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

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    i CONTENTS PREFACE iii SUGGESTED COURSE OUTLINES iv Chapter 1 The Nature of Econometrics and Economic Data 1 Chapter 2 The Simple Regression Model 6 Chapter 3 Multiple Regression Analysis: Estimation 1 9 Chapter 4 Multiple Regression Analysis: Inference 34 Chapter 5 Multiple Regression Analysis: OLS Asymptotics 4 8 Chapter 6 Multiple Regression Analysis: Further Issues 54 Chapter 7 Multiple Regression Analysis w ith Qualitative 71 Information: Binary (or Dummy) Variables Chapter 8 Heteroskedasticity 89 Chapter 9 More on Specification and Data Problems 103 Chapter 10 Basic Regression Analysis w ith Time Series Data 117 Chapter 11 Further Issues in U sing OLS w ith Time Series Data 129 Chapter 12 Serial Correlation and Heteroskedasticity in 143 Time Series Regressions Chapter 13 Pooling Cross Sections Across Time . Simple 156 Panel Data Methods Chapter 14 Advanced Panel Data Methods 172 Chapter 15 Instrumental Variables Estimation and Two Stage 187 Least Squares Chapter 16 Simultaneous Equations Models 20 5 Chapter 17 Limited Dependent Variable Models and Sample 219 Selection Corrections

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    ii Chapter 18 Advanced Time Series Topics 243 Chapter 19 Carrying Out an Em pirical Project 259 Appendix A Basic Mathematical Tools 260 Appendix B Fundamentals of Probability 263 Appendix C Fundamentals of Mathema tical Statistics 265 Appendix D Summary of Matrix Algebra 269 Appendix E The Linear Regression Model in Matrix Form 271

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    iii PREFACE This manual contains suggested course outlines, teaching notes, and detailed solutions to all of the problems and computer exercises in Introductory Econometrics: A Modern Approach , 5 e . For several problems , I have added additional notes about interesting asides or suggestions for how to modify or extend the problem. Some of the answers given here are subjective, and you may want to supplement or repl ace them with your own answers. I wrote all solutions as if I were preparing them for the students, so you may find some solutions a bit tedious (if not bordering on an insult to your intelligence ) . This way, if you prefer, you can distribute my answers to some of the even - numbered prob lems directly to the students. (The student study guide contains answers to all odd - numbered problems.) Many of the equations in the Word files were created using MathType, and the equations will not look quite right without MathType. Some equations I have cr eated using the equation editor in Word 2007. I solved the computer exercises using various versions of Stata, starting with version 4.0 and running through version 12 .0. Nevertheless, almost all of the estimation methods covered in the text have been standardized, and different econometrics or statistical packages should give the same answers. There can be differences when applying more advanced techniques, as conventions sometimes differ on how to choose or estimate auxiliary parameters. (Examples include heteroskedasticity - robust standard errors, estimates of a random effects model, and corrections for sample selection bias.) While I have endeavored to make the solutions mistake - free, some errors may have crept in. I would appreciate hearing from you if you find mistakes. I will update the manual occasionally and correct any mistakes that have been found. I heard from many of you regarding the earlier edition s of the text, and I incorporated many of your suggestions. I welcome any comments that will help me make improvements to future editions. I can be reached v ia e - mail at wooldri1@.msu.edu . The fifth edition of the text drops the chapter numbers preceding the problems and computer exercises. I have kept the chapter numbers in the solutions manual so that it is easy to keep track of where one is. For example, the solution to problem 4 in chapter 3 is labeled 3.4 and computer exercise 6 in chapter 8 is labeled C8.6. I hope you find this instructor’s manual useful, and I look forward to hearing your reactions to the fifth edition. Jeffrey M. Wooldridge Department of Economics Michigan State University 486 W. Circle Drive 110 Mar s hall - Adams Hall East Lansing, MI 48824 - 1038

