Eco311, Final Exam, Fall 2018. Prof. Bill Even...Eco311, Final Exam, Fall 2018. Prof. Bill Even 9 21. (6 points) Assume that you have an appropriate instrumental variable for milk

Eco311, Final Exam, Fall 2018. Prof. Bill Even

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Your Name (Please print) ________________________________________

Directions

Place all answers in the space provided below or within each question. Round all numerical answers to the nearest 100th (e.g. 1.23) unless told otherwise. Each question is worth 4 points unless otherwise indicated. A separate document with regression tables, statistical tables for the normal, F- and Chi-squared distribution are attached. You must turn in both the exam and the tables when you finish.


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Using data from the 2017 Current Population Survey, I estimated a linear probability model to investigate how the probability of being out of the labor force (i.e. not working and not looking for work) varies with a person’s age, sex, and schooling. The sample is restricted to people between the ages of 55 and 79. The results are in table 1 of the tables document.

In specification (1), a dummy variable for whether a person is out of the labor force is regressed on age dummies for different age ranges and years of schooling. In specification (2), a female dummy is added as a control along with interactions between the female dummy and all of the controls that were originally in specification (1).

1. Using specification (1), what is the probability that a 60 year old with 12 years of schooling is out of the labor force?

2. If specification (1) was re-estimated but with the reference group changed to 75-79 year olds, what would be the new estimate of

a. The coefficient on the constant term?

b. The coefficient on the dummy for the 60-64 year old age group?

3. Using specification (1), what is the difference in the probability of being out of the labor force for a 67 year old as compared to a 62 year old? Be sure to indicate whether the difference is positive or negative.

4. (2 points each) In specification (2), a female dummy variable is interacted with all of the age categories. Based on this specification, what is the difference between the probability of being out of the labor force for

a. a 67 year old and a 62 year old male?

b. a 67 year old and a 62 year old female?

c. a 62 year old male and a 62 year old female?

Be sure to indicate whether the differences in 4a-c are positive or negative.


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5. Suppose that you want to estimate the above linear probability model with weighted least squares. If p is the predicted probability of a yes obtained from the LPM, what is the weight that should be attached to each observation? [Recall that in Stata, weights should be inversely proportional to the variance of the residual].

A regression analysis of the loans received by applicants (loanamt – measured in 1000s of $) is provided in table 2. The control variables include the applicant’s annual income (appinc), dummy variables for whether the applicant was a single male (sgl_male) or single female (sgl_female). Married applicants are the reference group. You should be able to interpret the second regression below based on the Stata commands that follow the first regression.

6. (2 points each) Based on the regressions in table 2, a. the value of the Chi-squared test statistic for the null hypothesis that the residuals in the loan

equation are homoscedastic is ____________

b. the test statistic has a Chi-squared distribution with _________ degrees of freedom.

c. The null hypothesis that the residuals in the loan equation are homoskedastic is rejected at the .01 level if the chi-squared statistic is (greater, less) than _________.

7. (2 points each) If you want to perform a Breusch-Pagan test for heteroskedasticity in the loan regression,

a. what regression command would you execute in Stata (e.g. reg y x – use the relevant variable names from table 2 to define what variables would be used for y and x and define any additional variables that are required).

b. What test statement would you use to generate the relevant F-statistic for the Breusch-Pagan test? Use the appropriate variable names based on the information provided in table 2.

c. Using the table for the F-distribution, what is the critical value for rejecting the null hypothesis of homoskedasticity at the .05 level of significance?


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8. Based on the regressions in table 2, we can conclude that the variance of the residuals in the loan equation

a. Is always greater for those with larger predicted loan amounts b. Is always less for those with larger predicted loan amounts c. Initially rises with predicted loan amounts, but eventually falls d. Initially falls with predicted loan amounts, but eventually rises

9. (2 points each) Indicate whether each of the following statements is true or false (T or F) if the loan equation has heteroskedasticity,

a. The standard OLS estimates of the coefficients are biased _____

b. The standard OLS estimates of the standard errors for the coefficients are incorrect _____

c. The standard OLS estimates of the coefficients are no longer efficient ________

10. (2 points each) Suppose that you use Stata to estimate the loan regression with OLS and then you choose to estimate robust standard errors. Compared to conventional OLS, indicate whether each of the following statements is true or false (T/F). The use of robust standard errors will

a. Would cause all of the estimated coefficients to change _____

b. Will cause all the standard errors to rise ______


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11. Suppose you have data on total employment by county and you estimate the following regression:

