{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T06:49:09Z","timestamp":1783666149958,"version":"3.55.0"},"reference-count":62,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2022,6,7]],"date-time":"2022-06-07T00:00:00Z","timestamp":1654560000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of China","award":["12001105"],"award-info":[{"award-number":["12001105"]}]},{"name":"Natural Science Foundation of China","award":["2019M660156"],"award-info":[{"award-number":["2019M660156"]}]},{"name":"Natural Science Foundation of China","award":["2021J01662"],"award-info":[{"award-number":["2021J01662"]}]},{"name":"Natural Science Foundation of China","award":["19YJC790051"],"award-info":[{"award-number":["19YJC790051"]}]},{"name":"Postdoctoral Science Foundation of China","award":["12001105"],"award-info":[{"award-number":["12001105"]}]},{"name":"Postdoctoral Science Foundation of China","award":["2019M660156"],"award-info":[{"award-number":["2019M660156"]}]},{"name":"Postdoctoral Science Foundation of China","award":["2021J01662"],"award-info":[{"award-number":["2021J01662"]}]},{"name":"Postdoctoral Science Foundation of China","award":["19YJC790051"],"award-info":[{"award-number":["19YJC790051"]}]},{"name":"Natural Science Foundation of Fujian Province","award":["12001105"],"award-info":[{"award-number":["12001105"]}]},{"name":"Natural Science Foundation of Fujian Province","award":["2019M660156"],"award-info":[{"award-number":["2019M660156"]}]},{"name":"Natural Science Foundation of Fujian Province","award":["2021J01662"],"award-info":[{"award-number":["2021J01662"]}]},{"name":"Natural Science Foundation of Fujian Province","award":["19YJC790051"],"award-info":[{"award-number":["19YJC790051"]}]},{"name":"Humanities and Social Sciences Youth Foundation of Ministry of Education of China","award":["12001105"],"award-info":[{"award-number":["12001105"]}]},{"name":"Humanities and Social Sciences Youth Foundation of Ministry of Education of China","award":["2019M660156"],"award-info":[{"award-number":["2019M660156"]}]},{"name":"Humanities and Social Sciences Youth Foundation of Ministry of Education of China","award":["2021J01662"],"award-info":[{"award-number":["2021J01662"]}]},{"name":"Humanities and Social Sciences Youth Foundation of Ministry of Education of China","award":["19YJC790051"],"award-info":[{"award-number":["19YJC790051"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper deals with spatial data that can be modelled by partially linear varying coefficient spatial autoregressive models with Bayesian P-splines quantile regression. We evaluate the linear and nonlinear effects of covariates on the response and use quantile regression to present comprehensive information at different quantiles. We not only propose an empirical Bayesian approach of quantile regression using the asymmetric Laplace error distribution and employ P-splines to approximate nonparametric components but also develop an efficient Markov chain Monte Carlo technique to explore the joint posterior distributions of unknown parameters. Monte Carlo simulations show that our estimators not only have robustness for different spatial weight matrices but also perform better compared with quantile regression and instrumental variable quantile regression estimators in finite samples at different quantiles. Finally, a set of Sydney real estate data applications is analysed to illustrate the performance of the proposed method.<\/jats:p>","DOI":"10.3390\/sym14061175","type":"journal-article","created":{"date-parts":[[2022,6,13]],"date-time":"2022-06-13T02:01:44Z","timestamp":1655085704000},"page":"1175","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Bayesian P-Splines Quantile Regression of Partially Linear Varying Coefficient Spatial Autoregressive Models"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9247-5019","authenticated-orcid":false,"given":"Zhiyong","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2832-3616","authenticated-orcid":false,"given":"Minghui","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2107-2305","authenticated-orcid":false,"given":"Fangyu","family":"Ju","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Fujian Normal University, Fuzhou 350117, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2022,6,7]]},"reference":[{"key":"ref_1","unstructured":"Cliff, A.D., and Ord, J.K. (1973). Spatial Autocorrelation, Pion Ltd."