Solution Manual For Introductory Econometrics: A Modern Approach, 6th Edition

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1CHAPTER 1The Nature of Econometrics and Economic DataTable of ContentsTeaching notes2Solutions to Problems3Solutions to Computer Exercises4

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2TEACHING NOTESYou have substantial latitude about what to emphasize in Chapter 1. I find it useful to talk aboutthe economics of crime example (Example 1.1) and the wage example (Example 1.2) so thatstudents see, at the outset, that econometrics is linked to economic reasoning, even if theeconomics is not complicated 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 afirst-semester course. It is probably a good idea to mention the growing importance of data setsthat have both a cross-sectional and a time dimension.I spend almost an entire lecture talking about the problems inherent in drawing causal inferencesin the social sciences. I do this mostly through the agricultural yield, return to education, andcrime examples. These examples also contrast experimental and nonexperimental (observational)data. Students studying business and finance tend to find the term structure of interest ratesexample more relevant, although the issue there is testing the implication of a simple theory, asopposed to inferring causality. I have found that spending time talking about these examples, inplace of a formal review of probability and statistics, is more successful in teaching the studentshow econometrics can be used. (And, it is more enjoyable for the students and me.)I do not use counterfactual notation as in the modern “treatment effects” literature, but I dodiscuss causality using counterfactual reasoning. The return to education, perhaps focusing onthe return to getting a college degree, is a good example of how counterfactual reasoning iseasily incorporated into the discussion of causality.

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3SOLUTIONS TO PROBLEMS1.1(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 in class sizes (subject, of course, to ethical considerations and resource constraints).(ii)A negative correlation means that a larger class size is associated with lowerperformance. We might find a negative correlation because a larger class size actually hurtsperformance. However, with observational data, there are other reasons we might find anegative relationship. For example, children from more affluent families might be more likely toattend schools with smaller class sizes, and affluent children generally might score better onstandardized tests. Another possibility is that, within a school, a principal might assign thebetter students to smaller classes. Or, some parents might insist their children to be placed insmaller classes, and these same parents tend to be more involved in their children’s education.(iii)Given the potential for confounding factors – some of which are listed in (ii) – finding anegative correlation 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.1.2(i) Here is one way to pose the question: If two firms, sayAandB, are identical in allrespects except that firmAsupplies job training one hour per worker more than firmB, by howmuch would firmA’s output differ from firmB’s?(ii)Firms are likely to choose job training depending on the characteristics of workers. Someobserved characteristics are years of schooling, years in the workforce, 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.Moreover, 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 have different outputs ifthey use different amounts of capital or technology. The quality of managers would also have aneffect.(iv)No, unless the amount of training is randomly assigned. The many factors listed in parts(ii)and (iii) can contribute to finding a positive correlation betweenoutputandtrainingeven ifjob training does not improve worker productivity.1.3It 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, such as maximizing utility subject to the

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4constraint 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, which wecover starting in Chapter 2. But we would not be claiming that one variable “causes” the other.They are both choice variables of the student.1.4 (i) Experimental data have to be collected to undertake a statistical analysis.(ii) Yes, it is feasible to do a controlled experiment. The factors such as consumption, investment,net exports, and so on, would be required for a controlled experiments.(iii) No, the correlation analysis between GSP growth and tax rates is not likely to be convincingas the tax rates have a significant negative effect on gross state products even after controllingfactors like expenditure, fluctuations in the business, control in the supply of money, and so on.SOLUTIONS TO COMPUTER EXERCISESC1.1(i) The average ofeducis about 12.6 years. There are two people reporting zero years ofeducation and 19 people reporting 18 years of education.(ii)The average ofwagein the sample is about $5.90, which seems low.(iii)Using Table B-60 in the 2004Economic Report of the President, the CPI was 56.9 in1976 and 233 in 2013.(iv)To convert 1976 dollars into 2013 dollars, we use the ratio of the CPIs, which is233 / 56.94.09. Therefore, the average hourly wage in 2013 dollars is roughly4.09($5.90)$24.13, which is a reasonable figure.(v)The sample contains 252 women (the number of observations withfemale= 1) and 274men.C1.2(i) There are 1,388 observations in the sample. Tabulating the variablecigsshows that 212women havecigs> 0.(ii)The average ofcigsis about 2.09, but this includes the 1,176 women who did notsmoke. Reporting just the average masks the fact that almost 85 percent of the women did notsmoke. It makes more sense to say that the “typical” woman does not smoke during pregnancy;indeed, the median number of cigarettes smoked is zero.(iii)The average ofcigsover the women withcigs> 0 is about 13.7. Of course, this ismuch higher than the average over the entire sample because we are excluding 1,176 zeros.(iv)The average offatheducis about 13.2. There are 196 observations with a missing

