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1、CHAPTER 11.11.1(i)Ideally,we could randomly assign students to classes of different sizes.That is,eachstudent is assigned a different class size without regard to any student characteristics such asability and family background.For reasons we will see in Chapter 2,we would like substantialvariation

2、in class sizes(subject,of course,to ethical considerations and resource constraints).(ii)A negative correlation means that larger class size is associated with lower performance.We might find a negative correlation because larger class size actually hurts performance.However,with observational data,

3、there are other reasons we might find a negative relationship.For example,children from more affluent families might be more likely to attend schools withsmaller class sizes,and affluent children generally score better on standardized tests.Anotherpossibility is that,within a school,a principal migh

4、t assign the better students to smaller classes.Or,some parents might insist their children are in the smaller classes,and these same parentstend to be more involved in their childrens education.(iii)Given the potential for confounding factors some of which are listed in(ii)finding anegative correla

5、tion would not be strong evidence that smaller class sizes actually lead to betterperformance.Some way of controlling for the confounding factors is needed,and this is thesubject of multiple regression analysis.1.21.2(i)Here is one way to pose the question:If two firms,say A and B,are identical in a

6、llrespects except that firm A supplies job training one hour per worker more than firm B,by howmuch would firm As output differ from firm Bs?(ii)Firms are likely to choose job training depending on the characteristics of workers.Someobserved characteristics are years of schooling,years in the workfo

7、rce,and experience in aparticular job.Firms might even discriminate based on age,gender,or race.Perhaps firmschoose to offer training to more or less able workers,where“ability”might be difficult toquantify but where a manager has some idea about the relative abilities of different employees.Moreove

8、r,different kinds of workers might be attracted to firms that offer more job training onaverage,and this might not be evident to employers.(iii)The amount of capital and technology available to workers would also affect output.So,two firms with exactly the same kinds of employees would generally hav

9、e different outputs ifthey use different amounts of capital or technology.The quality of managers would also havean effect.(iv)No,unless the amount of training is randomly assigned.The many factors listed inparts(ii)and(iii)can contribute to finding a positive correlation between output and training

10、even if job training does not improve worker productivity.1.31.3 It does not make sense to pose the question in terms of causality.Economists would assumethat students choose a mix of studying and working(and other activities,such as attending class,leisure,and sleeping)based on rational behavior,su

11、ch as maximizing utility subject to theconstraint that there are only 168 hours in a week.We can then use statistical methods tomeasure the association between studying and working,including regression analysis that wecover starting in Chapter 2.But we would not be claiming that one variable“causes”

12、the other.They are both choice variables of the student.CHAPTER 22.12.1(i)Income,age,and family background(such as number of siblings)are just a fewpossibilities.It seems that each of these could be correlated with years of education.(Incomeand education are probably positively correlated;age and ed

13、ucation may be negatively correlatedbecause women in more recent cohorts have,on average,more education;and number of siblingsand education are probably negatively correlated.)(ii)Not if the factors we listed in part(i)are correlated with educ.Because we would liketo hold these factors fixed,they ar

14、e part of the error term.But if u is correlated with educ thenE(u|educ)0,and so SLR.4 fails.2.22.2 In the equation y=0+1x+u,add and subtract0 from the right hand side to get y=(0+0)+1x+(u 0).Call the new error e=u 0,so that E(e)=0.The new intercept is0+0,but the slope is still1.2.32.3(i)Let yi=GPAi,

15、xi=ACTi,and n=8.Thenx=25.875,y=3.2125,(xix)(yii1n=y)=5.8125,and(xix)2=56.875.From equation(2.9),we obtain the slope as1i1n=y5.8125/56.875.1022,rounded to four places after the decimal.From(2.17),0 x3.2125 (.1022)25.875.5681.So we can write1GPA=.5681+.1022 ACTn=8.The intercept does not have a useful

16、interpretation because ACT is not close to zero for thepopulation of interest.If ACT is 5 points higher,GPAincreases by.1022(5)=.511.(ii)The fitted values and residuals rounded to four decimal places are given along withthe observation number i and GPA in the following table:GPA uiGPA12.823.433.043.

17、553.663.072.783.72.71433.02093.32753.5319.0857.3791.1725.06813.2253.22533.1231.12313.1231.42313.6341.0659You can verify that the residuals,as reported in the table,sum to.0002,which is pretty close tozero given the inherent rounding error.(iii)When ACT=20,GPA=.5681+.1022(20)2.61.(iv)The sum of squar

18、ed residuals,ui1n2i,is about.4347(rounded to four decimal places),and the total sum of squares,regression isi1n(yiy)2,is about 1.0288.So the R-squared from theR2=1 SSR/SST1 (.4347/1.0288).577.Therefore,about 57.7%of the variation in GPA is explained by ACT in this small sample ofstudents.2.42.4(i)Wh

