'Multivariate linear regression fits the model Y n m X n k1 B k1 m E n m where

'Multivariate linear regression fits the model Y

ðn · mÞ

¼ X

ðn · kþ1Þ

B

ðkþ1 · mÞ

þ E

ðn · mÞ

where Y is a matrix of response variables; X is a model matrix (just as in the univariate linear model); B is a matrix of regression coefficients, one column per response variable; and E is a matrix of errors. The least-squares estimator of B is Bb ¼ ðX0 XÞ
&1 X0 Y (equivalent to what one would get from separate least squares regressions of each Y on the Xs). See Section 9.5 for a discussion of the multivariate linear model.

(a) Show how Bb can be computed from the means of the variables, µbY and µb X , and from their covariances, Sb XX and Sb XY (among the Xs and between the Xs and Ys, respectively).

(b) The fitted values from the multivariate regression are Yb ¼ XBb. It follows that the fitted values Ybij and Ybij0 for the ith observation on response variables j and j 0 are both linear combinations of the ith row of the model matrix, x0 i . Use this fact to find an expression for the covariance of Ybij and Ybij0 .

(c) Show how this result can be used in Equation 20.7 (on page 618), which applies the EM algorithm to multivariate-normal data with missing values.