# How does multivariate regression consider the correlation among response when estimating the parameters?

I have read a few posts about multivariate regression, where the response is a matrix instead of a vector. However, when I really look into the literatures, there is nothing.

I have read the multivariate books (the books recommended by people answering similar questions), all they talk about is multivariate distribution, Wishart distribution, MANOVA, PCA, factor analysis, etc. None of them discussed how to take the dependency among y into account.

Is there any literature about how to do the 'multivariate regression' instead of talking about how to deal with multivariate data, etc.?

• Need to convert multivariate into univariate at first. – user158565 Jul 24 '19 at 0:58
• I don't know what you are asking here. Your title does not match with your body. Can you be more precise about what it is that you want to do and why. – user2974951 Jul 24 '19 at 8:17

Assume your multivariate regression means multivariate linear model. Multivariate linear model is $$Y=X\beta + \epsilon$$ where the dimensions of these matrices are: $$Y: n \times k$$, $$X: n \times p$$, $$\beta: p \times k$$, $$\epsilon: n \times k$$. Let $$Z_i$$ be the row vector of $$i$$-th row in matrix $$Z$$. According to the assumptions of multivariate linear model, $$\epsilon_i' \sim N_k(0_k, \Sigma)$$ and $$\epsilon_i$$ and $$\epsilon_j$$ are independent for $$i \ne j$$. Let reshape the matrices as following:
$$Y_R = (Y_1, Y_2,... Y_n)'$$ $$X_R = \left(\begin{matrix} X_1 & 0_k&...&0_k \\0_k &X_1 & ... &0_k\\...&...&...\\0_k&0_k&...&X_1\\ ...&...&...&...\\X_n & 0_k&...&0_k \\0_k &X_n & ... &0_k\\...&...&...\\0_k&0_k&...&X_n \end {matrix} \right)$$ $$\epsilon_R = (\epsilon_1, \epsilon_2,... \epsilon_n)'$$ $$\beta_R= (\beta_{11}, \beta_{21}, ..., \beta_{p1},\beta_{12}, \beta_{22}, ..., \beta_{p2},...... \beta_{1k}, \beta_{2k}, ..., \beta_{pk})'$$ We have univaraite linear model: $$Y_R = X_R\beta_R +\epsilon_R$$ Here $$\epsilon_R \sim N_{kn}(0_{kn}, I_n \times\Sigma)$$ The $$\times$$ here means Kronecker product, should have a circle.
• $X_R$ and $\beta_R$ were added. – user158565 Jul 26 '19 at 3:56