eco311, second midterm exam, spring 2018. prof. …...eco311, second midterm exam, spring 2018....

Eco311, Second Midterm Exam, Spring 2018. Prof. Bill Even

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Your Name (Please print) ________________________________________

Your section (Please circle) 8:30 10:05 Your row (Please circle) 1 2 3 4 5

Directions

Place your answers to all questions in the space provided. Clearly circle your answer to each question so that it can easily be distinguished from your work. Round all numerical answers to the nearest 100th (e.g. 1.23) unless specifically told otherwise. Each question is worth 4 points unless otherwise indicated.

The formula sheet and tables with the standard normal CDF and F-distribution are attached to the end of the exam.


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Table 1 (provided in the attachment) provides the results of a regression of student grades in courses at Miami University between 2008 and 2017. In addition to coefficients and standard errors, the model’s total and residual sum of squares are provided along with the number of observations. The grade variable is the standard numeric value assigned to letter grades and ranges from 0 (F) to 4 (A or A+) The control variables used in table 1 include:

Female: dummy variable that equals one if the student is female; 0 otherwise. Resident: dummy variable that equals one if the student is a resident of Ohio; 0 otherwise High School GPA: The student’s grade point average in high school. ACT Score: The student’s composite ACT score. 1) Using specification (1), construct a 90% confidence interval for the coefficient on the Ohio resident

dummy variable. -.010±1.645 ∗ .005 = (−.𝟎𝟎𝟎𝟎,−.𝟎𝟎𝟎𝟎)

2) What is R2 in specification (1)?

1 −703772779560

= .097 ≅ ,10

3) (6 points) If another control variable is added to the regression, indicate whether each of the following

statements is true (T) or false (F). a) R2 cannot increase. F

b) Adjusted R2 will decrease unless the new variable has a t-statistic with an absolute value less than

one. F

c) Adjusted R2 will be less than R2 T

4) If the t-statistic for a coefficient is 2.00, what is the associated p-value for the null hypothesis that the coefficient is zero? Give answer to nearest .01. (Note: the table for the standard normal distribution is included attachments.) .05

5) Based on specification 1 in table 1, construct a t-statistic for the null hypothesis that Ohio residents and non-residents earn the same grades. a) What is the value of the t-statistic? -.010/.005=-2.0

b) Can you reject the null hypothesis at the i) .10 level of significance? yes

ii) .05 level of significance? yes


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6) In specification (2), a control for the person’s ACT score is added to the regression. You should also note that the coefficient on the female dummy variable rises relative to the estimate in specification (1). Based on this and how omitted variable bias works, it must be the case that: a) On average, women have higher ACT scores than men. b) On average, women have lower ACT scores than men. c) Neither of the above is implied by the change in the estimated coefficient on the female dummy

variable.

7) Using specification (2), what is the predicted course grade for a non-resident male student who had a 3.0 high school GPA, and a 25 on the ACT? Give your answer to the nearest .01. .6894+3(.483)+25(.023) ≅ 𝟎𝟎.𝟕𝟕𝟎𝟎

8) In specification (3), an interaction term between the female dummy variable and the ACT Score is added. Based on this specification, what is the marginal effect of a one point increase in the ACT score for a) a female student? .021 b) a male student? .024

9) Based on specification (3), we can conclude that as the ACT score rises, the predicted grade earned a) Rises for men, but falls for women. b) Rises for both men and women, but more for men than women. c) Rises for both men and women, but more for women than men.

10) Based on specification (3), we can conclude that, holding the other controls variables constant, a) Women have higher grades than men b) The female grade advantage is greater among students with higher ACT scores. c) Both a and b d) Neither a or b.

11) Using the results in specification (3), calculate the t-statistic to test the null hypothesis that the marginal effect of ACT scores on grades is identical for men and women. a) What is the t-statistic?

3.00

b) Based on the t-statistic, can you reject the null hypothesis that the marginal effect of ACT scores on grades is identical for men and women at i) The .05 level_Yes______ ii) the .01 level_yes_______


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12) In specification (4), the number of semesters of college experience and its square are added to the

regression. Use the information in the table to construct an F-statistic for the null hypothesis that the coefficients on experience and experience2 are jointly equal to zero. a) What is the value of the F-statistic? 403.52

b) How many degrees of freedom does the F-statistic have i) in the numerator? _2____ Ii) in the denominator? __1,036,303_____

ii) Given the F-statistic, can you reject the null hypothesis at the .05 level? (Note: a table for the F-distribution at the .05 level is provided in the attachments.) yes

13) Based on specification 4, indicate whether each of the following is true (T) or false (F). a) Other things being the same, grades for second semester seniors (8 semesters of experience) are

worse than those of first semester seniors. T

b) Other things being the same, grades for students are better in their second semester (experience=2) than in their first semester. T

