{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,4]],"date-time":"2026-09-04T17:59:39Z","timestamp":1788544779016,"version":"build-2803163510"},"reference-count":29,"publisher":"The Econometric Society","issue":"2","license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/legalcode"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["QE"],"abstract":"<jats:p>In this paper, we analyze regressions with observations collected at small time intervals over a long period of time. For the formal asymptotic analysis, we assume that samples are obtained from continuous time stochastic processes, and let the sampling interval \n            <jats:italic>\u03b4<\/jats:italic> shrink down to zero and the sample span \n            <jats:italic>T<\/jats:italic> increase up to infinity. In this setup, we show that the standard Wald statistic diverges to infinity and the regression becomes spurious as long as \n            <jats:italic>\u03b4<\/jats:italic>\u2009\u2192\u20090 sufficiently fast relative to \n            <jats:italic>T<\/jats:italic>\u2009\u2192\u2009\u221e. Such a phenomenon is indeed what is frequently observed in practice for the type of regressions considered in the paper. In contrast, our asymptotic theory predicts that the spuriousness disappears if we use the robust version of the Wald test with an appropriate long\u2010run variance estimate. This is supported, strongly and unambiguously, by our empirical illustration using the regression of long\u2010term on short\u2010term interest rates.\n          <\/jats:p>","DOI":"10.3982\/qe2055","type":"journal-article","created":{"date-parts":[[2025,6,11]],"date-time":"2025-06-11T06:57:46Z","timestamp":1749625066000},"page":"405-457","source":"Crossref","is-referenced-by-count":1,"title":["Understanding regressions with observations collected at high frequency over long span"],"prefix":"10.3982","volume":"16","author":[{"given":"Yoosoon","family":"Chang","sequence":"first","affiliation":[{"name":"Department of Economics, Indiana University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ye","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Economics, University of Sydney"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Joon Y.","family":"Park","sequence":"additional","affiliation":[{"name":"Department of Economics, Indiana University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"2359","published-online":{"date-parts":[[2025]]},"reference":[{"issue":"2","key":"quan200376-bib-0001","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1016\/j.jfineco.2005.10.006","volume":"83","author":"A\u00eft-Sahalia","year":"2007","journal-title":"Journal of Financial Economics"},{"key":"quan200376-bib-0002","doi-asserted-by":"crossref","first-page":"817","DOI":"10.2307\/2938229","volume":"59","author":"Andrews","year":"1991","journal-title":"Econometrica"},{"key":"quan200376-bib-0003","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1016\/S0304-3932(03)00032-1","volume":"50","author":"Ang","year":"2003","journal-title":"Journal of Monetary Economics"},{"key":"quan200376-bib-0004","doi-asserted-by":"crossref","first-page":"943","DOI":"10.1257\/aer.110.4.943","volume":"110","author":"Bernanke","year":"2020","journal-title":"American Economic Review"},{"issue":"2","key":"quan200376-bib-0005","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1007\/BF00531822","volume":"60","author":"Bhattacharya","year":"1982","journal-title":"Zeitschrift f\u00fcr Wahrscheinlichkeitstheorie und verwandte Gebiete"},{"key":"quan200376-bib-0006","doi-asserted-by":"crossref","first-page":"495","DOI":"10.2307\/2298008","volume":"58","author":"Campbell","year":"1991","journal-title":"Review of Economic Studies"},{"key":"quan200376-bib-0007","doi-asserted-by":"crossref","first-page":"889","DOI":"10.3982\/QE251","volume":"7","author":"Chang","year":"2016","journal-title":"Quantitative Economics"},{"key":"quan200376-bib-0008","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.jeconom.2015.02.048","volume":"192","author":"Choi","year":"2016","journal-title":"Journal of Econometrics"},{"key":"quan200376-bib-0009","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1086\/322834","volume":"109","author":"Cochrane","year":"2001","journal-title":"Journal of Political Economy"},{"key":"quan200376-bib-0010","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1016\/0304-405X(84)90013-8","volume":"13","author":"Fama","year":"1984","journal-title":"Journal of Financial Economic"},{"key":"quan200376-bib-0011","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/0304-4076(74)90034-7","volume":"2","author":"Granger","year":"1974","journal-title":"Journal of Econometrics"},{"key":"quan200376-bib-0012","doi-asserted-by":"crossref","first-page":"2291","DOI":"10.1016\/j.jmoneco.2007.06.029","volume":"54","author":"G\u00fcrkaynak","year":"2007","journal-title":"Journal of Monetary Economics"},{"issue":"2","key":"quan200376-bib-0013","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1093\/rfs\/6.2.327","volume":"6","author":"Heston","year":"1993","journal-title":"Review of Financial Studies"},{"key":"quan200376-bib-0014","unstructured":"Jeong, Minsoo and Joon Y. 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