Spatial autocorrelation and spatial heterogeneity are fundamental properties of geo-referenced data. Spatial autoregressive varying coefficient models have been proposed to simultaneously deal with spatial autocorrelation in the response variable and spatial heterogeneity in the regression relationship. Nevertheless, the identification of spatial heterogeneity in the spatial lag term and in the regression relationship remains to be studied, which is essential to model selection and deep understanding of the intrinsical characteristics of the regression relationship. In this article, based on the two-stage least squares estimation of the spatial autoregressive varying coefficient model and the related null models, two residual sums of squares based statistics are constructed and the bootstrap tests are proposed to detect spatial heterogeneity in the spatial lag term and in the regression relationship, respectively. Then, simulation studies are conducted to assess the performance of the tests. The results show that both tests are of valid size, satisfactory power, and robustness to the model error distribution. Furthermore, a real-life example based on the Boston housing price data is given to demonstrate the application of the proposed tests.

空间自相关和空间异质性是地理参考数据的基本属性。已经提出空间自回归变系数模型来同时处理响应变量中的空间自相关和回归关系中的空间异质性。然而，空间滞后项和回归关系中空间异质性的识别仍有待研究，这对于模型选择和深入理解回归关系的内在特征至关重要。在本文中，基于空间自回归变系数模型和相关零模型的两阶段最小二乘估计，构建了两个基于残差平方和的统计数据，并提出了自举检验来分别检测空间滞后项和回归关系中的空间异质性。然后，进行模拟研究以评估测试的性能。结果表明，这两个测试都具有有效的大小、令人满意的功效和对模型误差分布的鲁棒性。此外，还给出了一个基于波士顿房价数据的真实例子来展示所提出的测试的应用。