9 Multivariate Multiple Regression

In this section we consider the problem of modeling the relationship between \(m\) responses \((Y_1,Y_2,\ldots ,Y_m)\) and a single set of predictor variables \((Z_1,Z_2,\ldots ,Z_r)\) each response is assumed to follow its own regression model so that

\begin {align*} Y_1 & = \beta _{01} + \beta _{11}Z_1 + \cdots + \beta _{n1}Z_n + \varepsilon _1\\ Y_2 & = \beta _{02} + \beta _{12}Z_2 + \cdots + \beta _{n2}Z_n + \varepsilon _2\\ \vdots &\\ Y_m & = \beta _{0m} + \beta _{1m}Z_{m} + \cdots + \beta _{nm}Z_{n} + \varepsilon _{m}\\ \end {align*}

Let the all the equation be (1)

The error term \(\underline {\varepsilon }= \begin {bmatrix} \varepsilon _1 & \varepsilon _2 & \cdots& \varepsilon _n\\ \end {bmatrix}' \) has \(E(\underline {\varepsilon })=\underline {0}\) and \(var(\underline {\varepsilon })=\Sigma \).

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