Chapter

Advanced econometrics

Chapter 1 : Mathematics

By Lund University

All the mathematics that we need to get started. Most of this will be known to you. We begin by reviewing matrix algebra and functions of several variables. We then look at some multivariable calculus. In particular, how to differentiate a function with respect to a vector. We apply these derivatives to linear functions and quadratic forms.

Matrix algebra, review

All of this material is probably known to you. I am collecting all the stuff we need from matrix algebra that we need to get started with the linear regression in matrix form so that you can check that you know this. If you need to do some exercises based on this material, check out chapter 8 of the Mathematics for Economists course.

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

All of this material is probably known to you as well. I am collecting all the stuff we need from functions of several variables including partial derivatives, Hessian and optimization results that we need to get started with the linear regression in matrix form so that you can check that you know this. If you need to do some exercises based on this material, check out Functions of several variables and the first section of Optimization, several variables.

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

In this section, we look at multivariable calculus. We begin with the definitions of definite matrices. We then look at linear functions and quadratic form of an arbitrary number of variables and how to write them using vectors. Next, we look at the derivatives of a function with respect to a vector (gradient) and the second derivatives (Hessian). we also look at vector-valued functions and their derivatives, called Jacobians.

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