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5value forfatheduc, and those observations are necessarily excluded in computing the average.(v)The average and standard deviation offamincare about 29.027 and 18.739,respectively, butfamincis measured in thousands of dollars. So, in dollars, the average andstandard deviation are $29,027 and $18,739.C1.3(i) The largest is 100, the smallest is 0.(ii)289 out of 1,823, or about 15.85 percent of the sample.(iii)17(iv)The average ofmath4is about 71.9 and the average ofread4is about 60.1. So, at leastin 2001, the reading test was harder to pass.(v)The sample correlation betweenmath4andread4is about .843, which is a very highdegree of (linear) association. Not surprisingly, schools that have high pass rates on one testhave a strong tendency to have high pass rates on the other test.(vi)The average ofexpppis about $5,194.87. The standard deviation is $1,091.89, whichshows rather wide variation in spending per pupil. [The minimum is $1,206.88 and themaximum is $11,957.64.](vii) The percentage by which school A outspends school B is(vii)The percentage by which school A outspends school B is(6,000 −5,500)100∙5,500≈9.09%.

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6When we use the approximation based on the difference in the natural logs we get a somewhatsmaller number:100 ∙[log(6,000)−log(5,500)] ≈8.71%.C1.4(i) 185/445≈.416 is the fraction of men receiving job training, or about 41.6%.(ii)For men receiving job training, the average ofre78is about 6.35, or $6,350. For mennot receiving job training, the average ofre78is about 4.55, or $4,550. The difference is$1,800, which is very large. On average, the men receiving the job training had earnings about40% higher than those not receiving training.(iii)About 24.3% of the men who received training were unemployed in 1978; the figureis 35.4% for men not receiving training. This, too, is a big difference.(iv)The differences in earnings and unemployment rates suggest the training programhad strong, positive effects. Our conclusions about economic significance would be strongerif we could also establish statistical significance (which is done in Computer Exercise C9.10in Chapter 9).C1.5(i) The smallest and largest values ofchildrenare 0 and 13, respectively. The average isabout 2.27.(ii)Out of 4,358 women, only 611 have electricity in the home, or about 14.02 percent.(iii)The average ofchildrenfor women without electricity is about 2.33, and for those withelectricity it is about 1.90. So, on average, women with electricity have .43 fewer children thanthose who do not.(iv)We cannot infer causality here. There are many confounding factors that may berelated to the number of children and the presence of electricity in the home; householdincome and level of education are two possibilities. For example, it could be that women withmore education have fewer children and are more likely to have electricity in the home (thelatter due to an income effect).C1.6(i) There are 2,197 counties in the dataset. Of these, 1051 counties have zero murders.The percentage of counties having zero executions is 98.6%.(ii) The largest number of murders is 1403. The largest number of executions is 3. The averagenumber of executions is 0.0159, which is small because most of the counties have zeroexecutions.(iii) The correlation betweenmurdersandexecsis 0.21. There is very low positive relationship

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7between them.(iv) No, more executions do not cause more murders to occur. 21% percentage of murdersoccur by executions that took place of people sentenced to death in the given county.C1.7(i) The percentage of men in the sample report abusing alcohol is 9.9. The employmentrate is 24.3.(ii) The employment rate of men who abuse alcohol is 22.6.(iii) The employment rate who do not abuse alcohol is 24.5.(iv) The employment rates of men who abuse alcohol and who do not are 22.6 and 24.5,respectively. The difference in these employment rates is very less, which means that alcoholabuse does not cause unemployment.

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8CHAPTER 2The Simple Regression ModelTable of ContentsTeaching notes9Solutions to Problems10Solutions to Computer Exercises17

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9TEACHING NOTESThis 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, Iderive the variance. At a minimum, I talk about the factors affecting the variance. To simplifythe notation, after I emphasize the assumptions in the population model, and assume randomsampling, 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) byindependent sampling. I find that students are able to focus on the key assumption SLR.4 andsubsequently take my word about how conditioning on the independent variables in the sample isharmless. (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 theinterpretive differences between simple and multiple regression.)You might notice how, compared with most other texts, I use relatively few assumptions toderive 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, onceSLR.4 is assumed, nothing further about the relationship betweenuandxis needed to obtain theunbiasedness of OLS under random sampling.Incidentally, one of the uncomfortable facts about finite-sample analysis is that there is adifference between an estimator that is unbiased conditional on the outcome of the covariates andone that is unconditionally unbiased. If the distribution of the𝑥𝑥𝑖𝑖is such that they can all equalthe same value with positive probability – as is the case with discreteness in the distribution –then the unconditional expectation does not really exist. Or, if it is made to exist, then theestimator is not unbiased. I do not try to explain these subtleties in an introductory course, but Ihave had instructors ask me about the difference.

