Bayesian Inference for Regression Copulas

We propose a new semiparametric distributional regression smoother that is based on a copula decomposition of the joint distribution of the vector of response values. The copula is high-dimensional and constructed by inversion of a pseudo regression, where the conditional mean and variance are semiparametric functions of covariates modeled using regularized basis functions. By integrating out the basis coefficients, an implicit copula process on the covariate space is obtained, which we call a “regression copula.” We combine this with a nonparametric margin to define a copula model, where the entire distribution—including the mean and variance—of the response is a smooth semiparametric function of the covariates. The copula is estimated using both Hamiltonian Monte Carlo and variational Bayes; the latter of which is scalable to high dimensions. Using real data examples and a simulation study, we illustrate the efficacy of these estimators and the copula model. In a substantive example, we estimate the distribution of half-hourly electricity spot prices as a function of demand and two time covariates using radial bases and horseshoe regularization. The copula model produces distributional estimates that are locally adaptive with respect to the covariates, and predictions that are more accurate than those from benchmark models. Supplementary materials for this article are available online.

本文提出一种全新的半参数分布回归平滑器，其构建基于响应值向量联合分布的Copula函数（copula）分解。该Copula函数为高维形式，通过伪回归的逆变换构建，其中条件均值与条件方差为协变量的半参数函数，且采用正则化基函数对协变量进行建模。通过对基系数进行积分消去，可得到协变量空间上的隐式Copula过程，我们将其命名为“回归Copula”。我们将该过程与非参数边缘分布相结合，构建出Copula模型：响应变量的完整分布（包含均值与方差）均为协变量的平滑半参数函数。该Copula函数可通过哈密顿蒙特卡洛（Hamiltonian Monte Carlo）与变分贝叶斯（Variational Bayes）两种方法进行估计，其中后者可适配高维场景。通过真实数据集示例与模拟研究，我们验证了上述估计器与Copula模型的有效性。在一项具体实证案例中，我们采用径向基函数与马蹄形正则化（horseshoe regularization），基于负荷需求与两个时间协变量，对半小时级电力现货价格的分布进行估计。该Copula模型可生成针对协变量的局部自适应分布估计，且其预测精度优于基准模型。本文的补充材料可在线获取。