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Managing Energy Risk: An Integrated View on Power and Other Energy Markets by Gero Schindlmayr, Bernhard Graeber, Markus Burger

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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) xu…,xn of the form

image

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 xi. If those observations are made at different times t, the given data is yt and xti for i = 1,…n and t = 1,…N and the linear relation becomes

image

In vector notation, using y = (y1…,yN)T, β = (β1,…βn)T ∊ = (∊1,…,∊N)T, and X for the N × n-matrix (xti), this is written as

image

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

image

The solution to this problem is ...

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