Multivariate linear regression fits the model 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...

Multivariate linear regression fits the model

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 = (X’ X)
^-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, µ_Y
and µ
_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 Y = XB. It follows that the fitted values Y_ij
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, x’
_i
. Use this fact to find an expression for the covariance of Y_ij
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.