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    iv SUGGESTED COURSE OUTLINES For an introductory, one - semester course, I like to cover most of the material in Chapters 1 through 8 and Chapters 10 through 12, as well as parts of Chapter 9 (but mostly through examples). I do not typically cover all sections or subsections within each chapter. Under the chapter headings listed below, I provide some comments on the material I find most relevant for a first - semester course. An alternative course ignores time series applications altogether, while delving into some of the more advanced methods that are particularly useful for policy analysis. This would consist of Chapters 1 through 8, much of Chapter 9, and the first four sections of Chapter 13. Chapter 9 discusses the practically important topics of proxy variables, measurement error, outlying observations, and stratified sampling. In addition, I have written a more careful description of the method of least absolute deviations, including a discussion of its strengths and weaknesses. Chapter 13 covers, in a straightforward fashion, methods for pooled cross sections (including the so - called “natural experiment” approach) and two - period panel data analysis. The basic cross - sectional treatment of instrumental variables in Chapter 15 is a natural topic for cross - sectional, policy - oriented courses. For an accelerated course, the nonlinear methods used for cross - sectional analysis in Chapter 17 can be covered. I typically do not begin with a review of basic algebra, probability, and statistics. In my experience, this takes too l ong and the payoff is minimal. (Students tend to think that they are taking another statistics course and start to drift away from the material. ) Instead, when I need a tool (such as the summation or expectations operator), I briefly review the necessary d efinitions and key properties. Statistical inference is not more difficult to describe in the context of multiple regression than in test ing about mean a mean from a population, and so I briefly review the principles of statistical inference during multiple regression analysis. Appendices A, B, and C are fairly extensive. When I cover asymptotic properties of OLS, I provide a brief discussion of the main definitions and limit theorems. If students need more than the brief review provided in class, I point them to the appendices. For a master ’ s level course, I include a couple of lectures on the matrix approach to linear regression. This could be integrated into Chapters 3 and 4 or covered after Chapter 4. Again, I do not summarize mat rix algebra before proceeding. Instead, the material in Appendix D can be reviewed as it is needed in covering Appendix E. A second semester course, at either the undergraduate or master ’ s level, could begin with some of the material in Chapter 9, particularly with the issues of proxy va riables and measurement error. Least absolute deviations and, more generally, quantile regression are used more and more in empirical work, and the fifth edition has sections that can be used as an introduction to quantile regression. The advanced chapters, starting with Chapter 13, are particularly useful for students with a n interest in policy analysis. The pooled cross section and panel data chapters (Chapters 13 and 14) emphasize how these data structures can be used, in conjunction with econometric m ethods, for policy evaluation. Chapter 15, which introduces the method of instrumental variables, is also important for policy analysis. Most modern IV applications are used to address the problems of omitted variables (unobserved heterogeneity) or measurement

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    v error. I have intentionally separated out the conceptually more difficult topic of simultaneous equations models in Chapter 16. Chapter 17, in particular the material on probit, logit, Tobit, and Poisson regression models, is a good introduction to nonlinear econometric methods. Specialized courses that emphasize applications in labor economics can use the material on sample selection corrections. Duration models are also briefly covered as an example of a censored regression model. Chapter 18 is much different from the other advanced chapters, as it focuses on more advanced or recent developments in time seri es econometrics. Combined with some of the more advanced topics in Chapter 12, it can serve as the basis for a second semester course in time series topics, including forecasting. Most second semester courses would include an assignment to write an original empirical paper, and Chapter 19 should be helpful in this regard.

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    1 CHAPTER 1 TEACHING NOTES You have substantial latitude about w hat 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, even if the economics 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 cov er 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 infe rences 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 (observational) 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 o pposed 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 in teaching the students how econometrics can be used. ( A nd , it is more enj oyable for the students and me.) I do not use counterfactual notation as in the modern “treatment effects” literature, but I do discuss causality using counterfactual reasoning. The return to education, perhaps focusing on the return to getting a college degree, is a good example of how counterfactual reasoning is easily incorporated into the discussion of causality.

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    2 SOLUTIONS TO PROBLEMS 1.1 (i) Ideally, we could randomly assign students to classes of different sizes. That is, each student is assigned a different class size without regard to any student characteristics such as ability and family background. For reasons we will see in Chapter 2, we would like subst antial variation 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, there are other reasons we might find a negative relationship. For example, children from more affluent families might be more likely to attend schools with smaller class sizes, and affluent children generally score better on standardized tests. Another possibility is that, within a school, a principal might assign the better students to smaller classes. Or, some parents might insist their children are in the smaller 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 a negative correlation would not be strong evidence that smaller class sizes actually lead to better performance. Some way of controlling for the confounding factors is needed, and this is the subject of multiple regression analysis. 1.2 (i) Here i s one way to pose the question: If two firms, say A and B , are identical in all respects except that firm A supplies job training one hour per worker more than firm B , by how much would firm A ’s output differ from firm B ’s? (ii) Firms are likely to choose job training depending on the characteristics of workers. Some observed characteristics are years of schooling, years in the workforce, and experience in a particular job. Firms might even discriminate based on age, gender , or race. Perhaps f irms choose to offer training to more or less able workers, where “ability” might be difficult to quantify but where a manager has some idea about the relative abilit ies of different employees. Moreover, different kinds of workers might be attracted to firms that offer more job training on average, and this might not be evident to employers. (iii) The amount of capital and technology available to wor kers would also affect output. So, two firms with exactly the same kinds of employees would generally have different outputs if they use different amo unts of capital or technology. The quality of managers would also have an effect. (iv) No, unless the amount of training is randomly assigned. The many factors list ed in parts (ii) and (iii) can contribute to finding a positive correlation between output and training even if job training does not improve worker productivity. 1.3 It does not make sense to pose the question in terms of causality. Economists would assume that students choose a mix of studying and working (and other activities, such as attending class , leisure, and sleeping) based on rational behav ior, such as maximizing utility subject to the const r aint that there are only 168 hours in a week. We can then use statistical methods to

