{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T04:49:46Z","timestamp":1767156586860,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,24]],"date-time":"2019-04-24T00:00:00Z","timestamp":1556064000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Retailers need demand forecasts at different levels of aggregation in order to support a variety of decisions along the supply chain. To ensure aligned decision-making across the hierarchy, it is essential that forecasts at the most disaggregated level add up to forecasts at the aggregate levels above. It is not clear if these aggregate forecasts should be generated independently or by using an hierarchical forecasting method that ensures coherent decision-making at the different levels but does not guarantee, at least, the same accuracy. To give guidelines on this issue, our empirical study investigates the relative performance of independent and reconciled forecasting approaches, using real data from a Portuguese retailer. We consider two alternative forecasting model families for generating the base forecasts; namely, state space models and ARIMA. Appropriate models from both families are chosen for each time-series by minimising the bias-corrected Akaike information criteria. The results show significant improvements in forecast accuracy, providing valuable information to support management decisions. It is clear that reconciled forecasts using the Minimum Trace Shrinkage estimator (MinT-Shrink) generally improve on the accuracy of the ARIMA base forecasts for all levels and for the complete hierarchy, across all forecast horizons. The accuracy gains generally increase with the horizon, varying between 1.7% and 3.7% for the complete hierarchy. It is also evident that the gains in forecast accuracy are more substantial at the higher levels of aggregation, which means that the information about the individual dynamics of the series, which was lost due to aggregation, is brought back again from the lower levels of aggregation to the higher levels by the reconciliation process, substantially improving the forecast accuracy over the base forecasts.<\/jats:p>","DOI":"10.3390\/e21040436","type":"journal-article","created":{"date-parts":[[2019,4,25]],"date-time":"2019-04-25T03:02:59Z","timestamp":1556161379000},"page":"436","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Assessing the Performance of Hierarchical Forecasting Methods on the Retail Sector"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8516-6418","authenticated-orcid":false,"given":"Jos\u00e9 Manuel","family":"Oliveira","sequence":"first","affiliation":[{"name":"INESC Technology and Science, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal"},{"name":"Faculty of Economics, University of Porto, Rua Dr. Roberto Frias, 4200-464 Porto, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0959-8446","authenticated-orcid":false,"given":"Patr\u00edcia","family":"Ramos","sequence":"additional","affiliation":[{"name":"INESC Technology and Science, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal"},{"name":"School of Accounting and Administration of Porto, Polytechnic Institute of Porto, Rua Jaime Lopes Amorim, 4465-004 S. Mamede de Infesta, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,24]]},"reference":[{"key":"ref_1","unstructured":"Fildes, R., Ma, S., and Kolassa, S. (2019, April 24). Retail forecasting: Research and practice. Working paper. Available online: http:\/\/eprints.lancs.ac.uk\/128587\/."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2745","DOI":"10.1287\/mnsc.2015.2259","article-title":"The sum and its parts: Judgmental hierarchical forecasting","volume":"62","author":"Kremer","year":"2016","journal-title":"Manag. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"412","DOI":"10.1016\/j.ejor.2017.04.047","article-title":"Integrated hierarchical forecasting","volume":"263","author":"Pennings","year":"2017","journal-title":"Eur. J. Oper. Res."},{"key":"ref_4","first-page":"773","article-title":"Data aggregation and information loss","volume":"58","author":"Orcutt","year":"1968","journal-title":"Am. Econ. Rev."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1080\/01621459.1976.10481478","article-title":"Aggregate versus subaggregate models in local area forecasting","volume":"71","author":"Dunn","year":"1976","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1287\/mnsc.25.6.594","article-title":"Aggregation and proration in forecasting","volume":"25","author":"Shlifer","year":"1979","journal-title":"Manag. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/0304-4076(82)90087-2","article-title":"When is an aggregate of a time series efficiently forecast by its past?","volume":"18","author":"Kohn","year":"1982","journal-title":"J. Econom."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1002\/for.3980090304","article-title":"Disaggregation methods to expedite product line forecasting","volume":"9","author":"Gross","year":"1990","journal-title":"J. Forecast."