By Darrell A. Turkington
The statistical versions confronting econometricians are advanced in nature so it really is no effortless job to use the approaches suggested via classical statisticians to such types. This booklet offers the reader with mathematical instruments drawn from matrix calculus and zero-one matrices and demonstrates how using their instruments drastically allows such functions in a series of linear econometric versions of accelerating statistical complexity. The booklet differs from others in that the matrix calculus effects are derived from a couple of easy principles that are generalizations of the foundations utilized in traditional calculus. in addition the homes of a number of new zero-one matrices are investigated.
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Extra resources for Matrix Calculus and Zero-One Matrices: Statistical and Econometric Applications
A square matrix A, n x n, is upper triangular if [ ain l azn all a12 a a22 A= 0 ann and lower triangular if all a21 a22 A= 0 an d „n1 If in addition is 0 for i = 1, . . , n, A is said to be strictly triangular. Working with triangular matrices is relatively easy as their mathematical properties are simple. For example, if A is upper (lower) triangular then A' is lower (upper) triangular. The determinant of a triangular matrix is the product of its main diagonal elements. The product of a finite number of upper (lower) triangular matrices is also upper (lower) triangular, and if one of the matrices in the product is strictly upper (lower) triangular the product itself is strictly upper (lower) triangular.
As a special case, let x and y be q x 1 and n x 1 vectors, respectively; then (x' 0 y')t = x 0 y'. 2. From the basic relationship linking vecnand devecngiven by Eq. 10) we have (At)' = (AT. 14) 3. 2 and for D an n x n matrix we have a. (C A)" = (1p 0C)At , b. A(a 0 In) = (a' 0 Ip)At , c. [C A(I p 0D)]t = (1p 0C)At D, d. [A(E 0 D)]t = (Iq 0 A) (vec E 0 D). 4. From the same theorem, if we set D = x, an n x 1 vector, so that s = 1 we have e. vec A(I p 0 x) = At x, f. vec[CA(/p 0 x)] = (Ip 0C)At x.
Then we define vec2 as the mp x 2 matrix given by vec2 A = al a a3 a2 a a4 a2p-1 a2p that is, to form vec2A, we stack columns of A under each other, taking two at a time. More generally, if A is the m x np matrix A = (ai• • • anp) then vec. 1. Vec Operations K 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Vec Operators Performable on A 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 3 2 4 5 2 3 6 7 2 4 8 3 9 2 5 10 11 2 3 4 6 12 13 2 7 3 2 14 5 15 4 8 16 17 2 3 6 9 18 19 2 4 5 10 20 For a given m x K matrix A, the number of generalized vec operations that can be performed on A clearly depends on the number of columns K of A.