Generalized Vectorization, Cross-Products, and Matrix Calculus [Paperback]

This book studies the mathematics behind matrix calculus and the applications of matrix calculus in statistics and econometrics.Matrix calculus is an efficient procedure for obtaining many derivatives at once, used in statistics and econometrics. This book studies different concepts of matrix derivatives. A large portion of this book studies the particular brand of mathematics behind matrix calculus, which includes special matrices whose elements are all zero or one. The last chapter looks at applications of matrix calculus in statistics and econometrics.Matrix calculus is an efficient procedure for obtaining many derivatives at once, used in statistics and econometrics. This book studies different concepts of matrix derivatives. A large portion of this book studies the particular brand of mathematics behind matrix calculus, which includes special matrices whose elements are all zero or one. The last chapter looks at applications of matrix calculus in statistics and econometrics.This book presents the reader with new operators and matrices that arise in the area of matrix calculus. The properties of these mathematical concepts are investigated and linked with zero-one matrices such as the commutation matrix. Elimination and duplication matrices are revisited and partitioned into submatrices. Studying the properties of these submatrices facilitates achieving new results for the original matrices themselves. Different concepts of matrix derivatives are presented and transformation principles linking these concepts are obtained. One of these concepts is used to derive new matrix calculus results, some involving the new operators and others the derivatives of the operators themselves. The last chapter contains applications of matrix calculus, including optimization, differentiation of log-likelihood functions, iterative interpretations of maximum likelihood estimators, and a Lagrangian multiplier test for endogeneity.1. Mathematical prerequisites; 2. Zero-one matrices; lS(