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"146","DOI":"10.1016\/j.ijforecast.2008.07.004","article-title":"Hierarchical forecasts for Australian domestic tourism","volume":"25","author":"Athanasopoulos","year":"2009","journal-title":"Int. J. Forecast."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/0169-2070(92)90121-O","article-title":"Top-down or bottom-up: Aggregate versus disaggregate extrapolations","volume":"8","author":"Dangerfield","year":"1992","journal-title":"Int. J. Forecast."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.ijpe.2008.08.013","article-title":"Forecasting aggregate demand: An analytical evaluation of top-down versus bottom-up forecasting in a production planning framework","volume":"118","author":"Widiarta","year":"2009","journal-title":"Int. J. Prod. Econ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ejor.2015.11.010","article-title":"Supply chain forecasting: Theory, practice, their gap and the future","volume":"252","author":"Syntetos","year":"2016","journal-title":"Eur. J. Oper. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2579","DOI":"10.1016\/j.csda.2011.03.006","article-title":"Optimal combination forecasts for hierarchical time series","volume":"55","author":"Hyndman","year":"2011","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.csda.2015.11.007","article-title":"Fast computation of reconciled forecasts for hierarchical and grouped time series","volume":"97","author":"Hyndman","year":"2016","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Wickramasuriya, S.L., Athanasopoulos, G., and Hyndman, R.J. (2018). Optimal forecast reconciliation for hierarchical and grouped time series through trace minimization. J. Am. Stat. Assoc.","DOI":"10.1080\/01621459.2018.1448825"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/978-3-319-18732-7_15","article-title":"Game-Theoretically Optimal Reconciliation of Contemporaneous Hierarchical Time Series Forecasts","volume":"Volume 217","author":"Antoniadis","year":"2015","journal-title":"Modeling and Stochastic Learning for Forecasting in High Dimensions"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/j.trpro.2017.03.026","article-title":"Modified top down approach for hierarchical forecasting in a beverage supply chain","volume":"22","author":"Mircetic","year":"2017","journal-title":"Transplant. Res. Procedia"},{"key":"ref_18","unstructured":"Hyndman, R.J., and Athanasopoulos, G. (2019, April 24). Forecasting: Principles and Practice; Online Open-access Textbooks, 2018. Available online: https:\/\/OTexts.com\/fpp2\/."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Hyndman, R.J., Koehler, A.B., Ord, J.K., and Snyder, R.D. (2008). Forecasting with Exponential Smoothing: The State Space Approach, Springer.","DOI":"10.1007\/978-3-540-71918-2"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1214\/aos\/1176344136","article-title":"Estimating the dimension of a model","volume":"6","author":"Schwarz","year":"1978","journal-title":"Ann. Stat."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/j.rcim.2014.12.015","article-title":"Performance of state space and ARIMA models for consumer retail sales forecasting","volume":"34","author":"Ramos","year":"2015","journal-title":"Robot. Comput. Integr. Manuf."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Ramos, P., and Oliveira, J.M. (2016). A procedure for identification of appropriate state space and ARIMA models based on time-series cross-validation. Algorithms, 9.","DOI":"10.3390\/a9040076"},{"key":"ref_23","unstructured":"Box, G.E.P., Jenkins, G.M., Reinsel, G.C., and Ljung, G.M. (2015). Time Series Analysis: Forecasting and Control, John Wiley & Sons, Inc.. [5th ed.]."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1111\/j.2517-6161.1964.tb00553.x","article-title":"An analysis of transformations","volume":"26","author":"Box","year":"1964","journal-title":"J. R. Stat. Soc."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1080\/07350015.1995.10524598","article-title":"Are seasonal patterns constant over time? A test for seasonal stability","volume":"13","author":"Canova","year":"1985","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/0304-4076(92)90104-Y","article-title":"Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root?","volume":"54","author":"Kwiatkowski","year":"1992","journal-title":"J. Econom."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Hamilton, J. (1994). Time Series Analysis, Princeton University Press.","DOI":"10.1515\/9780691218632"},{"key":"ref_28","unstructured":"Theil, H. (1974). Linear Aggregation of Economic Relations, North-Holland."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1002\/1099-131X(200009)19:5<457::AID-FOR761>3.0.CO;2-6","article-title":"A note on aggregation, disaggregation and forecasting performance","volume":"19","author":"Zellner","year":"2000","journal-title":"J. Forecast."