In spatial and social network analysis, multivariate spatial autoregressive (MSAR) models are effective tools for analyzing network data with multivariate responses. When the dimension of the response is divergent, however, the number of unknown parameters in an MSAR increases at a rate that is proportional to the square of the dimensionality, which poses significant challenges to the model estimation process. To address this issue, we propose a novel reduced-rank MSAR model by imposing a low-rank structure on the spatial influence matrix of the multivariate responses. The proposed model achieves substantial dimensionality reduction and offers insightful interpretations. To mitigate the high computational cost of the quasi-maximum likelihood estimator (QMLE), we propose a least squares estimator (LSE) for estimating the unknown parameters. Furthermore, we establish the asymptotic nature of the LSE when both the network size and the dimensionality of the responses diverge to infinity. To determine the rank, we propose an information criterion estimator and demonstrate the consistency of its rank selection process. Extensive numerical simulations validate the proposed model and parameter estimates. Finally, a dataset derived from Shouqianba, one of the largest aggregate payment platforms, is analyzed for illustration purposes.

在目标人群中存在未观测混杂且缺乏工具变量时，因果效应的估计较为困难。本文利用来自辅助人群的工具变量，通过数据融合的方法来识别目标人群中的因果效应。文章首先讨论了基于条件平均处理效应同质性的传统思路，但该方法在处理效应存在异质性时可能产生偏倚。为此，本文提出了等混杂（equal-confounding）假设，即在调整观测协变量后，不同人群中的未观测混杂偏倚相同，但允许处理效应在不同人群之间存在差异。在此基础上，可以利用辅助人群识别混杂偏倚，并据此校正目标人群中处理与结果之间的关联性，从而恢复因果效应。文章进一步提出了多稳健估计方法及去偏机器学习方法，并通过模拟研究和真实数据分析验证了方法的有效性。