Anon

350 The Basics of financial economeTrics

Student’s t-distribution has a similar shape to the normal distribution, but
with thicker tails. For large degrees of freedom n, the Student’s t-distribution
does not significantly differ from the standard normal distribution. As a
matter of fact, for n ≥ 50, it is practically indistinguishable from N(0,1).
Figure B.4 shows the Student’s t-density function for various degrees of
freedom plotted against the standard normal density function. The same is
done for the distribution function in Figure B.5.
In general, the lower the degrees of freedom, the heavier the tails of the
distribution, making extreme outcomes much more likely than for greater
degrees of freedom or, in the limit, the normal distribution. This can be seen
by the distribution function that we depicted in Figure B.5 for n = 1 and
n = 5 against the standard normal cumulative distribution function (cdf).
For lower degrees of freedom such as n = 1, the solid curve starts to rise
earlier and approach 1 later than for higher degrees of freedom such as n = 5
or the N(0,1) case.
This can be understood as follows. When we rescale X by dividing by
Sn/ as in equation (B.5), the resulting XS//n obviously inherits ran-
domness from both X and S. Now, when S is composed of few Xi, only, say
n = 3, such that XS//n has three degrees of freedom, there is a lot of

FigURe B.4 Density Function of the t-Distribution for Various Degrees of Freedom
n Compared to the Standard Normal Density Function N(0,1)

−5^0 −4 −3 −2 −1 0 1 2 3 4 5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x

n = 1

n = 5

N(0,1)

f(x

)