GMM inference for the spatial autoregressive kink model with an unknown threshold

Abstract

This paper considers spatial autoregressive kink models with an unknown threshold, where the impact of a specific explanatory variable on the response variable is piecewise linear but differs below and above this threshold. To address the endogeneity issue, the paper presents the modified generalized method of moments (GMM) that consistently estimates the threshold location and slope changes. Asymptotic properties, including the consistency and asymptotic normality of the GMM estimators, and the limiting distribution of the Sup-Wald statistic, are established under a set of regularity assumptions. In view of the nonstandard asymptotic null distribution, we use a multiplier bootstrap to approximate the $p$-value of the Sup-Wald statistic to detect the presence of the threshold. Simulation study illustrates that the estimators and inference are well-behaved in finite samples. An empirical application to the secondary industrial structure data of 280 Chinese prefecture-level cities further highlights the practical merits of our methods.