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    3 measure the association between studying and working, including regression analysis , which w e cover starting in Chapter 2. But we would not be claiming that one variable “causes” the other. They are both choice variables of the student. SOLUTIONS TO COMPUTER EXERCISES C1.1 (i) The average of educ is about 12.6 years. There are two people reporting zero years of education, and 19 people reporting 18 years of education. (ii) The average of wage is about $5.90, which seems low in the year 200 8 . (iii) Using Table B - 60 in the 2004 Economic Report of the President , the CPI was 56.9 in 1976 and 184.0 in 2003. (iv) To convert 1976 dollars into 2003 dollars, we use the ratio of the CPIs, which is 184/ 56.9 3.23  . Therefore, t he average hourly wage in 2003 dollars is roughly 3.23($5.90) $19.06  , which is a reasonable figure. (v) The sample contains 252 women (the number of observations with female = 1) and 274 men. C1.2 (i) There are 1,3 88 observations in the sample. Tabulating the variable cigs shows that 212 women have cigs > 0. (ii) The average of cigs is about 2.09, but this includes the 1,176 women who did not smoke. Reporting just the average masks the fact that almost 85 perce nt of the women did not smoke. It makes more sense to say that the “typical” woman does not smoke during pregnancy; indeed, the median numbe r of cigarettes smoked is zero. (iii) The average of cigs over the women with cigs > 0 is about 13.7. Of course this is much higher than the average over the entire sample because we are excluding 1,176 zeros. (iv) The average of fatheduc is about 13.2. There are 196 observations with a missing value for fatheduc , and those observations are necessarily excluded in computing the average. (v) The average and standard deviation of faminc are about 29.02 7 and 18.739 , respectively, but faminc is measured in thousands of dollars. So, in dollars, the average and standard deviation are $29,027 and $18,739. C1.3 (i) The l argest is 100, the smallest is 0. (ii) 38 out of 1,823 , or about 2.1 percent of the sample. (iii) 17

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    4 (iv) The average of math4 is about 71.9 and the average of read4 is about 60.1. So, at least in 2001, the reading test was harder to pass. (v) The sample correlation between math4 and read4 is about .843, which is a very high degree of (linear) association. Not surprisingly, schools that have high pass ra tes on one test have a strong tendency to have high pass rates on the other test. (vi) The average of exp p p is about $ 5 , 194.8 7 . The standard deviation is $ 1 , 091.89 , which shows rather wide variation in s pending per pupil . [The minimum is $ 1 , 206.88 and the maximum is $1 1 , 957.64 . ] (vii) The percentage by which school A outspends school B is 100 ∙ ( 6 , 000 − 5 , 500 ) 5 , 500 ≈ 9 . 09% When we use the approximation based on the difference of the natural logs we get a somewhat smaller 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 of re78 is about 6.35, or $6,350. For men not receiving job training, the average of re78 is 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 about 40% higher than those not receiving training. (iii) About 24.3% of the men who received training were unemployed in 1978; the figure is 35.4% for men not receiving training. This, too, is a big difference. (iv) The differences in earnings and unemployment rates suggest the training program had strong, positive effects. Our conclusions about economic significance would be stronger if we could also establish statistical significance (which is done in Computer Exercise C9.10 in Chapter 9). C1.5 (i) The smallest and largest values of children are 0 and 13, respectively. The average is about 2.27. (ii) Out of 4,358 women, only 611 have electricity in the home, or about 14.02 percent. (iii) The average of children for women without electricity is about 2.33, and for those with electricity it is about 1.90. So, on average, women with electricity have .43 fewer children than those who do not. (iv) We cannot infer causality here. There are many confounding factors that may be related to the number of children and the presence of electricity in the home; household income and level of education are two possibilities. For example, it could be that women with more

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    5 education have fewer children and are more likely to have electricity in the home (the latter due to an income effect).

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    6 CHAPTER 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 us ually I derive the variance. At a minimum, I talk about the f actors 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( u i | x 1 , x 2 ,…, x n ) = E( u i | x i ) by independent sampling. I find that students are able to focus on the key assumption SLR.4 and subsequently take my word about how conditioning on the independent variab les in the sample is harmless. (If you prefer, the appendix to Chapter 3 does the con ditioning argument carefully.) Because statistical inference is no more difficult in multiple regression than in simple regression, I postpone inference until Chap ter 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 redunda nt or unnecessary assumptions. For example, once SLR. 4 is assumed, nothing further about the relationship between u and x is needed to obtain the unbiasedness of OLS under random sampling. Incidentally, one of the uncomfortable facts about finite - sample analysis is that there is a difference between an estimator that is unbiased conditional on the outcome of the covariates and one that is unconditionally unbiased. If the distribution of the 𝑥 𝑖 is such that they can all equal the 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 the estimator is not unbiased. I do not try to explain these subtleties in an introductory course, but I have had instructors ask me about the difference.

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    7 SOLUTIONS 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. 4 fails. 2.2 In the equation y =  0 +  1 x + u , add and subtract  0 from the right hand side to get y = (  0 +  0 ) +  1 x + ( 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 y i = GPA i , x i = ACT i , and n = 8. Then x = 25.875, y = 3.2125, 1 n i =  ( x i – x )( y i – y ) = 5.8125, and 1 n i =  ( x i – x ) 2 = 56.875. From equation (2.9), we obtain the slope as 1 ˆ  = 5.8125/56.875  .1022, rounded to four places after the decimal. From (2.17), 0 ˆ  = y – 1 ˆ  x  3.2125 – (.1022)25.875  .5681. So we can write GPA = .5681 + .1022 ACT 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, GPA increases by .1022(5) = .511. (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:

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    8 i GPA GPA ˆ u 1 2.8 2.7143 .0857 2 3.4 3.0209 .3791 3 3.0 3.2253 – .2253 4 3.5 3.3275 .1725 5 3.6 3.5319 .0681 6 3.0 3.1231 – .1231 7 2.7 3.1231 – .4231 8 3.7 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. (iii) When ACT = 20, GPA = .5681 + .1022(20)  2.61. (iv) The sum of squared residuals, 2 1 ˆ n i i u =  , is about .4347 (rounded to four decimal places), and the total sum of squares, 1 n i =  ( y i – y ) 2 , is about 1.0288. So the R - squared from the regression is R 2 = 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, bwght = 109.49. This is about an 8.6% drop. (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.

