16 The Basics of financial economeTrics
However, we admit that the variation of y will be influenced by other
quantities. Thus, we allow for some additional quantity representing a resid-
ual term that is uncorrelated with x, which is assumed to account for any
movement of y unexplained by equation (2.1). Since it is commonly assumed
that these residuals are normally distributed, and that x and the residuals
are jointly normally distributed, assuming that residuals are uncorrelated is
equivalent to assuming that the residuals are independent of x. (Note that x
and the residuals are defined as joint normal when any linear combination
of x and residuals has a normal distribution.) Hence, we obtain a relation-
ship as modeled by the following equation
yf=+()x ε (2.2)
where the residual or error is given by ε.
In addition to being independent of anything else, the residual is mod-
eled as having zero mean and some constant variance denoted by σe^2. A
disturbance of this sort is considered to be some unforeseen information or
shock. Assume a linear functional relationship,
fx()=+αβx (2.3)
where the population parameters α and β are the vertical axis intercept and
slope, respectively. With this assumption, equation (2.2) is called a simple
linear regression or a univariate regression. We refer to the simple linear
regression as a univariate regression because there is only one independent
variable whereas a multiple linear regression (the subject of later chapters)
is a regression with more than one independent variable. In the regression
literature, however, a simple linear regression is sometimes referred to as a
bivariate regression because there are two variables, one dependent and one
independent.
The parameter β determines how much y changes with each unit change
in x. It is the average change in y dependent on the average change in x one
can expect. This is not the case when the relationship between x and y is
not linear.
Distributional Assumptions of the Regression Model
The independent variable can be a deterministic quantity or a random vari-
able. The first case is typical of an experimental setting where variables are
controlled. The second case is typical in finance where we regress quantities