USC, Econ 513, Fall 2005
Lecture 11: Maximum Likelihood Estimation: Testing
After estimating the exponential model for the unemployment durations, Lancaster con
siders an extension. Consider the
hazard function
or
escape rate
h
(
y

x,θ
) = lim
Δ
→
0
Pr
(
y
≤
y < y
+ Δ

y
≤
y,X
)
/
Δ =
f
(
y

x,θ
)
1

F
(
y

x,θ
)
.
The hazard function is just another way of characterizing a distribution, like the density
function, the distribution function, the survivor function, the moment generating function
or the characteristic function. It is just a particularly convenient and interpretable way of
describing a distribution or durations. Given the hazard you can calculate the distribution
function as
F
(
y

x,θ
) = 1

exp
±

Z
y
0
h
(
s

x,θ
)
ds
²
,
and hence the density function. The exponential model implies that the hazard function
stays constant over the duration of the spell, equal to exp(
x
0
β
) in our previous speciﬁcation.
To see what this means, take a person and look at their chances of ﬁnding a job on the ﬁrst
day of being unemployed. These chances are the same as the chances that this same person
would ﬁnd a job on the ﬁfthieth day given that he has been unsuccessful in ﬁnding work in
the ﬁrst forty–nine days. This may be reasonable, but it might also be something you do
not wish to impose from the outset. Lancaster therefore considers an extension allowing the
hazard function to either increase, stay constant, or decrease over time. This extension is
known as the
Weibull distribution
:
h
(
y

x,β,α
) = (
α
+ 1)
·
y
α
exp(
x
0
β
)
.
Note that this reduces to the exponential distribution if
α
= 0. The implied density function
for the Weibull distribution is
f
(
y

x,β,α
) = (
α
+ 1)
·
y
α
exp(
x
0
β
) exp
±

y
α
+1
exp(
x
0
β
)
²
.
The moments of this distribution are
E
[
y
k

X
] = exp
±

k
·
x
0
β
.
(
α
+ 1)
²
·
Γ
³
k
+ 1
α
+ 1
´
.
1
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View Full Document(Note that for the case with
α
= 0 this reduces to the exponential case with
E
[
y
k

X
] = exp(

k
·
x
0
β
)
·
Γ(1 +
k
)
,
and thus with
k
= 1 the mean of the exponential distribution is
E
[
y

X
] = exp(

x
0
β
).)
The log likelihood function for this model is
L
(
α,β
) =
N
X
i
=1
±
ln(
α
+ 1) +
α
ln
y
i
+
x
0
i
β

y
(
i
α
+ 1)
·
exp(
x
0
i
β
)
²
.
One can estimate this model using any of the numerical methods described before (Newton
Raphson, DavidonFletcherPowell). The one (minor) complication is that numerical algo
rithms have to take account of the restriction that
α >

1; with
α
=

1 the density is
degenerate and all probability mass piles up at
y
= 0.
Table 1 presents the maximum likelihood results for both the exponential and Weibull
models. The standard errors are based on the second derivatives, evaluated at the maximum
likelihood estimates (see discussion below).
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 Fall '07
 Rashidian
 Econometrics, Normal Distribution, Unemployment, Maximum likelihood, Wald, log likelihood function

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