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By Chordia, Sarkar

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Proof Should z = 1 be either a simple or a second order pole of A~^ (z), then Theorem 3 trivially applies. The proof for a higher order pole follows along the same lines. D Two additional results, whose role will become apparent later on, are worth stating. 5 Matrix Polynomials and Difference Equation Systems F[z=0 29 (29) where j ^ and z^ are as determined by (17) and (24) of Theorems 2 and 3, respectively. Proof The proof is simple and is omitted. D Theorem 6 Let z = 1 be a second order pole of A~\z) and N, = HK' (30) a rank factorization of A^2- Then the following hold Ky = H',N,Y^g,+ H',M{L)g, H[z,= H[N^c (31) (32) where J, and z, are as determined by (18) and (25) of Theorem 2 and 3, respectively.

To prove ii) observe that, in view of (8), the null row-spaces of N^ and Cj_ are the same. Hence, insofar as the r columns of C form a basis for the null row-space of Ci, they span the null row-space of A^^ as claimed. Finally, expansion (13) follows by premultiplying the right-hand side of (2) by C in light of (8). Given that C is orthogonal to N^ the term in (1 - z)~' vanishes, thus removing the pole once located at z= 1, and the matrix function CA'Xz) is analytical about z = 1. 1. 1. 7 Closed-Forms of Laurent Expansion Coefficient Matrices 41 Proof of iv) follows from (4), in view of (3), by straightforward computations.

1. 5i)x, its columns span the row-kernel of A^2 ^^ claimed. Finally, expansion (59) follows by premultiplying the right-hand side of (21) by {C^SJ^ in light of (31). Given that {C^Sj^ is orthogonal to A^^. the term in (1 -z)~^ of {C^SJ'^A~\z) vanishes, a pole order-reduction occurs and the matrix function {C^SJ\A~\z) exhibits a simple pole at z = 1. 1. ii) Should r,

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An Empirical Analysis Of Stock And Bond Market Liquidity by Chordia, Sarkar

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