High-Dimensional Spatial Autoregression with Latent Factors by Diversified Projections
针对响应变量和协变量维度都发散的多元空间自回归模型，提出因子增强空间自回归模型，用多样化投影估计固定维度的潜在因子，再对每个响应分量用SCAD惩罚估计，实现高维下的变量选择。
We study one particular type of multivariate spatial autoregression (MSAR) model with diverging dimensions in both responses and covariates. This makes the usual MSAR models no longer applicable due to the high computational cost. To address this issue, we propose a factor-augmented spatial autoregression (FSAR) model. FSAR is a special case of MSAR but with a novel factor structure imposed on the high-dimensional random error vector. The latent factors of FSAR are assumed to be of a fixed dimension. Therefore, they can be estimated consistently by the diversified projections method (Fan and Liao, 2022), as long as the dimension of the multivariate response is diverging. Once the fixed-dimensional latent factors are consistently estimated, they are then fed back into the original SAR model and serve as exogenous covariates. This leads to a novel FSAR model. Thereafter, different components of the high-dimensional response can be modeled separately. To handle the high-dimensional feature, a smoothly clipped absolute deviation (SCAD) type penalized estimator is developed for each response component. We show theoretically that the resulting SCAD estimator is uniformly selection consistent, as long as the tuning parameter is selected appropriately. For practical selection of the tuning parameter, a novel BIC method is developed. Extensive numerical studies are conducted to demonstrate the finite sample performance of the proposed method.