{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T21:59:38Z","timestamp":1790805578020,"version":"4.1.0"},"reference-count":50,"publisher":"The Econometric Society","issue":"6","funder":[{"DOI":"10.13039\/100017045","name":"Research Computing Center, University of Chicago","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100017045","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100004316","name":"International Business Machines Corporation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100004316","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007618","name":"Booth School of Business, University of Chicago","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100007618","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100004318","name":"Microsoft","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100004318","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["ECTA"],"abstract":"<jats:p>\n                    We conduct inference on volatility with noisy high\u2010frequency data. We assume the observed transaction price follows a continuous\u2010time It\u00f4\u2010semimartingale, contaminated by a discrete\u2010time moving\u2010average noise process associated with the arrival of trades. We estimate volatility, defined as the quadratic variation of the semimartingale, by maximizing the likelihood of a misspecified moving\u2010average model, with its order selected based on an information criterion. Our inference is uniformly valid over a large class of noise processes whose magnitude and dependence structure vary with sample size. We show that the convergence rate of our estimator dominates\n                    <jats:italic>n<\/jats:italic>\n                    <jats:sup>1\/4<\/jats:sup>\n                    as noise vanishes, and is determined by the selected order of noise dependence when noise is sufficiently small. Our implementation guarantees positive estimates in finite samples.\n                  <\/jats:p>","DOI":"10.3982\/ecta15593","type":"journal-article","created":{"date-parts":[[2021,11,8]],"date-time":"2021-11-08T03:34:12Z","timestamp":1636342452000},"page":"2787-2825","source":"Crossref","is-referenced-by-count":41,"title":["When Moving\u2010Average Models Meet High\u2010Frequency Data: Uniform Inference on Volatility"],"prefix":"10.3982","volume":"89","author":[{"given":"Rui","family":"Da","sequence":"first","affiliation":[{"name":"Booth School of Business, University of Chicago"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dacheng","family":"Xiu","sequence":"additional","affiliation":[{"name":"Booth School of Business, University of Chicago"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"2359","published-online":{"date-parts":[[2021]]},"reference":[{"key":"ecta200356-bib-0001","volume-title":"High Frequency Financial Econometrics","author":"A\u00eft-Sahalia","year":"2014"},{"key":"ecta200356-bib-0002","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.jeconom.2016.05.002","volume":"194","author":"A\u00eft-Sahalia","year":"2016","journal-title":"Journal of Econometrics"},{"key":"ecta200356-bib-0003","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1016\/j.jeconom.2018.12.013","volume":"211","author":"A\u00eft-Sahalia","year":"2019","journal-title":"Journal of Econometrics"},{"key":"ecta200356-bib-0004","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1093\/rfs\/hhi016","volume":"18","author":"A\u00eft-Sahalia","year":"2005","journal-title":"Review of Financial Studies"},{"key":"ecta200356-bib-0005","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1016\/j.jeconom.2010.03.028","volume":"160","author":"A\u00eft-Sahalia","year":"2011","journal-title":"Journal of Econometrics"},{"key":"ecta200356-bib-0006","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1111\/1468-0262.00418","volume":"71","author":"Andersen","year":"2003","journal-title":"Econometrica"},{"issue":"5","key":"ecta200356-bib-0007","doi-asserted-by":"crossref","first-page":"2153","DOI":"10.3982\/ECTA9456","volume":"80","author":"Andrews","year":"2012","journal-title":"Econometrica"},{"issue":"2","key":"ecta200356-bib-0008","doi-asserted-by":"crossref","first-page":"496","DOI":"10.1016\/j.jeconom.2020.04.027","volume":"218","author":"Andrews","year":"2020","journal-title":"Journal of