Chapter 12 - Limited Dependent Variables
CHAPTER 12
Answers to End of Chapter Problems
12.1
OLS is not the most appropriate estimator because it the slope is constant across all
values of the independent variable. A more preferred estimator (the one is blue), has low
values for the slope at low values of the independent variable, the slope is large for
intermediate values of the independent variable, and the slope is low again for high
values of the independent variables. Four other reasons the OLS is not preferred are
(1) The OLS estimates are heteroskedastic.
(2) OLS estimates probabilities below 0.
(3) OLS estimates probabilities above 1.
(4) The R-squared is not an appropriate measure of goodness of fit.
12.2
The multinomial logit assumes independence of irrelevant alternatives while the
ordered probit is sometimes difficult to estimate but is the preferred alternative if
the iia assumption fails and the categories are ordered. The logit and probit can’t
be used because they only allow for two alternatives to the dependent variable
while the multinomial logit and ordered probit allow for more than two options
for the dependent variable.
12.3
a. For whites, age (negatively related), age squared (positively related), education
(negatively related), farm status (negatively related), south (positively related),
expected family earnings (positively related), and family composition (positively
related) are all statistically significant. Alternatively, for non-whites, age is not
statistically significant, age squared is not statistically significant, education
(negatively related and significant), farm status not statistically significant, south
(negatively related and significant), expected family earnings (positively related
and significant), and family composition (positively related and significant). It
seems that these coefficients have the expected signs but it is interesting that all
variables are statistically significant for whites while only a few variables are
12-1
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Chapter 12 - Limited Dependent Variables
statistically significant for non-whites and some of the signs are different. Age is
entered in level and in squares to allow for a quadratic effect.
b. To obtain the predicted probabilities the function is
For whites
For non-whites
The difference in predicted probabilities is 3.8%. The reason that whites and nonwhites were estimated separately is to allow for differences in coefficients.
c.
This is quite different from the predicted probability of 0.8563 (almost a 10%
difference) that we found above.
d. In STATA, the marginal effects estimated at the means is obtained by typing
the command mfx after estimating either the logit or the probit model. The reason
the marginal effects are preferred is that they are what economists are looking for,
i.e. if the independent variable increases one unit from the mean then the
dependent variable changes by the marginal effect holding all other independent
variables constant. For the logit model, the coefficient estimates are the change in
the log odds of the dependent variable being 1 when the independent variable
increases by 1 unit, holding other independent variables constant.
e. Probit and Logit are preferred over OLS (the linear probability model) because
it the LPM constrains the slope to be constant across all values of the independent
variable. A more preferred estimator (the Probit or Logit), has low values for the
slope at low values of the independent variable, the slope is large for intermediate
values of the independent variable, and the slope is low again for high values of
the independent variables.
12.4
a. OLS still provides unbiased estimates, especially for the mean values of the
data. The model will be heteroskedastic. The drawback of using OLS is not the
most appropriate estimator because the slope is constant across all values of the
independent variable. A more preferred estimator (the one is blue), has low
values for the slope at low values of the independent variable, the slope is large
for intermediate values of the independent variable, and the slope is low again for
high values of the independent variables.
12-2
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Chapter 12 - Limited Dependent Variables
b. If the logit or probit model are used then the coefficient estimates do not accurately
reflect the marginal effects but the marginal effects can be easily calculated (especially in
an advanced statistical program). The logit and probit models will never predict values
below 0 or above 1 and the marginal effects depend on the value of income.
Answers to End of Chapter Exercises
E12.1. a.
From this model we see that those who are married, divorced, hsonly, and live in
south central have a higher probability of hunting. Those who have a ba, some
post graduate degree, post graduate degree, black, other have a lower probability
of hunting. This is not the preferred estimator because he slope is constant across
all values of the independent variable. A more preferred estimator (the one is
blue), has low values for the slope at low values of the independent variable, the
slope is large for intermediate values of the independent variable, and the slope is
low again for high values of the independent variables.
12-3
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Chapter 12 - Limited Dependent Variables
b.
12-4
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Chapter 12 - Limited Dependent Variables
The results are relatively similar to the results that we found with OLS. All the same
variables are statistically significant and they have the same sign except hs only, which is
only marginally statistically significant in this model. It is important to remember the
these marginal effects were calculated at the mean values, which is where OLS and probit
estimates tend to be the most similar. If the marginal effects were calculated at other
points the results would be very different.
c.
12-5
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Chapter 12 - Limited Dependent Variables
12-6
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Chapter 12 - Limited Dependent Variables
Similar to what we found in part c, the results are relatively similar to the results that we
found with OLS and the probit. All the same variables are statistically significant and
they have the same sign (although the for hs only are not the same between probit and
logit). It is important to remember the these marginal effects were calculated at the mean
values, which is where OLS and logit estimates tend to be the most similar.
d. Because the results between probit and logit are almost identical, you are likely
indifferent between these two specifications. The OLS results were also very similar. In
general, if the marginal effects were calculated at other points the results would be very
different. However, in this specification, all the independent variables are binary as well,
which means that differences between the three models are not really an issue because the
marginal effects go from 0 to 1 for the variables.
E12.2 a.
It looks like there may be a small time trend but there is a strong seasonal component.
12-7
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Chapter 12 - Limited Dependent Variables
b.
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Chapter 12 - Limited Dependent Variables
c.
12-9
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Chapter 12 - Limited Dependent Variables
d.
E12.3
12-10
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Chapter 12 - Limited Dependent Variables
Marginal Effects
12-11
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Chapter 12 - Limited Dependent Variables
E12.4
12-12
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Chapter 12 - Limited Dependent Variables
12-13
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