In this paper, we develop a multi-step estimation procedure to simultaneously estimate the varying-coefficient functions using a local-linear generalized method of moments (GMM) based on continuous moment conditions. To incorporate spatial dependence, the continuous moment conditions are first projected onto eigen-functions and then combined by weighted eigen-values, thereby, solving the challenges of using an inverse covariance operator directly. We propose an optimal instrument variable that minimizes the asymptotic variance function among the class of all local-linear GMM estimators, and it outperforms the initial estimates which do not incorporate the spatial dependence. Our proposed method significantly improves the accuracy of the estimation under heteroskedasticity and its asymptotic properties have been investigated. Extensive simulation studies illustrate the finite sample performance, and the efficacy of the proposed method is confirmed by real data analysis.

翻译：本文提出了一种多步估计方法，基于连续矩条件，采用局部线性广义矩估计（GMM）同时估计变系数函数。为纳入空间依赖性，连续矩条件首先投影到特征函数上，然后通过加权特征值进行组合，从而解决了直接使用逆协方差算子所带来的挑战。我们提出了一种最优工具变量，其在所有局部线性GMM估计量类别中最小化渐近方差函数，其性能优于未考虑空间依赖性的初始估计。我们提出的方法显著提高了异方差下估计的准确性，并研究了其渐近性质。大量的模拟研究展示了有限样本下的性能，实际数据分析证实了所提方法的有效性。