Anon

410 The Basics of financial economeTrics

■ (^) Median. The median Med(X) is defined as that value that occupies a
central position in sample order statistics:

=

+









+

+

x

xx

N

N

Med(X)

(( )/2)

if isodd

if iseven

N

NN

(( 1)/2)

(/2) (/21)

Robust Estimators of the Spread

The variance is a classical estimator of the spread but it is not robust. Robust
estimators of the spread are the following:

■ (^) Median absolute deviation. The median absolute deviation (MAD) is
defined as the median of the absolute value of the difference between a
variable and its median, that is,
MAD = MED|X – MED(X)|
■ (^) Interquartile range. The interquartile range (IQR) is defined as the dif-
ference between the highest and lowest quartile:
IQR = Q(0.75) – Q(0.25)
where Q(0.75) and Q(0.25) are the 75th and 25th percentiles of the
data.
■ (^) Mean absolute deviation. The mean absolute deviation (MeanAD) is
defined as follows:
∑ −
N =
xX

1

j MED( )

j

N

1

■ (^) Winsorized standard deviation. The Winsorized standard deviation is
the standard deviation of Winsorized data, that is,
σ=
σ
()UL− /N
W
N
NN

Illustration of Robust Statistics

To illustrate the effect of robust statistics, consider the series of daily returns
of Nippon Oil in the period 1986 through 2005 depicted in Figure F.1. If