86 The Basics of financial economeTrics
explained by the second independent variable to the total of unexplained
variation before x 2 was included. Formally, we have
SSE SSE
SSE
12
1− (4.4)
where SSE 1 = the variation left unexplained by x 1
SSE 2 = the variation left unexplained after both x 1 and x 2 have been
included
This is equivalent to requiring that the additional variable is to be the
one that provides the largest coefficient of determination once included in
the regression. After the inclusion, an F-test with
(^) F
n
=
−
−
SSE SSE
SSE
12
1
2(4.5)
is conducted to determine the significance of the additional variable.
The addition of independent variables included in some set of candidate
independent variables is continued until either all independent variables are
in the regression or the additional contribution to explain the remaining
variation in y is not significant anymore. Hence, as a generalization to equa-
tion (4.5), we compute
Fn=−()SSE SSEii+ 1 /()SSEi ×−()i− 1
after the inclusion of the i + 1st variable and keep it included only if F is
found to be significant. Accordingly, SSEi denotes the sum of square residu-
als with i variables included while SSEi+ 1 is the corresponding quantity for
i + 1 included variables.
Stepwise Exclusion Regression Method
The stepwise exclusion regression method mechanically is basically
the opposite of the stepwise inclusion method. That is, one includes
all independent variables at the beginning. One after another of the
insignificant variables are eliminated until all insignificant independent
variables have been removed. The result constitutes the final regression
model. In other words, we include all k independent variables into the