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10SOLUTIONS TO PROBLEMS2.1(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 education 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 witheduc. Because we would like tohold these factors fixed, they are part of the error term. But ifuis correlated witheduc, thenE(u|educ)≠0, and so SLR.4 fails.2.2In the equationy=β0+β1x+u, add and subtractα0from the right hand side to gety= (α0+β0) +β1x+ (u−α0). Call the new errore=u−α0, so that E(e) = 0. The new intercept isα0+β0, but the slope is stillβ1.2.3(i) Letyi=GPAi,xi=ACTi, andn= 8. Thenx= 25.875,y= 3.2125,1ni=∑(xi–x)(yi–y) =5.8125, and1ni=∑(xi–x)2= 56.875. From equation (2.19), we obtain the slope as1ˆβ=5.8125/56.875≈.1022, rounded to four places after the decimal. From (2.17),0ˆβ=y–1ˆβx≈3.2125 – (.1022)25.875≈.5681. So we can writeGPA= .5681 + .1022ACTn= 8.The intercept does not have a useful interpretation becauseACTis not close to zero for thepopulation of interest. IfACTis 5 points higher,GPAincreases by .1022(5) = .511.(ii) The fitted values and residuals — rounded to four decimal places — are given along withthe observation numberiandGPAin the following table:

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11iGPAGPAˆu12.82.7143.085723.43.0209.379133.03.2253–.225343.53.3275.172553.63.5319.068163.03.1231–.123172.73.1231–.423183.73.6341.0659You can verify that the residuals, as reported in the table, sum to−.0002, which is pretty close tozero given the inherent rounding error.(iii) WhenACT= 20,GPA= .5681 + .1022(20)≈2.61.(iv) The sum of squared residuals,21ˆniiu=∑, is about .4347 (rounded to four decimal places),and the total sum of squares,1ni=∑(yi–y)2, is about 1.0288. So theR-squared from the regressionisR2= 1 – SSR/SST≈1 – (.4347/1.0288)≈.577.Therefore, about 57.7% of the variation inGPAis explained byACTin this small sample ofstudents.2.4(i) Whencigs= 0, predicted birth weight is 119.77 ounces. Whencigs= 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 during birth. Also, something such as caffeine consumption can affect birthweight, and might also be correlated with cigarette smoking.(iii) If we want a predictedbwghtof 125, thencigs= (125 – 119.77)/( –.524)≈–10.18, orabout –10 cigarettes. This is nonsense, of course, and it shows what happens when we are tryingto predict something as complicated as birth weight with only a single explanatory variable. Thelargest predicted birth weight is necessarily 119.77. Yet, almost 700 of the births in the samplehad a birth weight higher than 119.77.

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12(iv) 1,176 out of 1,388 women did not smoke while pregnant, or about 84.7%. Because weare using onlycigsto explain birth weight, we have only one predicted birth weight atcigs= 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.2.5(i) The intercept implies that wheninc= 0,consis predicted to be negative $124.84. This, ofcourse, cannot be true, and reflects the 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:cons= –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).2.6(i) Yes. If living closer to an incinerator depresses housing prices, then being farther awayincreases housing prices.(ii) If the city chooses 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 OLS estimation is biased.inc1000100002000030000.7.728.853APCMPC.9APCMPC

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13(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 withdist[and log(dist)].2.7(i) When we condition onincin computing an expectation,incbecomes a constant. SoE(u|inc) = E(inc⋅e|inc) =inc⋅E(e|inc) =inc⋅0 because E(e|inc) = E(e) = 0.(ii) Again, when we condition onincin computing a variance,incbecomes a constant. SoVar(u|inc) = Var(inc⋅e|inc) = (inc)2Var(e|inc) =2eσincbecause Var(e|inc) =2eσ.(iii) Families with low incomes do not have much discretion about spending; typically, alow-income family must spend on food, clothing, housing, and other necessities. Higher-incomepeople have more discretion, and some might choose more consumption while others moresaving. This discretion suggests wider variability in saving among higher income families.2.8(i) From equation (2.66),1β=1niiix y=∑/21niix=∑.Plugging inyi=β0+β1xi+uigives1β=011()niiiixxuββ=++∑/21niix=∑.After standard algebra, the numerator can be written as201111innniiiiiixxx uββ===++∑∑∑.Putting this over the denominator we can write1βas1β=β01niix=∑/21niix=∑+β1+1niiix u=∑/21niix=∑.Conditional on thexi, we haveE(1β) =β01niix=∑/21niix=∑+β1