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    9 (iv) 1,176 out of 1,388 women did not smoke while pregnant, or about 84.7%. Because we are 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 at cigs = 0, and so we will under predict high birth rates. 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: 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 is negative, the smallest APC in the sample is positive. The graph starts at an annual income level of $1,000 (in 1970 dollars). 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. 4 , and OLS estimation is biased. inc 1000 10000 20000 30000 .7 .728 .853 APC MPC .9 APC MPC

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    10 (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( inc  e | inc ) = inc  E( e | inc ) = inc  0 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( inc  e | inc ) = ( inc ) 2 Var( e | inc ) = 2 e  inc because Var( e | inc ) = 2 e  . (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), 1  = 1 n i i i x y =        / 2 1 n i i x =        . Plugging in y i =  0 +  1 x i + u i gives 1  = 0 1 1 ( ) n i i i i x x u   =   + +      / 2 1 n i i x =        . After standard algebra, the numerator can be written as 2 0 1 1 1 1 i n n n i i i i i i x x x u   = = = + +    . Putting this over the denominator shows we can write 1  as 1  =  0 1 n i i x =        / 2 1 n i i x =        +  1 + 1 n i i i x u =        / 2 1 n i i x =        . Conditional on the x i , we have E( 1  ) =  0 1 n i i x =        / 2 1 n i i x =        +  1

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    11 because E( u i ) = 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 1 n i i x =  = 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 x i , Var( 1  ) = 2 2 1 n i i x − =        Var 1 n i i i x u =        = 2 2 1 n i i x − =        2 1 Var( ) n i i i x u =        = 2 2 1 n i i x − =        2 2 1 n i i x  =        = 2  / 2 1 n i i x =        . (iii) From (2.57), Var( 1 ˆ  ) =  2 / 2 1 ( ) n i i x x =   −      . From the hint, 2 1 n i i x =   2 1 ( ) n i i x x = −  , and so Var( 1  )  Var( 1 ˆ  ). A more direct way to see this is to write 2 1 ( ) n i i x x = −  = 2 2 1 ( ) n i i x n x = −  , which is less than 2 1 n i i x =  unless x = 0. (iv) For a given sample size, the bias in 1  increases as x increases (holding the sum of the 2 i x fixed). But as x increases, the variance of 1 ˆ  increases relative to Var( 1  ). The bias in 1  is also small when 0  is small. Therefore, whether we prefer 1  or 1 ˆ  on a mean squared error basis depends on the sizes of 0  , x , and n (in addition to the size of 2 1 n i i x =  ). 2.9 (i) We follow the hint, noting that 1 c y = 1 c y (the sample average of 1 i c y is c 1 times the sample average of y i ) and 2 c x = 2 c x . When we regress c 1 y i on c 2 x i (including an intercept) we use equation (2.19) to obtain the slope: 2 2 1 1 1 2 1 1 1 2 2 2 2 2 2 1 1 1 1 1 1 2 2 2 1 ( )( ) ( )( ) ( ) ( ) ( )( ) ˆ . ( ) n n i i i i i i n n i i i i n i i i n i i c x c x c y c y c c x x y y c x c x c x x x x y y c c c c x x   = = = = = = − − − − = = − − − − =  = −      

