Robust Regressions 161
iteratively the weights through an iterative reweighted least squares (RLS)
procedure. Clearly the iterative procedure depends numerically on the
choice of the weighting functions. Two commonly used choices are the
Huber weighting function, wH(e), defined as
week
H ke ek()
/
=
≤
>
1for
forand the Tukey bisquare weighting function, wT(e), also referred to as biweight
function, defined as
week ek
T ek()
((/))
=
−≤
>
1
0
(^22) for
for
where k is a tuning constant often set at 1.345 × (standard deviation of
errors) for the Huber function, and at 4.685 × (standard deviation of errors)
for the Tukey function. (Note that w = 1 [constant function] recovers the
case of ordinary least squares.)
ILLUSTRATION: ROBUSTNESS OF THE
CORPORATE BOND YIELD SPREAD MODEL
To illustrate robust regressions, let’s continue with our illustration of the
spread regression used in Chapter 6 to show how to incorporate dummy
variables into a regression model. Recall that there are 200 issues in the
sample studied. Table 8.1 shows the diagonal elements of the hat matrix
called leverage points. These elements are all very small, much smaller than
the safety threshold 0.2. We therefore expect that the robust regression does
not differ much from the standard regression.
We ran two robust regressions with the Huber and Tukey weighting
functions. The tuning parameter k is set as suggested earlier. The estimated
coefficients of both robust regressions were identical to the coefficients of
the standard regression. In fact, with the Huber weighting function, we
obtained the parameters estimates shown in the second column of Table 8.2.
The tuning parameter was set at 160, that is, 1.345 the standard deviation
of errors. The algorithm converged at the first iteration.
With the Tukey weighting function, we obtained the beta parameters
shown in the third column of Table 8.2, with the tuning parameter set at
550, that is, 4.685 the standard deviation of errors. The algorithm con-
verged at the second iteration.