Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition

Preview (16 of 317 Pages)

100%

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 1 preview image

Loading page ...

1© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, inwhole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on apassword-protected website or school-approved learning management system for classroom use.CHAPTER 1The Nature of Econometrics and Economic DataTable of ContentsTeaching notes2Solutions to Problems3Solutions to Computer Exercises4

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 2 preview image

Loading page ...

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 3 preview image

Loading page ...

2© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, inwhole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on apassword-protected website or school-approved learning management system for classroom use.TEACHINGNOTESYou havesubstantial latitude about what to emphasize in Chapter 1. I find it useful to talkaboutthe economics of crime example (Example 1.1) and the wage example (Example 1.2) sothatstudents see, at the outset, that econometrics is linked to economic reasoning, even iftheeconomics is not complicatedtheory.I like to familiarize students with the important data structures that empirical economistsuse,focusing primarily on cross-sectional and time series data sets, as these are what I cover inafirst-semester course.Itis probably a good idea to mention the growing importance of datasetsthat have both a cross-sectional andatimedimension.I spend almost an entire lecture talking about the problems inherent in drawing causalinferencesin the social sciences. I do this mostly through the agricultural yield, return to education,andcrimeexamples. Theseexamples also contrast experimental and nonexperimental(observational)data. Students studying business and finance tend to find the termstructure of interestratesexample 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 thestudentshow econometrics can be used. (And, it is more enjoyable for the students andme.)I do not use counterfactual notation as in the modern “treatment effects” literature, but Idodiscuss causality using counterfactual reasoning. The return to education, perhaps focusingonthe return to getting a college degree, is a good example of how counterfactual reasoningiseasily incorporated into the discussion ofcausality.

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 4 preview image

Loading page ...

3© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, inwhole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on apassword-protected website or school-approved learning management system for classroom use.SOLUTIONS TOPROBLEMS1.1(i) Ideally, we could randomly assign students to classes of different sizes. That is,eachstudent isassigned a different class size without regard to any student characteristics suchasability and family background. For reasons we will see in Chapter 2, we would likesubstantialvariation in class sizes (subject, of course, to ethicalconsiderations and resourceconstraints).(ii)A negative correlation means thatalarger class size is associated with lowerperformance.We might find a negative correlation becausealarger class size actually hurtsperformance.However, with observational data, there are other reasons we might find anegativerelationship. For example, children from more affluent families might be more likely toattend schoolswith smaller class sizes, and affluent children generallymightscore better onstandardized tests.Another possibility is that, within a school, a principal might assign thebetter students to smallerclasses. Or, some parents might insist their childrento be placed insmaller classes, and these sameparents tend to be more involved in their children’seducation.(iii)Given the potential for confounding factors–some of which are listed in (ii)–findinganegative correlation would not be strong evidence that smaller class sizes actually lead tobetterperformance. Some way of controlling for the confounding factors is needed, and this isthesubject of multiple regressionanalysis.1.2(i) Here is one way to pose the question: If two firms, sayAandB, are identical inallrespects except that firmAsupplies job training one hour per worker more than firmB, byhowmuch 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 inaparticular job. Firms might even discriminate based on age, gender, or race. Perhapsfirmschoose to offer training to more or less able workers, where “ability” might be difficulttoquantify but where a manager has some idea about the relative abilities of differentemployees.Moreover, different kinds of workers might be attracted to firms that offer more job trainingonaverage, and this might not be evident toemployers.(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 havedifferent outputsifthey use different amounts of capital or technology. The quality of managers would also haveaneffect.(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 betweenoutputandtrainingevenifjob training does not improve workerproductivity.1.3Itdoes not make sense to pose the question in terms of causality. Economists wouldassumethatstudents choose a mix of studying and working (and other activities, such as attendingclass,leisure, and sleeping) based on rational behavior, such as maximizing utility subject tothe

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 5 preview image

Loading page ...

4© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, inwhole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on apassword-protected website or school-approved learning management system for classroom use.constraint that there are only 168 hours in a week. We can then use statistical methodstomeasure the association between studying and working, including regression analysis, whichwecover starting in Chapter 2. But we would not be claiming that one variable “causes” theother.They are both choice variables of thestudent.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 fora controlled experiments.(iii) No,the correlation analysis between GSP growth and tax ratesisnot likely to be convincingasthe tax rateshave a significant negative effect on gross state products even after controllingfactors like expenditure, fluctuations in the business, control in the supply of money, andso on.SOLUTIONS TO COMPUTEREXERCISESC1.1(i) The average ofeducis about 12.6 years. There are two people reporting zero yearsofeducation and 19 people reporting 18 years ofeducation.(ii)The average ofwagein the sampleis about $5.90, which seems low.(iii)Using Table B-60 in the 2004Economic Report of the President, the CPI was 56.9in1976 and233in2013.(iv)To convert 1976 dollars into 2013 dollars, we use the ratio of theCPIs, whichis233/ 56.94.09.Therefore, the average hourly wage in2013dollars isroughly4.09($5.90)$24.13, which is a reasonablefigure.(v)The sample contains 252 women (the number of observations withfemale= 1) and274men.C1.2(i) There are 1,388 observations in the sample. Tabulating the variablecigsshows that212women havecigs>0.(ii)The average ofcigsis about 2.09, but thisincludes the 1,176 women who didnotsmoke. Reporting just the average masks the fact that almost 85 percent of the women didnotsmoke.Itmakes more sense to say that the “typical” woman does not smoke duringpregnancy;indeed, the median number ofcigarettes smoked iszero.(iii)The average ofcigsover the women withcigs> 0 is about 13.7. Of course,thisismuch higher than the average over the entire sample because we are excluding 1,176zeros.(iv)The average offatheducis about 13.2. There are196 observations with amissing