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    12 From (2.17), we obtain the intercept as 0  = (c 1 y ) – 1  ( c 2 x ) = ( c 1 y ) – [( c 1 / c 2 ) 1 ˆ  ]( c 2 x ) = c 1 ( y – 1 ˆ  x ) = c 1 0 ˆ  ) because the intercept from regressing y i on x i is ( y – 1 ˆ  x ). (ii) We use the same approach from part (i) along with the fact that 1 ( ) c y + = c 1 + y and 2 ( ) c x + = c 2 + x . Therefore, 1 1 ( ) ( ) i c y c y + − + = ( c 1 + y i ) – ( c 1 + y ) = y i – y and ( c 2 + x i ) – 2 ( ) c x + = x i – x . So c 1 and c 2 entirely drop out of the slope formula for the regression of ( c 1 + y i ) on ( c 2 + x i ), and 1  = 1 ˆ  . The intercept is 0  = 1 ( ) c y + – 1  2 ( ) c x + = ( c 1 + y ) – 1 ˆ  ( c 2 + x ) = ( 1 ˆ y x  − ) + c 1 – c 2 1 ˆ  = 0 ˆ  + c 1 – c 2 1 ˆ  , which is what we wanted to show. (iii) We can simply apply part (ii) because 1 1 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 set c 2 = 0. (iv) Again, we can apply part (ii) with c 1 = 0 and replacing c 2 with log( c 2 ) and x i with log( x i ). If 0 1 ˆ ˆ and   are the original intercept and slope, then 1 1 ˆ   = and 0 0 2 1 ˆ ˆ log( ) c    = − . 2.10 (i) This derivation is essentially done in equation (2.52), once (1/ SST ) x is brought inside the summation (which is valid because SST x does not depend on i ). Then, just define / SST i i x w d = . (ii) Because 1 1 1 ˆ ˆ Cov( , ) E[( ) ] , u u    = − we show that the latter is zero. But, from part (i), ( ) 1 1 1 1 ˆ E[( ) ] =E E( ). n n i i i i i i u w u u w u u   = =   − =       Because the i u are pairwise uncorrelated (they are independent), 2 2 E( ) E( / ) / i i u u u n n  = = (because E( ) 0, i h u u i h =  ). Therefore, (iii) The formula for the OLS intercept is 0 ˆ ˆ y x   = − and, plugging in 0 1 y x u   = + + gives 0 0 1 1 0 1 1 ˆ ˆ ˆ ( ) ( ) . x u x u x        = + + − = + − − (iv) Because 1 ˆ and u  are uncorrelated, 2 2 2 2 2 2 2 0 1 ˆ ˆ Var( ) Var( ) Var( ) / ( / SST ) / / SST x x u x n x n x       = + = + = + , which is what we wanted to show. (v) Using the hint and substitution gives ( ) 2 2 0 ˆ Var( ) [ SST / ]/ SST x x n x   = + ( ) ( ) 2 1 2 2 2 2 1 2 1 1 / SST / SST . n n i x i x i i n x x x n x   − − = =   = − + =       2 2 1 1 1 E( ) ( / ) ( / ) 0. n n n i i i i i i i w u u w n n w   = = = = = =   

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    13 2.11 (i) We would want to randomly assign the number of hours in the preparation course 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 {( , ) : 1,..., } i i sat hours i 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 in i hours as is feasible. (ii) Here are three factors: innate ability, family income, and general health on the day of the exam. If we think students with higher native intelligence think they do not need to prepare for the SAT, then ability and hours will be negatively correlated. Family income would probably be positively correlated with hours , because higher income families can more easily afford preparation courses. Ruling out chronic health problems, health on the day of the exam should be roughly uncorrelated with hours spent in a preparation course. (iii) If preparation courses are effective, 1  should be positive: other factors equal, an increase in hours should increase sat . (iv) The intercept, 0  , has a useful interpretation in this example: because E( u ) = 0, 0  is the average SAT score for students in the population with hours = 0. 2.12 (i) I will show the result without using calculus. Let 𝑦 ̅ be the sample average of the 𝑦 𝑖 and write 2 2 0 0 1 1 2 2 0 0 1 1 1 2 2 0 0 1 1 2 2 0 1 ( ) [( ) ( )] ( ) 2 ( )( ) ( ) ( ) 2( ) ( ) ( ) ( ) ( ) n n i i i i n n n i i i i i n n i i i i n i i y b y y y b y y y y y b y b y y y b y y n y b y y n y b = = = = = = = = − = − + − = − + − − + − = − + − − + − = − + −         where we use the fact (see Appendix A) that 1 ( ) 0 n i i y y = − =  always. The first term does not depend on 0 b and the second term, 2 0 ( ) n y b − , which is nonnegative, is clearly minimized when 0 b y = . (ii) If we define i i u y y = − then 1 1 ( ) n n i i i i u y y = = = −   and we already used the fact that this sum is zero in the proof in part (i). SOLUTIONS TO COMPUTER EXERCISES

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    14 C 2.1 (i) The average prate is about 87.36 and the average mrate is about .732. (ii) The estimated equation is prate = 83.05 + 5.86 mrate n = 1,534, R 2 = .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 ˆ 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. C 2. 2 (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 log( ) salary = 6.51 + .0097 ceoten n = 177, R 2 = .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%. C2.3 (i) The estimated equation is sleep = 3,586.4 – .151 totwrk n = 706, R 2 = .103.

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    15 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 sleep  = – .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  = – .151(300) = – 45.3 minutes, or about five minutes a night. C2.4 (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. (ii) This calls for a level - level model: wage = 116.99 + 8.30 IQ n = 935, R 2 = .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, R 2 = .099. If  IQ = 15 then log( ) wage  = .0088(15) = .132, which is the (approximate) proportionate change in predicted wage. The percentage increase is therefore approximately 13.2. C2. 5 (i) The constant elasticity 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 is log( ) rd = – 4.105 + 1.076 log( sales ) n = 32, R 2 = .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%.

