{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,5]],"date-time":"2026-10-05T02:56:01Z","timestamp":1791168961526,"version":"4.1.0"},"reference-count":34,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2012,10,12]],"date-time":"2012-10-12T00:00:00Z","timestamp":1350000000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["1804 03"],"award-info":[{"award-number":["1804 03"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2013,3,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>In conditionally heteroscedastic models, the optimal prediction of powers, or logarithms, of the absolute value has a simple expression in terms of the volatility and an expectation involving the independent process. A natural procedure for estimating this prediction is to estimate the volatility in the first step, for instance by Gaussian quasi-maximum-likelihood or by least absolute deviations, and to use empirical means based on rescaled innovations to estimate the expectation in the second step. The paper proposes an alternative one-step procedure, based on an appropriate non-Gaussian quasi-maximum-likelihood estimator, and establishes the asymptotic properties of the two approaches. Asymptotic comparisons and numerical experiments show that the differences in accuracy can be important, depending on the prediction problem and the innovations distribution. An application to indices of major stock exchanges is given.<\/jats:p>","DOI":"10.1111\/j.1467-9868.2012.01045.x","type":"journal-article","created":{"date-parts":[[2012,10,12]],"date-time":"2012-10-12T06:30:28Z","timestamp":1350023428000},"page":"345-367","source":"Crossref","is-referenced-by-count":35,"title":["Optimal Predictions of Powers of Conditionally Heteroscedastic Processes"],"prefix":"10.1093","volume":"75","author":[{"given":"Christian","family":"Francq","sequence":"first","affiliation":[{"name":"Centre de Recherche en Economie et Statistique , Malakoff , and , France"},{"name":"Universit\u00e9 Lille 3 , Malakoff , and , France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jean-Michel","family":"Zako\u00efan","sequence":"additional","affiliation":[{"name":"Centre de Recherche en Economie et Statistique , Malakoff , and , France"},{"name":"Universit\u00e9 Lille 3 , Malakoff , and , France"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2012,10,12]]},"reference":[{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"885","DOI":"10.2307\/2527343","article-title":"Answering the skeptics: yes, standard volatility models do provide accurate forecasts","volume":"39","author":"Andersen","year":"1998","journal-title":"Int. Econ. Rev."},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"655","DOI":"10.1111\/j.1467-9868.2009.00696.x","article-title":"Splines for financial volatility","volume":"71","author":"Audrino","year":"2009","journal-title":"J. R. Statist. Soc. B"},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1016\/0304-4076(92)90066-Z","article-title":"Prediction in dynamic models with time-dependent conditional variance","volume":"52","author":"Baillie","year":"1992","journal-title":"J. Econmetr."},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"2730","DOI":"10.1214\/08-AOS674","article-title":"Asymptotic normality of the Quasi-maximum likelihood estimator for multidimensional causal processes","volume":"37","author":"Bardet","year":"2009","journal-title":"Ann. Statist."},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1214\/009053604000000120","article-title":"The efficiency of the estimators of the parameters in GARCH processes","volume":"32","author":"Berkes","year":"2004","journal-title":"Ann. Statist."},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"201","DOI":"10.3150\/bj\/1068128975","article-title":"GARCH processes: structure and estimation","volume":"9","author":"Berkes","year":"2003","journal-title":"Bernoulli"},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/0304-4076(92)90067-2","article-title":"Stationarity of GARCH processes and of some non-negative time series","volume":"52","author":"Bougerol","year":"1992","journal-title":"J. Econmetr."},{"key":"2023031109352574300_","first-page":"399","article-title":"Autoregressive conditional kurtosis","volume":"3","author":"Brooks","year":"2005","journal-title":"J. Finan. Econmetr."},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/0304-4149(92)90142-D","article-title":"M-estimation for autoregressions with infinite variance","volume":"40","author":"Davis","year":"1992","journal-title":"Stoch. Processes Appl."},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/0927-5398(93)90006-D","article-title":"A long memory property of stock market returns and a new model","volume":"1","author":"Ding","year":"1993","journal-title":"J. Empir. Finan."},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/07474938608800095","article-title":"Modeling the persistence of conditional variances","volume":"5","author":"Engle","year":"1986","journal-title":"Econmetr. Rev."