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.

翻译：多元线性回归模型常面临由多个解释变量引起的异方差性问题。传统基于单变量依赖权重的加权最小二乘估计在构建权重函数方面存在局限。为此，本文提出一种多元依赖加权最小二乘估计方法。通过构建解释变量的线性组合并最大化其与绝对残差值的Spearman秩相关系数，结合极大似然法刻画异方差性，能够全面反映随机误差的方差变化趋势并提升模型精度。本文证明该算法获得的异方差波动最优线性组合指数估计量具有一致性与渐近正态性。在模拟实验中设计了三种异方差场景，对比表明所提方法在参数估计与模型预测方面均优于单变量依赖加权方法。在实际数据应用中，将所提方法应用于中国居民消费支出与波士顿房价两个真实数据集，从MAE、RSE、交叉验证及拟合性能等角度，验证了其在模型预测、区间估计与泛化能力方面的准确性与稳定性。此外，该方法在波动较大数据的拟合中展现出相对优势。本研究为处理多元线性回归中的异方差问题提供了有效的新途径。