volume":"3","author":"Cai","year":"2018","journal-title":"J. Manag. Sci. Eng."},{"key":"ref_29","first-page":"2176","article-title":"The semiparametric varying-coefficient composite expectile regression model in risk measurement and its application","volume":"40","author":"Liu","year":"2020","journal-title":"Syst. Eng.-Theory Pract."},{"key":"ref_30","first-page":"1377","article-title":"Semiparametric varying-coefficient expectile model for estimating value at risk on dependent samples","volume":"51","author":"Wang","year":"2021","journal-title":"Sci. China Math."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Syuhada, K., Hakim, A., and Nur\u2019aini, R. (2021). The expected-based value-at-risk and expected shortfall using quantile and expectile with application to electricity market data. Commun. -Stat.-Simul. Comput., 1\u201318. just-accepted.","DOI":"10.1080\/03610918.2021.1928191"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Davison, A.C., Padoan, S.A., and Stupfler, G. (2022). Tail risk inference via expectiles in heavy-tailed time series. J. Bus. Econ. Stat., 1\u201334. just-accepted.","DOI":"10.1080\/07350015.2022.2078332"},{"key":"ref_33","first-page":"345","article-title":"Single-index expectile models for estimating conditional value at risk and expected shortfall","volume":"20","author":"Jiang","year":"2022","journal-title":"J. Financ. Econom."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"522","DOI":"10.1080\/07350015.2020.1833890","article-title":"Prediction of extremal expectile based on regression models with heteroscedastic extremes","volume":"40","author":"Xu","year":"2022","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"987","DOI":"10.2307\/1912773","article-title":"Autoregressive conditional heteroskedasticity with estimates of the U.K. inflation","volume":"50","author":"Engle","year":"1982","journal-title":"Econometrica"},{"key":"ref_36","unstructured":"Tong, H. (1978). Pattern Recognition and Signal Procession, Sijthoff & Noordhoff."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.1214\/aos\/1176343648","article-title":"Descriptive statistics for nonparametric models. III. Dispersion","volume":"4","author":"Bickel","year":"1976","journal-title":"Ann. Stat."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1214\/aos\/1176344124","article-title":"Tests for heteroscedasticity, nonlinearity","volume":"6","author":"Bickel","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Carroll, R., and Ruppert, D. (1988). Transformations and Weighting in Regression, Chapman and Hall.","DOI":"10.1007\/978-1-4899-2873-3"},{"key":"ref_40","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_41","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1002\/1099-131X(200103)20:2<111::AID-FOR786>3.0.CO;2-N","article-title":"Robust modelling of ARCH models","volume":"20","author":"Jiang","year":"2001","journal-title":"J. Forecast."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"269","DOI":"10.1111\/j.1467-9892.1986.tb00494.x","article-title":"On the consistency of least squares estimators for a thershold AR(1) model","volume":"7","author":"Petruccelli","year":"1986","journal-title":"J. Time Ser. Anal."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"520","DOI":"10.1214\/aos\/1176349040","article-title":"Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model","volume":"21","author":"Chan","year":"1993","journal-title":"Ann. Stat."},{"key":"ref_44","unstructured":"Liu, X. (2014). The Statistical Analysis of Financial Risk Measurement Time Series Models and Their Application. [Ph.D. Thesis, Shanghai University of Finance and Economics]."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1111\/j.1467-9868.2009.00725.x","article-title":"Local composite quantile regression smoothing: An efficient and safe alternative to local polynomial regression","volume":"72","author":"Kai","year":"2010","journal-title":"J. R. Stat. Soc. Ser. B (Methodological)"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1214\/10-AOS842","article-title":"New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models","volume":"39","author":"Kai","year":"2011","journal-title":"Ann. Stat."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1108","DOI":"10.1214\/07-AOS507","article-title":"Composite quantile regression and the oracle model selection theory","volume":"36","author":"Zou","year":"2008","journal-title":"Ann. Stat."},{"key":"ref_48","first-page":"1479","article-title":"Oracle model selection for nonlinear models based on weighted composite quantile regression","volume":"22","author":"Jiang","year":"2012","journal-title":"Stat. Sin."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1272","DOI":"10.1017\/S0266466614000176","article-title":"Efficient regressions via optimally combining quantile information","volume":"30","author":"Zhao","year":"2014","journal-title":"Econom. Theory"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1080\/713665670","article-title":"Empirical properties of asset returns: Stylized facts and statistical issues","volume":"1","author":"Cont","year":"2001","journal-title":"Quant. Financ."},{"key":"ref_51","first-page":"283","article-title":"A heavy-tailed distribution for ARCH residuals with application to volatility prediction","volume":"5","author":"Politis","year":"2004","journal-title":"Ann. Econ. Financ."},{"key":"ref_52","unstructured":"Hong, Y., Lee, J.C., Ding, G., and Heavy-Tailed Distributions, GARCH Model and the Stock Market Returns in South Korea (2023, June 14). Available Ssrn 3014472. Available online: https:\/\/papers.ssrn.com\/sol3\/papers.cfm?abstract-id=3014472."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"186","DOI":"10.1017\/S0266466600004394","article-title":"Asymptotics for least absolute deviation regression estimators","volume":"7","author":"Pollard","year":"1991","journal-title":"Econom. Theory"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"123","DOI":"10.2307\/2951780","article-title":"The limiting distribution of the maximum rank correlation estimator","volume":"61","author":"Sherman","year":"1993","journal-title":"Econometrica"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/8\/1204\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:32:44Z","timestamp":1760128364000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/8\/1204"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,13]]},"references-count":54,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["e25081204"],"URL":"https:\/\/doi.org\/10.3390\/e25081204","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,13]]}}}