Question 1- Draw a picture that explains why OLS is not the most appropriate estimator when the dependent variable only takes on two values. List four other reasons the Logit or Probit is preferred to OLS in this circumstance.
Question 2- How are the multinomial logic and the ordered probit different? Why are the logit or probit not appropriate models in this circumstance?
Question 3- A study tried to find the determinants of the increase in the number of households headed by a female. Using 1940 and 1960 historical census data, a logit model was estimated to predict whether a woman is the head of a household (living on her own) or whether she is living within another's household. The limited dependent variable takes on a value of 1 if the female lives on her own and is 0 if she shares housing. The results for 1960 using 6.051 observations on prime-age whites and 1.294 on nonwhites were as shown in the accompanying table:
| Regression |
(1) White |
(2) Nonwhite |
| Regression Model |
Logit |
Logit |
| Constant |
1.459 |
-2.874 |
| (0.685) |
(1.423) |
| Age |
-0.275 |
0.084 |
| (0.037) |
(0.068) |
| Age Squared |
0.00463 |
0.00021 |
| (0.00044) |
(0.00081) |
| Education |
-0.171 |
-0.127 |
| (0.026) |
(0.038) |
| Farm Status |
-0.687 |
-0.498 |
| (0.173) |
(0.346) |
| South |
0.376 |
0.520 |
| (0.098) |
(0.18) |
| Expected Family Earning |
0.0018 |
0.0011 |
| (0.00019) |
(0.00024) |
| Family Composition |
4.123 |
2.751 |
| (0.294) |
(0.345) |
| Pseudo-R2 |
0.266 |
0.189 |
| Percent Correctly Predicted |
82.0 |
83.4 |
where Age is measured in years. Education is years of schooling of the family head, Farm status is a binary variable taking the value of one if the family head lived on a farm, South is a binary variable for living in a certain region of the country, Expected family earnings was generated from a separate OLS regression to predict earnings from a set of regressors, and Family composition refers to the number of family members under the age of 18 divided by the total number in the family.
The mean values for the variables were as shown in the next table.
| Variable |
(1) White Mean |
(2) Nonwhite Mean |
| Age |
46.1 |
42.9 |
| Age squared |
2,263.50 |
1,965.60 |
| Education |
12.6 |
10.4 |
| Farm status |
0.03 |
0.02 |
| South |
0.3 |
0.5 |
| Expected family earnings |
2,336.40 |
1,507.30 |
| Family composition |
0.2 |
0.3 |
a. Interpret the results. Do the coefficients have the expected signs? Why do you think age was entered both in levels and in squares?
b. Calculate the difference in the predicted probability between whites and non-whites and the sample mean values of the explanatory variables. Why do you think the study did not combine the observations and allow for a nonwhite binary variable to enter?
c. What would be the effect on the probability of a nonwhite woman living on her own, if Education and Family composition were changed from their current mean to the mean of whites, while all other variables were left unchanged at the nonwhite mean values?
d. How would you calculate marginal effects in a more advanced statistical pack-age? Why are marginal effects more informative than the coefficient estimates printed in this problem?
e. What are the primary reasons probit or logit models are used instead of the linear probability model?