Spatial autoregressive (SAR) models are important tools for studying network effects. However, with an increasing emphasis on data privacy, data providers often implement privacy protection measures that make classical SAR models inapplicable. In this study, we introduce a privacy-protected SAR model with noise-added response and covariates to meet privacy-protection requirements. However, in this scenario, the traditional quasi-maximum likelihood estimator becomes infeasible because the likelihood function cannot be directly formulated. To address this issue, we first consider an explicit expression for the likelihood function with only noise-added responses. Then, we develop techniques to correct the biases for derivatives introduced by noise. Correspondingly, a Newton-Raphson-type algorithm is proposed to obtain the estimator, leading to a corrected likelihood estimator. To further enhance computational efficiency, we introduce a corrected least squares estimator based on the idea of bias correction. These two estimation methods ensure both data security and the attainment of statistically valid estimators. Theoretical analysis of both estimators is carefully conducted, statistical inference methods and model extensions are discussed. The finite sample performances of different methods are demonstrated through extensive simulations and the analysis of a real dataset.

翻译：空间自回归（SAR）模型是研究网络效应的重要工具。然而，随着对数据隐私的日益重视，数据提供者通常会实施隐私保护措施，这使得经典SAR模型不再适用。在本研究中，我们引入了一种带有加噪响应变量和协变量的隐私保护SAR模型，以满足隐私保护要求。然而，在此情境下，传统的拟极大似然估计量变得不可行，因为似然函数无法直接构建。为解决这一问题，我们首先考虑了仅对响应变量加噪时似然函数的显式表达式。接着，我们开发了技术来校正由噪声引入的导数偏差。相应地，我们提出了一种牛顿-拉夫森型算法来获得估计量，从而得到校正似然估计量。为了进一步提高计算效率，我们基于偏差校正的思想引入了一种校正最小二乘估计量。这两种估计方法既确保了数据安全性，又获得了统计上有效的估计量。我们对两种估计量进行了细致的理论分析，讨论了统计推断方法和模型扩展。通过大量模拟实验和一个真实数据集的分析，展示了不同方法在有限样本下的表现。