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1、Introduction to Econometrics, 3e (Stock)Chapter 6 Linear Regression with Multiple Regressors6.1 Multiple Choice1) In the multiple regression model, the adjusted R2, r2A) cannot be negative.B) will never be greater than the regression R2.C) equals the square of the correlation coefficient r.D) cannot
2、 decrease when an additional explanatory variable is added.Answer: B2) Under imperfect multicollinearityA) the OLS estimator cannot be computed.B) two or more of the regressors are highly correlated.C) the OLS estimator is biased even in samples of n 100.D) the error terms are highly, but not perfec
3、tly, correlated.Answer: B3) When there are omitted variables in the regression, which are determinants of the dependent variable, thenA) you cannot measure the effect of the omitted variable, but the estimator of your included variable(s) is (are) unaffected.B) this has no effect on the estimator of
4、 your included variable because the other variable is not included.C) this will always bias the OLS estimator of the included variable.D) the OLS estimator is biased if the omitted variable is correlated with the included variable.Answer: D4) Imagine you regressed earnings of individuals on a consta
5、nt, a binary variablewhich takes onthe value 1 for males and is 0 otherwise, and another binary variable Female) which takes on the value 1 for females and is 0 otherwise. Because females typically earn less than males, you would expectA) the coefficient for Male to have a positive sign, and for Fem
6、ale a negative sign.B) both coe仔icients to be the same distance from the constant, one above and the other below.C) none of the OLS estimators to exist because there is perfect multicollinearity.D) this to yield a difference in means statistic.Answer: C5) When you have an omitted variable problem, t
7、he assumption that E(uj | Xf) = 0 is violated. This implies thatA) the sum of the residuals is no longer zero.B) there is another estimator called weighted least squares, which is BLUE.C) the sum of the residuals times any of the explanatory variables is no longer zero.D) the OLS estimator is no lon
8、ger consistent.Answer: D .2工=33.33; X1j = 2.025; X2/ = 17.3137=1z=I/=I2 X2 = 8.3103; .二.0122;= 66422/=1i=i=Z = -0.2304; Z yix2i = L5676; xlzx2/ = -0.0520i=li=lz=l(a) What are your expected signs for the regression coefficient? Calculate the coefficients and see if their signs correspond to your intu
9、ition.(b) Find the regression R , and interpret it. What other factors can you think of that might have an influence on productivity?Answer:AA(a) You expect 01 0 with no prior expectation on the intercept. Substituting the aboveAA八numbers into the equations for the regression coe仔icients results in
10、01 = -12.95, 02 = L39, and (30 =0.34.(b) R2 =- =0.62. 62 percent of the variation in relative productivity is explainedEx2i=by the regression. There is a vast literature on the subject and students* answers will obviously vary.Some may focus on additional economic variables such as the initial level
11、 of productivity and the inflation rate during the sample period. Others may emphasize institutional variables such as whether or not the country was democratic over the sample period, or had political stability, etc.4) A subsample from the Current Population Survey is taken, on weekly earnings of i
12、ndividuals, their age, and their gender. You have read in the news that women make 70 cents to the $1 that men earn. To test this hypothesis, you first regress earnings on a constant and a binary variable, which takes on a value of 1 for females and is 0 otherwise. The results were:Ea = 570.70 - 170
13、.72 x Female, r2=0.084, SER = 282.12.(a) There are 850 females in your sample and 894 males. What are the mean earnings of males and females in this sample? What is the percentage of average female income to male income?(b) You decide to control for age (in years) in your regression results because
14、older people, up to a point, earn more on average than younger people. This regression output is as follows:Earn = 323.70 - 169.78 x Female + 5.15 x Age, R2=0.135, SER = 274.45.Interpret these results carefully. How much, on average, does a 40-year-old female make per year in your sample? What about
15、 a 20-year-old male? Does this represent stronger evidence of discrimination against females?Answer:(c) Males earn $570.70, females $399.98. Percentage of average female income to male income is 70.1% in the sample.(d) As individuals become one year older, they earn $5.15 more, on average. Females e
16、arn significantly less money on average and for a given age. 13.5 percent of the earnings variation is explained by the regression. A 40-year-old female earns $359.92, while a 20-year-old male makes $426.70. There is somewhat more evidence here, since age has been added as a regressor. However, many