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 6 preview image

Loading page ...

5© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, inwhole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on apassword-protected website or school-approved learning management system for classroom use.value forfatheduc, and those observations are necessarily excluded in computing theaverage.(v)The average and standard deviation offamincare about 29.027 and18.739,respectively, butfamincis measured in thousands of dollars. So, in dollars, the averageandstandard deviation are $29,027 and$18,739.C1.3(i) The largest is 100, the smallest is0.(ii)289out of 1,823, or about15.85percent of thesample.(iii)17(iv)The average ofmath4is about 71.9 and the average ofread4is about 60.1. So, atleastin 2001, the reading test was harder topass.(v)The sample correlation betweenmath4andread4is about .843, which is a veryhighdegree of (linear) association. Notsurprisingly, schools that have high pass rates on onetesthave a strong tendency to have high pass rates on the othertest.(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 andthemaximum is$11,957.64.](vii) The percentage by which school A outspends school B is(vii)The percentage by which school A outspends school Bis(6,000 −5,500)100∙5,500≈9.09%.

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 7 preview image

Loading page ...

6© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, inwhole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise onapassword-protected website or school-approved learning management system for classroom use.When we use the approximation based on the differenceinthe natural logs we get asomewhatsmallernumber: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 receivingtraining.(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 bigdifference.(iv)The differences in earnings and unemployment rates suggest the training programhad strong, positive effects. Our conclusions about economic significance would be strongerifwe could also establish statisticalsignificance (which is done in Computer Exercise C9.10in Chapter9).C1.5(i) The smallest and largest values ofchildrenare 0 and 13, respectively. The averageisabout2.27.(ii)Out of 4,358 women, only 611 have electricity in the home, or about 14.02percent.(iii)The average ofchildrenfor women without electricity is about 2.33, and for thosewithelectricity it is about 1.90. So, on average, women with electricity have .43 fewer childrenthanthose who donot.(iv)We cannot infer causality here. Thereare many confounding factors that may berelatedto the number of children and the presence of electricity in the home; householdincomeand level of education are two possibilities. For example, it could be that women withmoreeducation have fewer children and are more likely to have electricity in the home (thelatterdue to an incomeeffect).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 is3. 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

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 8 preview image

Loading page ...

7© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website, inwhole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise onapassword-protected website or school-approved learning management system for classroom use.between them.(iv)No, more executionsdonot 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. Theemploymentrate is 24.3.(ii) Theemploymentrateof menwho abuse alcohol is 22.6.(iii) Theemploymentrate who do not abuse alcohol is24.5.(iv)The employment ratesof men who abuse alcohol and who do not are 22.6 and 24.5,respectively. The differenceintheseemployment rates is very less, which means that alcoholabuse does not cause unemployment.C1.8(i) There are 856 students in the sample. Of these, 317 students report having taken aneconomics course in high school.(ii) For those who took an economics course in high school, the average score is 72.08. Forthose who did not take an economics course in high school, the average score is 72.91.(iii) Comparing averages tells us very little about the causal effect of having taken a economicscourse in high school on performance in university level economics. There are many factorsthat influence a student’s performance in an economics course and some of these factors arelikely correlated with whether or not they took economics in high school (and subsequentlychose to take economics in university). Without controlling for these factors, we cannot reallysay anything about the causal effect of highschool economics exposure on university leveleconomics performance. Though there is a small (negative) correlation between the twovariables, there is little evidence of causation from simply comparing performance betweenthese two groups.(iv) Ideally we would have an experiment in which we could observe the same individuals intwo different scenarios: one in which they took high school economics and one in which theydid not. As this is not feasible, we would like to randomly select students into two groups: acontrol group that did not take economics in high school and a treatment group that did takeeconomics in high school. Even this would be challenging as it would require selection intotreatment and control when the students were in high school. Thus, the most feasibleexperiment would be to compare the difference in scores for students that are as similar aspossible in every way except for their exposure to high school economics. Effectively, wewould need to control for all of the other factors that could influence both performance inuniversity level economics and exposure to high school economics.

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 9 preview image

Loading page ...

8© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website,in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwiseon a password-protectedwebsite or school-approved learning management system for classroom use.CHAPTER 2The Simple Regression ModelTable of ContentsTeaching notes9Solutions to Problems10Solutions to Computer Exercises17

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 10 preview image

Loading page ...

9© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website,in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwiseon a password-protectedwebsite or school-approved learning management system for classroom use.TEACHING NOTESThis is the chapter where I expectstudents 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 the𝑥𝑖is 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.

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 11 preview image

Loading page ...

10© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website,in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwiseon a password-protectedwebsite or school-approved learning management system for classroom use.SOLUTIONS TO PROBLEMS2.1(i) Income, age, and family background (such as number of siblings) are just a fewpossibilities. It seems that each ofthese 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=0+1x+u, add and subtract0from the right hand side to gety= (0+0) +1x+ (u0). Call the new errore=u0, so that E(e)= 0. The new intercept is0+0, but the slope is still1.2.3(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 as1ˆ=5.8125/56.875.1022, rounded to four places after the decimal. From (2.17),0ˆ=y–1ˆx3.2125–(.1022)25.875.5681. So we can writeGPA= .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:

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 12 preview image

Loading page ...

11© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website,in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwiseon a password-protectedwebsite or school-approved learning management system for classroom use.iGPAGPAˆu12.82.7143.085723.43.0209.379133.03.2253–.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.

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 13 preview image

Loading page ...

12© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website,in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwiseon a password-protectedwebsite or school-approved learning management system for classroom use.(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 reflectsthefact 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 citychoosesto 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

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 14 preview image

Loading page ...

13© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website,in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwiseon a password-protectedwebsite or school-approved learning management system for classroom use.(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)=2eincbecause Var(e|inc)=2e.(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),1=1niiix y/21niix.Plugging inyi=0+1xi+uigives1=011()niiiixxu/21niix.After standard algebra, the numerator can be written as201111innniiiiiixxx u.Putting this over the denominator we can write1as1=01niix/21niix+1+1niiix u/21niix.Conditional on thexi, we haveE(1) =01niix/21niix+1

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 15 preview image

Loading page ...

14© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website,in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwiseon a password-protectedwebsite or school-approved learning management system for classroom use.because E(ui) = 0 for alli. Therefore, the bias in1is given by the first term in this equation.This bias is obviously zero when0= 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 anintercept.(ii) From the last expression for1in part (i) we have, conditional on thexi,Var(1)=221niixVar1niiix u=221niix21Var()niiixu=221niix221niix=2/21niix.(iii) From (2.57), Var(1ˆ) =2/21()niixx. From the hint,21niix21()niixx, and soVar(1)Var(1ˆ). A more direct way to see this is to write21()niixx=221()niixn x,which is less than21niixunlessx= 0.(iv) For a given sample size, the bias in1increases asxincreases (holding the sum of the2ixfixed). But asxincreases, the variance of1ˆincreases relative to Var(1). The bias in1is also small when0is small. Therefore, whether we prefer1or1ˆon a mean squared errorbasis depends on the sizes of0,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:

Class Notes for Introductory Econometrics: A Modern Approach , 7th Edition - Page 16 preview image

Loading page ...

15© 2016 Cengage Learning®. May not be scanned, copied or duplicated, or posted to a publicly accessible website,in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwiseon a password-protectedwebsite or school-approved learning management system for classroom use.From (2.17), we obtain the intercept as0= (c1y)–1(c2x)= (c1y)–[(c1/c2)1ˆ](c2x)=c1(y–1ˆx)=c10ˆ) because the intercept from regressingyionxiis (y–1ˆx).(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), and1=1ˆ. The intercept is0=1()cy–12()cx= (c1+y)–1ˆ(c2+x)= (1ˆyx)+c1–c21ˆ=0ˆ+c1–c21ˆ, 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) andxiwith log(xi).If01ˆˆandare the original intercept and slope, then11ˆand0021ˆˆlog()c.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[()] ,uuwe show that the latter is zero. But, from part (i),1111ˆE[() ] =EE().nniiiiiiuw uuwu uBecause theiuare pairwise uncorrelated(they are independent),22E()E(/)/iiu uunn(becauseE()0,ihu uih). Therefore,(iii) The formula for the OLS intercept isand plugging in01yxugives0011011ˆˆˆ()() .xuxux22111E()(/)(/)0.nnniiiiiiiwu uwnnw

Preview Mode

This document has 317 pages. Sign in to access the full document!

Report

Study Now!

Document Details

Related Documents

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

Solution Manual for Introductory Econometrics: A Modern Approach , 7th Edition

Solution Manual for Introductory Econometrics: A Modern Approach, 5th Edition

Economics - Theory of the Firm

Economics - Theory of the Consumer

Economics - Perfect Competition

Economics - Monopoly

Economics - Monopolistic Competition and Oligopoly

Economics - Money and Banking

Economics - Labor Market

Company

Explore

Study Tools