14) If the error term in the regressions for table has heteroscedasticity, indicate whether each of the following statements would be true (T) or false (F). a) The coefficient estimates would be biased. _____F______ b) The estimated standard errors for the coefficients would be incorrect. ___T_______

For the next several questions, refer to table 2. This uses the same grade data as table 1 but adds dummy variables for the subject that the grade was earned in. Dummy variables are included for 5 subjects and all other subject areas (i.e. those not included in the 5 subjects listed) are lumped into a single category referred to as “other subjects”. The dummy for other subjects is the omitted dummy and therefore, other subjects are the reference group. 15) Based on the estimates in table 2, holding the other control variables constant, on average, grades in

Finance are __.10______ (higher, lower) than in Accounting.

16) Calculate the t-statistic for the null hypothesis that grades in Finance are the same as those in the reference group (“other subjects”).


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a) What is the value of the t-statistic? 31.4

b) Based on this t-statistic, can you reject the null hypothesis that grades in economics courses are the same as those in “other subjects” at the .01 level of significance? yes

17) If you are to test the null hypothesis that grades are equal across course subject areas (economics, finance, marketing, math, accounting, and “all other subjects”), what would be the precise test command you would use in Stata? i.e. test …….. test eco=fin=mkt=math=acc=0 (using abbreviatons for subject dummies)

18) If the dummy variables for “other subjects” was included and that for economics dropped, what would be the coefficient on a) The accounting dummy variable? .13

b) The constant (i.e. the intercept)? .12

In table 3, a linear probability model is estimated using the same data as in tables 1 and 2. The dependent variable is a dummy variable whenever the course taken is economics and is zero otherwise. It is important to notice that, other than the female dummy, all of the control variables are measured as deviations from sample means. 19) Given the estimated regression, what is the predicted probability that a course taken by a male is an

economics course if the student has the average ACT, high school GPA, and semesters at Miami? .042

20) Other things being the same, how much does a 10 point increase in a student’s ACT score affect the probability of taking an economics course? Be sure to indicate whether it increases or decreases the probability as well as by how much. .01 increase

21) Compared to men with average values for the control variables, indicate whether each of the following statements is true (T) or false (F). a) The probability that a woman takes an economics course is .021 lower. _T____

b) The probability that a woman takes an economics courses is 2.1 percentage points lower. _T____

c) The probability that a woman takes an economics course is 50 percent lower. __T___

In table 4, data from the 2016 American Community Survey is used to estimate a regression explaining the annual earnings of employed women. Controls include a dummy variable for being married, a


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dummy variable indicating whether a person attended college, the woman’s age and the square of age. In the first specification, annual earnings is the dependent variable. In the second specification, the natural log of annual earnings is the dependent variable.

21) Based on the information in table 4, holding other controls constant, compared to married women, single women earn $____344.56_________ (more, less) and __9.1______ percent (more, less).

22) Assume that a working woman’s age can range between 16 and 80. Based on specification (1) of table 4, a woman’s annual earnings are maximized when she reaches an age of _________. QUESTION OMITTED Answer any 2 of the next 3 questions. (8 points each)

23) Married women are more likely to work part-time than single women. If a new control variable was added to the regressions in table 4 indicating the number of hours worked per week, how would this affect the estimated coefficient on the married dummy variable? JUSTIFY YOUR ANSWER WITH REFERENCE TO WHAT DETERMINES THE DIRECTION OF OMITTED VARIABLE BIAS. The omission of hours worked creates an omitted variables bias on the estimated coefficient for the married dummy variable. The sign of the bias will be negative since the bias depends on the sign of two things (1) the cov(married, hours)<0 ; and (2) the coefficient on the omitted variable (hours worked) is positive. Since the bias is negative, adding a control for hours worked to the regression should eliminate that bias and cause the estimated coefficient on the married dummy to increase.


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24) Women may differ in their career goals. Some may want to work full time for their entire work-life

while others may want to work some years but reduce work effort when their children are young. Women who are more “career oriented” would receive a higher return to an investment in a college degree and therefore be more likely to go to college. Explain how these facts will likely cause the estimated coefficient on the college dummy to be biased. Describe the direction of the bias and explain how you know the direction. (Hint: think of this as college as a treatment and consider the nature of the selection decision into college.) Since “career oriented” women are more likely to work longer hours, their earnings are higher. Moreover, since this factor is not controlled for in the earnings regression, the effect of career orientation is in the error term and this will cause cov(coll, error)>0 if women who pursue college degrees are more career oriented. As a result, the college variable is endogenous and the estimated coefficient on the college dummy variable will be biased upward.

