Prob 3: Labor Force Participation
a
)
Y
i
Work
i
=
1 :
10
Educ
i
+
20
Exper
i
+
30
Exper
2
i
+
40
> u
i
:
,
X
i
o
> u
i
0 :
Otherwise
Here,
u
i
is an unobserved threshold for individual i, educ denotes educa
tion, exper denotes experience, and exper2 denotes the square of experience.
where X
i
[
Educ
i
Exper
i
Exper
2
i
1
] : 1
x
4
:
:Assume:
(
Y
i
X
i
u
i
)
is i.i.d. over i.
X
variables are bounded and exogenous
X
has full column rank
o
is in the interior of a compact set
u
i
is i.i.d. N(0,
±
2)
To interpret this equation, suppose that an individual works (Work = 1)
if Ln
(
Wage
i
)
exceed some reservation level, r
i
:
b
)
Work
i
= 1
,
Ln
(
Wages
i
)
> r
i
:
Assume that wages for individual i are given as :
c
)
Ln(
Wage
i
) =
²
1
0
Educ
i
+
²
20
Exper
i
+
²
30
Exper
2
i
+
²
40
+
"
i
:
Assume that all explanatory variables in
(c)
are independent of r
i
and of
"
i
:
With u
i
r
i
±
"
i
;
the model in
(a)
follows from
(bc).
Viewing the work
decision in this manner should suggest a priori signs for the coe¢ cients.
and then turn to an empirical application of the work decision.
1)
With
u
i
=±
having a standard normal density and with
as its (cu
mulative) distribution function, explain why:
Pr (
Y
i
1
j
X
i
X
i
o
=±
)
:
1
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For three individuals, suppose that Y
1
= 1
; Y
2
= 0
; Y
3
= 1
:
Letting
X
1
; X
2
;
and X
3
be the observed vectors of explanatory variables for these
three individuals, explain why the following joint probability statement is
correct:
Pr (
Y
1
= 1
; Y
2
= 0
; Y
3
= 1
j
X
1
; X
2
; X
3
X
1
o
) [1
X
2
o
X
3
o
)
o
±
±
0
=²
With N observations, show that in general the averaged log of the above joint
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 Spring '10
 ChanShen
 Econometrics, Covariance and contravariance of vectors, explanatory variables, Covariance matrix

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