Anon

282 The Basics of financial economeTrics

The first term is a constant, the second and the third terms are negative;
therefore, maximizing the log-likelihood is equivalent to finding the solution
of the following minimization problem:

βσ σ
σ

(),a= rgminlogny()+−()ikββ−−xx−β k

1

2 2 011



22

i 1

n

=

∑











^ (13.18)

The analytic solution to this problem requires equating to zero the par-
tial derivatives with respect to the arguments. Computing the derivatives of
equation (13.15) yields the following expressions:

∂−()−−−













∂

=

=

=

∑yxx

n

y

ikk

i

n

j

ββ β

β

σ

011

2

1

2

0

1



iikk

i

n

()−−xx−−

=

∑ ββ 011 β

2

1



The first condition is exactly the same condition as that obtained with the
OLS method. The second expression states that the variance σ^2 is estimated
by the empirical variance of the sample residuals. We find the important
result that, if variables are normally distributed, OLS and MLE estimation
methods yield exactly the same estimators.

application of Mle to Factor Models

The same reasoning applies to factor models. Consider the factor model
equation (12.3) from Chapter 12:

yatt=+Bf+=εt, tT, ...,1

The variables y are the only observable terms. If we assume that the vari-
ables yt are normally distributed, we can write

Py()t =Na(),Σ

where a is the vector of averages and Σ is the covariance matrix of the yt.
Because the logarithm is a monotone function, maximizing the likelihood is
equivalent to maximizing the log-likelihood

loglLPog yt

t

T

()= ()()

=

∑

1