ECON 21020– Summer 2023 Problem Set #4
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ECON 21020– Summer 2023
Problem Set #4 - due on 07/09 at 11:59pm Central Time
Instructions: submit your answers in pdf through Gradescope before the deadline. I encourage you to work with other students, but submit your own answers. Also include the names of people that worked with you in your file. Exercises marked with a * do not need to be uploaded with your answers, but they are strongly recommended as well.
1. Stock and Watson, Exercises 6.6, 6.9, *6.11, 6.12, *7.4, *7.8.(a)-(b)
2. Suppose
Y = β0 + β1 X + U ,
where Y is a binary random variable. Suppose further that E[U|X] = 0 and 0 < Var[X] < ∞ .
(a) What is E[Y |X]? What is P{Y = 1|X}?
(b) What is Var[Y |X]?
(c) What is Var[U|X]? Is the model homoskedastic or heteroskedastic?
(d) Let (Y1 ,X1 ), . . . , (Yn ,Xn ) be a i.i.d. sample from (Y,X). In addition to the assumptions above, suppose that E[X4] < ∞ . Assume that the sample size, n, is large.
i. How would you test the null hypothesis that β 1 = 0 versus the alternative that β 1 
0 at the 5% significance level?
ii. How would you compute the p-value for the test in part (i)?
iii. How would you construct a (two-sided) confidence interval for β 1 at the 5% significance level?
3. Consider the following regression model in the population
Yi = X
β + Ui
where Yi is a scalar (1x1 vector) that represents the outcome for observation i, Xi = (1,X1i,X2i , ...,Xki )\ is a (k+1)x1 vector that contains the k regressors plus one intercept (represented by the value 1), and Ui is a scalar that represents the error term. Moreover β = (β0 ,β 1 , ...,βk )\ is a (k+1)x1 vector that contains all the population parameters in the BLP(Y |X).
(a) Show that the equation above is equivalent to
Yi = β0 + β1 X1i + ... + βk Xki + Ui
(b) Multiply the original model by the vector Xi , such that
Xi Yi = Xi X
β + Xi Ui
Perform the multiplication of the matrices to show each element that is inside the vectors: Xi Yi , Xi X
and Xi Ui .
(c) Now assume E[Xi Ui] = 0. Notice this is a set of k+1 equations. Show that it implies that E[Ui] = 0 and cov(Xji ,Ui ) = 0, when j = 1, ...,k .
(d) Show that if E[Xi Ui] = 0, then this system of equations have the solution β = E[Xi X ]
−1E[Xi Yi]. Which additional assumption have you used in this question?
4. Consider the following model of the determinants of wages:
log wage
= β0 + β1 educ + β2pareduc × educ + β3 experience + β4 experience2 + U ,
where
experience = years of individual work experience
pareduc = sum of the mother’s education and father’s years of education
educ = years of own education .
(a) What is the percent return on wages from another year of own education? Does it depend on the level of own education? Does it depend on parents’ education? Does it depend on the level of work experience?
(b) What is the percent return on wages from another year of work experience? Does it depend on the level of own education? Does it depend on parents’ education? Does it depend on the level of work experience?
(c) Using 722 observations, the following OLS estimates were obtained: βˆ0 = 5.65, βˆ1 = 0.047, βˆ2 = 0.00078, βˆ3 = 0.22, and βˆ4 = −0.005.
i. Compute the estimated return to an additional year of own education if both parents have high school education (pareduc = 24), and if both parents have college degrees (pareduc = 32). Use these results to interpret the estimated interaction between own education and parents’ education.
ii. Compute the estimated return to an additional year of experience if the individual has zero years of experience ( experience = 0), and if the individual has 20 years of experience (experience = 20). Use these results to interpret βˆ4 .
iii. How would the OLS estimates of β3 and β4 change if experience was measured in months instead of being measured in years?
5. Levitt and Venkatesh investigate Chicago street gangs by modelling the determinants of the wages paid to “foot soldiers” in the gangs. They estimated the following regression,
wage = β0 + β1 war + β2 large + U ,
where
wage = the hourly wage paid to the foot soldiers
war =
1 if the gang is currently involved in a gang war
0 otherwise
1 if the gang is “large” ,
0 otherwise
and found that
βˆ0 = 1.83, βˆ1 = 1.3, βˆ2 = 4.07
(a) According to their estimates, what is the average wage paid to a foot soldier in a small gang that is not at war?
(b) Why did they include a dummy variable for the gang being large but not also for the gang being small?
(c) How would you modify their regression to allow for the effects of a gang war on wages to be different for members of large versus small gangs? What is the reference group in your suggested model?
(d) Explain the interpretation of each of the coefficients in your model.
6. Prove the missing steps of the FWL theorem from our lecture:
(a) cov(X
,Xl ) = 0, Al 
j
(b) cov(X
,Xj ) = Var(X
)
(c) cov(X
,u) = 0
where X
= Xj − BLP(Xj |X−j ) and u = Y − BLP(Y |X)
7. * Show why the adjusted R2 is always smaller or equal to the R2 .
8. * Download the dataset for this problem from the course webpage. The dataset has 4 variables in the following order: college (an indicator for completing college), hs (an indicator for completing high school, but not college), wage (average hourly wage) and fem (an indicator for female).
Consider the following model of the determinants of income:
wage = β0 + β1fem + β2 college + β3 hs + U .
(a) Show that the regressors are not perfect colinear in this dataset. (Hint: regress one regressor on the other regressors).
(b) Assume the model is causal. What is the interpretation of U? Is it uncorrelated with the regres- sors? Interpret each of the coefficients in the model.
(c) Assume the model above is the BLP(wages|fem,college,hs). What is the interpretation of U?
Is it uncorrelated with the regressors? Interpret each of the coefficients in the model. For the rest of this question, assume you are estimating the BLP model.
(d) Estimate a model that allows the partial correlation (coefficients of the BLP) between college and wages to depend on the gender, and the partial correlation between high-school completion and wages to depend on the gender. Report your table of estimates.
(e) Test at 5% significance level whether the partial corrrelation between college and wages depend on the gender. Explain your results.
(f) Test at 5% significance level whether the partial correlation between high-school or college and wages depend on the gender. Explain your results.
2023-07-11