52 The Basics of financial economeTrics
Aggregate Bond Index.^10 For each of these indexes the dependent variable
is the monthly return in the value of the index. The time period covered is
from October 1989 to November 2003 (170 observations) and the monthly
return observations are given in the last three columns of Table 3.2.^11
Let’s begin with just one independent variable, an interest rate index.
We will use the monthly change in the U.S. Treasury yield index as measured
by the Lehman Treasury Index as the relevant interest rate variable. The
monthly values are given in the second column of Table 3.2. Notice that
the data are reported as the percentage difference between two months. So,
if in one month the value of the Treasury yield index is 7.20% and in the
next month it is 7.70%, the value for the observation is 0.50%. In finance, a
basis point is equal to 0.0001 or 0.01% so that 0.50% is equal to 50 basis
points. A 100 basis point change in interest rates is 1% or 1.00. We’ll need
to understand this in order to interpret the regression results.
The simple linear regression model (i.e., the univariate case) is
y = b 0 + b 1 x 1 + e
where y = the monthly return of an index
x 1 = the monthly change in the Treasury yield
The estimated regression coefficient b 1 is the empirical duration. To
understand why, if we substitute 100 basis points in the above equation
for the monthly change in the Treasury yield, the regression coefficient b 1
tells us that the estimated change in the monthly return of an index will
be b 1. This is precisely the definition of empirical duration: the approx-
imate change in the value of an asset for a 100 basis point change in
interest rates.
The estimated regression coefficient and other diagnostic values are
reported in Table 3.3. Notice that negative values for the estimated empiri-
cal duration are reported. In practice, however, the duration is quoted as a
positive value. Let’s look at the results for all three assets.
For the electric utility sector, the estimated regression coefficient for b 1
is –4.5329, suggesting that for a 100 basis point change in Treasury yields,
the percentage change in the value of the stocks comprising this sector will
(^10) The Lehman U.S. Aggregate Bond Index is now the Barclays Capital U.S. Aggre-
gate Bond Index.
(^11) The data for this illustration were supplied by David Wright of Northern Illinois
University.