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By Randall L. Eubank

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The Fundamental Covariance Structure 51 The smaller root is strictly smaller than W0 except when one of H, W0 or Q0 is zero. Thus, except for these exceptional cases, R(∞) is F 2 W0 + H 2 Q0 + W0 2 + (F 2 W0 + H 2 Q0 + W0 )2 − 4F 2 W 2 0 . 0779. 1 Introduction In this chapter we will develop algorithms for computing the matrices L and L−1 that arise in the Cholesky factorization of Var(y), where y is the vector of responses from a state-space model. There are several reasons for considering this particular problem.

4 with a new recursion that computes above diagonal entries to evaluate the entire matrix ΣXε . 5 This algorithm computes R(t), S(t|t), S(t|t − 1), t = 1, . , n and σXε (t, j), t, j = 1, . , n. 4 An example To illustrate the results of the previous section consider the state-space model where H(t), F (t), Q(t) and W (t) are time independent. In this case y(t) = Hx(t) + e(t) and x(t + 1) = F x(t) + u(t) for known matrices H and F . 3 we also have Var(e(t)) = W0 , Var(u(t − 1)) = Q0 for t = 1, . , n and S(0|0) = 0 so that x(0) = 0.

4 as Cov(x(t + © 2006 by Taylor & Francis Group, LLC The Fundamental Covariance Structure 33 1), ε(t + 1)) = S(t + 1|t)H T (t + 1). The following provides an algorithmic implementation of these ideas. 2 This algorithm computes the diagonal and below diagonal blocks of ΣXε . 2 is an example of a forward recursion in the sense that it works its way from the upper left hand corner of ΣXε down to the lower right hand block of the matrix. We should also note that there is nothing special about evaluation of ΣXε on a row-by-row basis.

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