𝑒𝑒𝑒𝑒𝑝𝑝𝑖𝑖 = 𝛼𝛼0 + 𝛼𝛼1𝑝𝑝𝑝𝑝𝑝𝑝𝑖𝑖 + 𝛼𝛼2(𝑝𝑝𝑝𝑝𝑝𝑝𝑖𝑖 ∗ 𝑎𝑎𝑎𝑎𝑒𝑒𝑖𝑖) + 𝑒𝑒𝑖𝑖 where the subscript i indexes the county, emp represents total employment in the county, pop represents the total population, and age is the average age of people living in the county. The residuals (e)are likely to be

a. Heteroskedastic and their variance will be greater in counties with larger populations. b. Homoskedastic and their variance will be greater counties with larger populations c. Heteroskedastic and their variance will be greater in counties with smaller populations. d. Homoskedastic and their variance will be greater counties with smaller populations

To answer the next 4 questions, suppose that a group of states pass a law in 2010 requiring “pay for performance” (PFP) where school districts are required to tie their teachers’ pay to their student test scores. This would mean that in any given year, teachers who had students with higher test scores would receive larger pay increases. Suppose you have the following data on test scores. PFP states are those that pass PFP laws in 2010. Non-PFP states never have PFP laws.

Group Mean test score PFP states in 2008 76 PFP states in 2012 78 Non-PFP states in 2008 74 Non-PFP states in 2012 72

12. What is the diff-in-diff estimate of the impact of the PFP laws on test scores? Be sure to indicate whether the effect is positive or negative.

13. Define the following variables: 𝑇𝑇𝑆𝑆𝑖𝑖𝑖𝑖 as the average test scores for district i in period t (t=2008, 2012); 𝑃𝑃𝑃𝑃𝑃𝑃𝑖𝑖 as dummy variable that equals 1 for all states that pass PFP in 2010 and is zero otherwise; PFPit is a dummy variable that equals one in period t for state i if state i has a PFP law in year t, and is zero otherwise 𝑌𝑌2012𝑖𝑖 is a dummy that equals 1 for all states in 2012 and is zero otherwise. Write out a regression equation that allows you to obtain the diff-in-diff estimate of the effect of PFP on test scores. Be sure to indicate which coefficient represents the diff-in-diff estimate of the impact of PFP on test scores.


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14. Suppose that the group of states that passed the PFP laws in 2010 had a large immigration of families that had better than average students, whereas those that did not pass the PFP had no immigration. This immigration pattern would cause:

a. the diff-in-diff estimator of the PFP effect to be biased upward b. the diff-in-diff estimator of the PFP effect to be biased downward c. No bias in the diff-in-diff estimator because it will difference out the effect of such trends.

To answer the next 3 questions, recall that In class, we discussed a study examining examined the effect of the Headstart program on student achievement in grade school. The simple regression model estimated was similar to the one below

𝑇𝑇𝑒𝑒𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑝𝑝𝑇𝑇𝑒𝑒𝑖𝑖𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1𝐻𝐻𝑆𝑆𝑖𝑖𝑖𝑖 + 𝛽𝛽2𝑀𝑀𝑝𝑝𝑒𝑒𝑀𝑀𝑑𝑑𝑖𝑖𝑖𝑖 + 𝛽𝛽3𝑃𝑃𝑎𝑎𝑒𝑒𝐹𝐹𝐹𝐹𝑇𝑇𝑖𝑖𝑖𝑖 + 𝑢𝑢𝑖𝑖𝑖𝑖 where i indexes the child and j indexes the family, testscore is score of academic ability administered several years after the age children could be enrolled in the Headstart program, HS is a dummy variable indicating whethe the student was enrolled in Headstart, MomEd is the number of years of schooling for the child’s mother, and Faminc is the family income for the child.

15. (6 points) Precisely define what it means for HS to be “endogenous” in the testscore regression.

16. (6 points) Which direction is the OLS estimate of 𝛽𝛽1 likely to be biased? WHY? Be sure that you explain how the non-random assignment of students to Headstart could lead to this bias.


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You must answer 4 of the next 6 questions. Write the word SKIPPED in the answer space for the two questions you choose not to answer. If you answer more than 4 questions, I will grade your first 4 answers.

17. (6 points) How did the authors of the Headstart study re-estimate the testscore regression model to remove the bias you describe in the previous question? Write out the regression model and clearly define the variables and subscripts that you use.

18. (6 points) Why couldn’t the authors include MomEd (defined earlier) using the estimation approach that you described in your answer to the prior question?


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To answer the next 3 questions, consider the following demand equation for milk in state i.

𝑄𝑄𝑖𝑖𝐷𝐷 = 𝛽𝛽0 + 𝛽𝛽1𝑃𝑃𝑖𝑖 + 𝛽𝛽2𝐴𝐴𝑎𝑎𝑒𝑒𝑖𝑖 + 𝛽𝛽3𝐼𝐼𝐹𝐹𝑇𝑇𝑖𝑖 + 𝑢𝑢𝑖𝑖

where i indicates the variable is measured for state I, 𝑄𝑄𝑖𝑖𝐷𝐷 is the per capita number of gallons of milk demanded monthly, Pi is the price of a gallon of milk, Agei is the average person’s age in the state, and Inc is average income in the state. 19. (6 points) Given that prices are determined by both supply and demand, explain why a simple OLS

regression would likely result in a biased estimate of 𝛽𝛽1. Be sure to indicate whether the bias is positive or negative and how you were able to determine the direction of the bias.

