{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T07:04:02Z","timestamp":1781507042951,"version":"3.54.1"},"reference-count":35,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,4]],"date-time":"2023-01-04T00:00:00Z","timestamp":1672790400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)","doi-asserted-by":"publisher","award":["437270842"],"award-info":[{"award-number":["437270842"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In a time series context, the study of the partial autocorrelation function (PACF) is helpful for model identification. Especially in the case of autoregressive (AR) models, it is widely used for order selection. During the last decades, the use of AR-type count processes, i.e., which also fulfil the Yule\u2013Walker equations and thus provide the same PACF characterization as AR models, increased a lot. This motivates the use of the PACF test also for such count processes. By computing the sample PACF based on the raw data or the Pearson residuals, respectively, findings are usually evaluated based on well-known asymptotic results. However, the conditions for these asymptotics are generally not fulfilled for AR-type count processes, which deteriorates the performance of the PACF test in such cases. Thus, we present different implementations of the PACF test for AR-type count processes, which rely on several bootstrap schemes for count times series. We compare them in simulations with the asymptotic results, and we illustrate them with an application to a real-world data example.<\/jats:p>","DOI":"10.3390\/e25010105","type":"journal-article","created":{"date-parts":[[2023,1,4]],"date-time":"2023-01-04T04:28:17Z","timestamp":1672806497000},"page":"105","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":18,"title":["Partial Autocorrelation Diagnostics for Count Time Series"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8739-6631","authenticated-orcid":false,"given":"Christian H.","family":"Wei\u00df","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Helmut Schmidt University, 22043 Hamburg, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Boris","family":"Aleksandrov","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Helmut Schmidt University, 22043 Hamburg, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maxime","family":"Faymonville","sequence":"additional","affiliation":[{"name":"Department of Statistics, TU Dortmund University, 44221 Dortmund, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7824-1697","authenticated-orcid":false,"given":"Carsten","family":"Jentsch","sequence":"additional","affiliation":[{"name":"Department of Statistics, TU Dortmund University, 44221 Dortmund, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1098\/rspa.1931.0069","article-title":"On periodicity in series of related terms","volume":"131","author":"Walker","year":"1931","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1098\/rsta.1927.0007","article-title":"On a method of investigating periodicities in disturbed series, with special reference to Wolfer\u2019s sunspot numbers","volume":"226","author":"Yule","year":"1927","journal-title":"Philos. Trans. R. Soc. Lond. Ser. A"},{"key":"ref_3","unstructured":"Box, G.E.P., and Jenkins, G.M. (1970). Time Series Analysis: Forecasting and Control, Holden-Day. [1st ed.]."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1214\/09-SS060","article-title":"The ARMA alphabet soup: A tour of ARMA model variants","volume":"4","author":"Holan","year":"2010","journal-title":"Stat. Surv."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Wei\u00df, C.H. (2018). An Introduction to Discrete-Valued Time Series, John Wiley & Sons, Inc.","DOI":"10.1002\/9781119097013"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1111\/1467-842X.00143","article-title":"Non-Gaussian conditional linear AR(1) models","volume":"42","author":"Grunwald","year":"2000","journal-title":"Aust. N. Z. J. Stat."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1111\/j.1467-9892.1991.tb00073.x","article-title":"The integer-valued autoregressive (INAR(p)) model","volume":"12","author":"Du","year":"1991","journal-title":"J. Time Ser. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1111\/j.1467-9892.2006.00496.x","article-title":"Integer-valued GARCH processes","volume":"27","author":"Ferland","year":"2006","journal-title":"J. Time Ser. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1715","DOI":"10.1007\/s00180-020-00980-6","article-title":"Models for autoregressive processes of bounded counts: How different are they?","volume":"35","author":"Kim","year":"2020","journal-title":"Comput. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1007\/s10182-010-0139-9","article-title":"Useful models for time series of counts or simply wrong ones?","volume":"95","author":"Jung","year":"2011","journal-title":"AStA Adv. Stat. Anal."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Brockwell, P.J., and Davis, R.A. (1991). Time Series: Theory and Methods, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4419-0320-4"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1080\/02331888.2014.974606","article-title":"Goodness-of-fit tests for binomial AR(1) processes","volume":"49","author":"Kim","year":"2015","journal-title":"Statistics"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/0304-4149(91)90047-G","article-title":"Independence of partial autocorrelations for a classical immigration branching process","volume":"37","author":"Mills","year":"1991","journal-title":"Stoch. Process. Their Appl."},{"key":"ref_14","first-page":"20180018","article-title":"Checking model adequacy for count time series by using Pearson residuals","volume":"12","author":"Scherer","year":"2020","journal-title":"J. Time Ser. Econom."