Abstract
The partially linear single-index spatial autoregressive models (PLSISARM) can be used to evaluate the linear and nonlinear effects of covariates on the response for spatial dependent data. With the nonparametric function approximated by free-knot splines, we develop a Bayesian sampling-based method which can be performed by facilitating efficient Markov chain Monte Carlo approach to analyze PLSISARM and design a Gibbs sampler to explore the joint posterior distributions. To obtain a rapidly-convergent algorithm, we improve the movement step of Bayesian splines with free-knots so that all the knots can be relocated each time instead of only one knot. We illustrate the performance of the proposed model and estimation method by a simulation study and analysis of a Boston housing price dataset.
Similar content being viewed by others
References
Anselin L (1988) Spatial econometrics: methods and models. Kluwer Academic Publishers, Dordrecht
Anselin L, Bera AK (1998) Spatial dependence in linear regression models with an introduction to spatial econometrics. Handbook of applied economics statistics. Marcel Dekker, New York
Antoniadis A, Grégoire G, McKeague IW (2004) Bayesian estimation in single-index models. Stat Sin 14:1147–1164
Basile R (2008) Regional economic growth in Europe: a semiparametric spatial dependence approach. Pape Reg Sci 87(4):527–544
Basile R (2009) Productivity polarization across regions in Europe: the role of nonlinearities and spatial dependence. Int Reg Sci Rev 32(1):92–115
Basile R, Gress B (2005) Semi-parametric spatial auto-covariance models of regional growth behaviour in Europe. Rég Dév 21:93–118
Basile R, Durbán M, Mínguez R, Montero JM, Mur J (2014) Modeling regional economic dynamics: spatial dependence, spatial heterogeneity and nonlinearities. J Econ Dyn Control 48:229–245
Bellman RE (1961) Adaptive control processes. Princeton University Press, Princeton
Carroll RJ, Fan J, Gijbels I, Wand MP (1997) Generalized partially linear single-index models. J Am Stat Assoc 92:477–489
Case AC (1991) Spatial patterns in householed demand. Econometrica 59(4):953–965
Chen M-H, Schmeiser BW (1996) General hit-and-run Monte Carlo sampling for evaluating multidimensional integrals. Oper Res Lett 19:161–169
Chen JQ, Wang RF, Huang YX (2015) Semiparametric spatial autoregressive model: a two-step Bayesian approach. Ann Public Health Res 2(1):1012
Cheng SL, Chen JB (2019) Estimation of partially linear single-index spatial autoregressive model. Stat Pap. https://doi.org/10.1007/s00362-019-01105-y
Cheng SL, Chen JB, Liu X (2019) GMM Estimation of partially linear single-index spatial autoregressive model. Spat Stat 31:100354
Cliff AD, Ord JK (1973) Spatial autocorrelation. Pion Ltd, London
Cressie N (1992) Statistics for spatial data. Terra Nova 4(5):613–617
de Boor C (1978) A practical guide to splines. Springer, New York
Denison DGT, Mallick BK, Smith AFM (1998) Automatic Bayesian curving fitting. J R Stat Soc Ser B 60:333–350
Dimatteo I, Genovese CR, Kass RE (2001) Bayesian curve fitting with free-knot splines. Biometrika 88:1055–1071
Du J, Sun XQ, Cao RY, Zhang ZZ (2018) Statistical inference for partially linear additive spatial autoregressive models. Spat Stat 25:52–67
Fan JQ, Gijbels I (1996) Local polynomial modelling and its applications. Chapman and Hall, New York
Friedman JH, Stuetzle W (1981) Projection pursuit regression. J Am Stat Assoc 76:817–823
Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences. Stat Sci 7:457–511
Green P (1995) Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82:711–732
Harrison DJ, Rubinfeld DL (1978) Hedonic housing prices and the demand for clean air. J Environ Econ Manag 5:81–102
Hastie TJ, Tibshirani RJ (1990) Generalized additive models. Chapman and Hall, London
Hastie TJ, Tibshirani RJ (1993) Varying-coefficient models. J R Stat Soc Ser B 55:757–796
Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97–109
Holmes CC, Mallick BK (2001) Bayesian regression with multivariate linear splines. J R Stat Soc Ser B 63:3–17
Holmes CC, Mallick BK (2003) Generalized nonlinear modeling with multivariate free-knot regression splines. J Am Stat Assoc 98:352–368
Hoshino T (2017) Semiparametric spatial autoregressive models with endogenous regressors: with an application to crime data. J Bus Econ Stat 36(1):160–172
Kakamu K, Wago H (2008) Small-sample properties of panel spatial autoregressive models: comparison of the Bayesian and maximum likelihood methods. Spat Econ Anal 3(3):305–319
Kazar BM, Celik M (2012) Spatial autoregressive model. Springer Press, New York
