{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,12]],"date-time":"2026-09-12T05:55:24Z","timestamp":1789192524519,"version":"build-2803163510"},"reference-count":17,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2025,3,6]],"date-time":"2025-03-06T00:00:00Z","timestamp":1741219200000},"content-version":"vor","delay-in-days":64,"URL":"https:\/\/academic.oup.com\/pages\/standard-publication-reuse-rights"}],"funder":[{"DOI":"10.13039\/100000001","name":"U.S. National Science Foundation","doi-asserted-by":"publisher","award":["DMS 2310973"],"award-info":[{"award-number":["DMS 2310973"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Columbia University\u2019s Shared Research Computing Facility"},{"DOI":"10.13039\/100000002","name":"NIH","doi-asserted-by":"publisher","award":["1G20RR030893-01"],"award-info":[{"award-number":["1G20RR030893-01"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"name":"New York State Empire State Development"},{"name":"Division of Science Technology and Innovation"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,3,16]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>A fundamental and often final step in time series modelling is to assess the quality of fit of a proposed model to the data. Since the underlying distribution of the innovations that generate a model is often not prescribed, goodness-of-fit tests typically take the form of testing the fitted residuals for serial independence. However, these fitted residuals are intrinsically dependent since they are based on the same parameter estimates, and thus standard tests of serial independence, such as those based on the autocorrelation function or auto-distance correlation function of the fitted residuals, need to be adjusted. The sample-splitting procedure of Pfister et al. (2018) is one such fix for the case of models for independent data, but fails to work in the dependent setting. In this article, sample splitting is leveraged in the time series setting to perform tests of serial dependence of fitted residuals using the autocorrelation function and auto-distance correlation function. The first $ f_{n} $ of the data points are used to estimate the parameters of the model and then, using these parameter estimates, the last $ l_{n} $ of the data points are used to compute the estimated residuals. Tests for serial independence are then based on these $ l_{n} $ residuals. As long as the overlap between the $ f_{n} $ and $ l_{n} $ data splits is asymptotically $ 1\/2 $, the autocorrelation function and auto-distance correlation function tests of serial independence often have the same limit distributions as when the underlying residuals are indeed independent and identically distributed. In particular, if the first half of the data is used to estimate the parameters and the estimated residuals are computed for the entire dataset based on these parameter estimates, then the autocorrelation function and auto-distance correlation function can have the same limit distributions as if the residuals were independent and identically distributed. This procedure ameliorates the need for adjustment in the construction of confidence bounds for both the autocorrelation function and the auto-distance correlation function in goodness-of-fit testing.<\/jats:p>","DOI":"10.1093\/biomet\/asaf017","type":"journal-article","created":{"date-parts":[[2025,3,2]],"date-time":"2025-03-02T23:15:19Z","timestamp":1740957319000},"source":"Crossref","is-referenced-by-count":4,"title":["Sample splitting and assessing goodness-of-fit of time series"],"prefix":"10.1093","volume":"112","author":[{"suffix":"By","given":"Richard A","family":"Davis","sequence":"first","affiliation":[{"name":"Department of Statistics, Columbia University , 1255 Amsterdam Avenue, New York, New York 10027,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Leon","family":"Fernandes","sequence":"additional","affiliation":[{"name":"Department of Statistics, Columbia University , 1255 Amsterdam Avenue, New York, New York 10027,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2025,3,5]]},"reference":[{"key":"2026030420012850400_asaf017-B1","doi-asserted-by":"crossref","first-page":"287","DOI":"10.2307\/1403689","article-title":"A Kolmogorov-Smirnov type statistic with application to test for nonlinearity in time series","volume":"59","author":"An","year":"1991","journal-title":"Int. Statist. Rev"},{"key":"2026030420012850400_asaf017-B2","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1017\/S0266466603194017","article-title":"Asymptotics