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.

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