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 linear regression models a linear relationship between a dependent variable y and a number of independent variables (regressors) xu…,xn 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 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
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
The ordinary least squares (OLS) estimator for β minimises the quadratic error
The solution to this problem is ...