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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.
This 2002 book presents the reader with mathematical tools taken
from matrix calculus and zero-one matrices and demonstrates how
these tools greatly facilitate the application of classical
statistical procedures to econometric models. The matrix calculus
results are derived from a few basic rules that are generalizations
of the rules of ordinary calculus. These results are summarized in
a useful table. Well-known zero-one matrices, together with some
newer ones, are defined, their mathematical roles explained, and
their useful properties presented. The basic building blocks of
classical statistics, namely the score vector, the information
matrix, and the Cramer-Rao lower bound, are obtained for a sequence
of linear econometric models of increasing statistical complexity.
From these are obtained interactive interpretations of maximum
likelihood estimators, linking them with efficient econometric
estimators. Classical test statistics are also derived and compared
for hypotheses of interest.
Recent advances in establishing the nature and scope of estimators
in econometrics have shed more light on the importance of
instrumental variables. In this book, the authors argue that such
methods may be regarded as a strong organizing principle for a wide
variety of estimation and hypothesis testing problems in
econometrics and statistics. In support of this claim they present
and develop the methodology of instrumental variables in its most
general and explanatory form. They show, for instance, that
techniques commonly used to handle simultaneity and related
problems can be reduced to one of two generic variables of
instrumental variables estimators, allowing them to explore further
the conditions under which different proposed estimators are
efficient.
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.
The statistical models confronting econometricians are complicated in nature so it is no easy task to apply the procedures recommended by classical statisticians to such models. This book presents the reader with mathematical tools drawn from matrix calculus and zero-one matrices and demonstrates how the use of their tools greatly facilitates such applications in a sequence of linear econometric models of increasing statistical complexity. The book differs from others in that the matrix calculus results are derived from a few basic rules which are generalizations of the rules used in ordinary calculus. Moreover the properties of several new zero-one matrices are investigated.
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