12.3 A study tried to find the determinants of the increase in the number of hou
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12.3 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 ving on her own) or whether she is living within another's household. The limited dependent variable takes on a value of l if the female lives on her own and is 0 if she shares housing. The results for 1960 using 6.05 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 Constant 1.459 2.874 (0.685) 1.423 Age 0.275 0.084 (0.068) (0.037) Age squared 0.00463 0.0002 (0,00044) (0.0008 Education (0,026) 0.038) Farm Status 0.687 0.498 (0.1730 0.346) South 0.376 0.520 (0,098) 0.180) Expected Family Earnings 0,0018 0.001 (0.000190 (0.00024) Family Composition 4.123 2.75 (0.294) (0.345) Pseudo-R2 0.266 Percent Correctly Predicted 82.0 83.4 where Age is measured in years. Educarion is years of schooling of the family head. Fann status is a binary variable taking the value of one if the family head lived on a farm, South sa binary variable for living in a region of the country, 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 lable. Variable (1) White Mean (2) Nonwhite Mean Age 46.1 42.9 Age squared 2,263.5 1,965.6 Education 2.6 04 Farm status 0,03 0.02 South 0.3 0.5 Expected family earnings 2,336.4 507.3 Family composition 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?Explanation / Answer
a. In a logistic regression, the odds of the probability of the dependent variable will increase with positive coefficient.
The coefficient of Age is negative for White woman, which means that the probability of head of household decrease with Age, which is not so logical. A senior member has large probability to be head of the household.
The coefficient of Age is positive for Non-whites woman, which means that the probability of head of household is increases with Age, which is also logical.
The coefficient of Age^2 is positive for both Whites and Non-whites woman, which means that the probability of head of household increases with Age, which is also logical.
The coefficient of Education is negative for both Whites and Non-whites woman, which means that the probability of head of household is decreases with Education. With increased education, the women have high probability of doing the jobs and the other partner will take care of the household.
The coefficient of Farm Status is highly negative for both Whites and Non-whites woman, which means that the probability of head of household is low for families in farm areas. In farm areas, usually mean are the head of the household.
The coefficient of South is positive for White woman, which means that the probability of head of household is high for whites family living in South region.
The coefficient of South is negatve for Non-White woman, which means that the probability of head of household is low for non-whites family living in South region.
The coefficient of Expected earnings is around 0 for both Whites and Non-whites woman, which means that the probability of head of household is almost constant (increase by small amount) with Expected earnings, which is also logical.
The coefficient of Family Composition is positive for both Whites and Non-whites woman, which means that the probability of head of household is high for large families, which is also logical.
Age was entered both in levels and in squares as it gives better fit in the logit model. In other words, the odds of the probability of the dependent variable is better fitted by the square of the Age rather than the Age.
See when Age-Squared is used, it correctly predicts that the probability of head of household is increases with Age for Whites.
b. Explanatory variables of Whites mean
Age - 46.1, Age Squared - 2263.5, Education - 12.6, Farm Status - 0.03, South - 0.3, Expected Family earnings - 2336.4, Family Composition - 0.2
Logit(p) = 1.459 - 0.275Age + 0.00463AgeSquared - 0.171Education - 0.687FarmStatus + 0.376South + 0.0018FamilyEarnings + 4.123FamilyComposition
Logit(p) = 1.459 - 0.275*46.1 + 0.00463*2263.5 - 0.171*12.6 - 0.687*0.03 + 0.376*0.3 + 0.0018*2336.4 + 4.123*0.2
Logit(p) = 2.229215
Explanatory variables of Non-Whites mean
Age - 42.9, Age Squared - 1965.6, Education - 10.4, Farm Status - 0.02, South - 0.5, Expected Family earnings - 1507.3, Family Composition - 0.3
Logit(p) = -2.874 + 0.0084Age + 0.00021AgeSquared - 0.127Education - 0.498FarmStatus - 0.520South + 0.0011FamilyEarnings + 2.751FamilyComposition
Logit(p) = -2.874 + 0.0084*42.9 + 0.00021*1965.6 - 0.127*10.4 - 0.498*0.02 - 0.520*0.5 + 0.0011*1507.3 + 2.751*0.3
Logit(p) = -1.208294
Difference in predicted probability = exp(2.229215)/(1+exp(2.229215)) - exp(-1.208294)/(1+exp(-1.208294)) = 0.6728
The study did not combine the observations because, the odds of the probability of head of household is higher for whites than non-whites women.
c. Explanatory variables of Non-Whites mean where the Education and Family composition changed.
Age - 42.9, Age Squared - 1965.6, Education - 12.6, Farm Status - 0.02, South - 0.5, Expected Family earnings - 1507.3, Family Composition - 0.2
Logit(p) = -2.874 + 0.0084Age + 0.00021AgeSquared - 0.127Education - 0.498FarmStatus - 0.520South + 0.0011FamilyEarnings + 2.751FamilyComposition
Logit(p) = -2.874 + 0.0084*42.9 + 0.00021*1965.6 - 0.127*12.6 - 0.498*0.02 - 0.520*0.5 + 0.0011*1507.3 + 2.751*0.2
Logit(p) = -1.762794
On changing the mean values of Education and Family composition, the odds of the probability of head of household for non-whites decreased.
d. Marginal effects are computed differently for discrete (i.e. categorical) and continuous variables. With binary independent variables, marginal effects measure discrete change, i.e. how do predicted probabilities change as the binary independent variable changes from 0 to 1. Marginal effects for continuous variables measure the instantaneous rate of change and often provide a good approximation to the amount of change in Y that will be produced by a 1-unit change in Xk.
Marginal effects can be an informative means for summarizing how change in a response is related to change in a covariate. For categorical variables, the effects of discrete changes are computed, i.e., the marginal effects for categorical variables show how P(Y = 1) is predicted to change as Xk changes from 0 to 1 holding all other Xs equal. This can be quite useful, informative, and easy to understand.
For continuous independent variables, the marginal effect measures the instantaneous rate of change. If the instantaneous rate of change is similar to the change in P(Y=1) as Xk increases by one, this too can be quite useful and intuitive. However, there is no guarantee that this will be the case; it will depend, in part, on how Xk is scaled.
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