50 The Basics of financial economeTrics

with n − k − 1 degrees of freedom. The value bj is the sample estimate of the jth
regression coefficient and sbj is the standard error of the coefficient estimate.
The standard error of each coefficient is determined by the estimate of
the variance matrix of the entire vector β, given by

(^) ()
−
sX^2 TX
1
(3.16)
which is a matrix multiplied by the univariate standard error of the regres-
sion, s^2. The latter is given by

=

−−

=

−−

s

ee

nknk 1

SSE

1

T

(^2) (3.17)
SSE was previously defined and the degrees of freedom are determined
by the number of observations, n, minus the number of independent param-
eters, k, and minus one degree of freedom lost on the constant term. Hence,
we obtain n − k − 1 degrees of freedom. The jth diagonal element of equa-
tion (3.16), then, is the standard error of the jth regression coefficient used
in equation (3.15).^8 This test statistic in equation (3.15) needs to be com-
pared to the critical values of the tabulated t-distribution with n − k − 1
degrees of freedom at some particular significance level α, say 0.05. So, if the
test statistic should exceed the critical value then the independent variable
is said to be statistically significant. Equivalently, the p-value of equation
(3.15) would then be less than α.
The F-Test for inclusion of Additional Variables
Suppose we have k − 1 independent variables in the regression. The
goodness-of-fit is given by R 12. If we want to check whether it is appropri-
ate to add another independent variable to the regression model, we need
a test statistic measuring the improvement in the goodness-of-fit due to the
additional variable. Let R^2 denote the goodness-of-fit of the regression after
the additional independent variable has been included into the regression.
Then the improvement in the explanatory power is given by RR^2 − 12 , which
is chi-square distributed with one degree of freedom. Because 1 − R^2 is chi-
square distributed with n − k − 1 degrees of freedom, the statistic

−

−

−−

F

RR

R

nk

1

1

1

2

1

2

2 (3.18)

(^8) Typically one does not have to worry about all these rather mathematical steps
because statistical software performs these calculations. The interpretation of that
output must be understood.