Heteroscedasticity as a Basis of Direction Dependence in Reversible Linear Regression Models

Heteroscedasticity is a well-known issue in linear regression modeling. When heteroscedasticity is observed, researchers are advised to remedy possible model misspecification of the explanatory part of the model (e.g., considering alternative functional forms and/or omitted variables). The present contribution discusses another source of heteroscedasticity in observational data: Directional model misspecifications in the case of nonnormal variables. Directional misspecification refers to situations where alternative models are equally likely to explain the data-generating process (e.g., <i>x</i> → <i>y</i> versus <i>y</i> → <i>x</i>). It is shown that the homoscedasticity assumption is likely to be violated in models that erroneously treat true nonnormal predictors as response variables. Recently, Direction Dependence Analysis (DDA) has been proposed as a framework to empirically evaluate the direction of effects in linear models. The present study links the phenomenon of heteroscedasticity with DDA and describes visual diagnostics and nine homoscedasticity tests that can be used to make decisions concerning the direction of effects in linear models. Results of a Monte Carlo simulation that demonstrate the adequacy of the approach are presented. An empirical example is provided, and applicability of the methodology in cases of violated assumptions is discussed.

异方差性（Heteroscedasticity）是线性回归建模中广为人知的一类问题。当观测到异方差性时，研究者通常需要修正模型解释部分可能存在的设定偏误（例如，考虑替代函数形式或遗漏变量）。本研究探讨了观测数据中异方差性的另一来源：非正态变量情形下的方向性模型设定偏误。方向性设定偏误指的是不同模型均能同等合理地解释数据生成过程的场景（例如，<i>x</i> → <i>y</i> 与 <i>y</i> → <i>x</i>）。研究表明，若错误地将真实非正态预测变量当作响应变量进行建模，则极有可能违背同方差性（Homoscedasticity）假设。近年来，方向依赖分析（Direction Dependence Analysis, DDA）被提出作为在线性模型中实证评估效应方向的分析框架。本研究将异方差性现象与DDA相结合，阐述了可用于判定线性模型效应方向的可视化诊断方法与九种同方差性检验。本文呈现了验证该方法适用性的蒙特卡洛模拟（Monte Carlo simulation）结果，并提供了一则实证案例，同时讨论了该方法在假设违背场景下的应用可行性。