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Imbens, Lecture Notes 6, ARE213 Spring ’06
1
ARE213
Econometrics
Spring 2006 UC Berkeley Department of Agricultural and Resource Economics
Maximum Likelihood Estimation I:
Basics and Likelihood Functions (W 13.1, 13.2)
When we looked at the linear regression model,
Y
i
=
X
±
i
β
+
ε
i
,
with
ε
i

X
i
∼N
(0
,σ
2
), we focused on least squares estimation:
ˆ
β
= arg min
β
N
±
i
=1
(
Y
i

X
±
i
β
)
2
,
leading to the estimator
ˆ
β
=
²
N
±
i
=1
X
i
X
±
i
³

1
²
N
±
i
=1
X
i
Y
i
³
.
We can motivate this estimator in a diﬀerent way, namely as a maximum likelihood estimator
,
or mle:
ˆ
β
= arg max
β,σ
2
L
(
β,σ
2
)
,
where
L
(
β,σ
2
)=
N
±
i
=1

1
2
ln(2
πσ
2
)

1
2
σ
2
(
Y
i

X
±
i
β
)
2
.
This leads to the same estimator for
β
, and to
ˆ
σ
2
=
1
N
(
Y
i

X
±
i
ˆ
β
)
2
.
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2
This approach is more general, allowing us to deal with more complex nonlinear models. We
will frst look at the construction oF the likelihood Function itselF in the setting oF a particular
model under various sampling schemes.
In general the likelihood Function is the joint density oF the data viewed as a Function
oF the parameters. Suppose we have independent and identically distributed random vari
ables
Z
i
,...,Z
N
, with common density
f
(
z
;
θ
). Then the likelihood Function given a sample
z
1
,...,z
n
is
L
(
θ
)=
N
±
i
=1
f
(
z
i
,θ
)
.
Its logarithm is reFerred to as the log likelihood Function:
L
(
θ
)=ln
L
(
θ
)=
N
²
i
=1
ln
f
(
z
i
,θ
)
.
Lancaster (1979) is interested in determining “the causes oF variation between unemployed
persons in the length oF time they are out oF work .
... bearing as it does upon the design and
eﬀect oF welFare policy.” He has data on unemployment durations oF 479 unskilled workers,
as well as some oF their individual characteristics such as age, the local unemployment rate
and the replacement ratio, measured as “how much they had coming in From all these sources
(unemployment beneft, supplementary beneft, and Family income supplement) during the
main period oF their unemployment”, divided by the answer to the question “how much did
you earn, aFter deductions, in your last job.” Especially the coeﬃcient on the last variable
is viewed as relevant For social policy.
The economic theory underlying Lancaster’s analysis is job search theory. An unemployed
individual is assumed to receive job oﬀers, arriving according to some rate
λ
(
t
), such that
the expected number oF job oﬀers arriving in a short interval oF length
dt
is
λ
(
t
)
dt
. Each oﬀer
consists oF some wage rate
w
, drawn independently oF previous wages, From some distribution
with distribution Function
F
(
w
). The oﬀer is compared to some reservation wage ¯
w
(
t
), and
iF the oﬀer is better than the reservation wage, that is with probability 1

F
(¯
w
(
t
)), the oﬀer
Imbens, Lecture Notes 6, ARE213 Spring ’06
3
is accepted. The reservation wage is set to maximize utility. Suppose that the arrival rate
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This note was uploaded on 08/01/2008 for the course ARE 213 taught by Professor Imbens during the Spring '06 term at University of California, Berkeley.
 Spring '06
 IMBENS

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