{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:30:09Z","timestamp":1772119809470,"version":"3.50.1"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T00:00:00Z","timestamp":1731456000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T00:00:00Z","timestamp":1731456000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2024,12]]},"DOI":"10.1007\/s11222-024-10532-3","type":"journal-article","created":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T18:42:14Z","timestamp":1731523334000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["On periodic $$\\log GARCH$$ model with empirical application"],"prefix":"10.1007","volume":"34","author":[{"given":"Abdelouahab","family":"Bibi","sequence":"first","affiliation":[]},{"given":"Fay\u00e7al","family":"Hamdi","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,11,13]]},"reference":[{"issue":"1","key":"10532_CR1","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1111\/j.1467-9892.2008.00598.x","volume":"30","author":"A Aknouche","year":"2009","unstructured":"Aknouche, A., Bibi, A.: Quasi-maximum likelihood estimation of periodic GARCH and periodic ARMA-GARCH processes. J. Time Ser. Anal. 30(1), 19\u201346 (2009)","journal-title":"J. Time Ser. Anal."},{"issue":"3","key":"10532_CR2","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1007\/s11203-017-9160-x","volume":"21","author":"A Aknouche","year":"2018","unstructured":"Aknouche, A., Al-Eid, E., Demouche, N.: Generalized quasi-maximum likelihood inference for periodic conditionally heteroskedastic models. Stat. Inference Stoch. Process. 21(3), 485\u2013511 (2018)","journal-title":"Stat. Inference Stoch. Process."},{"issue":"5B","key":"10532_CR3","doi-asserted-by":"publisher","first-page":"2730","DOI":"10.1214\/08-AOS674","volume":"37","author":"JM Bardet","year":"2009","unstructured":"Bardet, J.M., Wintenberger, O.: Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes. Ann. Stat. 37(5B), 2730\u20132759 (2009)","journal-title":"Ann. Stat."},{"issue":"6","key":"10532_CR4","doi-asserted-by":"publisher","first-page":"651","DOI":"10.1111\/1467-9892.00246","volume":"22","author":"IV Basawa","year":"2001","unstructured":"Basawa, I.V., Lund, R.: Large sample properties of parameter estimates for periodic ARMA models. J. Time Ser. Anal. 22(6), 651\u2013663 (2001)","journal-title":"J. Time Ser. Anal."},{"issue":"2","key":"10532_CR5","doi-asserted-by":"publisher","first-page":"201","DOI":"10.3150\/bj\/1068128975","volume":"9","author":"I Berkes","year":"2003","unstructured":"Berkes, I., Horvath, L., Kokoszka, P.: GARCH processes: structure and estimation. Bernoulli 9(2), 201\u2013227 (2003)","journal-title":"Bernoulli"},{"issue":"5","key":"10532_CR6","first-page":"788","volume":"12","author":"P Billingsley","year":"1961","unstructured":"Billingsley, P.: The Lindeberg-Levy theorem for martingales. Proc. Am. Math. Soc. 12(5), 788\u2013792 (1961)","journal-title":"Proc. Am. Math. Soc."},{"issue":"3","key":"10532_CR7","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1016\/0304-4076(86)90063-1","volume":"31","author":"T Bollerslev","year":"1986","unstructured":"Bollerslev, T.: Generalized autoregressive conditional heteroskedasticity. J. Econom. 31(3), 307\u2013327 (1986)","journal-title":"J. Econom."},{"issue":"2","key":"10532_CR8","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1080\/07350015.1996.10524640","volume":"14","author":"T Bollerslev","year":"1996","unstructured":"Bollerslev, T., Ghysels, E.: Periodic autoregressive conditional heteroskedasticity. J. Bus. Econ. Stat. 14(2), 139\u2013152 (1996)","journal-title":"J. Bus. Econ. Stat."},{"issue":"9","key":"10532_CR9","doi-asserted-by":"publisher","first-page":"1637","DOI":"10.1080\/00949655.2017.1401626","volume":"88","author":"N Boussaha","year":"2018","unstructured":"Boussaha, N., Hamdi, F.: On periodic autoregressive stochastic volatility models: structure and estimation. J. Stat. Comput. Simul. 88(9), 1637\u20131668 (2018)","journal-title":"J. Stat. Comput. Simul."},{"issue":"1","key":"10532_CR10","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1016\/0927-5398(93)90006-D","volume":"1","author":"Z Ding","year":"1993","unstructured":"Ding, Z., Granger, C.W., Engle, R.F.: A long memory property of stock market returns and a new model. J. Empir. Finance. 1(1), 83\u2013106 (1993)","journal-title":"J. Empir. Finance."