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    16 C2.6 (i ) It seems plausible that another dollar of spending has a larger effect for low - spending schools than for high - spending schools. At low - spending schools, more money can go toward purchasing more books, computers, and for hiring better qualified teachers. At high levels of spending, we would ex pend little, if any, effect because the high - spending schools already have high - quality teachers, nice facilities, plenty of books, and so on. (ii) If we take changes, as usual, we obtain 1 1 10 log( ) ( /100)(% ), math expend expend    =    just as in the second row of Table 2.3. S o, if % 10, expend  = 1 10 /10. math   = (iii) The regression results are 2 10 69.34 11.16 log( ) 408, .0297 math expend n R = − + = = (iv) If expend increases by 10 percent, 10 math increases by about 1.1 percentage points. This is not a huge effect, but it is not t rivial for low - spending schools, where a 10 percent increase in spending might be a fairly small dollar amount. (v) In this data set, the largest value of math10 is 66.7, which is not especially close to 100. In fact, the largest fitted values is only about 30.2. 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 (which presumably means that someone has been on the mailing list for at least four years) and the maximum value is 3.5. (iii) The estimated equation is 2 2.01 2.65 4,268, .0138 gift mailsyear n R = + = = (iv) The slope coefficient from part (iii) means that each mailing per year is associated with – perhaps even “causes” – an estimated 2.65 additional guilders, on average. Therefore, if each mailing costs one guilder, the expected profit from each mailing is estimated to be 1.65 guilders. This is only the average, however. Some mailings generate no contributions, or a contribution less than the mailing cost; other mailings generated much more than the mailing cost. (v) Because the smallest mailsyear in the sample is .25, the smallest predicted value of gifts is 2.01 + 2.65(.25)  2.67. Even if we look at the overall population, where some people have received no mailings, the smalles t predicted value is about two. So, with this estimated equation, we never predict zero charitable gifts.

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    17 C2.8 There is no “correct” answer to this question because all answers depend on how the random outcomes are generated. I used Stata 11 and, before generating the outcomes on the i x , I set the seed to the value 123. I reset the seed to 123 to generate the outcomes on the i u . Specifically, to answer parts (i) through (v), I used the sequence of commands set obs 500 set seed 123 gen x = 10*runiform() sum x set seed 123 gen u = 6*rnormal() sum u gen y = 1 + 2*x + u reg y x predict uh, resid gen x_uh = x*uh sum uh x_uh gen x_u = x*u sum u x_u (i) The sample mean of the i x is about 4.912 with a sample standard deviation of about 2.874. (ii) The sample average of the i u is about .221, which is pretty far from zero. We do not get zero because this is just a sample of 500 from a population with a zero mean. The current sample is “unlucky” in the sense that the sample average is far from the population average. The sample standard deviation is about 5.768, which is nontrivially below 6, the population value. (iii) After generating the data on i y and running the regression, I get, rounding to three decimal places, 0 ˆ 1.862  = and 1 ˆ 1.870  = The population values are 1 and 2, respectively. Thus, the estimated intercept based on this sample of data is well above the population value. The estimated slope is somewhat below the population value, 2. When we sample from a population our estimates contain sampling error; that is why the estimates differ from the population values. (iv) When I use the command sum uh x_uh and multipl y by 500 I get, using scientific notation, sums equal to 4.181e - 06 and .00003776 , respectively. These are zero for practical purposes, and differ from zero only due to rounding inherent in the machine imprecision (which is unimportant).

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    18 (v) We already computed the sample average of the i u in part (ii). When we multiply by 500 the sample average is about 110.74. The sum of i i x u is about 6.46. Neither is close to zero, and nothing says they should be particularly close. (vi) For this part I set the seed to 789. The sample average and standard deviation of the i x are about 5.030 and 2.913; those for the i u are about .077 − and 5.979. When I generated the i y and run the regression I get 0 ˆ .701  = and 1 ˆ 2.044  = These are different from those in part (iii) because they are obtained from a different random sample. Here, for both the intercept and slope, we get estimates that are much closer to the population values. Of course, in practice we would never know that.

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    19 CHAPTER 3 TEACHING NOTES For undergraduates, I do not work through 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 consequences of having highly correlated independent variables. But this is often beyond our control, except that we can ask less of our multiple regression analysis. If two or more explanatory variables are highly correlated in the sample, we should not expect to precisely estimate their ceteris paribus effects in the population. I find extensive treatments of multicollinearity, where one “tests” or somehow “solves” the multicollinearity probl em, to be misleading, at best. Even the organization of some texts gives the impression that imperfect collinearity is somehow a violation of the Gauss - Markov assumptions . In fact, they include multicollinearity in a chapter or part of the book devoted to “violation of the basic assumptions,” or something like that. I have noticed that master’s students who have had some undergraduate econometrics are often confused on the multicollinearity issue. It is very important that students not confuse multicollinearity among the included explanatory variables in a regression model with the bias caused by omitting an important variable. I do not p rove the Gauss - Markov theorem. Instead, I emphasize its implications. Sometimes, and certainly for advanced beginners, I put a special case of Problem 3.12 on a midterm exam, where I make a particular choice for the function g ( x ). Rather than have the students directly compare

