  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 dependence
among units. 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 and we derive its limiting
covariance. We consider two types of applications. The first is when the true
covariate is imprecisely measured with replicated measurements or 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 two real datasets; i) peer
influence in GPA for middle school students in New Jersey and ii) county-level
death rates from the COVID-19 pandemic.