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/1926089","article-title":"Is aggregation necessarily bad?","volume":"42","author":"Grunfeld","year":"1960","journal-title":"Rev. Econ. Stat."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1080\/07350015.1984.10509388","article-title":"Forecasting contemporaneously aggregated vector ARMA processes","volume":"2","author":"Lutkepohl","year":"1984","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_32","unstructured":"McLeavey, D.W., and Narasimhan, S. (1974). Production Planning and Inventory Control, Allyn and Bacon Inc."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.1016\/S0305-0548(99)00017-9","article-title":"An investigation of aggregate variable timesSeries forecast strategies with specific subaggregate time series statistical correlation","volume":"26","author":"Fliedner","year":"1999","journal-title":"Comput. Oper. Res."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.ejor.2017.02.046","article-title":"Forecasting with temporal hierarchies","volume":"262","author":"Athanasopoulos","year":"2017","journal-title":"Eur. J. Oper. Res."},{"key":"ref_35","first-page":"151","article-title":"A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics","volume":"4","author":"Strimmer","year":"2005","journal-title":"Stat. Appl. Genet. Mol. Biol."},{"key":"ref_36","unstructured":"R Development Core Team (2019). R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing."},{"key":"ref_37","first-page":"1","article-title":"Automatic time series forecasting: the forecast package for R","volume":"26","author":"Hyndman","year":"2008","journal-title":"J. Stat. Softw."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1007\/s11600-018-0120-7","article-title":"Predictability of monthly temperature and precipitation using automatic time series forecasting methods","volume":"66","author":"Papacharalampous","year":"2018","journal-title":"Acta Geophys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1186\/s40562-018-0111-1","article-title":"One-step ahead forecasting of geophysical processes within a purely statistical framework","volume":"5","author":"Papacharalampous","year":"2018","journal-title":"Geosci. Lett."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Papacharalampous, G., Tyralis, H., and Koutsoyiannis, D. (2019). Comparison of stochastic and machine learning methods for multi-step ahead forecasting of hydrological processes. Stoch. Environ. Res. Risk Assess.","DOI":"10.20944\/preprints201710.0133.v3"},{"key":"ref_41","unstructured":"Hyndman, R., Lee, A., Wang, E., and Wickramasuriya, S. (2019, April 24). hts: Hierarchical and Grouped Time Series, 2018. R package Version 5.1.5. Available online: https:\/\/pkg.earo.me\/hts\/."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"510","DOI":"10.1016\/j.ijforecast.2012.09.002","article-title":"Measuring forecasting accuracy: The case of judgmental adjustments to SKU-level demand forecasts","volume":"29","author":"Davydenko","year":"2013","journal-title":"Int. J. Forecast."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1692","DOI":"10.1016\/j.jbusres.2015.03.028","article-title":"Simple versus complex selection rules for forecasting many time series","volume":"68","author":"Fildes","year":"2015","journal-title":"J. Bus. Res."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"218","DOI":"10.1145\/5666.5673","article-title":"How not to lie with statistics: The correct way to summarize benchmark results","volume":"29","author":"Fleming","year":"1986","journal-title":"Commun. ACM"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1016\/j.annals.2019.02.001","article-title":"Cross-temporal coherent forecasts for Australian tourism","volume":"75","author":"Kourentzes","year":"2019","journal-title":"Ann. Tourism Res."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Hollander, M., Wolfe, D.A., and Chicken, E. (2015). Nonparametric Statistical Methods, John Wiley & Sons, Inc.","DOI":"10.1002\/9781119196037"},{"key":"ref_47","doi-asserted-by":"crossref","unstructured":"Kourentzes, N., Svetunkov, I., and Schaer, O. (2019, April 24). tsutils: Time Series Exploration, Modelling and Forecasting, 2019. R package Version 0.9.0. Available online: https:\/\/rdrr.io\/cran\/tsutils\/.","DOI":"10.32614\/CRAN.package.tsutils"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/4\/436\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:46:54Z","timestamp":1760186814000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/4\/436"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,24]]},"references-count":47,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,4]]}},"alternative-id":["e21040436"],"URL":"https:\/\/doi.org\/10.3390\/e21040436","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2019,4,24]]}}}