},{"issue":"4","key":"10532_CR11","doi-asserted-by":"publisher","first-page":"987","DOI":"10.2307\/1912773","volume":"50","author":"RF Engle","year":"1982","unstructured":"Engle, R.F.: Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50(4), 987\u20131007 (1982)","journal-title":"Econometrica"},{"key":"10532_CR12","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1016\/j.eneco.2018.05.017","volume":"74","author":"A Escribano","year":"2018","unstructured":"Escribano, A., Sucarrat, G.: Equation-by-equation estimation of multivariate periodic electricity price volatility. Energy Econ. 74, 287\u2013298 (2018)","journal-title":"Energy Econ."},{"issue":"2","key":"10532_CR13","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1080\/07350015.2013.840239","volume":"32","author":"J Fan","year":"2014","unstructured":"Fan, J., Qi, L., Xiu, D.: Quasi-maximum likelihood estimation of GARCH models with heavy-tailed likelihoods. J. Bus. Econ. Stat. 32(2), 178\u2013191 (2014)","journal-title":"J. Bus. Econ. Stat."},{"issue":"2","key":"10532_CR14","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1016\/j.jeconom.2011.08.001","volume":"165","author":"C Francq","year":"2011","unstructured":"Francq, C., Lepage, G., Zako\u00efan, J.M.: Two-stage non Gaussian QML estimation of GARCH models and testing the efficiency of the Gaussian QMLE. J. Econom. 165(2), 246\u2013257 (2011)","journal-title":"J. Econom."},{"issue":"1","key":"10532_CR15","first-page":"129","volume":"16","author":"C Francq","year":"2018","unstructured":"Francq, C., Sucarrat, G.: An exponential chi-squared QMLE for log-GARCH models via the ARMA representation. J. Financ. Econom. 16(1), 129\u2013154 (2018)","journal-title":"J. Financ. Econom."},{"issue":"1","key":"10532_CR16","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1016\/j.jeconom.2013.05.004","volume":"177","author":"C Francq","year":"2013","unstructured":"Francq, C., Wintenberger, O., Zakoian, J.M.: GARCH models without positivity constraints: exponential or logGARCH? J. Econ. 177(1), 34\u201346 (2013)","journal-title":"J. Econ."},{"issue":"4","key":"10532_CR17","doi-asserted-by":"publisher","first-page":"605","DOI":"10.3150\/bj\/1093265632","volume":"10","author":"C Francq","year":"2004","unstructured":"Francq, C., Zakoian, J.M.: Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10(4), 605\u2013637 (2004)","journal-title":"Bernoulli"},{"issue":"5","key":"10532_CR18","doi-asserted-by":"publisher","first-page":"483","DOI":"10.1080\/096031000416352","volume":"10","author":"PH Franses","year":"2000","unstructured":"Franses, P.H., Paap, R.: Modeling day-of-the-week seasonality in the S &P 500 index. Appl. Financ. Econ. 10(5), 483\u2013488 (2000)","journal-title":"Appl. Financ. Econ."},{"issue":"1","key":"10532_CR19","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1080\/07474938608800097","volume":"5","author":"J Geweke","year":"1986","unstructured":"Geweke, J.: Modeling the persistence of conditional variances: a comment. Econ. Rev. 5(1), 57\u201361 (1986)","journal-title":"Econ. Rev."},{"issue":"3","key":"10532_CR20","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/s40304-019-00193-4","volume":"9","author":"A Ghezal","year":"2021","unstructured":"Ghezal, A.: QMLE for periodic time-varying asymmetric log GARCH models. Commun. Math. Stat. 9(3), 273\u2013297 (2021)","journal-title":"Commun. Math. Stat."},{"key":"10532_CR21","first-page":"385","volume":"2","author":"EG Gladyshev","year":"1961","unstructured":"Gladyshev, E.G.: Periodically correlated random sequences. Soviet. Math. 2, 385\u2013388 (1961)","journal-title":"Soviet. Math."},{"key":"10532_CR22","doi-asserted-by":"crossref","unstructured":"Hamdi, F., Souam, S.: Mixture periodic GARCH models: applications to exchange rate modeling. Paper presented at: 5th international conference on modeling, simulation and applied optimization (ICMSAO), IEEE, 28-30; Hammamet, Tunisia (2013 Apr)","DOI":"10.1109\/ICMSAO.2013.6552570"},{"issue":"4","key":"10532_CR23","doi-asserted-by":"publisher","first-page":"1925","DOI":"10.1007\/s00181-017-1348-9","volume":"55","author":"F Hamdi","year":"2018","unstructured":"Hamdi, F., Souam, S.: Mixture periodic GARCH models: theory and applications. Empir. Econ. 55(4), 1925\u20131956 (2018)","journal-title":"Empir. Econ."