17、 attributes, which could potentially explain this difference, are still omitted.5) You have collected data from Major League Baseball (MLB) to find the determinants of winning. You have a general idea that both good pitching and strong hitting are needed to do well. However, you do not know how much
18、 each of these contributes separately. To investigate this problem, you collect data for all MLB during 1999 season. Your strategy is to first regress the winning percentage on pitching quality (Team ERA1), second to regress the same variable on some measure of hitting (OPS - On-base Plus Slugging p
19、ercentage1), and third to regress the winning percentage on both.Summary of the Distribution of Winning Percentage, On Base plus Slugging Percentage, and Team Earned Run Average for MLB in 1999AverageStandard deviationPercentile10%25%40%50% (median)60%75%90%TeamERA4.710.533.844.354.724.784.915.065.2
20、5OPS0.7780.0340.7200.7540.7690.7800.7900.7980.820Winning Percentage0.500.080.400.430.460.480.490.590.60The results are as follows:Wbipci = 0.94 - 0.100 x teamera, R? = 0.49, SER = 0.06.VVhipct = -0.68 + 1.513 x ops, r2=0.45, SER = 0.06.Wmpct = -0.19 - 0.099 x teamera + 1.490 x ops, =0,92, SER = 0.02
21、.(a) Interpret the multiple regression. What is the effect of a one point increase in team ERA? Given that the Atlanta Braves had the most wins that year, wining 103 games out of 162, do you find this effect important? Next analyze the importance and statistical significance for the OPS coefficient.
22、 (The Minnesota Twins had the minimum OPS of 0.712, while the Texas Rangers had the maximum with 0.840.) Since the intercept is negative, and since winning percentages must lie between zero and one, should you rerun the regression through the origin?(b) What are some of the omitted variables in your
23、 analysis? Are they likely to affect the coefficient on Team ERA and OPS given the size of the R and their potential correlation with the included variables?Answer:(a) A single point increase in team ERA lowers the winning percentage by approximately 10 percent. A 0.1 increase in OPS results roughly
24、 in an increase of 15 percent. Given that there are no observations close to the origin, you should not interpret the intercept. The multiple regression explains 92 percent of the variation in winning percentage. The Atlanta Braves only won 63.6 percent of their games. Given that this represents the
25、 best record during that season, a 10 percentage point drop is important. Although the intercept cannot be interpreted, it anchors the regression at a certain level and should therefore not be omitted.(b) The quality of the management and coaching comes to mind, although both may be reflected in the
26、 performance statistics, as are salaries. There are other aspects of baseball performance that are missing, such as the fielding percentage of the team.6) In the process of collecting weight and height data from 29 female and 81 male students at your university, you also asked the students for the n
27、umber of siblings they have. Although it was not quite clear to you initially what you would use that variable for, you construct a new theory that suggests that children who have more siblings come from poorer families and will have to share the food on the table. Although a friend tells you that t
28、his theory does not pass the straight-face1 test, you decide to hypothesize that peers with many siblings will weigh less, on average, for a given height. In addition, you believe that the muscle/fat tissue composition of male bodies suggests that females will weigh less, on average, for a given hei
29、ght. To test these theories, you perform the following regression:,Studentw = -229.92 - 6.52 x Female + 0.51 x Sibs+ 5.58 x Height, R=0.50, SER = 21.08where Studentw is in pounds, Height is in inches, Female takes a value of 1 for females and is 0 otherwise, Sibs is the number of siblings.Interpret
30、the regression results.Answer: For every additional inch in height, students weigh, on average, roughly 5.5 pounds more. For a given height and number of siblings, female students weigh approximately 6.5 pounds less. For every additional sibling, the weight of students increases by half a pound. Sin
31、ce there are no observations close to the origin, you cannot interpret the intercept. The regression explains half of the variation in student weight.7) You have collected data for 104 countries to address the difficult questions of the determinants for differences in the standard of living among th
32、e countries of the world. You recall from your macroeconomics lectures that the neoclassical growth model suggests that output per worker (per capita income) levels are determined by, among others, the saving rate and population growth rate. To test the predictions of this growth model, you run the
