Weighted least squares estimation by multivariate-dependent weights for linear regression models
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线性回归中依赖多个解释变量的加权最小二乘估计
- 研究问题
- 当异方差由多个解释变量共同驱动时，如何构造比单变量依赖权重更灵活的权重函数，并评估参数估计和预测表现？
- 方法与思路
- 作者寻找与绝对残差具有最大 Spearman 秩相关的解释变量线性组合，再结合最大似然刻画异方差，构造多变量依赖权重。
- 材料陈述的贡献
- 摘要称相关指数估计量具有一致性和渐近正态性，并在三类异方差模拟及消费、房价数据中展示相对单变量权重的改善。
- 阅读关注点
- 应核对权重函数与指数参数的定义、残差和权重的估计步骤，以及理论条件、区间估计是否计入权重估计的不确定性。
- 阅读边界
- 所据为 arXiv 预印本，同行评审或期刊接收未核实；摘要未列明一致性条件、比较数值及对权重函数误设的稳健性。
原文摘要
Multivariate linear regression models often face the problem of heteroscedasticity caused by multiple explanatory variables. The weighted least squares estimation with univariate-dependent weights has limitations in constructing weight functions. Therefore, this paper proposes a multivariate dependent weighted least squares estimation method. By constructing a linear combination of explanatory variables and maximizing their Spearman rank correlation coefficient with the absolute residual value, combined with maximum likelihood method to depict heteroscedasticity, it can comprehensively reflect the trend of variance changes in the random error and improve the accuracy of the model. This paper demonstrates that the optimal linear combination exponent estimator for heteroscedastic volatility obtained by our algorithm possesses consistency and asymptotic normality. In the simulation experiment, three scenarios of heteroscedasticity were designed, and the comparison showed that the proposed method was superior to the univariate-dependent weighting method in parameter estimation and model prediction. In the real data applications, the proposed method was applied to two real-world datasets about consumer spending in China and housing prices in Boston. From the perspectives of MAE, RSE, cross-validation, and fitting performance, its accuracy and stability were verified in terms of model prediction, interval estimation, and generalization ability. Additionally, the proposed method demonstrated relative advantages in fitting data with large fluctuations. This study provides an effective new approach for dealing with heteroscedasticity in multivariate linear regression.