25) Ignore the issues raised in the last 2 questions for this question. There is some research that suggests that earnings rise faster with age for those who have a college degree. Explain how you could test the null hypothesis that earnings growt at the same rate for women with and without a college degree. Write out the precise model you would estimate and describe what hypothesis you would test regarding the coefficients in your model (e.g. test b1=b2=0). To test this hypothesis, I would add an interaction term between the college dummy and age to the regression. That is, I would estimate the following regression:

𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒔𝒔𝒆𝒆 = 𝜷𝜷𝟎𝟎 + 𝜷𝜷𝟏𝟏𝒎𝒎𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒅𝒅𝒆𝒆 + 𝜷𝜷𝟎𝟎𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 + 𝜷𝜷𝟑𝟑𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 + 𝜷𝜷𝟒𝟒(𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 ∗ 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆) The wage growth that occurs as a person ages is given by the following derivative

𝝏𝝏𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒔𝒔𝝏𝝏𝒆𝒆𝒆𝒆𝒆𝒆

= 𝜷𝜷𝟑𝟑 + 𝜷𝜷𝟒𝟒 ∗ 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒆𝒆𝒆𝒆𝒆𝒆

If 𝜷𝜷𝟒𝟒 = 𝟎𝟎, the marginal effect of age (i.e. earnings growth) is the same for people with and without a college degree. So the appropriate null hypothesis is 𝜷𝜷𝟒𝟒 = 𝟎𝟎 which can be tested with the t-statistic associated with 𝜷𝜷𝟒𝟒.


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Your name______________________________

REGRESSION TABLES, FORMULA SHEET, AND STATISTICAL DISTRIBUTIONS


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Table 1. Determinants of Course Grades

(1) (2) (3) (4) grade grade grade grade Ohio resident -0.010 0.007 0.007 0.006 (0.005) (0.005) (0.005) (0.005) High School GPA 0.570 0.483 0.484 0.484 (0.006) (0.007) (0.007) (0.007) Female 0.168 0.198 0.275 0.275 (0.005) (0.005) (0.038) (0.038) ACT score 0.023 0.024 0.025 (0.001) (0.001) (0.001) Female*ACT score -0.003 -0.003 (0.001) (0.001) Number of semesters at Miami

0.021

(0.002) Squared number of semesters at Miami

-0.002

(0.000) Constant 1.009 0.694 0.652 0.604 (0.023) (0.023) (0.032) (0.033) R2 0.097 0.103 0.103 0.104 Residual Sum of Squares

703772 699104 699079 698535

Total sum of squares 779560 779560 779560 779560 Observations 1036311 1036311 1036311 1036311

Standard errors in parentheses Dependent variable is course grade which ranges from 0 (F) to 4 (A or A+).


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Table 2. Effect of Course Subject on Course Grades. (1) Course subject (all other courses are reference group)

Economics -0.543 (0.007) Finance -0.314 (0.010) Marketing 0.040 (0.005) Math -0.516 (0.007) Accounting -0.412 (0.007)

Female 0.165 (0.005) ACT score 0.024 (0.001) High School GPA 0.488 (0.007) Number of semesters at Miami 0.024 (0.002) Squared number of semesters at Miami -0.002 (0.000) Constant 0.666 (0.024) Observations 1,036,311

Standard errors in parentheses. Dependent variable is course grade which ranges from 0 (F) to 4 (A or A+). Course subject variables are dummy variables that equal one if the course is in that particular subject and zero otherwise.


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Table 3.Linear probability model of whether course taken is Economics. (1) eco Female -0.021 (0.001) ACT score (deviation from mean) 0.001 (0.000) High School GPA (deviation from mean) -0.005 (0.001) Number of semesters at Miami (deviation from mean) 0.001 (0.000) Constant 0.042 (0.000) Observations 1064772

Standard errors in parentheses. Dependent variable is dummy variable indicating whether course subject is Economics.


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Table 4. Determinants of Annual Earnings for Employed Married Women in 2016. (1) (2) Annual Earnings Log(Annual Earnings)

Married dummy variable -344.555 -0.091 (147.954) (0.003) College dummy variable 34810.450 0.660 (141.399) (0.002) Age 510.517 0.051 (5.416) (0.001) Age2 -0.000 (0.000) Constant 11672.460 8.828 (291.438) (0.014)

Observations 520263 519908

Standard errors in parentheses. Data source is 2016 American Community Survey.


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1.


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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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1. SST=SSR+SSE=sum of squared residuals + sum of squares explained

2.

3.

4.

5. SD

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eco311, second midterm exam, spring 2018. prof. …...eco311, second midterm exam, spring 2018....

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