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1899","DOI":"10.1111\/j.1468-0262.2004.00558.x","article-title":"Asymptotic Distribution of Quasi-Maximum Likelihood Estimators for Spatial Autoregressive Models","volume":"72","author":"Lee","year":"2004","journal-title":"Econometrica"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1016\/j.jeconom.2005.10.004","article-title":"GMM and 2SLS Estimation of Mixed Regressive Spatial Autoregressive Models","volume":"137","author":"Lee","year":"2007","journal-title":"J. Econom."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1080\/17421770802353725","article-title":"Small-sample properties of panel spatial autoregressive models: Comparison of the Bayesian and maximum likelihood methods","volume":"3","author":"Kakamu","year":"2008","journal-title":"Spat. Econ. Anal."},{"key":"ref_5","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_6","doi-asserted-by":"crossref","first-page":"527","DOI":"10.1111\/j.1435-5957.2008.00175.x","article-title":"Regional economic growth in Europe: A semiparametric spatial dependence approach","volume":"87","author":"Basile","year":"2008","journal-title":"Pap. Reg. Sci."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1177\/0160017608326944","article-title":"Productivity polarization across regions in Europe: The role of nonlinearities and spatial dependence","volume":"32","author":"Basile","year":"2008","journal-title":"Int. Reg. Sci. Rev."},{"key":"ref_8","first-page":"93","article-title":"Semi-parametric spatial auto-covariance models of regional growth behaviour in Europe","volume":"21","author":"Basile","year":"2005","journal-title":"Reg. Dev."},{"key":"ref_9","unstructured":"Paelinck, J.H.P., and Klaassen, L.H. (1979). Spatial Econometrics, Gower Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1214\/13-AOS1201","article-title":"A Semiparametric spatial dynamic model","volume":"42","author":"Sun","year":"2014","journal-title":"Ann. Stat."},{"key":"ref_11","first-page":"1012","article-title":"Semiparametric spatial autoregressive model: A two-step Bayesian approach","volume":"2","author":"Chen","year":"2015","journal-title":"Ann. Public Health Res."},{"key":"ref_12","unstructured":"Dai, X., Li, S., and Tian, M. (2016). Quantile regression for partially linear varying coefficient spatial autoregressive models. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1595","DOI":"10.1198\/016214508000000977","article-title":"Nonparametric quantiles estimations for dynamic smooth coefficients models","volume":"103","author":"Cai","year":"2008","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Bellman, R.E. (1961). Adaptive Control Processes, Princeton University Press.","DOI":"10.1515\/9781400874668"},{"key":"ref_15","unstructured":"Hastie, T.J., and Tibshirani, R.J. (1990). Generalized Additive Models, Chapman and Hall."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1080\/01621459.1981.10477729","article-title":"Projection Pursuit Regression","volume":"376","author":"Friedman","year":"1981","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1111\/j.2517-6161.1993.tb01939.x","article-title":"Varying-coefficient models","volume":"55","author":"Hastie","year":"1993","journal-title":"J. R. Stat. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1198\/016214501753168280","article-title":"Smoothing Spline Estimation for Varying Coefficient Models with Repeatedly Measured Dependent Variables","volume":"96","author":"Chiang","year":"2001","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1111\/j.1467-9868.2004.B5595.x","article-title":"Smoothing Spline Estimation in Varying-coefficient Models","volume":"66","author":"Eubank","year":"2004","journal-title":"J. R. Stat. Soc."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2249","DOI":"10.1080\/03610920801931887","article-title":"Penalized Spline Estimation for Varying-Coefficient Models","volume":"37","author":"Lu","year":"2008","journal-title":"Commun. Stat. Theory Methods"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1388","DOI":"10.1080\/01621459.1998.10473800","article-title":"Asymptotic confidence regions for kernel smoothing of a varying-coefficient model with longitudinal data","volume":"93","author":"Wu","year":"1998","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"888","DOI":"10.1080\/01621459.2000.10474280","article-title":"Efficient Estimation and Inferences for Varying Coefficient Models","volume":"451","author":"Cai","year":"2000","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_23","first-page":"18","article-title":"Two-Step Likelihood Estimation Procedure for Varying Coefficient Models","volume":"1","author":"Cai","year":"2002","journal-title":"J. Multivar. