计量经济学答案.pdf

CHAPTER 1 TEACHING NOTES You have substantial latitude about what to emphasize in Chapter 1.I find it useful to talk about the economics of crime example(Example 1.1)and the wage example(Example 1.2)so that students see,at the outset,that econometrics is linked to economic reasoning,if not economic theory.I like to familiarize students with the important data structures that empirical economists use,focusing primarily on cross-sectional and time series data sets,as these are what I cover in a first-semester course.It is probably a good idea to mention the growing importance of data sets that have both a cross-sectional and time dimension.I spend almost an entire lecture talking about the problems inherent in drawing causal inferences in the social sciences.I do this mostly through the agricultural yield,return to education,and crime examples.These examples also contrast experimental and nonexperimental data.Students studying business and finance tend to find the term structure of interest rates example more relevant,although the issue there is testing the implication of a simple theory,as opposed to inferring causality.I have found that spending time talking about these examples,in place of a formal review of probability and statistics,is more successful(and more enjoyable for the students and me).3CHAPTER 2 TEACHING NOTES This is the chapter where I expect students to follow most,if not all,of the algebraic derivations.In class I like to derive at least the unbiasedness of the OLS slope coefficient,and usually I derive the variance.At a minimum,I talk about the factors affecting the variance.To simplify the notation,after I emphasize the assumptions in the population model,and assume random sampling,I just condition on the values of the explanatory variables in the sample.Technically,this is justified by random sampling because,for example,E(ui|x1,x2,xn)=E(ui|xi)by independent sampling.I find that students are able to focus on the key assumption SLR.3 and subsequently take my word about how conditioning on the independent variables in the sample is harmless.(If you prefer,the appendix to Chapter 3 does the conditioning argument carefully.)Because statistical inference is no more difficult in multiple regression than in simple regression,I postpone inference until Chapter 4.(This reduces redundancy and allows you to focus on the interpretive differences between simple and multiple regression.)You might notice how,compared with most other texts,I use relatively few assumptions to derive the unbiasedness of the OLS slope estimator,followed by the formula for its variance.This is because I do not introduce redundant or unnecessary assumptions.For example,once SLR.3 is assumed,nothing further about the relationship between u and x is needed to obtain the unbiasedness of OLS under random sampling.4SOLUTIONS TO PROBLEMS 2.1(i)Income,age,and family background(such as number of siblings)are just a few possibilities.It seems that each of these could be correlated with years of education.(Income and education are probably positively correlated;age and education may be negatively correlated because women in more recent cohorts have,on average,more education;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 would like to hold these factors fixed,they are part of the error term.But if u is correlated with educ then E(u|educ)0,and so SLR.3 fails.2.2 In the equation y=0+1x+u,add and subtract 0 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 is 0+0,but the slope is still 1.2.3(i)Let yi=GPAi,xi=ACTi,and n=8.Then x=25.875,y=3.2125,(x1ni=i x)(yi y)=5.8125,and(x1ni=i x)2=56.875.From equation(2.9),we obtain the slope as 1=5.8125/56.875 .1022,rounded to four places after the decimal.From(2.17),0=y 1x 3.2125 (.1022)25.875 .5681.So we can write =.5681+.1022 ACT?GPA n=8.The intercept does not have a useful interpretation because ACT is not close to zero for the population of interest.If ACT is 5 points higher,increases by.1022(5)=.511.?GPA (ii)The fitted values and residuals rounded to four decimal places are given along with the observation number i and GPA in the following table:i GPA?GPA u12.8 2.7143.085723.4 3.0209.379133.0 3.2253.225343.5 3.3275.172553.6 3.5319.068163.0 3.1231.123172.7 3.1231.423183.7 3.6341.0659 You can verify that the residuals,as reported in the table,sum to.0002,which is pretty close to zero given the inherent rounding error.5 (iii)When ACT=20,=.5681+.1022(20)GPA 2.61.