Econometrics"},{"key":"ecta200356-bib-0009","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1111\/j.1467-937X.2008.00474.x","volume":"75","author":"Bandi","year":"2008","journal-title":"Review of Economic Studies"},{"key":"ecta200356-bib-0010","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.3982\/ECTA6495","volume":"76","author":"Barndorff-Nielsen","year":"2008","journal-title":"Econometrica"},{"issue":"1","key":"ecta200356-bib-0011","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1016\/j.jeconom.2010.03.031","volume":"160","author":"Barndorff-Nielsen","year":"2011","journal-title":"Journal of Econometrics"},{"key":"ecta200356-bib-0012","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1093\/restud\/rdt044","volume":"81","author":"Belloni","year":"2014","journal-title":"Review of Economic Studies"},{"issue":"3","key":"ecta200356-bib-0013","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1080\/07350015.2017.1356728","volume":"37","author":"Bibinger","year":"2019","journal-title":"Journal of Business & Economic Statistics"},{"key":"ecta200356-bib-0014","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1016\/0304-4076(95)01735-6","volume":"73","author":"Comte","year":"1996","journal-title":"Journal of Econometrics"},{"key":"ecta200356-bib-0015","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1111\/1467-9965.00057","volume":"8","author":"Comte","year":"1998","journal-title":"Mathematical Finance"},{"key":"ecta200356-bib-0016","doi-asserted-by":"crossref","unstructured":"Da, R., and D. Xiu (2021a): \u201cDisentangling Autocorrelated Intraday Returns,\u201d Discussion Paper, University of Chicago.","DOI":"10.2139\/ssrn.3857785"},{"key":"ecta200356-bib-0017","doi-asserted-by":"crossref","unstructured":"Da, R., and D. Xiu (2021b): \u201cSupplement to \u2018When Moving-Average Models Meet High-Frequency Data: Uniform Inference on Volatility\u2019,\u201d Econometrica Supplemental Material, 89, https:\/\/doi.org\/10.3982\/ECTA15593.","DOI":"10.3982\/ECTA15593"},{"key":"ecta200356-bib-0018","doi-asserted-by":"crossref","first-page":"829","DOI":"10.1016\/j.spa.2010.02.006","volume":"120","author":"Fukasawa","year":"2010","journal-title":"Stochastic Processes and Applications"},{"issue":"3","key":"ecta200356-bib-0019","first-page":"311","volume":"29","author":"Gardner","year":"1980","journal-title":"Journal of the Royal Statistical Society. Series C (Applied Statistics)"},{"issue":"2","key":"ecta200356-bib-0020","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1198\/073500106000000071","volume":"24","author":"Hansen","year":"2006","journal-title":"Journal of Business & Economic Statistics"},{"key":"ecta200356-bib-0021","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1080\/07474930701853640","volume":"27","author":"Hansen","year":"2008","journal-title":"Econometric Reviews"},{"key":"ecta200356-bib-0022","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780195301649.001.0001","volume-title":"Empirical Market Microstructure","author":"Hasbrouck","year":"2007"},{"issue":"2","key":"ecta200356-bib-0023","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1080\/07350015.2012.754313","volume":"31","author":"Hautsch","year":"2013","journal-title":"Journal of Business & Economic Statistics"},{"key":"ecta200356-bib-0024","volume-title":"Discretization of Processes","author":"Jacod","year":"2011"},{"key":"ecta200356-bib-0025","doi-asserted-by":"crossref","first-page":"2249","DOI":"10.1016\/j.spa.2008.11.004","volume":"119","author":"Jacod","year":"2009","journal-title":"Stochastic Processes and Their Applications"},{"issue":"4","key":"ecta200356-bib-0026","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.3982\/ECTA13085","volume":"85","author":"Jacod","year":"2017","journal-title":"Econometrica"},{"key":"ecta200356-bib-0027","doi-asserted-by":"crossref","first-page":"80","DOI":"10.1016\/j.jeconom.2018.09.006","volume":"208","author":"Jacod","year":"2019","journal-title":"Journal of Econometrics"},{"key":"ecta200356-bib-0028","doi-asserted-by":"crossref","first-page":"1478","DOI":"10.1214\/09-AOS756","volume":"38","author":"Jacod","year":"2010","journal-title":"Annals of Statistics"},{"issue":"3","key":"ecta200356-bib-0029","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1017\/S0266466600009403","volume":"11","author":"Kabaila","year":"1995","journal-title":"Econometic