sociology 362
mock exam 4
1. Let Yibe a Bernoulli (binary) variable such that Yi= 1 if a student drops out of high school and
Yi= 0 if the student completes high school. Let P(Yi= 1) = πiand P(Yi= 0) = (1 −πi). Let the
regression model fit to individual-level data be:
Yi=α+βxXi+βzZi+ui(1)
where αis the intercept, βxand βzare slope parameters, uis a disturbance with E(u) = 0, X is years of
schooling of the students father and Z is a dummy variable coded 1 if student worked while in school,
zero otherwise.
a. Are the least-squares estimators of the slope parameters βxand βzunbiased? Explain.
b. Are the least-squares estimators of βxand βzefficient? Explain.
c. What makes the linear model an unsuitable response function for describing the relation between a
binary 0/1 dependent variable and one or more quantitative regressors?
2. Suppose the model of question 1 is estimated with data and yields the following fitted function:
ˆπi=.15 −.012X+.16Z(2)
a. Interpret the coefficients of X and Z.
b. What is the expected probability that a student who works and whose father has 16 years of schooling
will drop out of high school?
c. What is the expected variance of Yi(=1 if drop out; 0 otherwise) among students who work and whose
father has 10 years of schooling?
3. Consider the following grouped data on age (A) and diabetes for six groups of patients:
Ajnjrj
10 80 4
14 150 8
28 140 12
46 30 5
where njis the size of the jth patient group and rjis the number diagnosed as having diabetes.
a. Suppose we wished to fit by least-squares the linear probability model to these data. Give the value of
the dependent variable for the oldest age group.
b. Suppose we wished to fit by least-squares the logit model to these data. For the 28 year-old group,
give the value of the dependent variable.
4. Fitting the logit model by maximum likelihood to the grouped data above (problem 4) yields an
estimate of ˆ
β=.08 (SE( ˆ
β) = .034) for the age coefficient and -5.2 for the intercept.
a. According to the model, what is the difference in the logit for diabetes of someone age 30 compared to
someone age 38.
1
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