Econometrics Problem Set 6

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1 Econometrics Problem Set 6 WISE, Xiamen University Spring Conceptual Questions 1. This question refers to the estimated regressions shown in Table 1 computed using data for 1988 from the CPS. The data set consists of information on 4000 full-time full-year workers. The highest educational achievement for each worker was either a high school diploma or a bachelor s degree. The worker s ages ranged from 25 to 34 years. The data set also contained information on the region of the country where the person lived, marital status, and number of children. For the purposes of these exercises let AHE = average hourly earnings (in 1998 dollars) College = binary variable (1 if college, 0 if high school) F emale = binary variable (1 if female, 0 if male) Age = age (in years) N ortheast = binary variable (1 if Region = Northeast, 0 otherwise) M idwest = binary variable (1 if Region = Midwest, 0 otherwise) South = binary variable (1 if Region = South, 0 otherwise) W est = binary variable (1 if Region = West, 0 otherwise) (a) (SW 7.1) Add * (5%) and ** (1%) to Table 1 to indicate statistical significance of the coefficients. (b) (SW 7.2) Using the regression results in column (1): i. Is the college-high school earnings difference estimated from this regression statistically significant at the 5% level? Construct a 95% confidence interval of the difference. ii. Is the male-female earnings difference estimated from this regression statistically significant at the 5% level? Construct a 95% confidence interval for the difference. (c) (SW 7.3) Using the regression results in column (2): i. Is age an important determinant of earnings? Use an appropriate statistical test and/or confidence interval to explain your answer. ii. Sally is a 29-year-old female college graduate. Betsy is a 34-year-old female college graduate. Construct a 95% confidence interval for the expected difference between their earnings. (d) (SW 7.4) Using the regression results in column (3): i. Do there appear to be important regional differences? Use an appropriate hypothesis to explain your answer.

2 ii. Juanita is a 28-year-old female college graduate from the South. Molly is a 28-yearold female college graduate from the West. Jennifer is a 28-year-old female college graduate from the Midwest. α) Construct a 95% confidence interval for the difference in expected earnings between Juanita and Molly. β) Explain how you would construct a 95% confidence interval for the difference in expected earnings between Juanita and Jennifer. Dependent variable: average hourly earnings (AHE). Regressor (1) (2) (3) College(X 1 ) (0.21) (0.21) (0.21) F emale(x 2 ) (0.20) (0.20) (0.20) Age(X 3 ) (0.04) (0.04) Northeast(X 4 ) 0.69 (0.30) Midwest(X 5 ) 0.60 (0.28) South(X 6 ) (0.26) Intercept (0.14) (1.05) (1.06) Summary Statistics F -statistic for regional effects = SER R R 2 n Table 1: Results of Regressions of Average Hourly Earnings on Gender and Education Binary Variables and Other Characteristics Using 1988 Data from the Current Populations Survey (e) (SW 7.5) The regression shown in column (2) was estimated again, this time using data from 1992 (4000 observations selected at random from the March 1993 CPS, converted into 1998 dollars using the consumer price index). The results are ÂHE = College 2.59 F emale Age, SER = 5.85, R 2 = (0.98) (0.20) (0.18) (0.03) Comparing this regression to the regression for 1998 shown in column (2) was there a statistically significant change in the coefficient on College? 2. (SW 7.7) Data were collected from a random sample of 220 home sales from a community in Let P denote the selling price (in $1000), BDR denote the number of bedrooms, Page 2

