Anon

280 The Basics of financial economeTrics

principle is applied to determine the parameters of a distribution. We will
again use our sample in Table 13.1 and assume that sample data y are random
draws from a normal distribution with mean μ and variance σ^2. Therefore,
the normal distribution computed on the sample data will have the form:

Py

y

()i =−i i

 ()−









 =

1

2 2

110

2

2

σπ

μ

σ

exp,..., (13.12)

As the data are assumed to be independent random draws, their likeli-
hood is the product

LPy

y

i

i

i

i

= ()=−

 ()−











==

∏

1

10 2

2

1

(^101)
σπ 2 2
μ
σ
∏∏ exp^ (13.13)
We can simplify the expression for the likelihood by taking its loga-
rithm. Because the logarithm is a monotonically growing function, those
parameters that maximize the likelihood also maximize the logarithm of
the likelihood and vice versa. The logarithm of the likelihood is called the
log-likelihood. Given that the logarithm of a product is the sum of the loga-
rithms, we can write
∏∑
∑
∑∑
∑

()()

()

()

()

==

=

σπ

−

−μ

σ





























=

σπ











−

−μ

σ

=− π− σ−

σ

−μ

==

=

==

=

LPyPy

y

y

y

loglog log

log

1

2

exp

2

log

1

2 2

10log2 10log

1

2

i

i

i

i

i

i

i

i

i

i

i

1

10

1

10

2

2

1

10

1

10 2

2

1

10

2

2

1

10

(13.14)

We can explicitly compute the log-likelihood of the sample data:

loglL og log

..

=− −

−

()− +−()+

10 210

1

2

07 13

2

22

πσ

σ

μμ 111 17 16

14 16

222

2

...

..

()− +−()+−()

+−()+−(

μμμ

μμ)) +−()+−()+−()















(^2217) ..μμ 21 2224. μ 
This is the expression that needs to be maximized in order to determine the
parameters of the normal distribution that fit our sample data y. Maximiza-
tion can be achieved either analytically by equating to zero the derivatives