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1、STUDENT SOLUTIONSMANUAL Jeffrey M.WooldridgeIntroductory Econometrics:A Modern Approach,4eCONTENTSPreface iv Chapter 1 Introduction 1 Chapter 2 The Simple Regression Model 3Chapter 3 Multiple Regression Analysis:Estimation 9 Chapter 4 Multiple RegressionAnalysis:Inference 17 Chapter 5 Multiple Regre
2、ssion Analysis:OLS Asymptotics 24Chapter 6 Multiple Regression Analysis:Further Issues 27 Chapter 7 MultipleRegression Analysis With Qualitative 34 Information:Binary(or Dummy VariablesChapter 8 Heteroskedasticity 42 Chapter 9 More on Specification and DataProblems 47 Chapter 10 Basic Regression Ana
3、lysis With Time Series Data 52 Chapter 11Further Issues in Using OLS With Time Series Data 58 Chapter 12 Serial Correlation andHeteroskedasticity in 65 Time Series RegressionsChapter 13 Pooling Cross Sections Across Time.Simple 71 Panel Data MethodsChapter 14 Advanced Panel Data Methods 78 Chapter 1
4、5 Instrumental VariablesEstimation and Two Stage 85 Least SquaresChapter 16 Simultaneous Equations Models 92 Chapter 17 Limited DependentVariable Models and Sample 99 Selection CorrectionsChapter 18 Advanced Time Series Topics 110iiAppendix A Basic Mathematical Tools 117 Appendix B Fundamentals ofPr
5、obability 119 Appendix C Fundamentals of Mathematical Statistics 120 Appendix DSummary of Matrix Algebra 122 Appendix E The Linear Regression Model in MatrixForm 123iiiPREFACEThis manual contains solutions to the odd-numbered problems and computerexercises in Introductory Econometrics:A Modern Appro
6、ach,4e.Hopefully,you willfind that the solutions are detailed enough to act as a study supplement to the text.Ratherthan just presenting the final answer,I usually provide detailed steps,emphasizing wherethe chapter material is used in solving the problems.Some of the answers given here are subjecti
7、ve,and you or your instructor may haveperfectly acceptable alternative answers or opinions.I obtained the solutions to the computer exercises using Stata,starting with version4.0and ending with version 9.0.Nevertheless,almost all of the estimation methodscovered in the text have been standardized,an
8、d different econometrics or statisticalpackages should give the same answers to the reported degree of accuracy.There can bedifferences when applying more advanced techniques,as conventions sometimes differon how to choose or estimate auxiliary parameters.(Examples include heteroskedasticity-robust
9、standard errors,estimates of a random effects model,and corrections for sampleselection bias.Any differences in estimates or test statistics should be practicallyunimportant,provided you are using a reasonably large sample size.While I haveendeavored to make the solutions free of mistakes,some error
10、s may have crept in.Iwould appreciate hearing from students who find mistakes.I will keep a list would alsolike to hear from students who have suggestions for improving either the solutions or theproblems themselves.I can be reached via e-mail at wooldril.msu.edu.I hope that youfind this solutions m
11、anual helpful when used in conjunction with the text.I look forwardto hearing from you.Jeffrey M.WooldridgeDepartment of EconomicsMichigan State University110 Marshall-Adams HallEast Lansing,MI 48824-1038ivCHAPTER 1SOLUTIONS TO PROBLEMS1.1 It does not make sense to pose the question in terms of caus
12、ality.Economistswould assume that students choose a mix of studying and working(and other activities,such as attending class,leisure,and sleeping based on rational behavior,such asmaximizing utility subject to the constraint that there are only 168 hours in a week.Wecan then use statistical methods
13、to measure the association between studying and working,including regression analysis that we cover starting in Chapter 2.But we would not beclaiming that one variable“causes“the other.They are both choice variables of thestudent.1.2(i Ideally,we could randomly assign students to classes of differen
14、t sizes.That is,each student is assigned a different class size without regard to any student characteristicssuch as ability and family background.For reasons we will see in Chapter 2,we wouldlike substantial variation in class sizes(subject,of course,to ethical considerations andresource constraint
15、s,(ii A negative correlation means that larger class size is associatedwith lower performance.We might find a negative correlation because larger class sizeactually hurts performance.However,with observational data,there are other reasons wemight find a negative relationship.For example,children fro
16、m more affluent familiesmight be more likely to attend schools with smaller class sizes,and affluent childrengenerally score better on standardized tests.Another possibility is that,within a school,aprincipal might assign the better students to smaller classes.Or,some parents might insisttheir child