3 Bath denote the number of bathrooms, Hsize denote the size of the house (in square feet), Lsize denote the lot size (in square feet), Age denote the age of the house (in years), and P r denote a binary variable that is equal to 1 if the condition of the house is reported as poor. An estimated regression yields ˆP = BDR Bath Hsize Lsize Age P r (23.9) (2.61) (8.94) (0.011) ( ) (0.311) (10.5) SER = 41.5, R2 = (a) Is the coefficient on BDR statistically significantly different from zero? (b) Typically five-bedroom houses sell for much more than two-bedroom houses. Is this consistent with your answer to (a) and with the regression more generally? (c) A homeowner purchases 2000 square feet from an adjacent lot. Construct a 99% confidence interval for the change in the value of her house. (d) Lot size is measured in square feet. Do you think that another scale might be more appropriate? Why or why not? (e) The F -statistic for omitting BDR and Age from the regression is F = Are the coefficients on BDR and Age statistically different from zero at the 10% level? 3. (SW 7.9) Consider the regression model Y i = β 0 + β 1 X 1i + β 2 X 2i + u i. Use the transform the regression approach discussed in class to transform the regression so that you can use a t-statistic to test (a) β 1 = β 2 ; (b) β 1 + aβ 2 = 0, where a is a constant; (c) β 1 + β 2 = 1; (Hint: You can redefine the dependent variable in the regression.) (d) β 1 + β 2 = a, where a is a constant. 4. (SW 7.10) Show that the following two formulas for the homoskedasticity-only F -statistic are equivalent. F = (SSR restricted SSR unrestricted )/q SSR unrestricted /(n k unrestricted 1) and F = (R 2 unrestricted R2 restricted )/q (1 R 2 unrestricted )/(n k unrestricted 1). 5. (SW 7.11) A school district undertakes an experiment to estimate the effect of class size on test scores in second-grade classes. The district assigns 50% of its previous years firstgraders to small second-grade classes (18 students per classroom) and 50% to regular-size classes (21 students per classroom). Students new to the district are handled differently: 20% are randomly assigned to small classes and 80% to regular-class sizes. At the end of the second-grade school year, each student is given a standardized exam. Let Y i denote the exam score for the i th student, X 1i denote a binary variable that equals 1 if the student is assigned to a small class, and X 2i denote a binary variable that equals 1 if the student is newly enrolled. Let β 1 denote the causal effect on test scores of reducing class size from regular to small. Page 3

4 (a) Consider the regression Y i = β 0 + β 1 X 1i + u i. Do you think that E(u i X 1i ) = 0? Is the OLS estimator of β 1 unbiased and consistent? Explain. (b) Consider the regression Y i = β 0 +β 1 X 1i +β 2 X 2i +u i. Do you think that E(u i X 1i, X 2i ) = 0 depends on X 1? Is the OLS estimator of β 1 unbiased and consistent? Explain. Do you think that E(u i X 1i, X 2i ) = 0 depends on X 2? Is the OLS estimator of β 2 unbiased and consistent? Explain. 6. The Bonferroni test of the joint hypothesis β 1 = β 1,0 and β 2 = β 2,0 based on the critical value c > 0 uses the following rule: Do not reject if t 1 c and if t 2 c; otherwise, reject, where t 1 and t 2 are the t-statistics that test the restriction on β 1 and β 2 respectively. For a significance level of 5% and two restrictions the Bonferroni critical value c equals For the following questions use a large sample approximation for your test statistics. (a) Using the above critical value, what is the probability of rejecting the null when the null is true i. when ρ ˆβ1, ˆβ 2 = 0. ii. when ρ ˆβ1, ˆβ 2 =.5. (Hint: If t 1 and t 2 are two jointly normally distributed random variables with correlation equal to 0.5 then Pr( t 1 < , t 2 < ) = ) iii. when ρ ˆβ1, ˆβ 2 = 1. (b) Comment on the size and power of the Bonferroni test as the correlation between β 1 and β 2 increases. Empirical Questions For these empirical exercises, the required datasets and a detailed description of them can be found at 7. (SW E7.3) The data set used in this empirical exercise (CollegeDistance) contains data from a random sample of high school seniors interviewed in 1980 and re-interviewed in In this exercise you will use these data to investigate the relationship between the number of completed years of education for young adults and the distance from each student s high school to the nearest four year college. (Proximity to college lowers the cost of education, so that students who live closer to a four-year college should, on average, complete more years of higher education.) (a) An education advocacy group argues that, on average, a person s educational attainment would increase by approximately 0.15 year if distance to the nearest college is decreased by 20 miles. Run a regression of years of completed education (ED) on distance to the nearest college (Dist). Is the advocacy groups claim consistent with the estimated regression? Explain. Page 4

5 (b) Other factors also affect how much college a person completes. Does controlling for these other factors change the estimated effect of distance on college years completed? To answer this question, construct a table like Table 7.1 in the textbook. Include a simple specification [constructed in (a)], a base specification (that includes a set of important control variables), and several modifications of the base specification. Discuss how the estimated effect of Dist on ED changes across specifications. (c) It has been argued that, controlling for other factors, blacks and Hispanics complete more college than whites. Is this result consistent with the regressions that you constructed in part (b)? (d) Graph a 95% joint confidence interval for the coefficients on blacks and Hispanics. Page 5

Econometrics Problem Set 3

Econometrics Problem Set 3 Conceptual Questions 1. This question refers to the estimated regressions in table 1 computed using data for 1988 from the U.S. Current Population Survey. The data set consists