Econometrics Problem Set 4

Transcription

1 Econometrics Problem Set 4 WISE, Xiamen University Spring Conceptual Questions 1. This question refers to the estimated regressions in 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 dataset 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) Midwest = 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 6.1) Compute the R 2 for each of the regressions. R 2 = 1 ( ) ( ) n 1 SSR n 1 n k 1 T SS = 1 (1 R 2 ) n k 1 Thus, the values of R 2 are , and respectively. (b) (SW 6.2) Using the regression results in column (1): i. Do workers with college degrees earn more, on average, than workers with only high school degrees? How much more? Workers with college degrees earn $5.46/hour more, on average, than workers with only high school degrees. ii. Do men earn more than women on average? How much more? Men earn $2.64/hour more, on average, than women. (c) (SW 6.3) Using the regression results in column (2): i. Is age an important determinant of earnings? Explain.

2 On average, a worker earns $0.29/hour more for each year he ages. ii. Sally is a 29-year-old female college graduate. Betsy is a 34-year-old female college graduate. Predict Sally s and Betsy s earnings. ÂHE Sally = (1) 2.62(1) (29) = Sallys earnings prediction is $15.67/hour. ÂHE Betsy = (1) 2.62(1) (34) = Sallys earnings prediction is $17.12/hour. The difference is $1.45/hour. (d) (SW 6.4) Using the regression results in column (3): i. Do there appear to be important regional differences? Workers in the Northeast earn $0.69 more per hour than workers in the West, on average, controlling for other variables in the regression. Workers in the Midwest earn $0.60 more per hour than workers in the West, on average, controlling for other variables in the regression. Workers in the South earn $0.27 less per hour than workers in the West, on average, controlling for other variables in the regression. ii. Why is the regressor W est omitted from the regression? What would happen if it was included? The regressor W est is omitted to avoid perfect multicollinearity. If W est is included, then the intercept could be written as a perfect linear function of the four regional regressors. Because of perfect multicollinearity, the OLS estimator could not be computed. iii. Juanita is a 28-year-old female college graduate from the South. Jennifer is a 28-year-old female college graduate from the Midwest. Calculate the expected difference in earnings between Juanita and Jennifer, ÂHE Juanita ÂHE Jennifer = = 0.87 The expected difference in earnings between Juanita and Jennifer is -$0.87/hour. 2. (SW 6.6) A researcher plans to study the causal effects of police on crime using data from a random sample of U.S. counties. He plans to regress the county s crime rate on the (per capita) size of the county s police force. Page 2

3 Dependent variable: average hourly earnings (AHE). Regressor (1) (2) (3) College(X 1 ) F emale(x 2 ) Age(X 3 ) Northeast(X 4 ) 0.69 Midwest(X 5 ) 0.60 South(X 6 ) Intercept Summary Statistics 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 (a) Explain why this regression is likely to suffer from omitted variable bias. Which variables would you add to the regression to control for important omitted variables? There are other important determinants of a country s crime rate, including demographic characteristics of the population. (b) Use your answer to (a) and the expression for omitted variable bias with a single regressor to determine whether the regression will likely over- or underestimate the effect of police on the crime rate. (That is, do you think that E( ˆβ 1 ) > β 1 or E( ˆβ 1 ) < β 1?) Suppose that the crime rate is positively affected by the fraction of young males in the population, and that counties with high crime rates tend to hire more police. In this case, the size of the police force is likely to be positively correlated with the fraction of young males in the population leading to a positive value for the omitted variable bias so that E( ˆβ 1 ) > β 1. Alternatively, suppose that the size of the police force is positively affected by average income in the county, and that counties with high average incomes tend to have lower crime rates. In this case, the size of the police force is likely to be positively correlated with average income leading to a negative value for the omitted variable bias so that E( ˆβ 1 ) < β (SW 6.10) (Y i, X 1,i, X 2,i ) satisfy the four multiple regression model least squares assumptions; in addition, var(u i X 1,i, X 2,i ) = 4 and var(x 1,i ) = 6. Under these conditions, the variance of Page 3

