Suppose the vector z = (x0

, y0

)0

, where x (p × 1) and y (q × 1) are jointly distributed with mean vectors µx and µy and with covariance matrix cov(z) = 

Σxx Σxy

Σyx Σyy

.

Consider projecting x on M = sp{1, y}, say, xb = b + By.

(a) Show the orthogonality conditions can be written as E(x − b − By) = 0, E[(x − b − By)y0

] = 0, leading to the solutions b = µx − Bµy and B = ΣxyΣ−1 yy .

(b) Prove the mean square error matrix is MSE = E[(x − b − By)x0

] = Σxx − ΣxyΣ−1 yy Σyx.

(c) How can these results be used to justify the claim that, in the absence of normality, Property 6.1 yields the best linear estimate of the state xt given the data Yt, namely, xt t, and its corresponding MSE, namely, Pt t ?