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得到广泛应用。其中应用最为广泛的潜变量模型为因子模型（factor model），其通过规模更小的一组潜变量（即因子），以线性关系刻画连续型观测变量间的内在关联。然而，复杂数据往往同时展现随机变量间的非对称相依、渐近相依以及正（负）相依特性，而线性假设、高斯分布（Gaussian distributions）及诸多现有分布均无法对这类特性进行有效建模。本文提出一种非线性因子模型（nonlinear factor model），其既可刻画上述变量相依特征，又保留了简洁的因子结构形式。本文所处理的随机变量边缘服从单位Fréchet分布（unit Fréchet distributions），可分解为两部分：一是基于高斯连接函数（Gaussian copula）变换得到的底层Fréchet特质风险（idiosyncratic risks）的最大线性函数，二是独立的共享外部Fréchet风险。通过允许随机变量共享带有随机影响参数的底层（潜）广域风险（pervasive risks），即可生成多样化的相依结构。该方法创新性地构建了一类具备简洁可解释性的分布族，为相关研究提供了极具潜力的技术路径。本文深入探究了所提模型的多元极值（multivariate extreme value）性质，并针对潜风险的影响参数研究了最大复合似然（maximum composite likelihood）估计方法，证明了估计量具备一致性。本文通过多组模拟数据对估计流程进行了演示，并对不同方案的性能表现开展了对比分析。在实际数据分析中，本文采用自助法（bootstrap method）获取标准误（standard errors）。针对金融数据的实际应用揭示了此前研究尚未发现的内在相依性，同时验证了模型对真实数据的解释能力。本文的补充材料可在线获取。