90 The Basics of financial economeTrics
and the assumptions about the error term. We discuss the implications of the
violation of these assumptions, some tests used to detect violations, and pro-
vide a brief explanation of how to deal with any violations. In some cases,
we discuss these violations in more detail in later chapters.
Tests for Linearity
To test for linearity, a common approach is to plot the regression residuals
on the vertical axis and values of the independent variable on the horizontal
axis. This graphical analysis is performed for each independent variable.
What we are looking for is a random scattering of the residuals around zero.
If this should be the case, the model assumption with respect to the residu-
als is correct. If not, however, there seems to be some systematic behavior in
the residuals that depends on the values of the independent variables. The
explanation is that the relationship between the independent and dependent
variables is not linear.
The problem of a nonlinear functional form can be dealt with by trans-
forming the independent variables or making some other adjustment to the
variables. For example, suppose that we are trying to estimate the relation-
ship between a stock’s return as a function of the return on a broad-based
stock market index such as the S&P 500. Letting y denote the return on
the stock and x the return on the S&P 500 we might assume the following
bivariate regression model:
y = b 0 + b 1 x + ε (4.8)
where ε is the error term.
We have made the assumption that the functional form of the relation-
ship is linear. Suppose that we find that a better fit appears to be that the
return on the stock is related to the return on the broad-based market index as
y = b 0 + b 1 x + b 2 x^2 + ε (4.9)
If we let x = x 1 and x^2 = x 2 and we adjust our table of observations accord-
ingly, then we can rewrite equation (4.9) as
y = b 0 + b 1 x 1 + b 2 x 2 + ε (4.10)
The model given by equation (4.10) is now a linear regression model despite
the fact that the functional form of the relationship between y and x is non-
linear. That is, we are able to modify the functional form to create a linear
regression model.