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，研究者需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）。本文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模型方向设定偏误。方向设定偏误指的是，存在多种备选模型均可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作为响应变量进行建模，很可能会违反同方差性（Homoscedasticity）假设。近年来，方向依赖分析（Direction Dependence Analysis, DDA）被提出，作为实证评估线性模型中效应方向的分析框架。本研究将异方差性现象与DDA相结合，阐述了可用于决策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。本文呈现了蒙特卡洛模拟（Monte Carlo simulation）的结果，以验证该方法的恰当性；同时提供了一个实证案例，并讨论了该方法在假设条件不满足场景下的适用性。