Network Varying Coefficient Model

We propose a novel network-varying coefficient model that extends traditional varying coefficient models to accommodate network data. The main idea is to model the regression coefficients as the functions of the latent “locations” of network nodes that drive formation of the network. To estimate the model, we identify the latent “locations” via the latent space model and then develop an iterative projected gradient descent algorithm by optimizing the network parameters and regression coefficients alternately. The non-asymptotic bounds of the estimated coefficient matrix are obtained. In addition, a Bayesian information criterion is proposed to select the dimension of the latent space. Moreover, we employ a penalized method to select covariates with varying coefficients that are significant to the response variable, and demonstrate the theoretical properties of selection. The utility of the proposed model is illustrated via simulation studies and a real-world application in the field of finance by analyzing the relationship between stock returns and their corresponding financial ratios from a network perspective.

本文提出一种新颖的网络变系数模型（network-varying coefficient model），该模型将传统变系数模型进行推广，以适配网络数据。其核心思路是将回归系数建模为驱动网络形成的网络节点潜在‘位置’的函数。为估计该模型，本文先通过潜在空间模型（latent space model）识别节点潜在‘位置’，随后提出一种交替优化网络参数与回归系数的迭代投影梯度下降算法。本文推导得到了估计系数矩阵的非渐近界。此外，本文提出贝叶斯信息准则以选取潜在空间的维度。进一步，本文采用惩罚方法筛选对响应变量具有显著变系数的协变量，并论证了该筛选方法的理论性质。最后，本文通过仿真实验与一则金融领域的实际应用案例验证了所提模型的实用性：该案例从网络视角分析了股票收益率与其对应财务比率之间的关联关系。