A Pairwise Difference Estimator for Partially Linear Spatial Autoregressive Models

Abstract

Su and Jin (2010) develop for partially linear spatial autoregressive (PL-SAR) model a profile quasimaximum likelihood based estimation procedure. More recently, Su (2011) proposes for this model a semiparametric GMM estimator. However, both of them can be computationally challenging for applied researchers and are not easy to implement in practice. In this article, we propose a computationally simple estimator for the PL-SAR model in the presence of either heteroscedastic or spatially correlated error terms. This estimator blends the essential features of both the GMM estimator for linear SAR model and the pairwise difference estimator for conventional partially linear model. Limiting distribution of the proposed estimator is established and consistent estimator for its asymptotic CV matrix is provided. Monte Carlo studies indicate that our estimator is attractive particularly when one is interested in estimating the finite-dimensional parameters in the model.

Semiparametric estimation and testing of smooth coefficient spatial autoregressive models

“…Third, we also consider two special cases of our model by allowing some of its parameter functions to be constant thus resulting in a partially linear specification. Our proposed estimators present an alternative to those by Su (2012) and Zhang (2013) who study a similar class of partially linear spatial models. Unlike their estimators, ours however preserves its consistency property if the true model is a pure spatial autoregression.…”

“…Such partially linear semiparametric models have been extensively studied for sampling with no spatial or cross-sectional dependence by, e.g., Ahmad, Leelahanon & Li (2005), Kai, Li & Zou (2011) and Cai & Xiao (2012). In the spatial autoregression literature, Su (2012) and Zhang (2013) both focus on the case when ρ (z i ) = ρ 0 over its domain, however, with varying assumptions about x i β (z i ) (in our notation). Zhang (2013) assumes that x i β (z i ) = x i β 1 + β 2 (z i ), whereas Su (2012) assumes that x i = 1.…”

“…In the spatial autoregression literature, Su (2012) and Zhang (2013) both focus on the case when ρ (z i ) = ρ 0 over its domain, however, with varying assumptions about x i β (z i ) (in our notation). Zhang (2013) assumes that x i β (z i ) = x i β 1 + β 2 (z i ), whereas Su (2012) assumes that x i = 1. The nonparametric GMM estimators proposed in these papers are however inconsistent if the true model is a pure spatial autoregressive model without other regressors.…”

“…While our "smooth coefficient" spatial autoregressive model closely relates to the family of partially linear semiparametric spatial models recently proposed in the literature (Su & Jin, 2010;Su, 2012;Zhang, 2013;Sun, Hongjia, Zhang & Lu, 2014), its distinct feature is that it permits the spatial autoregressive parameter to meaningfully vary across units. The latter may be highly desirable from a practitioner's point of view since it allows the identification of a neighborhoodspecific spatial dependence measure conditional on the vector of contextual variables.…”

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

“…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).…”