{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T00:36:26Z","timestamp":1760229386781,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,10]],"date-time":"2022-06-10T00:00:00Z","timestamp":1654819200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Statistical Science Research Project","award":["2021LY061","2019QDL039"],"award-info":[{"award-number":["2021LY061","2019QDL039"]}]},{"name":"Startup Foundation for Talents at Hangzhou Normal University","award":["2021LY061","2019QDL039"],"award-info":[{"award-number":["2021LY061","2019QDL039"]}]},{"name":"Statistical Science Research Project of Zhejiang Province","award":["2021LY061","2019QDL039"],"award-info":[{"award-number":["2021LY061","2019QDL039"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Heteroscedasticity is often encountered in spatial-data analysis, so a new class of heterogeneous spatial autoregressive models is introduced in this paper, where the variance parameters are allowed to depend on some explanatory variables. Here, we are interested in the problem of parameter estimation and the variable selection for both the mean and variance models. Then, a unified procedure via double-penalized quasi-maximum likelihood is proposed, to simultaneously select important variables. Under certain regular conditions, the consistency and oracle property of the resulting estimators are established. Finally, both simulation studies and a real data analysis of the Boston housing data are carried to illustrate the developed methodology.<\/jats:p>","DOI":"10.3390\/sym14061200","type":"journal-article","created":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T02:01:44Z","timestamp":1655085704000},"page":"1200","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Variable Selection of Heterogeneous Spatial Autoregressive Models via Double-Penalized Likelihood"],"prefix":"10.3390","volume":"14","author":[{"given":"Ruiqin","family":"Tian","sequence":"first","affiliation":[{"name":"School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1263-3457","authenticated-orcid":false,"given":"Miaojie","family":"Xia","sequence":"additional","affiliation":[{"name":"School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0903-9601","authenticated-orcid":false,"given":"Dengke","family":"Xu","sequence":"additional","affiliation":[{"name":"College of Economics, Hangzhou Dianzi University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,10]]},"reference":[{"key":"ref_1","unstructured":"Cliff, A., and Ord, J.K. (1973). Spatial Autocorrelation, Pion."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Anselin, L. (1988). Spatial Econometrics: Methods and Models, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-7799-1"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Ullah, A., and Giles, D.E.A. (1998). Spatial Dependence in Linear Regression Models with an Introduction to Spatial Econometrics. Handbook of Applied Economics Statistics, Marcel Dekker.","DOI":"10.1201\/9781482269901-36"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jeconom.2014.12.005","article-title":"A spatial autoregressive model with a nonlinear transformation of the dependent variable","volume":"186","author":"Xu","year":"2015","journal-title":"J. Econom."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.spasta.2018.05.001","article-title":"A penalized quasi-maximum likelihood method for variable selection in the spatial autoregressive model","volume":"25","author":"Liu","year":"2018","journal-title":"Spat. Stat."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1325","DOI":"10.1080\/03610926.2019.1649428","article-title":"Variable selection for spatial autoregressive models","volume":"50","author":"Xie","year":"2021","journal-title":"Commun.-Stat. Methods"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1125","DOI":"10.1007\/s00362-018-0984-2","article-title":"Variable selection for spatial autoregressive models with a diverging number of parameters","volume":"61","author":"Xie","year":"2020","journal-title":"Stat. Pap."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"18","DOI":"10.1016\/j.jeconom.2009.10.033","article-title":"Profile quasi-maximum likelihood estimation of partially linear spatial autoregressive models","volume":"157","author":"Su","year":"2010","journal-title":"J. Econom."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.spasta.2018.04.008","article-title":"Statistical inference for partially linear additive spatial autoregressive models","volume":"25","author":"Du","year":"2018","journal-title":"Spat. Stat."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1007\/s00362-019-01105-y","article-title":"Estimation of partially linear single-index spatial autoregressive model","volume":"62","author":"Cheng","year":"2021","journal-title":"Stat. Pap."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1016\/j.econmod.2017.04.015","article-title":"Statistical inference of partially linear varying coefficient spatial autoregressive models","volume":"64","author":"Wei","year":"2017","journal-title":"Econ. Model."