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14because E(ui) = 0 for alli. Therefore, the bias in1βis given by the first term in this equation.This bias is obviously zero whenβ0= 0. It is also zero when1niix=∑= 0, which is the same asx= 0. In the latter case, regression through the origin is identical to regression with an intercept.(ii) From the last expression for1βin part (i) we have, conditional on thexi,Var(1β) =221niix−=∑Var1niiix u=∑=221niix−=∑21Var()niiixu=∑=221niix−=∑221niixσ=∑=2σ/21niix=∑.(iii) From (2.57), Var(1ˆβ) =2/21()niixx=−∑. From the hint,21niix=∑≥21()niixx=−∑, and soVar(1β)≤Var(1ˆβ). A more direct way to see this is to write21()niixx=−∑=221()niixn x=−∑, whichis less than21niix=∑unlessx= 0.(iv) For a given sample size, the bias in1βincreases asxincreases (holding the sum of the2ixfixed). But asxincreases, the variance of1ˆβincreases relative to Var(1β). The bias in1βis also small when0βis small. Therefore, whether we prefer1βor1ˆβon a mean squared errorbasis depends on the sizes of0β,x, andn(in addition to the size of21niix=∑).2.9(i) We follow the hint, noting that1c y=1c y(the sample average of1ic yisc1times thesample average ofyi) and2c x=2c x. When we regressc1yionc2xi(including an intercept), weuse equation (2.19) to obtain the slope:

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15From (2.17), we obtain the intercept as0β= (c1y) –1β(c2x) = (c1y) – [(c1/c2)1ˆβ](c2x) =c1(y–1ˆβx) =c10ˆβ) because the intercept from regressingyionxiis (y–1ˆβx).(ii) We use the same approach from part (i) along with the fact that1()cy+=c1+yand2()cx+=c2+x. Therefore,11()()icycy+−+= (c1+yi) – (c1+y) =yi–yand (c2+xi) –2()cx+=xi–x. Soc1andc2entirely drop out of the slope formula for the regression of (c1+yi) on (c2+xi), and1β=1ˆβ. The intercept is0β=1()cy+–1β2()cx+= (c1+y) –1ˆβ(c2+x) = (1ˆyxβ−) +c1–c21ˆβ=0ˆβ+c1–c21ˆβ, which is what we wanted to show.(iii) We can simply apply part (ii) because11log()log()log()iic ycy=+. In other words,replacec1with log(c1), replaceyiwith log(yi), and setc2= 0.(iv) Again, we can apply part (ii) withc1= 0 and replacingc2with log(c2) andxiwithlog(xi). If01ˆˆandββare the original intercept and slope, then11ˆββ=and0021ˆˆlog()cβββ=−.2.10(i) This derivation is essentially done in equation (2.52), once(1/ SST )xis brought insidethe summation (which is valid becauseSSTxdoes not depend oni). Then, just define/ SSTiixwd=.(ii) Because111ˆˆCov(,)E[()] ,uuβββ=−we show that the latter is zero. But, from part (i),()1111ˆE[()] =EE().nniiiiiiuw uuwu uββ==−=∑∑Because theiuare pairwise uncorrelated(they are independent),22E()E(/)/iiu uunnσ==(becauseE()0,ihu uih=≠). Therefore,(iii) The formula for the OLS intercept isand plugging in01yxuββ=++gives0011011ˆˆˆ()() .xuxuxβββββββ=++−=+−−(iv) Because1ˆ anduβare uncorrelated,222222201ˆˆVar()Var()Var()/(/ SST )// SSTxxuxnxnxββσσσσ=+=+=+,which is what we wanted to show.(v) Using the hint and substitution gives()220ˆVar()[ SST /] / SSTxxnxβσ=+()()2122221211/ SST/ SST .nnixixiinxxxnxσσ−−===−+=∑∑22111E()(/)(/)0.nnniiiiiiiwu uwnnwσσ======∑∑∑

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