Managing Energy Risk: An Integrated View on Power and Other Energy Markets

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 x₁ = 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 = (y₁…,y_N)^T, β = (β₁,…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 ...