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Hu, Y.P., Wu, S.Y., Feng, S.Y., and Jin, J.L. (2020). Estimation in Partial Functional Linear Spatial Autoregressive Model. Mathematics, 8.","DOI":"10.3390\/math8101680"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.jeconom.2009.10.035","article-title":"GMM estimation of spatial autoregressive models with unknown heteroskedasticity","volume":"157","author":"Lin","year":"2010","journal-title":"J. Econom."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1423","DOI":"10.1007\/s00362-017-0880-1","article-title":"Bayesian Local Influence for Spatial Autoregressive Models with Heteroscedasticity","volume":"60","author":"Dai","year":"2019","journal-title":"Stat. Pap."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1007\/s00184-011-0352-x","article-title":"Variable selection for joint mean and dispersion models of the inverse Gaussian distribution","volume":"75","author":"Wu","year":"2012","journal-title":"Metrika"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1624","DOI":"10.1016\/j.spl.2013.02.023","article-title":"A semiparametric Bayesian approach to joint mean and variance models","volume":"83","author":"Xu","year":"2013","journal-title":"Stat. Probab. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1080\/02664763.2013.830284","article-title":"Variable selection for varying dispersion beta regression model","volume":"41","author":"Zhao","year":"2014","journal-title":"J. Appl. Stat."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1007\/s11424-016-5193-2","article-title":"Variable selection in joint location, scale and skewness models of the skew-normal distribution","volume":"30","author":"Li","year":"2017","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1080\/24754269.2021.1877961","article-title":"Bayesian quantile semiparametric mixed-effects double regression models","volume":"5","author":"Zhang","year":"2021","journal-title":"Stat. Theory Relat. Fields"},{"key":"ref_20","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"},{"key":"ref_21","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_22","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_23","doi-asserted-by":"crossref","first-page":"2062","DOI":"10.1080\/03610926.2019.1659367","article-title":"Variable selection for semiparametric varying-coefficient spatial autoregressive models with a diverging number of parameters","volume":"50","author":"Luo","year":"2021","journal-title":"Commun.-Stat. Methods"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1214\/009053607000000604","article-title":"Variable selection in semiparametric regression modeling","volume":"36","author":"Li","year":"2008","journal-title":"Ann. Stat."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1872","DOI":"10.1016\/j.jmva.2010.03.005","article-title":"Variable selection for semiparametric varying coefficient partially linear errorsin-variables models","volume":"101","author":"Zhao","year":"2010","journal-title":"J. Multivar. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.jmva.2014.07.015","article-title":"Penalized quadratic inference functions for semiparametric varying coefficient partially linear models with longitudinal data","volume":"132","author":"Tian","year":"2014","journal-title":"J. Multivar. Anal."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1899","DOI":"10.1111\/j.1468-0262.2004.00558.x","article-title":"Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models","volume":"72","author":"Lee","year":"2004","journal-title":"Econometrica"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"553","DOI":"10.1093\/biomet\/asm053","article-title":"Tuning parameter selectors for the smoothly clipped absolute deviation method","volume":"94","author":"Wang","year":"2007","journal-title":"Biometrika"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"333","DOI":"10.1023\/A:1007762613901","article-title":"Using the spatial configuration of the data to improve estimation","volume":"14","author":"Pace","year":"1997","journal-title":"J. Real Estate Financ. Econ."},{"key":"ref_30","unstructured":"Su, L., and Yang, Z. (2009). Instrumental Variable Quantile Estimation of Spatial Autoregressive Models, Singapore Management University. Working Paper."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/S0304-4076(01)00064-1","article-title":"On the asymptotic distribution of the Moran I test statistic with applications","volume":"104","author":"Kelejian","year":"2001","journal-title":"J. Econom."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"953","DOI":"10.2307\/2938168","article-title":"Spatial patterns in household demand","volume":"59","author":"Case","year":"1991","journal-title":"Econometrica"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1200\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:27:29Z","timestamp":1760138849000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1200"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,10]]},"references-count":32,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061200"],"URL":"https:\/\/doi.org\/10.3390\/sym14061200","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,6,10]]}}}