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1214\/aop\/1176994950","article-title":"Discrete analogues of self-decomposability and stability","volume":"7","author":"Steutel","year":"1979","journal-title":"Ann. Probab."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"467","DOI":"10.1111\/j.1467-9868.2008.00687.x","article-title":"Efficient estimation of auto-regression parameters and innovation distributions for semiparametric integer-valued AR(p) models","volume":"71","author":"Drost","year":"2009","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Faymonville, M., Jentsch, C., Wei\u00df, C.H., and Aleksandrov, B. (2022). Semiparametric estimation of INAR models using roughness penalization. Stat. Methods Appl., 1\u201336.","DOI":"10.1007\/s10260-022-00655-0"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1080\/07350015.1989.10509750","article-title":"Time series models for count or qualitative observations","volume":"7","author":"Harvey","year":"1989","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"496","DOI":"10.1016\/j.csda.2009.09.019","article-title":"Diagnostic checking integer-valued ARCH(p) models using conditional residual autocorrelations","volume":"54","author":"Zhu","year":"2010","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1016\/0167-9473(93)90157-O","article-title":"Approximate moments to O(n\u22122) for the sampled partial autocorrelations from a white noise process","volume":"16","author":"Anderson","year":"1993","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1111\/j.1467-9892.1993.tb00166.x","article-title":"Exact general-lag serial correlation moments and approximate low-lag partial correlation moments for Gaussian white noise","volume":"14","author":"Anderson","year":"1993","journal-title":"J. Time Ser. Anal."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1080\/1350485032000125245","article-title":"Sample partial autocorrelations and portmanteau tests for randomness","volume":"10","author":"Kwan","year":"2003","journal-title":"Appl. Econ. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.spl.2016.01.018","article-title":"Bias corrections for moment estimators in Poisson INAR(1) and INARCH(1) processes","volume":"112","author":"Schweer","year":"2016","journal-title":"Stat. Probab. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Hall, P. (1992). The Bootstrap and Edgeworth Expansion, Springer.","DOI":"10.1007\/978-1-4612-4384-7"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2359","DOI":"10.3150\/18-BEJ1057","article-title":"Bootstrapping INAR models","volume":"25","author":"Jentsch","year":"2019","journal-title":"Bernoulli"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1081\/ETC-120028836","article-title":"Automatic block-length selection for the dependent bootstrap","volume":"23","author":"Politis","year":"2004","journal-title":"Econom. Rev."},{"key":"ref_27","first-page":"1","article-title":"Bootstrap methods: Another look at the jackknife","volume":"73","author":"Efron","year":"1979","journal-title":"Ann. Stat."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1017\/S0266466612000655","article-title":"A warp-speed method for conducting Monte Carlo experiments involving bootstrap estimators","volume":"29","author":"Giacomini","year":"2013","journal-title":"Econom. Theory"},{"key":"ref_29","unstructured":"Freeland, R.K. (1998). Statistical Analysis of Discrete Time Series with Applications to the Analysis of Workers Compensation Claims Data. [Ph.D. Thesis, University of British Columbia]. Available online: https:\/\/open.library.ubc.ca\/cIRcle\/collections\/ubctheses\/831\/items\/1.0088709."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"284","DOI":"10.3390\/stats2020022","article-title":"INARMA modeling of count time series","volume":"2","author":"Feld","year":"2019","journal-title":"Stats"},{"key":"ref_31","first-page":"110","article-title":"Zur Transformation der Scharen bilinearer Formen von unendlichvielen Ver\u00e4nderlichen","volume":"1907","author":"Toeplitz","year":"1907","journal-title":"Nachrichten von der Ges. der Wiss. G\u00f6ttingen Math.-Phys. Kl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/BF01564502","article-title":"Zur Theorie der quadratischen und bilinearen Formen von unendlichvielen Ver\u00e4nderlichen. I. Teil: Theorie der L-Formen","volume":"70","author":"Toeplitz","year":"1911","journal-title":"Mathematische Annalen"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"233","DOI":"10.2307\/1401322","article-title":"The fitting of time-series models","volume":"28","author":"Durbin","year":"1960","journal-title":"Rev. Int. Stat. Inst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1002\/sapm1946251261","article-title":"The Wiener (root mean square) error criterion in filter design and prediction","volume":"25","author":"Levinson","year":"1946","journal-title":"J. Math. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1016\/0047-259X(73)90030-4","article-title":"On the parametrization of autoregressive models by partial autocorrelations","volume":"3","author":"Schou","year":"1973","journal-title":"J. Multivar. Anal."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/105\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:58:28Z","timestamp":1760119108000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/1\/105"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,4]]},"references-count":35,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["e25010105"],"URL":"https:\/\/doi.org\/10.3390\/e25010105","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,4]]}}}