This paper proposes methods for likelihood-based inference in multivariate linear regressions when the correlation matrix of the responses is separable; that is, it has a Kronecker product structure, but the variances are unrestricted. The methods are enabled by a block-coordinate ascent-like algorithm with closed-form updates that strictly increases the likelihood at every iteration until convergence. In the numerical experiments, the proposed algorithm is 300--2500 times faster than a general-purpose solver, making parametric bootstrap tests of correlation and covariance separability practical. Parameters are identifiable, and standard errors can therefore be obtained from the expected Fisher information, which can be computed efficiently using the Kronecker product structure. Simulations show that the proposed estimator has lower error than both separable covariance and unrestricted estimators when the model holds, and that bootstrap tests maintain nominal size where asymptotic tests fail. An application to dissolved oxygen data from the Mississippi River demonstrates that separable correlation captures location-specific variance patterns that separable covariance cannot.

翻译：本文提出了当响应变量的相关矩阵具有可分离性（即呈现克罗内克积结构，且方差无约束）时，多元线性回归中基于似然推断的方法。该方法通过一种闭式更新的块坐标上升类算法实现，该算法在每次迭代中严格增加似然值直至收敛。数值实验表明，所提算法比通用求解器快300–2500倍，使得相关性和协方差可分离性的参数自助检验具有实用性。参数可识别，因此可通过期望Fisher信息量获得标准误差，而该信息量可利用克罗内克积结构高效计算。模拟结果显示，当模型成立时，所提估计量的误差低于可分离协方差估计量及无约束估计量，且自助检验在渐近检验失效时仍能保持名义检验水平。对密西西比河溶解氧数据的应用表明，可分离相关性能捕捉可分离协方差无法解释的位置特异性方差模式。