19、en cigs=0,predicted birth weight is 119.77 ounces.When cigs=20,bwght=109.49.This is about an 8.6%drop.(ii)Not necessarily.There are many other factors that can affect birth weight,particularlyoverall health of the mother and quality of prenatal care.These could be correlated withcigarette smoking du

20、ring birth.Also,something such as caffeine consumption can affect birthweight,and might also be correlated with cigarette smoking.(iii)If we want a predicted bwght of 125,then cigs=(125 119.77)/(.524)10.18,orabout 10 cigarettes!This is nonsense,of course,and it shows what happens when we aretrying t

21、o predict something as complicated as birth weight with only a single explanatoryvariable.The largest predicted birth weight is necessarily 119.77.Yet almost 700 of thebirths in the sample had a birth weight higher than 119.77.(iv)1,176 out of 1,388 women did not smoke while pregnant,or about 84.7%.

22、Because weare using only cigs to explain birth weight,we have only one predicted birth weight at cigs=0.The predicted birth weight is necessarily roughly in the middle of the observed birth weights atcigs=0,and so we will under predict high birth rates.2.52.5(i)The intercept implies that when inc=0,

23、cons is predicted to be negative$124.84.This,of course,cannot be true,and reflects that fact that this consumption function might be a poorpredictor of consumption at very low-income levels.On the other hand,on an annual basis,$124.84 is not so far from zero.(ii)Just plug 30,000 into the equation:co

24、ns=124.84+.853(30,000)=25,465.16 dollars.(iii)The MPC and the APC are shown in the following graph.Even though the intercept isnegative,the smallest APC in the sample is positive.The graph starts at an annual income levelof$1,000(in 1970 dollars).MPCAPC.9MPC.853APC.728.71000100002000030000inc2.62.6(

25、i)Yes.If living closer to an incinerator depresses housing prices,then being farther awayincreases housing prices.(ii)If the city chose to locate the incinerator in an area away from more expensiveneighborhoods,then log(dist)is positively correlated with housing quality.This would violateSLR.4,and O

26、LS estimation is biased.(iii)Size of the house,number of bathrooms,size of the lot,age of the home,and quality ofthe neighborhood(including school quality),are just a handful of factors.As mentioned in part(ii),these could certainly be correlated with dist and log(dist).2.72.7(i)When we condition on

28、mes do not have much discretion about spending;typically,alow-income family must spend on food,clothing,housing,and other necessities.Higherincome people have more discretion,and some might choose more consumption while othersmore saving.This discretion suggests wider variability in saving among hig

29、her incomefamilies.2.82.8(i)From equation(2.66),nn21=xiyi/xi.i1i1Plugging in yi=0+1xi+ui givesnn21=xi(01xiui)/xi.i1i1After standard algebra,the numerator can be written as0 xi1x xiui.2i1i1innni1Putting this over the denominator shows we can write1asnn2nn21=0 xi/xi+1+xiui/xi.i1i1i1i1Conditional on th

30、e xi,we havenn2E(1)=0 xi/xi+1i1i1because E(ui)=0 for all i.Therefore,the bias in1is given by the first term in this equation.This bias is obviously zero when0=0.It is also zero whenx =0,which is the same asii1nx=0.In the latter case,regression through the origin is identical to regression with anint

31、ercept.(ii)From the last expression for1in part(i)we have,conditional on the xi,Var(1)nn2n2n2=xiVarxiui=xixiVar(ui)i1i1i1i1222n2n22n22=xixi=/xi.i1i1i1nn22(iii)From(2.57),Var(1)=/(xi x).From the hint,xi2i1i1(x x)ii1n2,and).A more direct way to see this is to writeso Var(1)Var(1which is less than(x x)

32、ii1n2=xi1n2in(x)2,xi1n2iunlessx=0.(iv)For a given sample size,the bias in1increases asxincreases(holding the sum ofincreases relative to Var().Thethex2fixed).But asxincreases,the variance ofi11on abias in1is also small when0is small.Therefore,whether we prefer1or1mean squared error basis depends on

33、the sizes of0,x,and n(in addition to the size ofxi1n2i).2.9 (i)We follow the hint,noting thatc1y=c1y(the sample average ofc1yiis c1 times thesample average of yi)andc2x=c2x.When we regress c1yi on c2xi(including an intercept)we use equation(2.19)to obtain the slope:1(c x c x)(c y c y)c c(x x)(y y)2i

34、n21i11 2niii1nni1(c x c x)2i2i1nn2c(x x)22ii12c1i1c2(xi x)(yi y)2i(x x)i1c11.c2(c2x)=From(2.17),we obtain the intercept as0=(c1y)1(c2x)=(c1y)(c1/c2)1)because the intercept from regressing yi on xi is(yx)=c1x).c1(y101(ii)We use the same approach from part(i)along with the fact that(c1 y)=c1+yand(c2 x

