Practice Final Exam
ECON 1150 Applied Econometrics
Fall Semester 2024
1. Information on new restaurants in Pittsburgh was collected. Specifically, the researcher checked whether the restaurants offered vegan items on the menu (V = 1) or not (V = 0) and whether they stayed in business for at least a year (B = 1) or not (B = 0).
(a) Using the definition of conditional probability, show that for any random variables Y, X,
Pr(Y = y, X = x) = P r(Y = y|X = x) × P r(X = x).
(b) Being sure that you know how conditional probabilities work and trying to test your patience, the re-searcher gave you only the following information:
❼ P r(V = 0) = 0.4
❼ P r(B = 0) = 0.5
❼ P r(V = 0|B = 0) = 0.6
❼ P r(B = 1|V = 0) = 0.25
Using this information, fill in the joint and marginal probabilities in the following table:
2. We collected enrollment data from elementary schools located in two counties in California. In Los Angeles County, we have a sample of 79 schools with a sample mean enrollment of 658.62 students and a sample standard deviation of 204.42 students. In San Diego County, we have a sample of 75 schools with a sample mean enrollment of 583.28 students and a sample standard deviation of 177.65 students.
(a) Test the null hypothesis that the population mean enrollment for Los Angeles County is equal to 600 at the 5% significance level.
(b) Test the null hypothesis that there is no difference in the population mean enrollment between Los Angeles County and San Diego County.
3. Suppose that the true population regression is Y = β0+β1D1+β2D2+u, where D1 and D2 are binary variables. However, in estimating our sample regression, we left out D2 by mistake and estimated Yˆ = βˆ
0+βˆ
1D1. Assume E(u|D1, D2) = 0. (Note: This means that cov(D1, u) = 0. You don’t have to prove this.)
(a) Show that βˆ
1 → β1 + β2 var(D1)/cov(D1,D2) as n → ∞.
(b) Show that var(D1) = Pr(D1 = 1) × (1 − Pr(D1 = 1)).
(c) Show that cov(D1, D2) = P r(D1 = 1, D2 = 1) − P r(D1 = 1) × P r(D2 = 1).
(d) If D1 and D2 are independent and β2 > 0, will βˆ
1 be biased? If so, in which direction?
4. Infant birth weight is an important indicator of their health. Suppose you’re interested in estimating the effect of smoking, marital status and education on infant birth weight using the following population regression:
birthweight = β0 + βe educ + βu unmarried + βs smoker + u,
where birthweight is infant birth weight (in grams), educ is years of education, unmarried = 1 if the individual is unmarried and = 0 otherwise, and smoker = 1 if the individual is a smoker and = 0 otherwise.
Using a sample of 3000 mothers, we estimate the following regression:
(a) What is the marginal change in birthweight associated with smoking? Is this effect significant at the 5% level?
(b) Suppose you ran the transformed regression:
where Z = smoker + unmarried.
What is the null hypothesis you’re trying to test? Can you reject this null hypothesis at the 5% level?
5. We’re interested in studying the determinants of a country’s level of democracy and collected data from 92 countries in 1985. We regressed their index of democracy (dem ind), which is a measure between 0 and 1 with larger values indicating higher levels of democracy, on the natural logarithm of their population (log(pop)), the natural logarithm of their GDP per capita (log(gdppc)) and average years of education (educ). We obtain the following sample regression estimate:
(a) Which of the coefficients on regressors are significantly different from zero at the 5% level?
(b) Formally test the hypothesis that the coefficients on educ and log(pop) are both equal to zero at the 5% level. You will need to use information from the following regression estimate:
(c) Interpret the coefficient on log(gdppc).
6. Suppose you’re interested in finding out how one’s demographic characteristics affect the probability that one is employed. You obtain data on 5,220 workers from the 2006 Current Population Survey and ran the following linear probability model regression:
where employed = 1 if the individual is employed and = 0 otherwise, female = 1 if the individual is female and = 0 otherwise, married = 1 if the individual is married and = 0 otherwise, and age is the individual’s age. You also included the squared term for age, age2
.
(a) From the linear probability model results, what is the probability of a married male aged 25 being em-ployed?
(b) Notice that the coefficient on age2
is negative. What does this say about the marginal effect of age? From what age will the marginal effect of age on the probability of being employed be negative?
(c) Using the same variables and data, you ran the logit model:
What is the probability of a married male aged 25 being employed? For such an individual, what is the marginal change in the probability of employed associated with a one year increase in age?