Assuming that A1, A2, . An are distinct, then (a)0≤A<1fori=1,2,…,m1and0≤<1forj=1,2,…,n (b)A= i for i=1,2,…,n; (c)K2yyk= In and A'ExxA= In (d)AΣxK=R, where 0 R 00 We may interpret the canonical correlations as follows. The first canonical variates nIt and Eit can be interpreted as those linear combination of yt and x respectively, such that the correlation between nit and Sit is as large as possible The variates n2t and E2t gives those linear combination of yt and xt that are un- correlated with nit and SIt and yield the largest remaining correlation between met and E2t, and so on 1.2 Sample Canonical Correlations The canonical correlations r: calculated by the procedure just described are pop- ulation parameters-they are functions of the population moments >yy, 2xy, and 2xx. To find their sample analogs, all we have to do is to start from the sample moment of∑yy,Sy,andx Suppose we have a sample of T observations on the(ni 1) vector yt and the (n2 x 1) vector xt, whose sample moment are given by Sy=(1/m∑yy ∑yx=(1T>yxAssuming that λ1, λ2, ..., λn are distinct, then (a) 0 ≤ λi < 1 for i = 1, 2, ..., n1 and 0 ≤ µj < 1 for j = 1, 2, ..., n2; (b) λi = µi for i = 1, 2, ..., n; (c) K′ΣyyK = In and A′ΣxxA = In; (d) A′ΣxyK = R, where R2 =         λ1 0 . . . 0 0 λ2 . . . 0 . . . . . . . . . . . . . . . . . . 0 0 . . . λn         . We may interpret the canonical correlations as follows. The first canonical variates η1t and ξ1t can be interpreted as those linear combination of yt and xt , respectively, such that the correlation between η1t and ξ1t is as large as possible. The variates η2t and ξ2t gives those linear combination of yt and xt that are un￾correlated with η1t and ξ1t and yield the largest remaining correlation between η2t and ξ2t , and so on. 1.2 Sample Canonical Correlations The canonical correlations ri calculated by the procedure just described are pop￾ulation parameters–they are functions of the population moments Σyy, Σxy, and Σxx. To find their sample analogs, all we have to do is to start from the sample moment of Σyy, Σxy, and Σxx. Suppose we have a sample of T observations on the (n1 ×1) vector yt and the (n2 × 1) vector xt , whose sample moment are given by Σˆ yy = (1/T) X T t=1 yty ′ t Σˆ yx = (1/T) X T t=1 ytx ′ t Σˆ xx = (1/T) X T t=1 xtx ′ t . 3