},{"key":"2023031109352574300_","volume-title":"Applied Time Series Analysis of Economic Data","author":"Engle","year":"1983"},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.2307\/2999632","article-title":"Autoregressive conditional duration: a new model for irregularly spaced transaction data","volume":"66","author":"Engle","year":"1998","journal-title":"Econometrica"},{"key":"2023031109352574300_","doi-asserted-by":"crossref","DOI":"10.2139\/ssrn.1540363","volume-title":"Quasi maximum likelihood estimation of GARCH models with heavy-tailed likelihoods","author":"Fan","year":"2010"},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/j.jeconom.2011.08.001","article-title":"Two-stage non Gaussian QML estimation of GARCH models and testing the efficiency of the Gaussian QMLE","volume":"165","author":"Francq","year":"2011","journal-title":"J. Econmetr."},{"key":"2023031109352574300_","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":"2023031109352574300_","doi-asserted-by":"crossref","DOI":"10.1002\/9780470670057","volume-title":"GARCH Models: Structure, Statistical Inference and Financial Applications","author":"Francq","year":"2010"},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"681","DOI":"10.2307\/1913471","article-title":"Pseudo maximum likelihood methods: theory","volume":"52","author":"Gouri\u00e9roux","year":"1984","journal-title":"Econometrica"},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"137","DOI":"10.2307\/2526988","article-title":"A class of non-linear ARCH models","volume":"33","author":"Higgins","year":"1992","journal-title":"Int. Econ. Rev."},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1111\/1467-9892.00241","article-title":"Prediction in ARMA models with GARCH in mean effects","volume":"22","author":"Karanasos","year":"2002","journal-title":"J. Time Ser. Anal."},{"key":"2023031109352574300_","volume-title":"Diagnostic Checks in Time Series","author":"Li","year":"2004"},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1111\/j.1467-9868.2004.00432.x","article-title":"Estimation and testing stationarity for double-autoregressive models","volume":"66","author":"Ling","year":"2004","journal-title":"J. R. Statist. Soc. B"},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1111\/j.1467-9868.2005.00507.x","article-title":"Self-weighted least absolute deviation estimation for infinite variance autoregressive models","volume":"67","author":"Ling","year":"2005","journal-title":"J. R. Statist. Soc. B"},{"key":"2023031109352574300_","volume-title":"Probability Theory, vol. I","author":"Lo\u00e8ve","year":"1977","edition":"4th edn"},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"1530","DOI":"10.1017\/S0266466608080602","article-title":"M-estimation in GARCH models","volume":"24","author":"Mukherjee","year":"2008","journal-title":"Econmetr. Theor."},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1017\/S0266466600005296","article-title":"Stationarity and persistence in the GARCH(1,1) model","volume":"6","author":"Nelson","year":"1990","journal-title":"Econmetr. Theor."},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"587","DOI":"10.2307\/2171754","article-title":"Asymptotic bias for quasi-maximum-likelihood estimators in conditional heteroscedasticity models","volume":"65","author":"Newey","year":"1997","journal-title":"Econometrica"},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"2293","DOI":"10.1016\/j.csda.2004.12.008","article-title":"Bootstrap prediction for returns and volatilities in GARCH models","volume":"50","author":"Pascual","year":"2005","journal-title":"Computnl Statist. Data Anal."},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1016\/j.ijforecast.2010.05.007","article-title":"Prediction intervals in conditionally heteroscedastic time series with stochastic components","volume":"27","author":"Pellegrini","year":"2012","journal-title":"Int. J. Forecast."},{"key":"2023031109352574300_","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":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/0304-4076(91)90078-R","article-title":"Testing for strong correlation and dynamic conditional heteroskedasticity in multiple regression","volume":"47","author":"Robinson","year":"1991","journal-title":"J. Econmetr."},{"key":"2023031109352574300_","first-page":"2449","article-title":"Quasi-maximum likelihood estimation in conditionally heteroscedastic Time Series: a stochastic recurrence equations approach","volume":"5","author":"Straumann","year":"2006","journal-title":"Ann. Statist."},{"key":"2023031109352574300_","volume-title":"Asset Price Dynamics, Volatility and Prediction","author":"Taylor","year":"2007"},{"key":"2023031109352574300_","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1016\/0165-1889(94)90039-6","article-title":"Threshold heteroskedastic models","volume":"18","author":"Zako\u00efan","year":"1994","journal-title":"J. Econ. Dynam. Control"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Fj.1467-9868.2012.01045.x","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/75\/2\/345\/49510383\/jrsssb_75_2_345.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/75\/2\/345\/49510383\/jrsssb_75_2_345.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,30]],"date-time":"2024-04-30T03:04:11Z","timestamp":1714446251000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/75\/2\/345\/7075428"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,12]]},"references-count":34,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2012,10,12]]},"published-print":{"date-parts":[[2013,3,1]]}},"URL":"https:\/\/doi.org\/10.1111\/j.1467-9868.2012.01045.x","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2013,3]]},"published":{"date-parts":[[2012,10,12]]}}}