Economics 245A
Bivariate Response Models
Overview
In °rstyear econometrics, you learned results for models in which the depen
dent variable is restricted in some way.
The °rst result is that not all restrictions
require special treatment. For a model with a positive dependent variable, such as
wages or income, the log transform is su¢ cient. The second result concerned mod
els with binary dependent variables, which we concentrate on today.
Recall that
a binary dependent variable captures the presence or absence of a nonnumeric
quantity.
For this reason, binary dependent variable models fall within the class
of
qualitative
(or discrete) dependent variable (response) models. As an example,
in a study of work commute choice the dependent variable is
Y
t
=
°
1
if individual
t
drives to work
0
if individual
t
takes public transit
:
In binary response models, interest is primarily in the response probability
p
(
X
)
°
P
(
Y
= 1
j
X
) =
P
(
Y
= 1
j
X
1
; : : : ; X
K
)
;
for various values of
X
.
For a continuous regressor
X
j
, the partial e/ect of
X
j
on
the response probability is simply
@P
(
Y
=1
j
X
)
@X
j
.
When multiplied by
°
X
j
(for small
°
X
j
), the partial e/ect yields the approximate change in
P
(
Y
= 1
j
X
)
when
X
j
increases by
°
X
j
holding all other regressors constant.
For a discrete regressor
X
K
, the partial e/ect is
P
(
Y
= 1
j
x
1
; : : : ; x
K
°
1
; X
K
= 1)
±
P
(
Y
= 1
j
x
1
; : : : ; x
K
°
1
; X
K
= 0)
:
Binary response models inherit the features of Bernoulli random variables
P
(
Y
= 0
j
X
)
=
1
±
p
(
X
)
E
(
Y
j
X
)
=
p
(
X
)
V
(
Y
j
X
)
=
p
(
X
) [1
±
p
(
X
)]
:
Single Index Models  Linear Probability Model
The most widely used binary response models are those for which the regressors
enter through a single index
p
(
X
t
) =
F
(
X
0
t
°
)
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where the scalar
X
0
t
°
is the single index. The leading case is the linear probability
model in which
F
(
z
) =
z
so
p
(
X
t
) =
X
0
t
°;
which arises from
Y
t
=
X
0
t
°
+
U
t
.
The coe¢ cient
°
k
captures the change in
p
(
X
t
)
for each one unit change in
X
k
holding the other regressors constant.
One can
estimate the model via OLS, but the presence of heteroskedasticity in
V
(
Y
j
X
)
indicates that WLS is e¢ cient.
To construct WLS, one must have
^
p
(
X
t
) =
X
0
t
b
for all
t
.
Because
b
and
X
t
are not restricted,
^
p
may lie outside
[0
;
1]
.
If the
estimated probability lies outside
[0
;
1]
for any point in the sample, then WLS
cannot be used as the estimated variance for the observation
^
p
(1
±
^
p
)
is negative.
Note that if the model is
saturated
, in that all interactions between regressor
values are included, then the °tted probabilities correspond to cell averages (recall
the 140A lecture on functional form) and so lie in
[0
;
1]
.
The test statistic for
H
0
:
°
1
=
² ² ²
=
°
K
= 0
can be accurately constructed from the OLSE, as under
H
0
the error is homoskedastic,
EU
2
t
=
°
0
(1
±
°
0
)
. Regardless of how the model is
estimated, the
R
±
square is not informative.
(Draw a graph with the points
clustered at 0 and 1 on the yaxis and a straight line attempting to °t
them.)
Example (Married Women±s Labor Force Participation)
In a survey of 753 women, 428 report working more than zero hours.
Also, 606
have no young children while 118 have exactly one young child.
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 Fall '08
 Staff
 Econometrics, Logit, Probit, Linear Probability Model, probit model, partial e¤ect

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