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Advanced econometrics
Chapter 1
Matrix algebra, review
Matrices, definitions
Special matrices
Row- and column vectors
Vectors
Matrix addition and scalar multiplication
Algebraic rules for matrix addition and scalar multiplication
Matrix transpose
Dot product
Matrix multiplication, dimension
Matrix multiplication, computation
Matrix multiplication, rules
Matrix Inverse
Linear systems of equations
Linear systems in matrix notation
Determinants
Function of several variables, review
Functions of two variables
Graph of a function of two variables
Functions of several variables
Partial derivatives, rough introduction
Partial derivatives
Hessian
Optimizing a function of 2 variables, necessary conditions
Optimizing a function of 2 variables, sufficient conditions
Optimizing a function of 2 variables, additional results
Multivariable optimization, n variables
Multivariable calculus
Definite matrices
Linear functions and quadratic forms
Derivatives in vector form (gradients)
Second order derivatives in matrix form (Hessian)
Jacobian matrix
Matrix multiplication with transpose
Matrix multiplication with transpose
Problem: Gradient of a linear function and a quadratic form
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Matrix multiplication, dimension
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Summary
If
\(A\)
is
\(n×m\)
and
\(B\)
is
\(k×l\)
then the product
\(AB\)
is defined only if
\(m=k\)
. In this case,
\(AB\)
has dimension
\(n×l\)
.