Answers to End of Chapter Problems

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 relatedand 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 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 non-whites were estimated separately is to allow for differences in coefficients.

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.

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.

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.

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.

E12.3

Marginal Effects

E12.4

12-1