Anon

160 The Basics of financial economeTrics

Thus far we have discussed methods to ascertain regression robustness.
Let’s now discuss methods to “robustify” the regression estimates, namely,
methods based on M-estimators and W-estimators.

Robust Regressions Based on M-Estimators

Let’s first discuss how to make robust regressions with Huber M-estimators.
The LS estimators are M-estimators because they are obtained by minimiz-
ing the sum of the squared residuals. However they are not robust. We can
generalize equation (8.3) by introducing the weighting function:

ρρ=−β













=

YXtj∑ tj

j

N

1

(8.6)

We rewrite the M-estimator as follows:

MYNt X

t

T

tjtj

j

N

t

ββ 0 ρε ρβ

11

(),,... = ()=−













==

∑∑

==

∑

1

T

And we generalize the LS by minimizing the M-estimator with respect to
the coefficients β. To determine the minimum, we equate to zero the partial
derivatives of the M-estimator. If we define the functions:

ψ

ρψ

x

dx

dx

wx

x

x

()=

()

()=

()

,

We can write the following conditions:

∂

∂

=

∂−









∂

=−

=

=

∑

∑

θ

β

ρβ

β

ψβ

k

tjtj

j

N

t k

T

tj

YX

YX

1

1

ttj

j

N

t

T

Xktk N

= =

∑∑











 ==

1 1

00 ,,..., 1

ψβYXtjtj XwYXβ

j

N

t

T

tk tjtj

j

N

−











 =−

===

∑∑∑

111











 −











 =

==

∑∑

t

T

tjtj

j

N

YXXtk

11

β 0

or, in matrix form

X′WXβ = X′WY

where W is a diagonal matrix.
The above is not a linear system because the weighting function is in
general a nonlinear function of the data. A typical approach is to determine