# econometrics problem set 3

1. 31 points You are interested in the extent to which removing the option of paid childcare has affected the share of time mothers compared to fathers spend with their children. The economic and health response to the pandemic has caused job losses, increased telecommuting, closed day care and schools, and placed state restrictions on in-home paid childcare – all of which may have affected the bargaining dynamic between parents in deciding how much time to spend caring for their children. The American Time Use Survey provides nationally representative estimates of how and with whom Americans spend their time, including hours spent on paid work and childcare. It is an annual cross-sectional survey asked during weeks when school is typically in session that is linked to the Current Population Survey (demographic questions were asked several months before the Time Use questions), thus you can observe demographic information and family relationships.

You have data from 2015-2019 and will have data in 2020 next year. You are crafting your econometric specification. Prior to 2020, all states allowed paid in-home care and had schools open. As of the time of the survey in 2020, all states had closed schools (shutdown of physical buildings and in-person instruction) but there was state variation in allowing paid in-home care for children less than 6 years old. That is, some, but not all, states deemed in-home childcare essential when stay-at-home orders went into effect in 2020 such that the availability of paid in-home care depended on the state you lived in. You want to learn the effect of disallowing paid childcare and closing schools on spousal allocation of childcare hours as measured by y, the ratio of hours mother spent on childcare to hours father spent on childcare. For example, y = 1 when mothers and fathers spent the same amount of time on childcare and y = 1.5 when mothers spent 50% more than fathers.

You restrict your sample to opposite sex, married couples with children where both parents worked in the prior year and both parents report positive hours of childcare. You can observe the following variables for each couple:

yit =hours mother spent on childcare divided by hours father spent in couple i in year t

x1it =1 if wife in couple i in year t is currently working for pay, 0 otherwise

x2it =wifeâ€™s paid work hours last week reported in year t for couple i

x3it =1 if husband in couple i in year t is currently working for pay, 0 otherwise

x4it =husbandâ€™s paid work hours last week reported in year t for couple i

x5it =1 if couple i has a child under 6 years old in year t, 0 otherwise

x6it =1 if couple iâ€™s youngest child is 6-17 years old in year t, 0 otherwise

x7i =1 if wife in couple i has years of education = husbandâ€™s, 0 otherwise

x8i =1 if wife in couple i has years of education > husbandâ€™s, 0 otherwise

x9i =age of husband – age of wife

x10i =1 if wifeâ€™s occupation is teaching, 0 otherwise

x11i =1 if husbandâ€™s occupation is teaching, 0 otherwise

pandemict=1 if year is after pandemic hit (2020 or later), 0 otherwise

notavailit =1 if paid in-home childcare was NOT available for couple i in year t, 0 otherwise

Letâ€™s simply refer to the effect of removing the option of paid childcare on the ratio of motherâ€™s to fatherâ€™s hours of childcare as the treatment effect of interest: TE.

(a) Since some but not all states allow paid in-home childcare in 2020 at the time of the survey, you could estimate TE with Ë†Î±3 by running ordinary least squares using only 2020 data with the following specification:

yË† = Ë†Î±0 + Ë†Î±1x5i + Ë†Î±2x6i + Ë†Î±3(notavaili âˆ— x5i) + Ë†Î±4x7i + Ë†Î±5x8i + Ë†Î±6x9i

i. 4 points Interpret Ë†Î±3.

ii. 4 points Interpret Ë†Î±5.

iii. 5 points Do you think all of the necessary assumptions hold for Ë†Î±3 to be an unbiased estimator for TE? Explain your reasoning.

(b) 4 points Instead, you assume that E[y|X] = Î²0 + Î²1x5it + Î²2x6it + Î²3(notavailit âˆ— x5it) + Î²4(pandemict âˆ— x6it) Interpret Î²3. 2

(c) 6 points If you ran an ols regression based on the specification above, what type of estimator is Î²Ë† 3? And what assumptions are necessary for Î²Ë† 3 to be unbiased?

d) 4 points Compare the advantages and disadvantages of the two potential ordinary least squares estimators for TE: Î²Ë† 3 from the specification above and Ë†Î³3 from assuming

E[y|X] = Î³0+Î³1x5it+Î³2x6it+Î³3(notavailitâˆ—x5it)+Î³4(pandemictâˆ—x6it)+Î³5x1it+Î³6x2it+Î³7x3it+Î³8x4it

(e) 4 points The treatment effect of removing the option of paid childcare may be different for families where one of the parents is themselves a teacher. How would you suggest modifying your specification and why?

2. 20 points To complete the Master of Science in Applied Economics at Johns Hopkins University, students must complete the core Economerics course (440.606) and one of the four advanced econometrics courses, among other requirements. Students receiving a B- or better on their final course grade receive credit for the course, while students with a C+ or below do not receive credit. Instructors of the core Econometrics course calculate a final number score for the course with cutoff values for assigning the letter grades. Failing to receive credit for a course that you attended and paid tuition for can be discouraging possibly affecting your enthusiasm for the subject matter. However, sucessfully retaking a course you struggled in may boost your confidence and enthusiasm. Suppose that you were interested in estimating the Treatment Effect of receiving a passing grade on the number of advanced econometric courses taken.

Let x0 be the cutoff value for receiving a B- in a core Econometrics course. You observe for each student:

yi =number of advanced econometrics courses student i enrolled in

x1i=core Econometrics course score for student i

x2i=fraction of students failed by student iâ€™s core Econometrics instructor

x3i=student iâ€™s mean GPA (excluding econometrics courses)

x4i=1 if student i received an A in their Statistics class, 0 otherwise

x5i=1 if student i took Mathematical Methods for Economists, 0 otherwise

(a) 4 points To estimate the average treatment effect of receiving a passing grade on number of advanced econometrics courses taken, would you use a sharp or fuzzy Regression Discontinuity design? Explain.

(b) 4 points Write down your model specification to estimate the average treatment effect.

(c) 6 points Describe two graphical tests that would be important to perform and explain why.

(d) 6 points How would you design your falsification test? Explain why you chose it.

3. 24 points Suppose that your goal was to estimate the effect of education on weekly hours worked for individuals approaching retirement. Assume that education is exogenous. The data set includes men over the age of 50 but less than 60 years old.

(a) Suppose the data set includes a categorical variable that equals 1 if the man works 0 hours (that is, does not work), 2 if he works more than 0 but less than 35 hours per week, and 3 if he works more than 35 hours per week. There are no missing values.

i. 4 points Which is the appropriate econometric model?

ii. 4 points Explain why you chose this model.

iii. 4 points Write the log likelihood function implied by your model choice.

(b) Instead, suppose that you had the actual number of hours worked per week (e.g. 0,1,2,. . .) and you observe that a substantial share of men donâ€™t work at all (have 0 hours of work).

i. 4 points Which is the appropriate econometric model?

ii. 4 points Explain why you chose this model.

iii. 4 points Write the log likelihood function implied by your model choice.