20. (6 points) If you wish to estimate the demand equation using an instrumental variables approach, would the price of cattle feed (PCF) be an appropriate instrument? (Note: milk comes from dairy cattle that must be fed.) In your answer, describe the two necessary conditions that PCF must satisfy in order to be an appropriate IV and indicate why you think PCF would or would not satisfy each of those properties.


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21. (6 points) Assume that you have an appropriate instrumental variable for milk prices named Z. Using the variable names described in the original regression for demand and Z, describe the two stage least squares regression process. Use the variable names defined above and clearly define any other variables that would be included in the regressions at each stage of the two stage regression process.

23 (6 points) Several car insurance companies now offer drivers a discount if they agree to install a tracking device in their cars and prove that they are a safe driver, for example, by driving at or below the speed limit. It is important to note that if the tracking device suggests the driver is an unsafe drive, the insurance company may raise the person’s rates. Suppose that the insurance company is interested in the effect of tracking devices on driver behavior. They have information on the number of speeding tickets for each of its customers in a given year (S), a dummy variable T that indicates whether customer agreed to have a tracking device installed; and each person’s age age and sex. The following model is estimated:

𝑆𝑆𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1𝑇𝑇𝑖𝑖 + 𝛽𝛽2𝑎𝑎𝑎𝑎𝑒𝑒𝑖𝑖 + 𝛽𝛽3𝑃𝑃𝑒𝑒𝑒𝑒𝑎𝑎𝐹𝐹𝑒𝑒𝑖𝑖 + 𝑒𝑒𝑖𝑖 Given that customers decide whether to have a tracking device installed, would you expect the OLS estimate of 𝛽𝛽1 to be biased upward or downward? WHY? [Hint: You should think about the type of driver that will agree to tracking.]


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Your name ________________________________

Tables for Eco311 Final Exam


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Table 1. Linear Probability Model of Whether a Person is Out of Labor Force. (1) (2) Out of Labor Force Out of Labor Force Age Dummies 55-59 0 0 60-64 0.150*** 0.154*** (77.92) (55.64) 65-69 0.382*** 0.396*** (190.87) (136.85) 70-74 0.504*** 0.526*** (233.32) (167.88) 75-79 0.576*** 0.606*** (233.64) (167.39) Years of School -0.0250*** -0.0244*** (-104.41) (-72.09) Female * Age 55-59 0.116*** (16.50) Female * Age 60-64 0.107*** (15.10) Female * Age 65-69 0.0887*** (12.27) Female * Age 70-74 0.0711*** (9.69) Female * Age 75-79 0.0563*** (7.49) Female * School -0.000630 (-1.32) Constant 0.639*** 0.575*** (180.86) (115.19) Observations 413844 413844


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TABLE 2.

n1=numerator degrees of freedom; n2=denominato degrees of freedom.

_cons -33162.71 4496.584 -7.38 0.000 -41981.28 -24344.14 yhat2 -.0688413 .1024685 -0.67 0.502 -.2697995 .1321169 yhat 277.7028 46.11293 6.02 0.000 187.2675 368.1381 ehat2 Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 8.8117e+11 1,970 447296080 Root MSE = 19080 Adj R-squared = 0.1861 Residual 7.1647e+11 1,968 364058674 R-squared = 0.1869 Model 1.6471e+11 2 8.2353e+10 Prob > F = 0.0000 F(2, 1968) = 226.21 Source SS df MS Number of obs = 1,971

. reg ehat2 yhat yhat2

. gen yhat2=yhat^2

. predict yhat, xb

. gen ehat2=ehat^2

. predict ehat, residual

_cons 118.8811 2.619157 45.39 0.000 113.7445 124.0177 sgl_female -31.18393 4.973165 -6.27 0.000 -40.93716 -21.43071 sgl_male -19.72992 4.048456 -4.87 0.000 -27.66964 -11.79021 appinc .3858079 .0186439 20.69 0.000 .349244 .4223719 loanamt Coef. Std. Err. t P>|t| [95% Conf. Interval]

Total 12856381 1,970 6526.08172 Root MSE = 71.923 Adj R-squared = 0.2073 Residual 10175210.8 1,967 5172.95921 R-squared = 0.2085 Model 2681170.23 3 893723.412 Prob > F = 0.0000 F(3, 1967) = 172.77 Source SS df MS Number of obs = 1,971

. reg loanamt appinc sgl_*


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Standard normal cumulative density function. Entries provide probability that standard normal variable takes value less than z (i.e. the area to the left of z under the standard normal density function).


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Eco311, Final Exam, Fall 2018. Prof. Bill Even...Eco311, Final Exam, Fall 2018. Prof. Bill Even 9 21. (6 points) Assume that you have an appropriate instrumental variable for milk