Quasi-Score Matching Estimation for Spatial Autoregressive Model with Random Weights Matrix and Regressors
Xuan Liang, Tao Zou
TL;DR
This work introduces a determinant-free quasi-score matching estimator for spatial autoregressive models with random weights and regressors, dramatically improving scalability for large n. By formulating a score-matching objective that eliminates the normalizing constant, the authors derive closed-form estimators for beta and sigma^2 and a lambda-search strategy, yielding O(n^2) computation (potentially linear under sparsity). They establish new LLN and CLT results for general linear-quadratic forms with random coefficient matrices, enabling consistent and asymptotically normal inference under random W and X. An efficiency-improved variant preserves computational gains while approaching QMLE efficiency, and simulations plus a real network case study demonstrate favorable accuracy versus speed, especially for large datasets. The approach broadens applicability of SAR-type models in settings where network structure and covariates are observed or random, offering a practical tool for scalable spatial econometrics with robust inference.
Abstract
With the rapid advancements in technology for data collection, the application of the spatial autoregressive (SAR) model has become increasingly prevalent in real-world analysis, particularly when dealing with large datasets. However, the commonly used quasi-maximum likelihood estimation (QMLE) for the SAR model is not computationally scalable to handle the data with a large size. In addition, when establishing the asymptotic properties of the parameter estimators of the SAR model, both weights matrix and regressors are assumed to be nonstochastic in classical spatial econometrics, which is perhaps not realistic in real applications. Motivated by the machine learning literature, this paper proposes quasi-score matching estimation for the SAR model. This new estimation approach is developed based on the likelihood, but significantly reduces the computational complexity of the QMLE. The asymptotic properties of parameter estimators under the random weights matrix and regressors are established, which provides a new theoretical framework for the asymptotic inference of the SAR-type models. The usefulness of the quasi-score matching estimation and its asymptotic inference is illustrated via extensive simulation studies and a case study of an anti-conflict social network experiment for middle school students.