18 The Basics of financial economeTrics
The distributions of all ε are identical. Typically, these distributions
are assumed to follow normal distributions.^2 Consequently, the error
terms are continuous variables that are normally distributed with zero
mean and constant variance.^3
Estimating the Regression Model
Even if we assume that the linear assumption in equation (2.2) is plausible,
in most cases we will not know the population parameters. We have to
estimate the population parameters to obtain the sample regression param-
eters. An initial approach might be to look at the scatter plot of x and y and
iteratively draw a line through the points until one believes the best line has
been found. This approach is demonstrated in Figure 2.2. We have five pairs
of bivariate data. While at first glance both lines appear reasonable, we do
not know which one is optimal.
There might very well exist many additional lines that will look equally
suited if not better. The intuition behind retrieving the best line is to balance
(^2) See Appendix B for a discussion of the normal distribution.
(^3) Formally, this is indicated by
εσ∼
iid
N(,^2 )0
FIGUre 2.1 Linear Functional Relationship between x and y with Distribution of
Disturbance Term
x 3
X
Y
α
ΔX
με= 0 β ⋅ ΔX
με= 0
με= 0x 1 x 2