Semiparametric estimation and testing of smooth coefficient spatial autoregressive models

Abstract

This paper considers a flexible semiparametric spatial autoregressive (mixed-regressive) model in which unknown coefficients are permitted to be nonparametric functions of some contextual variables to allow for potential nonlinearities and parameter heterogeneity in the spatial relationship. Unlike other semiparametric spatial dependence models, ours permits the spatial autoregressive parameter to meaningfully vary across units and thus allows the identification of a neighborhood-specific spatial dependence measure conditional on the vector of contextual variables. We propose several (locally) nonparametric GMM estimators for our model. The developed two-stage estimators incorporate both the linear and quadratic orthogonality conditions and are capable of accommodating a variety of data generating processes, including the instance of a pure spatially autoregressive semiparametric model with no relevant regressors as well as multiple partially linear specifications. All proposed estimators are shown to be consistent and asymptotically normal. We also contribute to the literature by putting forward two test statistics to test for parameter constancy in our model. Both tests are consistent.

Estimation and inference in functional-coefficient spatial autoregressive panel data models with fixed effects

“…However, if x it , g i and z it are all irrelevant or weak in predicting y it , q it is not going to be a good instrument. Without pre-testing the relevance of exogenous covariates in a purely cross-sectional version of model (1.2), Malikov & Sun (2017) show that combining both linear and quadratic moments can be used to consistently estimate unknown coefficient curves regardless of whether the exogenous covariates are relevant in predicting the dependent variable. We expect similar results to hold in our panel data setup.…”

“…We then use ρ(z| p) as the final estimator for ρ (z). Since the proofs of consistency and asymptotic normality of the "tilted" estimator are tedious and closely follow those given in Malikov & Sun (2017), we omit the details here.…”

“…Few existing nonparametric studies, all of which focus on a purely cross-sectional setup, include the works of Su & Jin (2010), Su (2012) and Zhang (2013), who consider a Robinson-type partially linear semiparametric SAR model, whereas Sun, Yan, Zhang & Lu (2014) and Malikov & Sun (2017) study fully and/or partially linear functional-coefficient SAR models. The spatial autoregressive models in which spatial weights are specified in the form of unknown nonparametric functions of some geographic or economic distance are examined by Pinkse, Slade & Brett (2002) and Sun (2016).…”

“…The proposed is, essentially, the test for spatial endogeneity. Our specification test belongs to the family of similar nonparametric residual-based tests considered for independent data (e.g., Zheng, 1996;Li & Wang, 1998), weakly dependent time series data (e.g., Fan & Li, 1999;Li, 1999), integrated time series data (e.g., Wang & Phillips, 2012;Sun, Cai & Li, 2015) and, more recently, for spatial data (e.g., Su & Qu, 2017;Malikov & Sun, 2017).…”

Time-Varying Coefficient Spatial Autoregressive Panel Data Model with Fixed Effects

“…For example, Su and Jin (2010) consider using QML method to estimate a partially linear SAR model, and Su (2012) proposes a semiparametric GMM estimation method for SAR models with a nonparametric regressor term. A functional-coefficient spatial model with nonparametric spatial weights is also studied in Sun (2016), and Malikov and Sun (2017) study semiparametric estimation and testing of smooth coefficient spatial autoregressive models. In terms of spatial panel data models, Zhang and Sun (2015) and Zhang and Shen (2015) discuss estimation of semiparametric varying-coefficient spatial panel data models with random-effects and partial specified dynamic spatial panel data models, respectively.…”