# Appendices

# Appendix A: Mathematical background

This appendix reviews some important mathematical concepts that are used throughout the book. However, it does not give mathematically exact formulations or proofs. For these we refer to specialised books on econometrics, stochastic analysis or financial mathematics.

## A.1 ECONOMETRIC METHODS

### A.1.1 Linear Regression

A *linear regression* models a linear relationship between a dependent variable *y* and a number of independent variables (regressors) *x*_{u}…,*x*_{n} of the form

where ∊ is an error term. Setting *x*_{1} *=* la constant term can be included in the model. The linear regression is used to find the coefficients of such a relationship based on a number of observations on *y* and *x*_{i}. If those observations are made at different times *t*, the given data is *y*_{t} and *x*_{ti} for *i* = 1,…*n* and *t* = 1,…*N* and the linear relation becomes

In vector notation, using **y** = (*y*_{1}…,*y*_{N})^{T}, *β* = (*β*_{1}*,…β*_{n})^{T} ∊ = (∊_{1},…,∊_{N})^{T}, and **X** for the *N* × *n*-matrix (*x*_{ti}), this is written as

The *ordinary least squares* (OLS) estimator for *β* minimises the quadratic error

The solution to this problem is ...