44 The Basics of financial economeTrics
that our total dimensionality is k + 1. The estimated model generates values
on a k-multidimensional hyperplane, which expresses the functional linear
relationship between the dependent and independent variables. The esti-
mated hyperplane is called a regression hyperplane. In the univariate case,
this is simply the regression line of the yˆ estimates stemming from the one
single independent variable x.^1
Each of the k coefficients determines the slope in the direction of the cor-
responding independent variable. In the direction of the k + 1st dimension of
the y-values, we extend the estimated errors, ey=−yˆ. At each y-value, these
errors denote the distance between the hyperplane and the observation of
the corresponding y-value.
To demonstrate this, we consider some variable y. Suppose, we also have
a two-dimensional variable x with independent components x 1 and x 2. Hence,
we have a three-dimensional space as shown in Figure 3.1. For y, we have
three observations, y 1 , y 2 , and y 3. The hyperplane for equation (3.7) formed
(^1) In general, the hyperplane formed by the linear combination of the x-values is
always one dimension less than the overall dimensionality.
FiGuRE 3.1 Vector Hyperplane and Residuals
6
4
2
0
–2
–4
–6
10
5
0
–5
–10 –10
–5
0
5
10
y
x 2 x 1
y 3
b 1
y 1
e 1
b 2
y 2
e 2
e 3
b 0