Automatic Structure Identification of Semiparametric Spatial Autoregressive Model Based on Smooth-Threshold Estimating Equation

Abstract

Issues concerning spatial dependence among cross-sectional units in econometrics have received more and more attention, while in statistical modeling, rarely can the analysts have a priori knowledge of the dependency relationship of the response variable with respect to independent variables. This paper proposes an automatic structure identification and variable selection procedure for semiparametric spatial autoregressive model, based on the generalized method of moments and the smooth-threshold estimating equations. The novel method is easily implemented without solving any convex optimization problems. Model identification consistency is theoretically established in the sense that the proposed method can automatically separate the linear and zero components from the varying ones with probability approaching to one. Detailed issues on computation and turning parameter selection are discussed. Some Monte Carlo simulations are conducted to demonstrate the finite sample performance of the proposed procedure. Two empirical applications on Boston housing price data and New York leukemia data are further considered.

Keywords

Semiparametric spatial autoregressive model / Generalized method of moments / Automatic structure recovery / Smooth-threshold estimating equation / Consistency / 62G35 / 62H11 / 62J07

References

[1] Ai C, Zhang Y. Estimation of partially specified spatial panel data models with fixed-effects. Economet. Rev., 2017, 36: 6-22

[2] Carroll RJ, Fan J, Gijbels I, Wand MP. Generalized partially linear single-index models. J. Am. Stat. Assoc., 1997, 92: 477-489

[3] Chen Y, Wang Q, Yao W. Adaptive estimation for varying coefficient models. J. Multivar. Anal., 2015, 137: 17-31

[4] Fan J, Huang T. Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Bernoulli, 2005, 11: 1031-1057

[5] Fan J, Li R. Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc., 2001, 96: 1348-1360

[6] Geyer CJ. On the asymptotics of constrained M-estimation. Ann. Stat., 1994, 22: 1993-2010

[7] Gilley OW, Pace RK. On the Harrison and Rubinfeld data. J. Environ. Econ. Manag., 1996, 31: 403-405

[8] Harrison D, Rubinfeld DL. Hedonic housing prices and the demand for clean air. J. Environ. Econ. Manag., 1978, 5: 81-102

[9] Hu T, Xia Y. Adaptive semi-varying coefficient model selection. Stat. Sin., 2012, 22: 575-599

[10] Huang J, Wu C, Zhou L. Varying-coefficient models and basis function approximation for the analysis of repeated measurements. Biometrika, 2002, 89: 111-128

[11] Huang J, Wei F, Ma S. Semiparametric regression pursuit. Stat. Sin., 2012, 22: 1403-1426

[12] Jencks C, Mayer S. The social consequences of growing up in a poor neighborhood. Inner-city poverty in the United States, 1990, Washington, National Academy

[13] Jeong H, Lee LF. Spatial dynamic models with intertemporal optimization: specification and estimation. J. Econom., 2020, 218: 82-104

[14] Jiang J, Zhou H, Jiang X, Peng J. Generalized likelihood ratio tests for the structure of semiparametric additive models. Can. J. Stat., 2007, 35: 381-398

[15] Kai B, Li R, Zou H. New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. Ann. Stat., 2011, 39: 305-332

[16] Kelejian HH, Prucha IR. A generalized moments estimator for the autoregressive parameter in a spatial model. Int. Econ. Rev., 1999, 40: 509-533

[17] Kelejian HH, Prucha IR. Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. J. Econom., 2010, 157: 53-67

[18] Lee LF. Asymptotic distribution of quasi-maximum likelihood estimators for spatial autoregressive models. Econometrica, 2004, 72: 1899-1925

[19] Lee LF. GMM and 2SLS estimation of mixed regressive, spatial autoregressive models. J. Econom., 2007, 137: 489-514

[20] Lee LF, Liu X. Efficient GMM estimation of high order spatial autoregressive models with autoregressive disturbances. Econom. Theor., 2010, 26: 187-230

[21] Li R, Liang H. Variable selection in semiparametric regression model. Ann. Stat., 2008, 36: 261-286

[22] Lian H. Semiparametric estimation of additive quantile regression models by two-fold penalty. J. Bus. Econ. Stat., 2012, 30: 337-350

[23] Lian H, Chen X, Yang JY. Identification of partially linear structure in additive models with an application to gene expression prediction from sequences. Biometrics, 2012, 68: 437-445

