ABSTRACT
This book is mainly concerned with the linear model https://www.w3.org/1998/Math/MathML">Y=XB+E, i.e., ytj=∑i=1kxtiβij+εtj (t=1,2,…,n),https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203180754/c73c8eaa-6e2d-45b3-bd8b-866a06ca944e/content/math_1_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> where X = [xti] is an n × k matrix of n observations on k independent or exogenous variables, Y = [ytj] is an n × m matrix of n observations on m jointly dependent or endogenous variables, and E = [ɛtj] is an n × m matrix of random errors with zero means and specified variances and covariances, where t = 1, 2,…, n. To allow for a constant term, the first column of X may be specified to be a column of 1s. We shall in fact for the most part in Chapters 2-7 concentrate on the special univariate case m = 1, returning to the multivariate case in Chapter 8. The purpose of this chapter is to embed the above model in a multivariate model in which the rows xt. = (xt1, xt2, …, xtk) of X and yt. = (yt1, yt2, …, ytm) of Y are specified to have a joint distribution, and to consider the problem of the best predictor of yt. given xt. The linearity of the above model emerges as a practical aspect of optimal prediction.
