Multivariate linear regression is a natural extension of multiple linear re-
gression in that both techniques try to interpret possible linear relation-
ships between certain input and output variables. Multiple regression is
concerned with studying to what extent the behavior of a single output
variable Y is influenced by a set of r input variables X =(X1, ··· ,Xr)τ .
Multivariate regression has s output variables Y =(Y1, ··· ,Ys)τ ,each
of whose behavior may be influenced by exactly the same set of inputs
X =(X1, ··· ,Xr)τ .
So, not only are the components of X correlated with each other, but in
multivariate regression, the components of Y are also correlated with each
other (and with the components of X). In this chapter, we are interested
in estimating the regression relationship between Y and X, taking into
account the various dependencies between the r-vector X and the s-vector
Y and the dependencies within X and within Y.