Multiple Linear Regression 59
case. Note also that the estimated value of the regression coefficients are
not much different than in the univariate case. As for our new indepen-
dent variable, x 2 , we see that it is statistically significant at the 1% level
of significance for all three asset indexes. While we can perform statistical
tests discussed earlier for the contribution of adding the new indepen-
dent variable, the contribution of the two stock sectors to explaining the
movement in the return in the sector indexes is clearly significant. The R^2
for the electric utility sector increased from around 7% in the univariate
case to 13% in the multiple linear regression case. The increase was obvi-
ously more dramatic for the commercial bank sector, the R^2 increasing
from 1% to 49%.
Next we analyze the regression of the Lehman U.S. Aggregate Bond
Index. Using only one independent variable, we have R 12 = 91.77%. If we
include the additional independent variable, we obtain the improved R^2 =
93.12%. For the augmented regression, we compute with n = 170 and k =
2 the adjusted R^2 as
()
()
=−−
−
−−
=−−
−
−−
=
RR
n
nk11
1
1
11 0.9312
170 1
170 21
0.9304
adj22Let’s apply the F-test to the Lehman U.S. Aggregate Bond Index to
see if the addition of the new independent variable increasing the R^2 from
91.77% to 93.12% is statistically significant.
From equation (3.18), we have
=
−
−
−−
=
−
−
−−
F =
RR
R
nk1
1
0.9312 0.9177
10.9312
170 21
1 32.7689
2
12
2This value is highly significant with a p-value of virtually zero. Hence,
the inclusion of the additional variable is statistically reasonable.
Predicting the 10-Year Treasury Yield^12
The U.S. Treasury securities market is the world’s most liquid bond mar-
ket. The U.S. Department of the Treasury issues two types of securities:
(^12) We are grateful to Robert Scott of the Bank for International Settlements for sug-
gesting this example and for providing the data.