{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:44:09Z","timestamp":1760150649133,"version":"build-2065373602"},"reference-count":20,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,20]],"date-time":"2023-12-20T00:00:00Z","timestamp":1703030400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["23CX03012A","2021YFA1000102","ZR2021MA025","ZR2021MA028"],"award-info":[{"award-number":["23CX03012A","2021YFA1000102","ZR2021MA025","ZR2021MA028"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["23CX03012A","2021YFA1000102","ZR2021MA025","ZR2021MA028"],"award-info":[{"award-number":["23CX03012A","2021YFA1000102","ZR2021MA025","ZR2021MA028"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Shandong Provincial Natural Science Foundation of China","award":["23CX03012A","2021YFA1000102","ZR2021MA025","ZR2021MA028"],"award-info":[{"award-number":["23CX03012A","2021YFA1000102","ZR2021MA025","ZR2021MA028"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>With the widespread application of spatial data in fields like econometrics and geographic information science, the methods to enhance the robustness of spatial econometric model estimation and variable selection have become a central focus of research. In the context of the spatial error model (SEM), this paper introduces a variable selection method based on exponential square loss and the adaptive lasso penalty. Due to the non-convex and non-differentiable nature of this proposed method, convex programming is not applicable for its solution. We develop a block coordinate descent algorithm, decompose the exponential square component into the difference of two convex functions, and utilize the CCCP algorithm in combination with parabolic interpolation for optimizing problem-solving. Numerical simulations demonstrate that neglecting the spatial effects of error terms can lead to reduced accuracy in selecting zero coefficients in SEM. The proposed method demonstrates robustness even when noise is present in the observed values and when the spatial weights matrix is inaccurate. Finally, we apply the model to the Boston housing dataset.<\/jats:p>","DOI":"10.3390\/axioms13010004","type":"journal-article","created":{"date-parts":[[2023,12,20]],"date-time":"2023-12-20T07:20:31Z","timestamp":1703056831000},"page":"4","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Robust Variable Selection with Exponential Squared Loss for the Spatial Error Model"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-9807-0301","authenticated-orcid":false,"given":"Shida","family":"Ma","sequence":"first","affiliation":[{"name":"College of Science, China University of Petroleum, Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yiming","family":"Hou","sequence":"additional","affiliation":[{"name":"College of Science, China University of Petroleum, Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yunquan","family":"Song","sequence":"additional","affiliation":[{"name":"College of Science, China University of Petroleum, Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1897-5677","authenticated-orcid":false,"given":"Feng","family":"Zhou","sequence":"additional","affiliation":[{"name":"College of Science, China University of Petroleum, Qingdao 266580, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Anselin, L. (1988). Spatial Econometrics: Methods and Models, Springer Science and Business Media.","DOI":"10.1007\/978-94-015-7799-1"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.2517-6161.1996.tb02080.x","article-title":"Regression shrinkage and selection via the lasso","volume":"58","author":"Tibshirani","year":"1996","journal-title":"J. R. Stat. Soc. Ser. B Stat. Methodol."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1348","DOI":"10.1198\/016214501753382273","article-title":"Variable selection via nonconcave penalized likelihood and its oracle properties","volume":"96","author":"Fan","year":"2001","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1418","DOI":"10.1198\/016214506000000735","article-title":"The adaptive lasso and its oracle properties","volume":"101","author":"Zou","year":"2006","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_5","unstructured":"Huber, P.J. (2004). Robust Statistics, John Wiley Sons."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"632","DOI":"10.1080\/01621459.2013.766613","article-title":"Robust variable selection with exponential squared loss","volume":"108","author":"Wang","year":"2013","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_7","first-page":"33","article-title":"Regression quantiles","volume":"46","author":"Koenker","year":"1978","journal-title":"Econom. J. Econom. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1108","DOI":"10.1214\/07-AOS507","article-title":"Composite quantile regression and the oracle model selection theory","volume":"36","author":"Zou","year":"2008","journal-title":"Ann. Statist."},{"key":"ref_9","first-page":"15","article-title":"Variable Selection for Spatial Error Models","volume":"46","author":"Liu","year":"2020","journal-title":"J. Yanbian Univ. Nat. Sci. Ed."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"110978","DOI":"10.1016\/j.econlet.2023.110978","article-title":"Modified harmonic mean method for spatial autoregressive models","volume":"223","year":"2023","journal-title":"Econ. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"107094","DOI":"10.1016\/j.csda.2020.107094","article-title":"Robust variable selection with exponential squared loss for the spatial autoregressive model","volume":"155","author":"Song","year":"2021","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Banerjee, S., Carlin, B.P., and Gelfand, A.E. (2014). Hierarchical Modeling and Analysis for Spatial Data, CRC Press.","DOI":"10.1201\/b17115"},{"key":"ref_13","first-page":"653","article-title":"A naive least squares method for spatial autoregression with covariates","volume":"30","author":"Ma","year":"2020","journal-title":"Stat. Sin."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1198\/073500106000000251","article-title":"Robust regression shrinkage and consistent variable selection through the LAD-Lasso","volume":"25","author":"Wang","year":"2007","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_15","unstructured":"Forsythe, G.E. (1977). Computer Methods for Mathematical Computations, Prentice-Hall."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1162\/08997660360581958","article-title":"The concave-convex procedure","volume":"15","author":"Yuille","year":"2003","journal-title":"Neural Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1137\/080716542","article-title":"A fast iterative shrinkage-thresholding algorithm for linear inverse problems","volume":"2","author":"Beck","year":"2009","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1198\/jasa.2009.0127","article-title":"Variable selection for partially linear models with measurement errors","volume":"104","author":"Liang","year":"2009","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Burnham, K.P., and Anderson, D.R. (2004). Model Selection and Multimodel Inference. A Practical Information-Theoretic Approach, Springer.","DOI":"10.1007\/b97636"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/0095-0696(78)90006-2","article-title":"Hedonic housing prices and the demand for clean air","volume":"5","author":"Harrison","year":"1978","journal-title":"J. Environ. Econ. Manag."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/1\/4\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:42:12Z","timestamp":1760132532000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/1\/4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,20]]},"references-count":20,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["axioms13010004"],"URL":"https:\/\/doi.org\/10.3390\/axioms13010004","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,12,20]]}}}