2023F_322_Midterm2_Solutions

Section 2. Multiple Choice Questions. 20 points in total, 2 points each 1. Assume that Y is distributed like a standard normal, N (0 , 1). Then, the prob- ability that Y is between -1.96 and 1.96 is: (a) 0.90 (b) 0.925 (c) 0.95 (d) 0.975 (e) None of the above 2. New Brunswick’s daily temperature has an expected value of 52F and a standard deviation of 11F. The formula to convert degrees Fahrenheit (F) to degrees Celsius (C) is: C = 5 9 ( F − 32) What is the expected value of New Brunswick’s daily temperature in Celsius (rounded to the nearest integer)? (a) 9C (b) 10C (c) 11C (d) 12C (e) None of the above 3. New Brunswick’s daily temperature has an expected value of 52F and a standard deviation of 11F. The formula to convert degrees Fahrenheit (F) to degrees Celsius (C) is: C = 5 9 ( F − 32) What is the variance of New Brunswick’s daily temperature in Celsius (rounded to the nearest integer)? (a) 37C (b) 42C (c) 47C (d) 49C (e) None of the above 3

Section 3. Short Answer Questions. 12 points in total 11. [6 points] The company Fashion Icon LLC is planning to increase its spending on the advertisement of their clothing items with the objective of increasing the visibility of the brand and, ultimately, increase their profits. Fashion Icon LLC has a bunch of data from previous years that contain information on two variables: revenue t , which denotes the total revenues from their sales in a given year t (in thousand $ ), and ad costs t , which denotes the total costs incurred by the company in the advertising campaign for year t (in thousand $ ). (a) [4 points] Using the two variables, write down a univariate regression equa- tion that the company could estimate where the slope coefficient β can be interpreted as the percentage change in total revenues that is associated with a 1 percent increase in advertising spending. (b) [2 points] If you estimate your regression equation in (a) using OLS, what would the distribution of ˆ β be if your sample is large enough and i.i.d? Why? 12. [6 points] Answer the following questions (round to two decimal places): (a) [3 points] In Metrics’ High School, the probability that a student takes a Statistics course is 0.27 and the probability that a student takes a Literature course is 0.12. Given that the joint probability that a student takes Literature but does not take Statistics is 0.62, what is the probability that a student takes Literature conditional on not taking a Statistics course? (b) [3 points] What is the probability that, when I roll a die twice, I get a two in the first roll and a five on the second roll? What is the probability that, when I roll a die twice, I get a five in each of the two rolls? Assume that the die is not rigged. Hint: does the second roll depend on what the realization of the first roll is? 6

Rutgers University Department of Economics Econometrics: 01:220:322:01 Midterm 2 Fall 2023 Instructor: Hector Blanco Exam Version: A Instructions - PLEASE READ ALL THESE INSTRUCTIONS FIRST • Write your name on the front page. Do it now! • This is a closed book exam –you may have one 3x5 notecard “cheatsheet” (both sides ok) • Calculators may be used (no cell phones as calculators!) • You have the full class period to complete the exam (approximately 1 hour and 20 minutes) • There are a total of 80 points • Use a 5% significance level for tests unless otherwise stated (critical value: 1.96) • Unless stated otherwise, please round all answers to 2 decimal places • Please fill all questions on this exam sheet. Do not unstaple it . If you need more paper there are extra sheets up front. Please clearly label all short/long answer responses with the number and/or letter of the question you are answering. You can write on the front and back of the sheet. Name: 1

4. What is the difference between an estimator and an estimate? (a) Both an estimator and an estimate are functions of a sample of data to be drawn randomly from a population. (b) An estimator is a function of a sample of data to be drawn randomly from a population whereas an estimate is the numerical value of the estimator when it is actually computed using data from a specific sample. (c) An estimate is a function of a sample of data to be drawn randomly from a population whereas an estimator is the numerical value of the estimator when it is actually computed using data from a specific sample. (d) Both an estimator and an estimate are numerical values computed using data from a specific sample. (e) None of the above 5. Consider the multivariate regression model Y i = θ 0 + θ 1 X 1 i + θ 2 X 2 i + ... + θ k X ki + ε i Ordinary Least Squares (OLS) estimates the parameters { θ 0 , θ 1 , ..., θ k } by min- imizing the following function: (a) ∑ n i =1 ( Y i − θ 0 − θ 1 X 1 i − θ 2 X 2 i − ... − θ k X ki − ε i ) 2 (b) ∑ n i =1 ( Y i + θ 0 + θ 1 X 1 i + θ 2 X 2 i + ... + θ k X ki + ε i ) 2 (c) ∑ n i =1 ( Y i − θ 0 − θ 1 X 1 i − θ 2 X 2 i − ... − θ k X ki ) 2 (d) ∑ n i =1 ( Y i + θ 0 + θ 1 X 1 i + θ 2 X 2 i + ... + θ k X ki ) 2 (e) ∑ n i =1 ( Y 2 i − θ 0 − θ 1 X 2 1 i − θ 2 X 2 2 i − ... − θ k X 2 ki ) 6. Imagine that you were told that the t-statistic for the slope coefficient of the regression line TestScore V = 698 . 8 + 2 . 28 StudentTeacherRatio was 4.38. What are the units of measurement for the t-statistic? (a) Points of the test score (b) Number of students per teacher (c) Points of the test score / Number of students per teacher (d) Standard deviations (estimated by the corresponding standard error) (e) Dollars 7. You estimate Y i = α + β 1 X i + β 2 X 2 i + ϵ i . How can you test for whether the relationship between Y i and X i is linear or quadratic? (a) Compute the t-statistic for β 2 to test the null hypothesis that β 2 = 0 (b) Compute the t-statistic for β 1 to test the null hypothesis that β 1 = 0 (c) Compute the F-statistic to test the null hypothesis that both β 1 = 0 and β 2 = 0 (d) Compare the t-statistic for β 1 when including X 2 in the regression to the t-statistic for β 1 when not including X 2 in the regression (e) None of the above 4