Theory"},{"key":"ecta200356-bib-0030","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/j.jeconom.2008.09.016","volume":"147","author":"Kalnina","year":"2008","journal-title":"Journal of Econometrics"},{"issue":"1","key":"ecta200356-bib-0031","first-page":"21","volume":"21","author":"Leeb","year":"2005","journal-title":"Econometic Theory"},{"issue":"2","key":"ecta200356-bib-0032","first-page":"338","volume":"24","author":"Leeb","year":"2008","journal-title":"Econometric Theory"},{"key":"ecta200356-bib-0033","doi-asserted-by":"crossref","first-page":"1613","DOI":"10.3982\/ECTA12306","volume":"84","author":"Li","year":"2016","journal-title":"Econometrica"},{"key":"ecta200356-bib-0034","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1017\/S0266466613000418","volume":"30","author":"Li","year":"2014","journal-title":"Econometric Theory"},{"issue":"2","key":"ecta200356-bib-0035","doi-asserted-by":"crossref","first-page":"536","DOI":"10.1016\/j.jeconom.2019.10.004","volume":"215","author":"Li","year":"2020","journal-title":"Journal of Econometrics"},{"key":"ecta200356-bib-0036","first-page":"19","volume":"64","author":"Mancini","year":"2001","journal-title":"Giornale dell'Istituto Italiano degli Attuari"},{"issue":"1","key":"ecta200356-bib-0037","doi-asserted-by":"crossref","first-page":"173","DOI":"10.3982\/ECTA9371","volume":"80","author":"Mikusheva","year":"2012","journal-title":"Econometrica"},{"issue":"2","key":"ecta200356-bib-0038","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1017\/S0266466600004382","volume":"7","author":"P\u00f6tscher","year":"1991","journal-title":"Econometic Theory"},{"key":"ecta200356-bib-0039","doi-asserted-by":"crossref","first-page":"772","DOI":"10.1214\/10-AOS855","volume":"39","author":"Rei\u00df","year":"2011","journal-title":"Annals of Statistics"},{"key":"ecta200356-bib-0040","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1016\/j.jeconom.2010.03.036","volume":"160","author":"Renault","year":"2011","journal-title":"Journal of Econometrics"},{"issue":"1","key":"ecta200356-bib-0041","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1214\/aos\/1176344897","volume":"8","author":"Shibata","year":"1980","journal-title":"The Annals of Statistics"},{"key":"ecta200356-bib-0042","doi-asserted-by":"crossref","first-page":"127","DOI":"10.2307\/3214348","volume":"23","author":"Shibata","year":"1986","journal-title":"Journal of Applied Probability"},{"issue":"6","key":"ecta200356-bib-0043","first-page":"1","volume":"33","author":"Varneskov","year":"2016","journal-title":"Econometric Theory"},{"key":"ecta200356-bib-0044","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/1912526","volume":"50","author":"White","year":"1982","journal-title":"Econometrica"},{"key":"ecta200356-bib-0045","volume-title":"Hypothesis Testing in Time Series Analysis","author":"Whittle","year":"1951"},{"issue":"1","key":"ecta200356-bib-0046","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/0706001","volume":"6","author":"Wilson","year":"1969","journal-title":"SIAM Journal on Numerica Analysis"},{"key":"ecta200356-bib-0047","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.jeconom.2010.07.002","volume":"159","author":"Xiu","year":"2010","journal-title":"Journal of Econometrics"},{"key":"ecta200356-bib-0048","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.3150\/bj\/1165269149","volume":"12","author":"Zhang","year":"2006","journal-title":"Bernoulli"},{"key":"ecta200356-bib-0049","doi-asserted-by":"crossref","first-page":"1394","DOI":"10.1198\/016214505000000169","volume":"100","author":"Zhang","year":"2005","journal-title":"Journal of the American Statistical Association"},{"key":"ecta200356-bib-0050","volume-title":"Trigonometric Series","volume":"1","author":"Zygmund","year":"2002"}],"container-title":["Econometrica"],"original-title":[],"language":"en","deposited":{"date-parts":[[2023,11,12]],"date-time":"2023-11-12T05:31:19Z","timestamp":1699767079000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.econometricsociety.org\/doi\/10.3982\/ECTA15593"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021]]},"references-count":50,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021]]}},"alternative-id":["10.3982\/ECTA15593"],"URL":"https:\/\/doi.org\/10.3982\/ecta15593","relation":{},"ISSN":["0012-9682"],"issn-type":[{"value":"0012-9682","type":"print"}],"subject":[],"published":{"date-parts":[[2021]]}}}