Max-Linear Competing Factor Models

Models incorporating “latent” variables have been commonplace in financial, social, and behavioral sciences. Factor model, the most popular latent model, explains the continuous observed variables in a smaller set of latent variables (factors) in a matter of linear relationship. However, complex data often simultaneously display asymmetric dependence, asymptotic dependence, and positive (negative) dependence between random variables, which linearity and Gaussian distributions and many other extant distributions are not capable of modeling. This article proposes a nonlinear factor model that can model the above-mentioned variable dependence features but still possesses a simple form of factor structure. The random variables, marginally distributed as unit Fréchet distributions, are decomposed into max linear functions of underlying Fréchet idiosyncratic risks, transformed from Gaussian copula, and independent shared external Fréchet risks. By allowing the random variables to share underlying (latent) pervasive risks with random impact parameters, various dependence structures are created. This innovates a new promising technique to generate families of distributions with simple interpretations. We dive in the multivariate extreme value properties of the proposed model and investigate maximum composite likelihood methods for the impact parameters of the latent risks. The estimates are shown to be consistent. The estimation schemes are illustrated on several sets of simulated data, where comparisons of performance are addressed. Bootstrap method is employed to obtain standard errors in real data analysis. Real application to financial data reveals inherent dependencies that previous work has not disclosed and demonstrates the model’s interpretability to real data. Supplementary materials for this article are available online.

引入潜变量（latent variables）的模型在金融、社会与行为科学领域已颇为常见。因子模型（Factor model）作为最常用的潜模型，通过线性关系以更精简的潜变量（因子）集合来阐释连续观测变量。然而，复杂数据往往同时呈现随机变量（random variables）间的非对称相依（asymmetric dependence）、渐近相依（asymptotic dependence）以及正（负）相依特征，而线性假设、高斯分布（Gaussian distributions）及诸多其他现有分布均无法对这类特征进行建模。本文提出一种非线性因子模型，既能够对上述变量相依特征进行建模，又保留了简洁的因子结构形式。边缘服从单位Fréchet分布（unit Fréchet distributions）的随机变量，可被分解为两部分：一是由高斯连接函数（Gaussian copula）变换得到的潜在Fréchet特质风险的极大线性函数，二是独立共享的外部Fréchet风险。通过允许随机变量共享带有随机影响参数的潜在（潜）普遍性风险，可生成各类相依结构。该方法创新性地提出了一种极具应用前景的新技术，能够生成具备简洁可解释性的分布族。本文深入研究了所提模型的多元极值性质（multivariate extreme value properties），并针对潜风险的影响参数探究了最大复合似然法（maximum composite likelihood methods）。研究证明该估计量具有相合性。本文通过多组仿真数据对估计流程进行了演示，并对模型性能展开了对比分析。在实际数据分析中，本文采用Bootstrap方法（Bootstrap method）获取标准误（standard errors）。针对金融数据的实际应用揭示了过往研究未被发掘的内在相依关系，同时验证了该模型对实际数据的可解释性。本文的补充材料可在线获取。