(iv)The sum of squared residuals,21niiu=,is about.4347(rounded to four decimal places),and the total sum of squares,(y1ni=i y)2,is about 1.0288.So the R-squared from the regression is R2 =1 SSR/SST 1 (.4347/1.0288).577.Therefore,about 57.7%of the variation in GPA is explained by ACT in this small sample of students.2.4(i)When cigs=0,predicted birth weight is 119.77 ounces.When cigs=20,=109.49.This is about an 8.6%drop.?bwght (ii)Not necessarily.There are many other factors that can affect birth weight,particularly overall health of the mother and quality of prenatal care.These could be correlated with cigarette smoking during birth.Also,something such as caffeine consumption can affect birth weight,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,or about 10 cigarettes!This is nonsense,of course,and it shows what happens when we are trying to predict something as complicated as birth weight with only a single explanatory variable.The largest predicted birth weight is necessarily 119.77.Yet almost 700 of the births 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%.2.5(i)The intercept implies that when inc=0,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 poor predictor 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:=124.84+.853(30,000)=25,465.16 dollars.?cons (iii)The MPC and the APC are shown in the following graph.Even though the intercept is negative,the smallest APC in the sample is positive.The graph starts at an annual income level of$1,000(in 1970 dollars).6 inc1000100002000030000.7.728.853APCMPC.9APCMPC 2.6(i)Yes.If living closer to an incinerator depresses housing prices,then being farther away increases housing prices.(ii)If the city chose to locate the incinerator in an area away from more expensive neighborhoods,then log(dist)is positively correlated with housing quality.This would violate SLR.3,and OLS estimation is biased.(iii)Size of the house,number of bathrooms,size of the lot,age of the home,and quality of the 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).2.7(i)When we condition on inc in computing an expectation,inc becomes a constant.So E(u|inc)=E(ince|inc)=incE(e|inc)=inc0 because E(e|inc)=E(e)=0.(ii)Again,when we condition on inc in computing a variance,inc becomes a constant.So Var(u|inc)=Var(ince|inc)=(inc)2Var(e|inc)=2einc because Var(e|inc)=2e.(iii)Families with low incomes do not have much discretion about spending;typically,a low-income family must spend on food,clothing,housing,and other necessities.Higher income people have more discretion,and some might choose more consumption while others more saving.This discretion suggests wider variability in saving among higher income families.2.8(i)From equation(2.66),71%=1niiix y=/21niix=.Plugging in yi=0+1xi+ui gives 1%=011()iniiixxu=+21nii/x=.After standard algebra,the numerator can be written as 201111innniiiiiixxx=+u.Putting this over the denominator shows we can write 1%as 1%=01niix=/21niix=+1+1niiixu=/21niix=.Conditional on the xi,we have E(1%)=01niix=/21niix=+1 because E(ui)=0 for all i.Therefore,the bias in 1%is given by the first term in this equation.This bias is obviously zero when 0=0.It is also zero when 1niix=0,which is the same as x=0.In the latter case,regression through the origin is identical to regression with an intercept.(ii)From the last expression for 1%in part(i)we have,conditional on the xi,Var(1%)=Var221niix=1niiixu=221niix=21Var()niiixu=221niix=221niix=2/21niix=.(iii)From(2.57),Var(1)=2/21()niixx=.From the hint,21niix=21(nii)xx=,and so Var(1%)Var(1).A more direct way to see this is to write 21()niixx=21()nii2xn x=,which is less than 21niix=unless x=0.8(iv)For a given sample size,the bias in 1%increases as x increases(holding the sum of the 2ix fixed).But as x increases,the variance of 1increases relative to Var(1%).The bias in 1%is also small when 0 is small.Therefore,whether we prefer 1%or 1 on a mean squared error basis depends on the sizes of 0,x,and n(in addition to the size of 21niix=).2.9(i)We follow the hint,noting that 1c y=1c y (the sample average of is c1ic y1 times the sample average of yi)and 2c x=2c x.When we regress c1yi on c2xi(including an intercept)we use equation(2.19)to obtain the slope:?22111 2111222221111112221()()()()()()().()nniiiiiinniiiiniiiniic xc x c yc yc c xxyyc xc xcxxxxyyccccxx=2)From(2.17),we obtain the intercept as 0%=(c1y)1%(c2x)=(c1y)(c1/c2)1(c2x)=c1(y 1x)=c10)because the intercept from regressing yi on xi is(y 1x).(ii)We use the same approach from part(i)along with the fact that 1(cy+)=c1+y and 2(cx+)=c2+x.Therefore,11()(icycy+)=(c1 +yi)(c1+y)=yi y and(c2+xi)2(cx+)=xi x.So c1 and c2 entirely drop out of the slope formula for the regression of(c1+yi)on(c2+xi),and 1%=1.The intercept is 0%=1()cy+1%2(cx)+=(c1+y)1(c2+x)=(1yx)+c1 c21=0+c1 c21,which is what we wanted to show.(iii)We can simply apply part(ii)because 11log()log()log()ic ycyi=+.