Anon

20 The Basics of financial economeTrics

Here, the minimum is obtained analytically by using differential calculus
(the first derivative to be more specific) with respect to α and β, respectively.
The resulting estimates are then given by

(^) b n
xxyy
n
xx
n
ii xy
i
n
i
i
n
ii
= i

()− ()−

()−

= =

=

∑

∑

1

1

1

1

2

1

==

=

∑

∑









−









−

1

2

1

(^12)
n
i
i
n
xy
n
xx

(2.6)

and

ay=−bx (2.7)

The OLS methodology provides the best linear unbiased estimate
approach in the sense that no other linear estimate has a smaller sum of
squared deviations. (See Appendix C for an explanation of best linear unbi-
ased estimate.) The line is leveled, meaning that

ei

i

n

=

∑ =

1

0

That is, the disturbances cancel each other out. The line is balanced on a
pivot point ()xy, like a scale.
If x and y were uncorrelated, b would be zero. Since there is no correla-
tion between the dependent variable, y, and the independent variable, x, all
variations in y would be purely random, that is, driven by the residuals, ε.
The corresponding scatter plot would then look something like Figure 2.3
with the regression line extending horizontally. This is in agreement with a
regression coefficient β = 0.

application to Stock returns

As an example, consider again the monthly returns from the S&P 500 index
(indicated by x) and the GE stock (indicated by y) from the period between
January 31, 1996, and December 31, 2003. We list the intermediate results
of regressing the index returns on the stock returns as follows:

x=0.0062

y=0.0159

1

(^961)
96
xyii
i=
∑ =0.0027