[24] Lian H, Liang H, Ruppert D. Separation of covariates into nonparametric and parametric parts in high-dimensional partially linear additive models. Stat. Sin., 2015, 25: 591-607

[25] Lin X, Lee LF. GMM estimation of spatial autoregressive models with unknown heteroskedasticity. J. Econom., 2010, 157: 34-52

[26] Liu X, Lee LF, Bollinger CR. An efficient GMM estimator of spatial autoregressive models. J. Econom., 2010, 159: 303-319

[27] Mack Y, Silverman B. Weak and strong uniform consistency of kernel regression estimates. Probab. Theory Relat. Fields, 1982, 61: 405-415

[28] Noh H, Van Keilegom I. Efficient model selection in semivarying coefficient models. Electron. J. Stat., 2012, 6: 2519-2534

[29] Olejnik J, Olejnik A. QML estimation with non-summable weight matrices. J. Geogr. Syst., 2020, 22: 469-495

[30] Ord JK. Estimation methods for models of spatial interaction. J. Am. Stat. Assoc., 1975, 70: 120-126

[31] Pace RK, Gilley OW. Using the spatial configuration of the data to improve estimation. J. Real Estate Finance Econ., 1997, 14: 333-340

[32] Paelinck JH, Klaassen LH. Spatial Econometrics, 1979, Aldershot, Gower Press

[33] Smirnov O, Anselin L. Fast maximum likelihood estimation of very large spatial autoregressive models: a characteristic polynomial approach. Comput. Stat. Data Anal., 2001, 35: 301-319

[34] Su L. Semiparametric GMM estimation of spatial autoregressive models. J. Econom., 2012, 167: 543-560

[35] Su L, Jin S. Profile quasi-maximum likelihood estimation of spatial autoregressive models. J. Econom., 2010, 157: 18-33

[36] Su, L., Yang, Z.: Instrumental variable quantile estimation of spatial autoregressive models. Working paper, Singapore Management University (2011)

[37] Sun Y, Wu Y. Estimation and testing for a partially linear single-index spatial regression model. Spat. Econ. Anal., 2018, 13: 473-489

[38] Sun Y, Yan H, Zhang W, Lu Z. A semiparametric spatial dynamic model. Ann. Stat., 2014, 42: 700-727

[39] Sun Y, Zhang Y, Huang JZ. Estimation of a semiparametric varying-coefficient mixed regressive spatial autoregressive model. Econom. Stat., 2019, 9: 140-155

[40] Tibshirani R. Regression shrinkage and selection via the Lasso. J. R. Stat. Soc. Ser. B (Methodology), 1996, 58: 267-288

[41] Ueki M. A note on automatic variable selection using smooth-threshold estimating equations. Biometrika, 2009, 96: 1005-1011

[42] Waller LA, Gotway CA. Applied Spatial Statistics for Public Health Data, 2004, Hoboken, John Wiley

[43] Wakefield J. Disease mapping and spatial regression with count data. Biostatistics, 2007, 8: 158-183

[44] Wang D, Kulasekera KB. Parametric component detection and variable selection in varying-coefficient partially linear models. J. Multivar. Anal., 2012, 112: 117-129

[45] Wang H, Leng C. Unified lasso estimation via least squares approximation. J. Am. Stat. Assoc., 2007, 102: 1039-1048

[46] Wang H, Xia Y. Shrinkage estimation of the varying coefficient model. J. Am. Stat. Assoc., 2009, 104: 747-757

[47] Wang HJ, Zhu Z, Zhou J. Quantile regression in partially linear varying coefficient models. Ann. Stat., 2009, 37: 3841-3866

[48] Wei H, Sun Y. Heteroskedasticity-robust semi-parametric GMM estimation of a spatial model with space-varying coefficients. Spat. Econ. Anal., 2017, 12: 113-128

[49] Xia Y, Zhang W, Tong H. Efficient estimation for semivarying-coefficient models. Biometrika, 2004, 91: 661-681

[50] Zhang HH, Cheng G, Liu Y. Linear or nonlinear? Automatic structure discovery for partially linear models. J. Am. Stat. Assoc., 2011, 106: 1099-1112

[51] Zou H. The adaptive LASSO and its oracle properties. J. Am. Stat. Assoc., 2006, 101: 1418-1429

RIGHTS & PERMISSIONS

School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature