38 The Basics of financial economeTrics
To linearize equation (2.14), we have the following natural logarithm
transformation of the y-values to perform:
lnlnyx=+αβ (2.15)
Linear regression of exponential Data
We estimate using OLS the ln y on the x-values to obtain ln a = 0.044 and
b = 0.993. Retransformation yields the following functional equation:
yaˆ==⋅⋅eebx⋅⋅1 045. 0.993x (2.16)
The estimated yˆ-values from equation (2.16) are represented by the +
symbol in Figure 2.5 and in most cases lie exactly on top of the original data
points. The coefficient of determination of the linearized regression is given
by approximately R^2 = 1 which indicates a perfect fit. Note that this is the
least squares solution to the linearized problem in equation (2.15) and not
the originally assumed functional relationship. The regression parameters
for the original problem obtained in some fashion other than via lineariza-
tion may provide an even tighter fit with an R^2 even closer to one.^10
Key Points
■ (^) Correlation or covariance is used to measure the association between
two variables.
■ (^) A regression model is employed to model the dependence of a variable
(called the dependent variable) on one (or more) explanatory variables.
■ (^) In the basic regression, the functional relationship between the depen-
dent variable and the explanatory variables is expressed as a linear
equation and hence is referred to as a linear regression model.
■ (^) When the linear regression model includes only one explanatory vari-
able, the model is said to be a simple linear regression.
■ (^) The error term, or the residual, in a simple linear regression model mea-
sures the error that is due to the variation in the dependent variable that
is not due to the explanatory variable.
■ (^) The error term is assumed to be normally distributed with zero mean
and constant variance.
(^10) As noted earlier, for functional relationships higher than of linear order, there is
often no analytical solution, the optima having to be determined numerically or by
some trial-and-error algorithms.