The Spatial AutoRegressive model (SAR) is commonly used in studies involving spatial and network data to estimate the spatial or network peer influence and the effects of covariates on the response, taking into account the spatial or network dependence. While the model can be efficiently estimated with a Quasi maximum likelihood approach (QMLE), the detrimental effect of covariate measurement error on the QMLE and how to remedy it is currently unknown. If covariates are measured with error, then the QMLE may not have the $\sqrt{n}$ convergence and may even be inconsistent even when a node is influenced by only a limited number of other nodes or spatial units. We develop a measurement error-corrected ML estimator (ME-QMLE) for the parameters of the SAR model when covariates are measured with error. The ME-QMLE possesses statistical consistency and asymptotic normality properties. We consider two types of applications. The first is when the true covariate cannot be measured directly, and a proxy is observed instead. The second one involves including latent homophily factors estimated with error from the network for estimating peer influence. Our numerical results verify the bias correction property of the estimator and the accuracy of the standard error estimates in finite samples. We illustrate the method on a real dataset related to county-level death rates from the COVID-19 pandemic.

翻译：空间自回归模型（SAR）常被用于空间数据与网络数据研究，旨在考虑空间或网络依赖性的前提下，估计空间/网络同伴影响以及协变量对响应变量的效应。尽管该模型可通过拟极大似然估计（QMLE）高效求解，但协变量测量误差对QMLE的负面影响及其修正方法目前尚不明确。当协变量存在测量误差时，即使节点仅受有限数量其他节点或空间单元影响，QMLE可能不再具有$\sqrt{n}$收敛性，甚至可能产生不一致估计。本文针对协变量存在测量误差的情形，提出一种测量误差校正的极大似然估计量（ME-QMLE）用于SAR模型参数估计。该估计量具有统计一致性和渐近正态性。我们考虑两类应用场景：其一是真实协变量无法直接测量，仅能观测到代理变量；其二是从网络中估计存在误差的潜在同质性因子以评估同伴影响。数值实验结果验证了该估计量的偏差校正特性及有限样本下标准误估计的准确性。我们通过一个与县级COVID-19疫情死亡率相关的真实数据集展示了该方法的实际应用。