FINANCE Corporate financial policy and R and D Management

Just as in the case of simple regression, the closeness of fit of the estimated
regression equation to the data is measured by the sum of squares of these
deviations:

(5.17)

where nis the number of observations in the sample. The larger the sum of
squares, the less closely the estimated regression equation fits; the smaller
the sum of squares, the more closely it fits. Thus, it seems reasonable to
choose the values of a, b 1 , and b 2 that minimize the expression in equation
(5.17). These estimates are least squares estimates, as in the case of simple
regression.

Multiple Coefficient of Determination

In a previous section we described how the coefficient of determination can
be used to measure how well a simple regression equation fits the data.
When a multiple regression is calculated, the multiple coefficient of deter-
mination, rather than the simple coefficient of determination discussed pre-
viously, is used for this purpose. The multiple coefficient of determination
is defined as:

(5.18)

(5.18)

where Y

^
iis the value of the dependent variable that is predicted from the
regression equation, which means that R^2 measures the proportion of the
total variation in the dependent variable that is explained by the regression
equation. The positive square root of the multiple coefficient of determina-
tion is called the multiple correlation coefficient and is denoted by R. It,
too, is sometimes used to measure how well a multiple-regression equation
fits the data.
If there are only two independent variables in a multiple regression, as
in equation (5.16), a relatively simple way to compute the multiple coeffi-
cient of determination is:

R

YY

YY

ii

i

n

i

i

n

2

2

1

2

1

=− 1

−

−

=

=

∑

∑

( ˆ)

()

(YYiiˆ)(Y abX bXi i i)

i

n

i

n

−= −− −

= =

∑ ∑

2

11 2 2

2

1 1

Multiple Coefficient of Determination 81