4 the least-squares estimator of β 1 is as follows. [ ] σ 2ˆβ1 = 1 1 σu 2. n 1 ρ 2 X 1,X 2 σx 2 1 A random sample of size n = 400 is drawn from the population. (a) Assume that X 1 and X 2 are uncorrelated. Compute the variance of ˆβ 1. When X 1 and X 2 are uncorrelated, ρ X1,X 2 = 0, ( ) var ˆβ1 = 1 [ ] (b) Assume that cor(x 1, X 2 ) = 0.5. Compute the variance of ˆβ 1. When ρ X1,X 2 = 0.5, ( ) var ˆβ1 = 1 [ ] (c) Comment on the following statements: When X 1 and X 2 are correlated, the variance of ˆβ 1 is larger than it would be if X 1 and X 2 were uncorrelated. Thus, if you are interested in β 1, it is best to leave X 2 out of the regression if it is correlated with X 1. The first part of the statement is correct. When X 1 and X 2 are correlated, the variance of ˆβ 1 is larger than it would be if X 1 and X 2 were uncorrelated. Intuitively, we estimate β 1 while holding X 2 fixed. If X 1 and X 2 are correlated then remaining variation in X 1 after controlling for X 2 is smaller, leading to larger variance of ˆβ 1. However, if you are interested in estimating the causal effect of X 1 on Y the second part of the statement is incorrect. In particular, if X 2 is a determinant of Y, then leaving X 2 out of the regression will lead to omitted variable bias in ˆβ (SW 6.11) Consider the regression model Y i = β 1 X 1i + β 2 X 2i + u i for i = 1,..., n. (Notice that there is no constant term in the regression). (a) Specify the least squares function that is minimized by OLS. (Y i b 1 X 1i b 2 X 2i ) 2 Page 4

5 (b) Compute the partial derivatives of the objective function with respect to b 1 and b 2. (Y i b 1 X 1i b 2 X 2i ) 2 b 1 (Y i b 1 X 1i b 2 X 2i ) 2 b 2 = 2 = 2 X 1i (Y i b 1 X 1i b 2 X 2i ) (1) X 2i (Y i b 1 X 1i b 2 X 2i ) (2) (c) Suppose that X 1iX 2i = 0. Show that ˆβ 1 = X 1iY i / X2 1i. From equation (1), ˆβ 1 satisfies or X 1i (Y i ˆβ 1 X 1i ˆβ 2 X 2i ) = 0 ˆβ 1 = X 1iY i ˆβ 2 X 1iX 2i X2 1i and the result follows immediately. (d) Suppose that X 1iX 2i 0. Derive an expression for ˆβ 1 as a function of the data (Y i, X 1i, X 2i ), i = 1,..., n. From equation (2) ˆβ 2 = X 2iY i ˆβ 1 X 1iX 2i n. X2 2i Substituting this into the expression for ˆβ 1 in (c) and doing some algebra yields ˆβ 1 = X2 2i X 1iY i X 1iX 2i X 2iY i n X2 1i X2 2i ( X. 1iX 2i ) 2 (e) Suppose that the model includes an intercept: Y i = β 0 + β 1 X 1i + β 2 X 2i + u i. Show that the least squares estimators satisfy ˆβ 0 = Ȳ ˆβ 1 X1 ˆβ 2 X2. The least squares objective function is (Y i b 0 b 1 X 1i b 2 X 2i ) 2 and the partial derivative with respect to b 0 is (Y i b 0 b 1 X 1i b 2 X 2i ) 2 b 0 = 2 (Y i b 0 b 1 X 1i b 2 X 2i ). Page 5