Krisztin T (2017) The determinants of regional freight transport: a spatial, semiparametric approach. Geogr Anal 49(3):268–308
Krisztin T (2018) Semi-parametric spatial autoregressive models in freight generation modeling. Transp Res Part E Logist Transp Rev 114:121–143
Lee LF (2004) Asymptotic distribution of quasi-maximum likelihood estimators for spatial autoregressive models. Econometrica 72:1899–1925
LeSage JP (1997) Bayesian estimation of spatial autoregressive models. Int Reg Sci Rev 20(1–2):113–129
LeSage JP (2000) Bayesian estimation of limited dependent variable spatial autoregressive models. Geogr Anal 32(1):19–35
LeSage JP, Parent O (2007) Bayesian model averaging for spatial econometric models. Geogr Anal 39(3):241–267
LeSage PJ, Pace RK (2009) Introduction to spatial econometrics. CRC Press, Boca Raton
LeSage PJ, Pace RK (2018) Spatial econometric Monte Carlo studies: raising the bar. Empir Econ 55(1):17–34
Li TT, Yang H, Wang JL, Xue LG, Zhu LX (2011) Correction on estimation for a partial-linear single-index model. Ann Stat 39(6):3441–3443
Liang H, Liu X, Li R, Tsai CL (2010) Estimation and testing for partially linear single-index models. Ann Stat 38(6):3811–3836
Lin X, Lee LF (2010) GMM estimation of spatial autoregressive models with unknown heteroskedasticity. J Econom 157(1):34–52
Lindstrom MJ (2002) Bayesian estimation of free-knot splines using reversible jump. Comput Stat Data Anal 41:255–269
Liu X, Chen JB, Cheng SL (2018) A penalized quasi-maximum likelihood method for variable selection in the spatial autoregressive mode. Spat Stat 25:86–104
Lv YZ, Zhang RQ, Zhao WH, Liu JC (2015) Quantile regression and variable selection of partial linear single-index model. Ann Inst Stat Math 67(2):375–409
Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equations of state calculations by fast computing machine. J Chem Phys 21:1087–1091
Paelinck JHP, Klaassen LH (1979) Spatial econometrics. Gower Press, Aldershot
Piribauer P, Crespo Cuaresma J (2016) Bayesian variable selection in spatial autoregressive models. Spat Econ Anal 11(4):457–479
Poon W-Y, Wang H-B (2013) Bayesian analysis of generalized partially linear single-index models. Comput Stat Data Anal 68:251–261
Su LJ, Jin SN (2010) Profile quasi-maximum likelihood estimation of partially linear spatial autoregressive models. J Econom 157(1):18–33
Su LJ, Yang ZL (2009) Instrumental variable quantile estimation of spatial autoregressive models. Working paper. Singapore Management University
Sun Y, Yan HJ, Zhang WY, Lu Z (2014) A semiparametric spatial dynamic model. Ann Stat 42(2):700–727
Sun Y (2017) Estimation of single-index model with spatial interaction. Reg Sci Urban Econ 62:36–45
Sun Y, Wu YQ (2018) Estimation and testing for a partially linear single-index spatial regression model. Spat Econ Anal 13(4):473–489
Tanner MA (1993) Tools for statistical inference: methods for the exploration of posterior distributions and likelihood functions, 2nd edn. Springer, New York
Tierney L (1994) Markov chains for exploring posterior distributions. Ann Stat 22:1701–1728
Wang JL, Xue LG, Zhu LX, Chong YS (2010) Estimation for a partial linear single index models. Ann Stat 38(1):246–274
Wei HJ, Sun Y (2016) Heteroskedasticity-robust semi-parametric GMM estimation of a spatial model with space-varying coefficients. Spat Econ Anal 12(1):113–128
Xia YC, Härdle W (2006) Semi-parametric estimation of partially linear single-index models. J Multivar Anal 97(5):1162–1184
Yu Y, Ruppert D (2002) Penalized spline estimation for partially linear single-index model. J Am Stat Assoc 97:1042–1054
Yu Y, Ruppert D (2004) Root-\(n\) consistency of penalized spline estimator for partially linear single-index models under general Euclidean space. Stat Sin 14(2):116–123
Yu Y, Wu C, Zhang Y (2017) Penalised spline estimation for generalised partially linear single-index models. Stat Comput 27(2):571–582
Zhu L-X, Xue L-G (2006) Empirical likelihood confidence regions in a partially linear single-index model. J R Stat Soc Ser B 68(3):549–570
Acknowledgements
This work was supported by the Natural Science Foundation of China (12001105), the Postdoctoral Science Foundation of China (2019M660156) and the Natural Science Foundation of Fujian Province (2018J05002, 2020J01170).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chen, Z., Chen, J. Bayesian analysis of partially linear, single-index, spatial autoregressive models. Comput Stat 37, 327–353 (2022). https://doi.org/10.1007/s00180-021-01123-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00180-021-01123-1