17、ren are in the smaller classes,and these same parents tend to be more involvedin their childrens education.(iii Given the potential fbr confounding factors-some of which are listed in(ii-finding a negative correlation would not be strong evidence that smaller class sizesactually lead to better perfo
18、rmance.Some way of controlling for the confounding factorsis needed,and this is the subject of multiple regression analysis.SOLUTIONS TO COMPUTER EXERCISESCl.l(i The average of educ is about 12.6 years.There are two people reporting zeroyears of education,and 19 people reporting 18 years of educatio
19、n.(ii The average of wage is about$5.90,which seems low in the year 2008.(iii Using Table B-60 in the 2004 Economic Report of the President,the CPI was56.9 in 1976 and 184.0 in 2003.(iv To convert 1976 dollars into 2003 dollars,we use the ratio of the CPIs,which is184/56.93.23Therefore,the average h
20、ourly wage in 2003 dollars is roughly3.23($5.90$19.06which is a reasonable figure.1(v The sample contains 252 women(the number of observations with female=1 and274 men.Cl.3(i The largest is 100,the smallest is 0.(ii 38 out of 1,823,or about 2.1 percent of the sample.(iii 17(iv The average of math4 i
21、s about 71.9 and the average of read4 is about 60.1.So,atleast in 2001,the reading test was harder to pass.(v The sample correlation between math4 and read4 is about.843,which is a veryhigh degree of(linear association.Not surprisingly,schools that have high pass rates onone test have a strong tende
22、ncy to have high pass rates on the other test.(vi The average of exppp is about$5,194.87.The standard deviation is$1,091.89,which shows rather wide variation in spending per pupil.The minimum is$1,206.88 andthe maximum is$11,957.64.2CHAPTER 2SOLUTIONS TO PROBLEMS2.2(i Let y i=GPA i,x i=ACT i,and n=8
23、.Then=25.875,=3.2125,1ni(x i-(y i-=5.8125,and 1ni(x i-2=56.875.From equation(2.9,we obtain the slope as 1八=5.8125/56.875 .1022,rounded to four places after the decimal.From(2.17,0八=_ 1=3.2125-(.102225.875.5681.So we can write GPA=.5681+.1022 ACTn=8.The intercept does not have a useful interpretation
24、 because ACT is not close to zerofor 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 givenalong with the observation number i and GPA in the following table:i GPA GPAu 12.8 2.7143.0857 23.4 3.0209.
25、3791 33.0 3.2253-.2253 43.5 3.3275.1725 53.63.5319.0681 63.0 3.1231-.1231 72.7 3.1231-.4231 83.73.6341.0659You can verify that the residuals,as reported in the table,sum to.0002,which ispretty close to zero given the inherent rounding error.(iii When ACT=20,GPA=.5681+.1022(20=2.61.(iv The sum of squ
26、ared residuals,21Xi i u,is about.4347(rounded to four decimal places,and the total sum of squares,1ni(y i-2,is about 1.0288.So the R-squared from theregression isR 2=1-SSR/SST=1 -(.4347/1.0288工.577.Therefore,about 57.7%of the variation in GPA is explained by ACT in this smallsample of students.2.3(i
27、 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.(Income and education are probably positively correlated;age and education may benegatively correlated because women in more recent cohorts
28、 have,on average,moreeducation;and number of siblings and education are probably negatively correlated.(ii Not if the factors we listed in part(i are correlated with educ.Because we wouldlike to hold these factors fixed,they are part of the error term.But if u is correlated witheduc then E(u|educ 0,
29、and so SLR.4 fails.2.4(i We would want to randomly assign the number of hours in the preparationcourse so that hours is independent of other factors that affect performance on the SAT.Then,we would collect information on SAT score for each student in the experiment,yielding a data set(,i i sat hours
30、 i n,where n is the number of students we can afford to havein the study.From equation(2.7,we should try to get as much variation in i hours as isfeasible.(ii Here are three factors:innate ability,family income,and general health on theday of the exam.If we think students with higher native intellig
31、ence think they do notneed to prepare for the SAT,then ability and hours will be negatively correlated.Familyincome would probably be positively correlated with hours,because higher incomefamilies can more easily affordpreparation courses.Ruling out chronic health problems,health on the day of theex
32、am should be roughly uncorrelated with hours spent in a preparation course.(iii If preparation courses are effective,1 should be positive:other factors equal,anincrease in hours should increase sat.(iv The intercept,0,has a useful interpretation in this example:because E(u=0,0 isthe average SAT scor
33、e for students in the population with hours=0.2.5(i When we condition on inc E(u line e line E(e|inc 0 because E(e line=E(e=0.(ii Again,when we condition on inc Var(u line e|inc 2Var(e|inc=2e inc becauseVar(e|inc=2e.(iii Families with low incomes do not have much discretion about spending;typically,
34、a low-income family must spend on food,clothing,housing,and othernecessities.Higher income people have more discretion,and some might choose moreconsumption while others more saving.This discretion suggests wider variability insaving among higher income families.2.8(i We follow the hint,noting that