6 The estimate ˆβ 0 is the value of b 0 that equates the above equation with zero. Solving for ˆβ 0 yields ˆβ 0 = Ȳ ˆβ 1 X1 ˆβ 2 X2. 5. Consider the population regression model Y i = β 0 + β 1 X 1i + β 2 X 21 + u i and the following 6 sample observations (Y i, X 1i, X 2i ). Observation Y i X 1i X 2i (a) The above data can be represented in matrix form as Y = Xβ + U. i. What is Y? 4 3 Y = ii. What is X? X = iii. What is β β 0 β = β 1. β 2 (b) What is X T X? Page 6

7 6 0 3 X T X = (c) What is (X T X) 1? (Hint: Use the function solve in R.) 125/372 1/ /372 (X T X) 1 = 21/ /2604 1/62 21/62 1/62 42/62 (d) Derive ˆβ. Recall that ˆβ = (X T X) 1 X T Y. 0 X T Y = Thus, 91/124 ˆβ = (X T X) 1 X T Y = 693/868 91/62 6. Suppose a sample of n = 20 households has the sample means and sample covariances below for a dependent variable and two regressors: (a) What is X T X? Sample Covariances Sample Means Y X 1 X 2 Y X X n X T X = n X 1i n X 2i X 1i X2 1i X 1iX 2i X 2i n X 1iX 2i X2 2i Page 7

8 X 1i =n X 1 = 20(7.24) = X 2i =n X 2 = 20(4) = 80 From the definition of sample variance s 2 Y = 1 n 1 (Y i Ȳ )2 = 1 n 1 Y 2 i n n 1Ȳ 2, which implies that Hence, Y 2 i = (n 1)s 2 Y + nȳ 2. Y i =nȳ = 20(6.39) = X1i 2 = 19(0.8) + 20(7.24) 2 = X2i 2 = 19(2.4) + 20(4) 2 = From the definition of sample covariance s XY = 1 n 1 (X i X)(Y i Ȳ ) = 1 n 1 X i Y i n n 1 XȲ, which implies that Hence, X i Y i = (n 1)s XY + n XȲ. X 1i X 2i = 19(0.28) + 20(7.24)(4) = Thus, X T X = Page 8

9 (b) What is (X T X) 1? (Hint: Use the function solve in R.) (X T X) 1 = (c) Derive ˆβ. Recall that ˆβ = (X T X) 1 X T Y, where n n X T X 1i X 2i X = n X 1i X2 1i X n 1iX 2i X and X T Y i Y = n n 2i X X 1iY i n 1iX 2i X 2iY i X2 2i Y i =nȳ = 20(6.39) = X 1i Y i = 19(0.22) + 20(7.24)(6.39) = X 2i Y i = 19(0.32) + 20(4)(6.39) = Hence, ˆβ = Empirical Questions For these empirical exercises, the required datasets and a detailed description of them can be found at 7. (SW E6.2) 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.) Page 9

10 The R code required for each question is listed within its respective solution. The code listed here initialises the software. # read data and attach data CD< read. csv ( D: /R/ C o l l e g e D i s t a n c e. csv ) # a t t a c h i n g a l l o w s you to d i r e c t l y a c c e s s v a r i a b l e names attach (CD) # add AER l i b r a r y f o r r e q u i r e d f u n c t i o n s l i b r a r y ( AER ) ## The following table shows the regression results that answer the questions that follow. Dependent variable: years of completed education (ED) Regressor (1) (2) Dist (0.0134) (0.0116) Bytest (0.0030) F emale (0.0504) Black (0.0675) Hispanic (0.0739) Incomehi (0.0619) Ownhome (0.0649) Dadcoll (0.0708) Cue (0.0093) Stwmf g (0.0197) Intercept (0.0378) (0.2413) Summary Statistics SER R R (a) Run a regression of years of completed education (ED) on distance to the nearest college (Dist). What is the estimated slope? Page 10