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    20 the variances, they should appeal to the Gauss - Markov theorem for the superiority of OLS over any other linear, unbiased estimator. SOLUTIONS TO PROBLEMS 3.1 (i) hsperc is defined so that the smaller it is, the lower the student’s standing in high school. Everything else equal, the worse the student’s standing in high school, the lower is his/her expected college GPA. (ii) Just plug these values into the equation: colgpa = 1.392 − .0135(20) + .00148(1050) = 2.676. (iii) The difference between A and B is simply 140 times the coefficient on sat , because hsperc is the same for both students. So A is predicted to have a score .00148(140)  .207 higher. (iv) With hsperc fixed, colgpa  = .00148  sat . Now, we want to find  sat such that colgpa  = .5, so .5 = .00148(  sat ) or  sat = .5/(.00148)  338. Perhaps not surprisingly, a large ceteris paribus difference in SAT score – almost two and one - half standard deviations – is needed to obtain a predicted difference in college GPA or a half a point. 3.2 (i) Yes. Because of budget constraints, it makes sense that, the more siblings there are in a family, the less education any one child in the family has. To find the increase in the number of siblings that reduces predicted education by one year, we solve 1 = .094(  sibs ), so  sibs = 1/.094  10.6. (ii) Holding sibs and feduc fixed, one more year of mother’s education implies .131 years more of predicted education. So if a mother has four more years of education, her son is predicted to have about a half a year (.524) more years of education. (iii) Since the number of siblings is the same, but meduc and feduc are both different, the coefficients on meduc and feduc both need to be accounted for. The predicted difference in education between B and A is .131(4) + .210(4) = 1.364. 3.3 (i) If adults trade off sleep for work, more work implies less sleep (other things equal), so 1  < 0. (ii) The signs of 2  and 3  are not obvious, at least to me. One could argue that more educated people like to get more out of life, and so, other things equal, they sleep less ( 2  < 0). The relationship between sleeping and age is more complicated than this model suggests, and economists are not in the best position to judge such things.

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    21 (iii) Since totwrk is in minutes, we must convert five hours into minutes:  totwrk = 5(60) = 300. Then sleep is predicted to fall by .148(300) = 44.4 minutes. For a week, 45 minutes less sleep is not an overwhelming change. (iv) More education implies less predicted time sleeping, but the effect is quite small. If we assume the difference between college and high school is four years, the college graduate sleeps about 45 minutes less per week, other things equal. (v) Not surprisingly, the three explanatory variables explain only about 11.3% of the variation in sleep . One important factor in the error term is general health. Another is marital status, and whether the person has children. Health (however we measure that), marital status, and number and ages of children would generally be correlated with totwrk . (For example, less healthy people would tend to work less.) 3.4 (i) A larger rank for a law school means that the school has less prestige; this lowers starting salaries. For example, a rank of 100 means there are 99 schools thought to be better. (ii) 1  > 0, 2  > 0. Both LSAT and GPA are measures of the quality of the entering class. No matter where better students attend law school, we expect them to earn more, on average. 3  , 4  > 0. The number of volumes in the law library and the tuition cost are both measures of the school quality. (Cost is less obvious than library volumes, but should reflect quality 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: a one percent increase in library volumes implies a .095% increase in predicted median starting salary, other things equal. (v) It is definitely better to attend a law school with a lower rank. If law school A has a ranking 20 less than law school B, the predicted difference in starting salary is 100(.0033)(20) = 6.6% higher for law school A. 3.5 (i) No. By definition, study + sleep + work + leisure = 168. Therefore, if we change study , we must change at least one of the other categories so that the sum is still 168. (ii) From part (i), we can write, say, study as a perfect linear function of the other independent variables: study = 168 − sleep − work − leisure . This holds for every observation, so MLR. 3 violated. (iii) Simply drop one of the independent variables, say leisure : GPA = 0  + 1  study + 2  sleep + 3  work + u .

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    22 Now, for example, 1  is interpreted as the change in GPA when study increases by one hour, where sleep , work , and u are all held fixed. If we are holding sleep and work fixed but increasing study by one hour, then we must be reducing leisure by one hour. The other slope parameters have a similar interpretation. 3.6 Conditioning on the outcomes of the explanatory variables, we have 1 E( )  = E( 1 ˆ  + 2 ˆ  ) = E( 1 ˆ  ) + E( 2 ˆ  ) =  1 +  2 = 1  . 3.7 Only (ii), omitting an important variable, can cause bias, and this is true only when the omitted variable is correlated with the included explanatory variables. Th e homoskedasticity assumption, MLR.5, played no role in showing that the OLS estimators are unbiased. (Homoskedasticity was used to obtain the usual variance formulas for the ˆ j  .) Further, the degree of collinearity between the explanatory variables in the sample, even if it is reflected in a correlation as high as .95, does not affect the Gauss - Markov assumptions. Only if there is a perfect linear relationship among two or more explanatory variables is MLR. 3 violated. 3.8 We can use Table 3.2. By definition, 2  > 0, and by assumption, Corr( x 1 , x 2 ) < 0. Therefore, there is a negative bias in 1  : E( 1  ) < 1  . This means that, on average across different random samples , the simple regression estimator underestimates the effect of the training program. It is even possible that E( 1  ) is negative even though 1  > 0. 3.9 (i) 1  < 0 because more pollution can be expected to lower housing values; note that 1  is the elasticity of price with respect to nox . 2  is probably positive because rooms roughly measures the size of a house. (However, it does not allow us to distinguish homes where each room is large from homes where each room is small.) (ii) If we assume that rooms increases with quality of the home, then log( nox ) and rooms are negatively correlated when poorer neighborhoods have more pollution, something that is often true. We can use Table 3.2 to determine the direction of the bias. If 2  > 0 and Corr( x 1 , x 2 ) < 0, the simple regression estimator 1  has a downward bias. But because 1  < 0, this means that the simple regression, on average, overstates the importance of pollution. [E( 1  ) is more negative than 1  .] (iii) This is what we expect from the typical sample based on our analysis in part (ii). The simple regression estimate, − 1.043, is more negative (larger in magnitude) than the multiple regression estimate, − .718. As those estimates are only for one sample, we can never know which is closer to 1  . But if this is a “typical” sample, 1  is closer to − .718. 3.10 (i) Because 1 x is highly correlated with 2 x and 3 x , and these latter variables have large partial effects on y , the simple and multiple regression coefficients on 1 x can differ by large

