94 The Basics of financial economeTrics
(^) K n
xx
n
xx
i
n
i
n
=
−
()−
==∑
∑
1
1
4
1
2
12()
(4.20)
for a sample of size n.
The expression in equation (4.19) is the skewness statistic of some dis-
tribution, and equation (4.20) is the kurtosis. Kurtosis measures the peak-
edness of the probability density function of some distribution around the
median compared to the normal distribution. Also, kurtosis estimates, rela-
tive to the normal distribution, the behavior in the extreme parts of the dis-
tribution (i.e., the tails of the distribution). For a normal distribution, K = 3.
A value for K that is less than 3 indicates a so-called light-tailed distribution
in that it assigns less weight to the tails. The opposite is a value for K that
exceeds 3 and is referred to as a heavy-tailed distribution. The test statistics
given by equation (4.18) are approximately distributed chi-square with two
degrees of freedom.
Analysis of Standardized Residuals Another means of determining the appro-
priateness of the normal distribution are the standardized residuals. Once
computed, they can be graphically analyzed in histograms. Formally, each
standardized residual at the ith observation is computed according to
en esnxx
siiei
x=
++
()−
⋅
()12
2(4.21)
where se is the estimated standard error (as defined in Chapter 3) and n is
the sample size. This can be done with most statistical software.
If the histogram appears skewed or simply not similar to a normal
distribution, the linearity assumption is very likely to be incorrect. Addi-
tionally, one might compare these standardized residuals with the normal
distribution by plotting them against their theoretical normal counterparts
in a normal probability plot. There is a standard routine in most statistical
software that performs this analysis. If the pairs lie along the line running
through the sample quartiles, the regression residuals seem to follow a nor-
mal distribution and, thus, the assumptions of the regression model stated
in Chapter 3 are met.