11 # simple l i n e a r r e g r e s s i o n model e s t i m a t i o n model 1< lm ( formula=yrsed d i s t, data=cd) # h e t e r o s k e d a s t i c i t y robust standard e r r o r s c o e f t e s t ( model 1, vcov=vcovhc( model 1, type= HC1 ) ) ## From column (1) in the above table, we know that the estimated slope is (b) Run a regression of ED on Dist, but include some additional regressors to control for characteristics of the student, the student s family and the local labor market. In particular, include as additional regressors Bytest, F emale, Black, Hispanic, Incomehi, Ownhome, DadColl, Cue80, and Stwmfg80. What is the estimated effect on Dist of ED? # m u l t i p l e l i n e a r r e g r e s s i o n model e s t i m a t i o n model 2< lm ( formula=yrsed d i s t+b y test+female+black+h i s p a n i c +incomehi+ownhome+d a d c o l l+cue80+stwmfg80, data=cd) # h e t e r o s k e d a s t i c i t y robust standard e r r o r s c o e f t e s t ( model 2, vcov=vcovhc( model 2, type= HC1 ) ) ## From column (2) in the above table, we know that the estimated effect on Dist of ED is (c) Is the estimated effect of Dist on ED in the regression in (b) substantially different from the regression in (a)? Based on this, does the regression in (a) seem to suffer from important omitted variable bias? The coefficient has fallen by more than 50%. Thus, it seems that the results in (a) did suffer from omitted variable bias. (d) Compare the fit of the regression in (a) and (b) using the regression standard errors, R 2 and R 2. Why are R 2 and R 2 so similar in regression (b)? # r squared and adjusted r squared summary( model 1 ) $r. squared summary( model 2 ) $r. squared # adjusted r squared summary( model 1 ) $adj. r. squared summary( model 2 ) $adj. r. squared # number o f o b s e r v a t i o n s Page 11

12 length ( yrsed ) ## The regression in (b) fits the data much better as indicated by the R 2, R 2, and SER. The R 2 and R 2 are similar because the number of observations is large (n = 3796). (e) What is the value of the coefficient on DadColl? What does this coefficient measure? The value of the coefficient on DadColl equals Students with a father who went to college complete years more of education, on average, than students with a father who did not go to college. (f) Explain why Cue80 and Swmfg80 appear in the regression. Are the signs of their estimated coefficients (+ or -) what you would have believed? Interpret the magnitude of these coefficients. These terms capture the cost of attending college. As Swmf g80 increases, the forgone wage of someone who studies increases, so that, on average, college attendance declines. The negative sign on the coefficient is consistent with this. As Cue80 increases, it is more difficult to find a job, which lowers the opportunity cost of attending college, so that college attendance increases. The positive sign on the coefficient is consistent with this. (g) Bob is a black male. His high school was 20 miles from the nearest college. His baseyear composite test score (Bytest) was 58. His family income in 1980 was $26,000, and his family owned a home. His mother attended college but his father did not. The unemployment rate in his county was 7.5%, and the state average manufacturing hourly wage was $9.75. Predict Bob s years of completed schooling using the regression in (b). # Bob p r e d i c t i o n p r e d i c t ( model 2, data. frame ( d i s t =2, b y t e s t =58, female =0, black =1, h i s p a n i c =0, incomehi =1,ownhome=1, d a d c o l l =0, cue80 =7.5, stwmfg80 =9.75)) ## Ŷ Bob = (2) (58) (0) (1) (0) (1) (1) (0) (7.5) (9.75) (h) Jim has the same characteristics as Bob except that his high school was 40 miles from the nearest college. Predict Jim s years of completed schooling using the regression in Page 12

13 (b). # Jim p r e d i c t i o n p r e d i c t ( model 2, data. frame ( d i s t =4, b y t e s t =58, female =0, black =1, h i s p a n i c =0, incomehi =1,ownhome=1, d a d c o l l =0, cue80 =7.5, stwmfg80 =9.75)) ## Ŷ Jim = (4) (58) (0) (1) (0) (1) (1) (0) (7.5) (9.75) Page 13