Section 4. Long Problem. 45 points in total 13. [21 points] Climate change is one of the biggest challenges of this century. Given this, suppose that we are interested in understanding to what extent economic progress has contributed to climate change. To examine this question, we collected data in 2023 for all counties in the United States. Counties are geographical areas similar to the size of an average metropolitan area. We have data for three variables: • AQI : Annual average of the Air Quality Index (AQI), which goes from 0 to 500 degrees (the lower, the better air quality). Air quality is affected by pollution, which is an important driver of climate change. • income capita : Income per capita in the county, in thousands of dollars, which is a proxy for economic progress. • urban : Dummy variable that takes value 1 if the county is mostly urban and takes value 0 if the county is mostly rural. The table below shows the results of estimating several regression equations by OLS, using AQI as the dependent variable: Variables (1) (2) (3) income capita 2.281 1.537 1.343 (0.493) (0.354) (0.295) urban 14.199 12.304 (3.769) (3.514) income capita × urban 0.189 (0.059) constant 39.5 28.1 26.1 (4.26) (4.09) (3.78) Obs 3,143 3,143 3,143 Note: Standard error in parentheses. This table is made up. Answer the questions below (round your answers to two decimal places): (a) [4 points] Column (1) regresses AQI on income per capita. Interpret the slope coefficient. Interpret the intercept (does it make sense in this case? why?). (b) [6 points] The regression in Column (1) omits variables that may bias the estimate of the effect of income per capita on AQI. Argue why the variable urban may introduce omitted variable bias (OVB). Clearly state the two necessary conditions for OVB and how they apply to this case. Column (2) adds urban to the regression. Was the coefficient in Column (1) upward or downward biased? (c) [6 points] Interpret the estimated coefficient of urban in Column (2). Is this coefficient different from zero at the 5% significance level? (d) [2 points] Column (3) adds an interaction term between the two main re- gressors. Interpret the coefficient on income capita × urban . (e) [3 points] Using the estimates from Column (3) compute the expected value of the AQI for an urban county with an income per capita of $ 25,000. 7

Answer: (a) We can estimate Equation (1). However, the interpretation of β does not make sense. It could mean the impact on GPA of changing classes from Busch to College Ave, or from Cook to Douglass. It is not readily interpretable. (b) The omitted group is students taking classes in College Ave. It is omitted because if adding a dummy variable for each of the five campuses would result in perfect multicollinearity. That is, we would be able to write one of the dummies as a perfect linear combination of the others because they are mutually exclusive, which violates the fourth assumption of multivariate regression. (c) ˆ α is the average GPA of students taking classes in College Ave. ˆ β 4 is the difference in average GPAs between students taking classes in Livingston and students taking classes in College Ave. (d) The joint null hypothesis and the alternative hypothesis are: H 0 : β 1 = 0 and β 2 = 0 and β 3 = 0 and β 4 = 0 H A : β 1 ̸ = 0 and/or β 2 ̸ = 0 and/or β 3 ̸ = 0 and/or β 4 ̸ = 0 You can also express H A as: at least one of the restriction in the joint null hypothesis does not hold. To test this, we would need to compute the F-statistic with 4 restrictions (degrees of freedom). To reject/fail to reject the null hypothesis, there are two alternatives: (1) compare the F- statistic to the critical value in the F-distribution table at the end of the textbook (if the F-statistic is higher than the critical value, we can reject the null), or (2) the statistical software will spit out a p-value associated with the F-statistic: if it is lower than 0.05, we can reject the null at the 5% significance value. (e) Yes. Since the β coefficients indicate the difference in the average GPA between the included groups and the omitted group, if we omit Busch and include College Ave, the coefficient for College Ave should be the negative of the coefficient for Busch, -0.221. 10