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1093\/biomet\/89.1.111","article-title":"Varying-coefficient models and basis functions approximations for the analysis of repeated measurements","volume":"89","author":"Huang","year":"2002","journal-title":"Biometrika"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1119","DOI":"10.1081\/STA-120029828","article-title":"Local asymptotics for B-spline estimators of the varying-coefficient model","volume":"33","author":"Lu","year":"2004","journal-title":"Commun. Stat."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1111\/j.1538-4632.2005.00577.x","article-title":"Unconditional Maximum Likelihood Estimation of Linear and Log-Linear Dynamic Models for Spatial Panels","volume":"37","author":"Elhorst","year":"2005","journal-title":"Geogr. Anal."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1016\/j.jeconom.2008.08.002","article-title":"Quasi-maximum likelihood estimators for spatial dynamic panel data with fixed effects when both n and t are large","volume":"146","author":"Yu","year":"2008","journal-title":"J. Econom."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1016\/j.jeconom.2009.08.001","article-title":"Estimation of spatial autoregressive panel data models with fixed effects","volume":"154","author":"Lee","year":"2010","journal-title":"J. Econom."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Chen, Z.Y., and Chen, J.B. (2021). Bayesian analysis of partially linear additive spatial autoregressive models with free-knot splines. Symmetry, 13.","DOI":"10.3390\/sym13091635"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"33","DOI":"10.2307\/1913643","article-title":"Regression Quantiles","volume":"46","author":"Koenker","year":"1978","journal-title":"Econometrica"},{"key":"ref_31","first-page":"8578156","article-title":"Multiple linear regressions by maximizing the likelihood under assumption of generalized Gauss\u2013Laplace distribution of the error","volume":"2016","year":"2016","journal-title":"Comput. Math. Methods Med."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1296","DOI":"10.1080\/01621459.1999.10473882","article-title":"Goodness of Hit and related inference processes for quantile regression","volume":"94","author":"Koenker","year":"1999","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1198\/016214503388619166","article-title":"On additive conditional quantiles with high-dimensional covariates","volume":"98","author":"Zerom","year":"2003","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/j.jeconom.2007.06.005","article-title":"Instrumental variable quantile regression: A robust inference approach","volume":"142","author":"Chernozhukov","year":"2008","journal-title":"J. Econom."},{"key":"ref_35","unstructured":"Su, L.J., and Yang, Z.L. (2009). Instrumental Variable Quantile Estimation of Spatial Autoregressive Models, Singapore Management University. Working Papers."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Koenker, B. (2005). Quantile Regression, Cambridge University Press.","DOI":"10.1017\/CBO9780511754098"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1016\/S0167-7152(01)00124-9","article-title":"Bayesian quantile regression","volume":"54","author":"Yu","year":"2001","journal-title":"Stat. Probab. Lett."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1007\/PL00022717","article-title":"A multivariate and asymmetric generalization of Laplace distribution","volume":"15","author":"Kozubowski","year":"2000","journal-title":"Comput. Stat."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1111\/j.1541-0420.2009.01269.x","article-title":"Bayesian quantile regression for longitudinal studies with nonignorable missing data","volume":"66","author":"Yuan","year":"2010","journal-title":"Biometrics"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1214\/10-BA521","article-title":"Bayesian regularized quantile regression","volume":"5","author":"Li","year":"2010","journal-title":"Bayesian Anal."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1214\/12-BA708","article-title":"Spatial quantile multiple regression using the asymmetric Laplace process","volume":"7","author":"Lum","year":"2012","journal-title":"Bayesian Anal."},{"key":"ref_42","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_43","first-page":"263","article-title":"Estimation Methods for Spatial Autoregressive Structures","volume":"8","author":"Anselin","year":"1980","journal-title":"Reg. Sci. Diss. Monogr. Ser."