for GARCH squared residual correlations","volume":"19","author":"Berkes","year":"2003","journal-title":"Economet. Theory"},{"key":"2026030420012850400_asaf017-B3","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":"2026030420012850400_asaf017-B4","doi-asserted-by":"crossref","first-page":"1509","DOI":"10.1080\/01621459.1970.10481180","article-title":"Distribution of residual autocorrelations in autoregressive-integrated moving average time series models","volume":"65","author":"Box","year":"1970","journal-title":"J. Am. Statist. Assoc"},{"key":"2026030420012850400_asaf017-B5","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-0320-4","volume-title":"Time Series: Theory and Methods","author":"Brockwell","year":"1991","edition":"2nd ed"},{"key":"2026030420012850400_asaf017-B6","doi-asserted-by":"crossref","first-page":"3087","DOI":"10.3150\/17-BEJ955","article-title":"Applications of distance correlation to time series","volume":"24","author":"Davis","year":"2018","journal-title":"Bernoulli"},{"key":"2026030420012850400_asaf017-B7","author":"Diebolt","year":"1998"},{"key":"2026030420012850400_asaf017-B8","first-page":"535","article-title":"Limiting distribution of weighted processes of residuals. application to parametric nonlinear autoregressive models","volume":"325","author":"Diebolt","year":"1997","journal-title":"C. R. Acad. Sci. I Math"},{"key":"2026030420012850400_asaf017-B9","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970586","volume-title":"Distribution Theory for Tests Based on the Sample Distribution Function","author":"Durbin","year":"1973"},{"key":"2026030420012850400_asaf017-B10","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/BFb0096877","volume-title":"Empirical Distributions and Processes","author":"Durbin","year":"1976"},{"key":"2026030420012850400_asaf017-B11","doi-asserted-by":"crossref","first-page":"204","DOI":"10.1214\/aos\/1018031108","article-title":"Nonparametric model checks for time series","volume":"27","author":"Koul","year":"1999","journal-title":"Ann. Statist."},{"key":"2026030420012850400_asaf017-B12","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1093\/biomet\/65.2.297","article-title":"On a measure of lack of fit in time series models","volume":"65","author":"Ljung","year":"1978","journal-title":"Biometrika"},{"key":"2026030420012850400_asaf017-B13","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1111\/rssb.12235","article-title":"Kernel-based tests for joint independence","volume":"80","author":"Pfister","year":"2018","journal-title":"J. R. Statist. Soc. B"},{"key":"2026030420012850400_asaf017-B14","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1111\/j.2517-6161.1978.tb01649.x","article-title":"On the half-sample method for goodness-of-fit","volume":"40","author":"Stephens","year":"1978","journal-title":"J. R. Statist. Soc. B"},{"key":"2026030420012850400_asaf017-B15","first-page":"97","volume-title":"Goodness-of-Fit Techniques","author":"Stephens","year":"1986"},{"key":"2026030420012850400_asaf017-B16","doi-asserted-by":"crossref","first-page":"2769","DOI":"10.1214\/009053607000000505","article-title":"Measuring and testing dependence by correlation of distances","volume":"35","author":"Sz\u00e9kely","year":"2007","journal-title":"Ann. Statist."},{"key":"2026030420012850400_asaf017-B17","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1016\/j.jeconom.2020.05.008","article-title":"Goodness-of-fit testing for time series models via distance covariance","volume":"227","author":"Wan","year":"2022","journal-title":"J. Economet"}],"container-title":["Biometrika"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/academic.oup.com\/biomet\/advance-article-pdf\/doi\/10.1093\/biomet\/asaf017\/62297940\/asaf017.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/112\/2\/asaf017\/62297940\/asaf017.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/biomet\/article-pdf\/112\/2\/asaf017\/62297940\/asaf017.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T01:01:39Z","timestamp":1772672499000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/biomet\/article\/doi\/10.1093\/biomet\/asaf017\/8056037"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025]]},"references-count":17,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,3,16]]}},"URL":"https:\/\/doi.org\/10.1093\/biomet\/asaf017","relation":{},"ISSN":["1464-3510"],"issn-type":[{"value":"1464-3510","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2025]]},"published":{"date-parts":[[2025]]},"article-number":"asaf017"}}