Model Estimation 281

of logL with respect to the mean μ and variance σ^2 or using commercial
software. We obtain the following estimates:

μ=

σ=

σ=

1.5600

0.4565

(^2) 0.2084
application of Mle to regression Models
We can now discuss how to apply the MLE principle to the estimation of
regression parameters. Consider first the regression equation (13.4). Assum-
ing samples are independent, the likelihood of the regression is the product
of the joint probabilities computed on each observation:
LPyxii xik
i
n
= ()

∏ ,,..., 1
1

(13.15)

Let’s assume that the regressors are deterministic. In this case, regressors
are known (probability equal to 1) and we can write

LPyxii xPik yx x
i

n

ii ik

i

= ()= ()

==

∏ ,,... 1 ,,...,

1

1

11

n

∏^ (13.16)

If we assume that all variables are normally distributed we can write

Py()tixx 10 , ..., ik ≈+Nx()ββ 11 ++... βσkkx,^2

We can write this expression explicitly as

Pyxx

yxx

ii ik

ikk

1

(^1011)
2
(), ...,e=−xp

( −− −−

σπ

 ββ  β ))











2

2 σ^2

and therefore:

LPyx x

yx

ii ik

i

n

i

= ()

=−

−− −

=

∏ 1

1

(^1011)
2

, ...,

exp

σπ

 ()ββ −











=

∏

β

σ

kk

i

n x^2

2

1 2

(13.17)

and

loglLn= og nylog i x









− ()−−−+

1

2

1

π^22011

σ

σ

()ββ −−

=

∑ βkk

i

n

x

2

1