35、1c y=1c(the sample average of li c y is c 1times the sample average of yi and 2=2c.When we regress c ly i on c 2x i(includingan intercept we use equation(2.19 to obtain the slope:2211121112222221111112221(八.(n ni iiii i iii i ni i i niic x c x c c c c x y c x c c x x y c c c c xFrom(2.17,we obtain t
36、he intercept as 0=(c 1 -1 (c 2=(c 1 -(c 1/c 2 IF(c2=c 1(-1=c 10because the intercept from regressing yi on xi is(-1(ii We use the same approach from part(i along with the fact that 1(=c 1 +and2(=c 2+.Therefore,11(i c y=(c l+y i-(c l+=yi-and(c2+x i-2(c x =x i-.So c 1 and c 2 entirely drop out of the
37、slope formula for the regression of(cl+y i on(c 2+x i,and 1=1八.The intercept is 0=1(c y-12(C X =(c 1 +-1八(c2+=(r+c 1 -c 2/=(T+c 1 -c 21八,which is what we wanted to show.(iii We can simply apply part(ii because 11 log(log(log(i i c y c y.In other words,replace c 1 with log(c 1,y i with log(y i,and se
38、t c 2=0.(iv Again,we can apply part(ii with c 1 =0 and replacing c 2 with log(c 2 and x iwith log(x i.If Or and are the original intercept and slope,then 1 T and 002 llog(c.2.9(i The intercept implies that when inc=0,cons is predicted to be negative$124.84.This,of course,cannot be true,and reflects
39、that fact that this consumptionfunction might be a poor predictor of consumption at very low-income levels.On theother hand,on an annual basis,$124.84 is not so far from zero.(ii Just plug 30,000 into the equation:cons=-124.84+.853(30,000=25,465.16 dollars.(iii The MPC and the APC are shown in the f
40、ollowing graph.Even though theintercept is negative,the smallest APC in the sample is positive.The graph starts at anannual income level of$1,000(in 1970 dollars.SOLUTIONS TO COMPUTER EXERCISESC2.1(i The average prate is about 87.36 and the average mrate is about.732.(ii The estimated equation ispra
41、te=83.05+5.86 mraten=1,534,R 2=.075.(iii The intercept implies that,even if mrate=0,the predicted participation rate is83.05 percent.The coefficient on mrate implies that a one-dollar increase in the matchrate-a fairly large increase-is estimated to increase prate by 5.86 percentage points.This assu
42、mes,ofcourse,that this change prate is possible(if,say,prate is already at 98,thisinterpretation makes no sense.(iv If we plug mrate=3.5 into the equation we get prate=83.05+5.86(3.5=103.59.This is impossible,as we can have at most a 100percent participation rate.This illustrates that,especially whe
43、n dependent variables arebounded,a simple regression model can give strange predictions for extreme values ofthe independent variable.(In the sample of 1,534 firms,only 34 have mrate 3.5.(v mrate explains about 7.5%of the variation in prate.This is not much,andsuggests that many other factors influe
44、nce 401(k plan participation rates.C2.3(i The estimated equation issleep=3,586.4-.151 totwrkn=706,R2=.l()3.The intercept implies that the estimated amount of sleep per week fbr someone whodoes not work is 3,586.4 minutes,or about 59.77 hours.This comes to about 8.5 hoursper night.(ii If someone work
45、s two more hours per week then totwrk=120(because totwrk ismeasured in minutes,and so sleep=-.151(120=-18.12 minutes.This is only a few minutes a night.If someone wereto work one more hour on each of five working days,sleep=-.151(300=-45.3 minutes,or about five minutes a night.C2.5(i The constant el
46、asticity model is a log-log model:log(rd=0+1 log(sales+u,where 1 is the elasticity of rd with respect to sales.(ii The estimated equation islog(rd=-4.105+1.076 log(salesn=32,R2=.91O.The estimated elasticity of rd with respect to sales is 1.076,which is just above one.A one percent increase in sales
47、is estimated to increase rd by about 1.08%.C2.7(i The average gift is about 7.44 Dutch guilders.Out of 4,268 respondents,2,561 did not give a gift,or about 60 percent.(ii The average mailings per year is about 2.05.The minimum value is.25(whichpresumably means that someone has been on the mailing li
48、st for at least four years andthe maximum value is 3.5.(iii The estimated equation is22.01 2.65 4,268,.0138giftmailsyear n R(iv The slope coefficient from part(iii means that each mailing per year is associatedwith-perhaps even causes”-an estimated 2.65 additional guilders,on average.Therefore,if ea
49、ch mailing costs one guilder,the expected profit from each mailing isestimated to be 1.65 guilders.This is only the average,however.Some mailings generateno contributions,or a contribution less than the mailing cost;other mailings generatedmuch more than the mailing cost.(v Because the smallest mail
50、syear in the sample is.25,the smallest predicted valueof gifts is 2.01+2.65(.25=2.67.Even if we look at the overall population,where somepeople have received no mailings,the smallest predicted value is about two.So,with thisestimated equation,we never predict zero charitable gifts.9CHAPTER 3SOLUTION