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0304-4076(98)00084-0","article-title":"GMM Estimation with Cross Sectional Dependence","volume":"92","author":"Conley","year":"1999","journal-title":"J. Econom."},{"key":"ref_45","first-page":"113","article-title":"Bayesian Estimation of Spatial Autoregressive Models","volume":"20","author":"LeSage","year":"1997","journal-title":"Int. Relations"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"385","DOI":"10.1080\/10485250500039049","article-title":"Approximate Bayesian inference for quantiles","volume":"17","author":"Dunson","year":"2005","journal-title":"J. Nonparametr. Stat."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"1138","DOI":"10.1016\/j.csda.2009.09.004","article-title":"Bayesian nonparametric quantile regression using splines","volume":"54","author":"Thompson","year":"1993","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_48","unstructured":"Boor, C.D. (1978). A Practical Guide to Splines, Springer."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/j.tre.2018.03.003","article-title":"Semi-parametric spatial autoregressive models in freight generation modeling","volume":"114","author":"Krisztin","year":"2018","journal-title":"Transp. Res. Part Logist. Transp. Rev."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1214\/ss\/1038425655","article-title":"Flexible smoothing with B-splines and penalties","volume":"11","author":"Eilers","year":"1996","journal-title":"Stat. Sci."},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"J\u00e4ntschi, L. (2019). A test detecting the outliers for continuous distributions based on the cumulative distribution function of the data being tested. Symmetry, 11.","DOI":"10.3390\/sym11060835"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1565","DOI":"10.1080\/00949655.2010.496117","article-title":"Gibbs sampling methods for bayesian quantile regression","volume":"81","author":"Kozumi","year":"2011","journal-title":"J. Stat. Comput. Simul."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1111\/1467-9868.00282","article-title":"Non-gaussian ornstein-uhlenbeck-based models and some of their uses in financial economics","volume":"63","author":"Shephard","year":"2001","journal-title":"J. R. Stat. Soc."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1080\/03610918908812785","article-title":"An easily implemented generalized inverse Gaussian generator","volume":"18","author":"Dagnapur","year":"1989","journal-title":"Commun. Stat. Simul. Comput."},{"key":"ref_55","first-page":"1701","article-title":"Markov chains for exploring posterior distributions","volume":"22","author":"Tierney","year":"1994","journal-title":"Ann. Stat."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1093\/biomet\/57.1.97","article-title":"Monte Carlo sampling methods using Markov Chains and their applications","volume":"57","author":"Hastings","year":"1970","journal-title":"Biometrika"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1063\/1.1699114","article-title":"Equations of state calculations by fast computing machine","volume":"21","author":"Metropolis","year":"1953","journal-title":"J. Chem. Phys."},{"key":"ref_58","unstructured":"Tanner, M.A. (1993). Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and lIkelihood Functions, Springer. [2nd ed.]."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"953","DOI":"10.2307\/2938168","article-title":"Spatial patterns in householed demand","volume":"59","author":"Case","year":"1991","journal-title":"Econometrica"},{"key":"ref_60","doi-asserted-by":"crossref","unstructured":"LeSage, J., and Pace, R.K. (2009). Introduction to Spatial Econometrics, Chapman and Hall\/CRC.","DOI":"10.1201\/9781420064254"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"457","DOI":"10.1214\/ss\/1177011136","article-title":"Inference from iterative simulation using multiple sequences","volume":"7","author":"Gelman","year":"1992","journal-title":"Stat. Sci."},{"key":"ref_62","doi-asserted-by":"crossref","unstructured":"Harezlak, J., Ruppert, D., and Wand, M.P. (2018). Semiparametric Regression with R, Springer.","DOI":"10.1007\/978-1-4939-8853-2"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1175\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:25:27Z","timestamp":1760138727000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/6\/1175"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,7]]},"references-count":62,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2022,6]]}},"alternative-id":["sym14061175"],"URL":"https:\/\/doi.org\/10.3390\/sym14061175","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,7]]}}}