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).If 0垐 and 1 are the original intercept and slope,then 11=%and 002垐log()c1=%.SOLUTIONS TO COMPUTER EXERCISES 2.10(i)The average prate is about 87.36 and the average mrate is about.732.(ii)The estimated equation is 9?prate=83.05+5.86 mrate n=1,534,R2=.075.(iii)The intercept implies that,even if mrate=0,the predicted participation rate is 83.05 percent.The coefficient on mrate implies that a one-dollar increase in the match rate a fairly large increase is estimated to increase prate by 5.86 percentage points.This assumes,of course,that this change prate is possible(if,say,prate is already at 98,this interpretation 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 100 percent participation rate.This illustrates that,especially when dependent variables are bounded,a simple regression model can give strange predictions for extreme values of the 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,and suggests that many other factors influence 401(k)plan participation rates.2.11(i)Average salary is about 865.864,which means$865,864 because salary is in thousands of dollars.Average ceoten is about 7.95.(ii)There are five CEOs with ceoten=0.The longest tenure is 37 years.(iii)The estimated equation is =6.51+.0097 ceoten?log()salary n=177,R2=.013.We obtain the approximate percentage change in salary given ceoten=1 by multiplying the coefficient on ceoten by 100,100(.0097)=.97%.Therefore,one more year as CEO is predicted to increase salary by almost 1%.2.12(i)The estimated equation is =3,586.4 .151 totwrk?sleep n=706,R2=.103.The intercept implies that the estimated amount of sleep per week for someone who does not work is 3,586.4 minutes,or about 59.77 hours.This comes to about 8.5 hours per night.(ii)If someone works two more hours per week then totwrk=120(because totwrk is measured in minutes),and so=.151(120)=18.12 minutes.This is only a few minutes a night.If someone were to work one more hour on each of five working days,=?sleep?sleep.151(300)=45.3 minutes,or about five minutes a night.2.13(i)Average salary is about$957.95 and average IQ is about 101.28.The sample standard deviation of IQ is about 15.05,which is pretty close to the population value of 15.10 (ii)This calls for a level-level model:=116.99+8.30 IQ?wage n=935,R2=.096.An increase in IQ of 15 increases predicted monthly salary by 8.30(15)=$124.50(in 1980 dollars).IQ score does not even explain 10%of the variation in wage.(iii)This calls for a log-level model:?log()wage=5.89+.0088 IQ n=935,R2=.099.If IQ=15 then =.0088(15)=.132,which is the(approximate)proportionate change in predicted wage.The percentage increase is therefore approximately 13.2.?log()wage 2.14(i)The constant elasticity model is a log-log model:log(rd)=0+1log(sales)+u,where 1 is the elasticity of rd with respect to sales.(ii)The estimated equation is =4.105+1.076 log(sales)?log()rd n =32,R2 =.910.The estimated elasticity of rd with respect to sales is 1.076,which is just above one.A one percent increase in sales is estimated to increase rd by about 1.08%.11CHAPTER 3 TEACHING NOTES For undergraduates,I do not do most of the derivations in this chapter,at least not in detail.Rather,I focus on interpreting the assumptions,which mostly concern the population.Other than random sampling,the only assumption that involves more than population considerations is the assumption about no perfect collinearity,where the possibility of perfect collinearity in the sample(even if it does not occur in the population)should be touched on.The more important issue is perfect collinearity in the population,but this is fairly easy to dispense with via examples.These come from my experiences with the kinds of model specification issues that beginners have trouble with.The comparison of simple and multiple regression estimates based on the particular sample at hand,as opposed to their statistical properties usually makes a strong impression.Sometimes I do not bother with the“partialling out”interpretation of multiple regression.As far as statistical properties,notice how I treat the problem of including an irrelevant variable:no separate derivation is needed,as the result follows form Theorem 3.1.I do like to derive the omitted variable bias in the simple case.This is not much more difficult than showing unbiasedness of OLS in the simple regression case under the first four Gauss-Markov assumptions.It is important to get the students thinking about this problem early on,and before too many additional(unnecessary)assumptions have been introduced.I have intentionally kept the discussion of multicollinearity to a minimum.This partly indicates my bias,but it also reflects reality.It is,of course,very important for students to understand the potential consequen