33、following regression:Rei Pers Inc = 0.339 - 12.894 x + 1.397 x Sk，R2=0.621, SER = 0.177where RelPersInc is GDP per worker relative to the United States, n is the average population growth rate, 1980-1990, and Sk is the average investment share of GDP from 1960 to 1990 (remember investment equals sav
34、ing).(a) Interpret the results. Do the signs correspond to what you expected them to be? Explain.(b) You remember that human capital in addition to physical capital also plays a role in determining the standard of living of a country. You therefore collect additional data on the average educational
35、attainment in years for 1985, and add this variable (Educ) to the above regression. This results in the modified regression output:HelPerslnc = 0.046 - 5.869 x + 0.738 x Sk + 0.055 x Educ, R2=0.775, SER = 0.1377How has the inclusion of Educ affected your previous results?(c) Upon checking the regres
36、sion output, you realize that there are only 86 observations, since data for Educ is not available for all 104 countries in your sample. Do you have to modify some of your statements in (d)?(d) Brazil has the following values in your sample: RelPersInc = 0.30, n = 0.021, Sk - 0.169, Educ = 3.5. Does
37、 your equation overpredict or underpredict the relative GDP per worker? What would happen to this result if Brazil managed to double the average educational attainment?Answer:(a) The Solow growth model predicts higher productivity with higher saving rates and lower population growth. The signs there
38、fore correspond to prior expectations. A 10 percent point increase in the saving rate results in a roughly 14 percent increase in per capita income relative to the United States. Lowering the population growth rate by 1 percent results in a 13 percent higher per capita income relative to the United
39、States. It is best not to interpret the intercept. The regression explains approximately 62 percent of the variation in per capita income among the 104 countries of the world.(b) The coefficient on the population growth rate is roughly half of what it was originally, while the coefficient on the sav
40、ing rate has approximately doubled. The regression R? has increased significantly, (c) When comparing results, you should ensure that the sample is identical, since comparisons are not valid otherwise.(d) The predicted value for Brazil is 0.240. Hence the regression underpredicts Brazils per capita
41、income. Increasing Educ to 7.0 would result in a predicted per capita income of 0.43, which is a substantial increase from both its current actual position and the previously predicted value.8) Attendance at sports events depends on various factors. Teams typically do not change ticket prices from g
42、ame to game to attract more spectators to less attractive games. However, there are other marketing tools used, such as fireworks, free hats, etc., for this purpose. You work as a consultant for a sports team, the Los Angeles Dodgers, to help them forecast attendance, so that they can potentially de
43、vise strategies for price discrimination. After collecting data over two years for every one of the 162 home games of the 2000 and 2001 season, you run the following regression:Attend = 15,005 + 201 x Temperat + 465 x DodgNetWin + 82 x OppNetWin+ 9647 x DFSaSu + 1328 x Drain + 1609 x D150m + 271 x D
44、Div - 978 x D2001;r2=o.416, SER = 6983where Attend is announced stadium attendance, Temperat it the average temperature on game day, DodgNetWin are the net wins of the Dodgers before the game (wins-losses), OppNetWin is the opposing teams net wins at the end of the previous season, and DFSaSu, Drain
45、, D150m, Ddiv, and D2001 are binary variables, taking a value of 1 if the game was played on a weekend, it rained during that day, the opposing team was within a 150 mile radius, the opposing team plays in the same division as the Dodgers, and the game was played during 2001, respectively.(a) Interp
46、ret the regression results. Do the coe仔icients have the expected signs?(b) Excluding the last four binary variables results in the following regression result:Atteml = 14,838 + 202 x Temperat + 435 x DodgNetWin + 90 x OppNetWin+ 10,472 x DFSaSu, R2 =0.410, SER = 6925According to this regression, wha
47、t is your forecast of the change in attendance if the temperature increases by 30 degrees? Is it likely that people attend more games if the temperature increases? Is it possible that Temperat picks up the effect of an omitted variable?(c) Assuming that ticket sales depend on prices, what would your
48、 policy advice be for the Dodgers to increase attendance?(d) Dodger stadium is large and is not often sold out. The Boston Red Sox play in a much smaller stadium, Fenway Park, which often reaches capacity. If you did the same analysis for the Red Sox, what problems would you foresee in your analysis?Answer:(a) 10 degree warmer temperature increases attendance by roughly 2,000. A 10 game net increase in wins results in approximately 4,600 more spectators. If the opponents net win is 10 games higher when compared to ano