},{"key":"10532_CR24","doi-asserted-by":"publisher","DOI":"10.1002\/9780470182833","volume-title":"Periodically correlated random sequences: spectral theory and practice","author":"HL Hurd","year":"2007","unstructured":"Hurd, H.L., Miamee, A.: Periodically correlated random sequences: spectral theory and practice. John Wiley & Sons Inc, Hoboken, New Jersey (2007)"},{"key":"10532_CR25","unstructured":"Milh\u00f8j, A.: A multiplicative parameterization of ARCH Models. Working Paper, Department of Statistics, University of Copenhagen. (1987)"},{"issue":"2","key":"10532_CR26","doi-asserted-by":"publisher","first-page":"347","DOI":"10.2307\/2938260","volume":"59","author":"DB Nelson","year":"1991","unstructured":"Nelson, D.B.: Conditional heteroskedasticity in asset returns: a new approach. Econometrica 59(2), 347\u2013370 (1991)","journal-title":"Econometrica"},{"key":"10532_CR27","doi-asserted-by":"crossref","unstructured":"Newey, W.K., Steigerwald, D.G.: Asymptotic bias for quasi-maximum-likelihood estimators in conditional heteroskedasticity models. Econometrica 65(3), 587\u2013599 (1997)","DOI":"10.2307\/2171754"},{"issue":"1","key":"10532_CR28","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1080\/07474938608800099","volume":"5","author":"SG Pantula","year":"1986","unstructured":"Pantula, S.G.: Modeling the persistence of conditional variances: a comment. Econom. Rev. 5(1), 71\u201374 (1986)","journal-title":"Econom. Rev."},{"issue":"7","key":"10532_CR29","doi-asserted-by":"publisher","first-page":"3733","DOI":"10.1080\/03610918.2020.1722833","volume":"51","author":"M Sadoun","year":"2022","unstructured":"Sadoun, M., Bentarzi, M.: On periodic EGARCH models. Commun. Stat. Simul. Comput. 51(7), 3733\u20133759 (2022)","journal-title":"Commun. Stat. Simul. Comput."},{"issue":"5","key":"10532_CR30","doi-asserted-by":"publisher","first-page":"2449","DOI":"10.1214\/009053606000000803","volume":"34","author":"D Straumann","year":"2006","unstructured":"Straumann, D., Mikosch, T.: Quasi-maximum-likelihood estimation in conditionally heteroscedastic time series: a stochastic recurrence equations approach. Ann. Stat. 34(5), 2449\u20132495 (2006)","journal-title":"Ann. Stat."},{"issue":"10","key":"10532_CR31","doi-asserted-by":"publisher","first-page":"809","DOI":"10.1080\/1351847X.2017.1336452","volume":"24","author":"G Sucarrat","year":"2018","unstructured":"Sucarrat, G., Escribano, A.: Estimation of log-GARCH models in the presence of zero returns. Eur. J. Financ. 24(10), 809\u2013827 (2018)","journal-title":"Eur. J. Financ."},{"issue":"1","key":"10532_CR32","doi-asserted-by":"publisher","first-page":"1539","DOI":"10.1515\/math-2017-0131","volume":"15","author":"H Xuan","year":"2017","unstructured":"Xuan, H., Song, L., Amin, M., Shi, Y.: Quasi-maximum likelihood estimator of Laplace (1, 1) for GARCH models. Open Math. 15(1), 1539\u20131548 (2017)","journal-title":"Open Math."},{"issue":"5","key":"10532_CR33","doi-asserted-by":"publisher","first-page":"931","DOI":"10.1016\/0165-1889(94)90039-6","volume":"18","author":"JM Zakoian","year":"1994","unstructured":"Zakoian, J.M.: Threshold heteroskedastic models. J. Econ. Dyn. Control 18(5), 931\u2013955 (1994)","journal-title":"J. Econ. Dyn. Control"}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-024-10532-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-024-10532-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-024-10532-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,25]],"date-time":"2024-11-25T07:15:12Z","timestamp":1732518912000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-024-10532-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,13]]},"references-count":33,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["10532"],"URL":"https:\/\/doi.org\/10.1007\/s11222-024-10532-3","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-3975303\/v1","asserted-by":"object"}]},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"value":"0960-3174","type":"print"},{"value":"1573-1375","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,11,13]]},"assertion":[{"value":"21 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 October 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 November 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"This article does not contain any studies with human participants performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Human Participants"}},{"value":"Informed consent was obtained from all individual participants included in the study.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Informed Consent"}}],"article-number":"220"}}