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While modeling using linear models and SARIMA models is well established in the literature, modeling using state-space models has been extended with the proposal of alternative estimation methods to the maximum likelihood. However, maximum likelihood estimation assumes, as a rule, that the errors are normal. This paper suggests implementing the bootstrap methodology, utilizing the model\u2019s innovation representation, to derive distribution-free estimates\u2014both point and interval\u2014of the parameters in the time-varying state-space model. Additionally, it aims to estimate the standard errors of these parameters through the bootstrap methodology. The simulation study demonstrated that the distribution-free estimation, coupled with the bootstrap methodology, yields point forecasts with a lower mean-squared error, particularly for small time series or when dealing with smaller values of the autoregressive parameter in the state equation of state-space models. In this context, distribution-free estimation with the bootstrap methodology serves as an alternative to maximum likelihood estimation, eliminating the need for distributional assumptions. The application of this methodology to real data showed that it performed well when compared to the usual maximum likelihood estimation and even produced prediction intervals with a similar amplitude for the same level of confidence without any distributional assumptions about the errors.<\/jats:p>","DOI":"10.3390\/forecast6010003","type":"journal-article","created":{"date-parts":[[2023,12,27]],"date-time":"2023-12-27T07:45:32Z","timestamp":1703663132000},"page":"36-54","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Bootstrapping State-Space Models: Distribution-Free Estimation in View of Prediction and Forecasting"],"prefix":"10.3390","volume":"6","author":[{"given":"Jos\u00e9 Francisco","family":"Lima","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Minho, 4710-057 Braga, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6545-2900","authenticated-orcid":false,"given":"Fernanda Catarina","family":"Pereira","sequence":"additional","affiliation":[{"name":"Centre of Mathematics, University of Minho, 4710-057 Braga, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8491-6048","authenticated-orcid":false,"given":"Arminda Manuela","family":"Gon\u00e7alves","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Minho, 4710-057 Braga, Portugal"},{"name":"Centre of Mathematics, University of Minho, 4710-057 Braga, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7686-2430","authenticated-orcid":false,"given":"Marco","family":"Costa","sequence":"additional","affiliation":[{"name":"Centre for Research and Development in Mathematics and Applications, \u00c1gueda School of Technology and Management, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"708","DOI":"10.1080\/07350015.2022.2060988","article-title":"Forecasting with Economic News","volume":"41","author":"Barbaglia","year":"2023","journal-title":"J. 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Forecasting: Principles and Practice, OTexts."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/j.camwa.2021.05.026","article-title":"Model and data reduction for data assimilation: Particle filters employing projected forecasts and data with application to a shallow water model","volume":"116","author":"Albarakati","year":"2022","journal-title":"Comput. Math. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"107062","DOI":"10.1016\/j.csda.2020.107062","article-title":"An algorithm for non-parametric estima- tion in state\u2013space models","volume":"153","author":"Chau","year":"2021","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1007\/s10687-017-0283-7","article-title":"Distribution-free inference in record series","volume":"20","author":"Hoayek","year":"2017","journal-title":"Extremes"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1257\/jep.15.4.87","article-title":"Applications of Generalized Method of Moments Estimation","volume":"15","author":"Wooldridge","year":"2001","journal-title":"J. Econ. Perspect."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1889","DOI":"10.1016\/j.jspi.2010.01.036","article-title":"Parameter estimation of state-space models for univariate observations","volume":"140","author":"Costa","year":"2010","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1021","DOI":"10.1007\/s00477-012-0640-7","article-title":"Predicting seasonal and hydro-meteorological impact in environmental variables modelling via Kalman filtering","volume":"27","author":"Costa","year":"2013","journal-title":"Stoch. Environ. Res. Risk Assess."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/07474930008800457","article-title":"Recent developments in bootstrapping time series","volume":"19","author":"Berkowitz","year":"2000","journal-title":"Econom. Rev."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1024","DOI":"10.1080\/01621459.1991.10475148","article-title":"Bootstrapping State-Space Models: Gaussian Maximum Likelihood Estimation and the Kalman Filter","volume":"86","author":"Stoffer","year":"1991","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1002\/jae.2843","article-title":"Bootstrap inference and diagnostics in state-space models: With applications to dynamic macro models","volume":"37","author":"Angelini","year":"2022","journal-title":"J. Appl. Econom."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1016\/j.iref.2013.09.002","article-title":"Purchasing and supply managers provide early clues on the direction of the US economy: An application of a new market-timing test","volume":"29","author":"Tsuchiya","year":"2014","journal-title":"Int. Rev. Econ. Financ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"117195","DOI":"10.1016\/j.eswa.2022.117195","article-title":"Mixed pooling of seasonality for time series forecasting: An application to pallet transport data","volume":"201","author":"Moon","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.jspi.2016.04.002","article-title":"Bias-correction of Kalman filter estimators associated to a linear state-space model with estimated parameters","volume":"176","author":"Costa","year":"2016","journal-title":"J. Stat. Plan. Inference"},{"key":"ref_21","first-page":"1","article-title":"Reconstructing the Kalman Filter for Stationary and Non Stationary Time Series","volume":"7","author":"Snyder","year":"2002","journal-title":"Stud. Nonlinear Dyn. Econom."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1111\/j.1467-9892.2008.00604.x","article-title":"Bootstrap prediction intervals in state\u2013space models","volume":"30","author":"Rodriguez","year":"2009","journal-title":"J. Time Ser. Anal."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1111\/j.1467-9892.2005.00448.x","article-title":"Bootstrap approximation to prediction MSE for state-space models with estimated parameters","volume":"26","author":"Pfeffermann","year":"2005","journal-title":"J. Time Ser. Anal."},{"key":"ref_24","unstructured":"Anderson, B.D.O., and Moore, J.B. (1979). Optimal Filtering, Prentice-Hall."},{"key":"ref_25","first-page":"1","article-title":"An assessment of the ism manufacturing price index for inflation forecasting","volume":"2018","author":"Bognanni","year":"2018","journal-title":"Econ. Comment."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"981","DOI":"10.1016\/j.jmoneco.2005.07.001","article-title":"The incredible volcker disinflation","volume":"52","author":"Goodfriend","year":"2005","journal-title":"J. Monet. Econ."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1080\/07350015.1995.10524599","article-title":"Comparing predictive accuracy","volume":"13","author":"Diebold","year":"1995","journal-title":"J. Bus. Econ. Stat."}],"container-title":["Forecasting"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2571-9394\/6\/1\/3\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:42:46Z","timestamp":1760132566000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2571-9394\/6\/1\/3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,27]]},"references-count":27,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["forecast6010003"],"URL":"https:\/\/doi.org\/10.3390\/forecast6010003","relation":{},"ISSN":["2571-9394"],"issn-type":[{"value":"2571-9394","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,12,27]]}}}