35、)=c2+x.Therefore,(c1 yi)(c1 y)=(c1+yi)(c1+y)=yiyand(c2+xi)(c2 x)=xix.So c1and c2 entirely drop out of the slope formula for the.The intercept is=(c y)regression of(c1+yi)on(c2+xi),and=1101x)+c1 c2+c1 c2(c2+x)=(y =,which is what1(c2 x)=(c1+y)10111we wantedto show.(iii)We can simply apply part(ii)beca

36、uselog(c1yi)log(c1)log(yi).In other words,replace c1 with log(c1),yi with log(yi),and set c2=0.(iv)Again,we can apply part(ii)with c1=0 and replacing c2 with log(c2)and xi with log(xi).垐垐If0 and1are the original intercept and slope,then11and00log(c2)1.2.102.10(i)This derivation is essentially done i

37、n equation(2.52),once(1/SSTx)is brought insidethe summation(which is valid becauseSSTxdoes not depend on i).Then,just definewi di/SSTx.垐(ii)BecauseCov(,u)E()u,we show that the latter is zero.But,from part(i),111nnwE(uu).Because theuare pairwise uncorrelated)u=EE(wuui11ii1iii1i(they are independent),

38、E(uiu)E(ui2/n)2/n(becauseE(uiuh)0,i h).Therefore,i1wiE(uiu)i1wi(2/n)(2/n)i1wi 0.垐 y xand,plugging iny x u(iii)The formula for the OLS intercept is010垐(x u)x u(?)x.gives0011011nnn and uare uncorrelated,(iv)Because12222222垐Var(0)Var(u)Var(1)x/n(/SSTx)x/nx/SSTx,which is what we wanted to show.)2SST/n x

39、2/SST(v)Using the hint and substitution givesVar(0 xx12222122nx x x/SST nx/SSTx.ixii1i12.112.11(i)We would want to randomly assign the number of hours in the preparation course so thathours is independent of other factors that affect performance on the SAT.Then,we wouldcollect information on SAT sco

40、re for each student in the experiment,yielding a data set(sati,hoursi):i 1,.,n,where n is the number of students we can afford to have in the study.From equation(2.7),we should try to get as much variation inhoursias is feasible.(ii)Here are three factors:innate ability,family income,and general hea

41、lth on the day ofthe exam.If we think students with higher native intelligence think they do not need to preparefor the SAT,then ability and hours will be negatively correlated.Family income wouldprobably be positively correlated with hours,because higher income families can more easilyafford prepar

42、ation courses.Ruling out chronic health problems,health on the day of the examshould be roughly uncorrelated with hours spent in a preparation course.(iii)If preparation courses are effective,1should be positive:other factors equal,anincrease in hours should increase sat.(iv)The intercept,0,has a us

43、eful interpretation in this example:because E(u)=0,0isthe average SAT score for students in the population with hours=0.CHAPTER 33.13.1(i)hsperc is defined so that the smaller it is,the lower the students standing in high school.Everything else equal,the worse the students standing in high school,th

44、e lower is his/herexpected college GPA.(ii)Just plug these values into the equation:nncolgpa=1.392 .0135(20)+.00148(1050)=2.676.(iii)The difference between A and B is simply 140 times the coefficient on sat,becausehsperc is the same for both students.So A is predicted to have a score.00148(140).207h

45、igher.(iv)With hsperc fixed,colgpa=.00148sat.Now,we want to find sat such thatcolgpa=.5,so.5=.00148(sat)or sat=.5/(.00148)338.Perhaps not surprisingly,alarge ceteris paribus difference in SAT score almost two and one-half standard deviations isneeded to obtain a predicted difference in college GPA o

46、r a half a point.3.23.2(i)Yes.Because of budget constraints,it makes sense that,the more siblings there are ina family,the less education any one child in the family has.To find the increase in the numberof siblings that reduces predicted education by one year,we solve 1=.094(sibs),so sibs=1/.09410.

47、6.(ii)Holding sibs and feduc fixed,one more year of mothers education implies.131 yearsmore of predicted education.So if a mother has four more years of education,her son ispredicted to have about a half a year(.524)more years of education.(iii)Since the number of siblings is the same,but meduc and

48、feduc are both different,thecoefficients on meduc and feduc both need to be accounted for.The predicted difference ineducation between B and A is.131(4)+.210(4)=1.364.3.33.3(i)If adults trade off sleep for work,more work implies less sleep(other things equal),so1 0.(ii)The signs of2and3are not obvio

49、us,at least to me.One could argue that moreeducated people like to get more out of life,and so,other things equal,they sleep less(2 0,2 0.Both LSAT and GPA are measures of the quality of the enteringclass.No matter where better students attend law school,we expect them to earn more,onaverage.3,4 0.T

50、he number of volumes in the law library and the tuition cost are bothmeasures of the school quality.(Cost is less obvious than library volumes,but should reflectquality of the faculty,physical plant,and so on.)(iii)This is just the coefficient on GPA,multiplied by 100:24.8%.(iv)This is an elasticity

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