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    23 amounts. We have not done this case explicitly, but given equation (3.46) and the discussion with a single omitted variable , the intuition is pretty straightforward. (ii) Here we would expect 1  and 1 ˆ  to be similar (subject, of course, to what we mean by “almost uncorrelated”). The amount of correlation between 2 x and 3 x does not directly effect the multiple regression estimate on 1 x if 1 x is essentially uncorrelated with 2 x and 3 x . (iii) In this case we are (unnecessarily) introducing multicollinearity into the regression: 2 x and 3 x have small partial effects on y and yet 2 x and 3 x are highly correlated with 1 x . Adding 2 x and 3 x like increases the standard error of the coefficient on 1 x substantially, so se( 1 ˆ  ) is likely to be much larger than se( 1  ). (iv) In this case, adding 2 x and 3 x will decrease the residual variance without causing much collinearity (because 1 x is almost uncorrelated with 2 x and 3 x ), so we should see se( 1 ˆ  ) smaller than se( 1  ). The amount of correlation between 2 x and 3 x does not directly affect se( 1 ˆ  ). 3.1 1 From equation (3.22) we have 1 1 1 2 1 1 ˆ , ˆ n i i i n i i r y r  = = =   where the 1 ˆ i r are defined in the problem. As usual, we must plug in the true model for y i : 1 0 1 1 2 2 3 3 1 1 2 1 1 ˆ ( . ˆ n i i i i i i n i i r x x x u r      = = + + + + =   The numerator of this expression simplifies because 1 1 ˆ n i i r =  = 0, 1 2 1 ˆ n i i i r x =  = 0, and 1 1 1 ˆ n i i i r x =  = 2 1 1 ˆ n i i r =  . These all follow from the fact that the 1 ˆ i r are the residuals from the regression of 1 i x on 2 i x : the 1 ˆ i r have zero sample average and are uncorrelated in sample with 2 i x . So the numerator of 1  can be expressed as

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    24 2 1 1 3 1 3 1 1 1 1 ˆ ˆ ˆ . n n n i i i i i i i i r r x r u   = = = + +    Putting these back over the denominator gives 1 3 1 1 1 1 1 3 2 2 1 1 1 1 ˆ ˆ . ˆ ˆ n n i i i i i n n i i i i r x ru r r    = = = = = + +     Conditional on all sample values on x 1 , x 2 , and x 3 , only the last term is random due to its dependence on u i . But E( u i ) = 0, and so 1 3 1 1 1 3 2 1 1 ˆ E( ) = + , ˆ n i i i n i i r x r    = =   which is what we wanted to show. Notice that the term multiplying 3  is the regression coefficient from the simple regression of x i 3 on 1 ˆ i r . 3.1 2 (i) The shares, by definition, add to one. If we do not omit one of the shares then the equation would suffer from perfect multicollinearity. The parameters would not have a ceteris paribus interpretation, as it is impossible to change one share while holding all of the other shares fixed. (ii) Because each share is a proportion (and can be at most one, when all other shares are zero), it makes little sense to increase share p by one unit. If share p increases by .01 – which is equivalent to a one percentage point increase in the share of property taxes in total revenue – holding share I , share S , and the other factors fixed, then growth increases by 1  (.01). With the other shares fixed, the excluded share, share F , must fall by .01 when share p increases by .01. 3.1 3 (i) For notational simplicity, define s zx = 1 ( ) ; n i i i z z x = −  this is not quite the sample covariance between z and x because we do not divide by n – 1, but we are only using it to simplify notation. Then we can write 1  as 1 1 ( ) . n i i i zx z z y s  = − = 

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    25 This is clearly a linear function of the y i : take the weights to be w i = ( z i − z )/ s zx . To show unbiasedness, as usual we plug y i = 0  + 1  x i + u i into this equation, and simplify: 0 1 1 1 0 1 1 1 1 1 ( )( ) ( ) ( ) ( ) n i i i i zx n n i zx i i i i zx n i i i zx z z x u s z z s z z u s z z u s       = = = = − + + = − + + − = − = +     where we use the fact that 1 ( ) n i i z z = −  = 0 always. Now s zx is a function of the z i and x i and the expected value of each u i is zero conditional on all z i and x i in the sample. Therefore, conditional on these values, 1 1 1 1 ( )E( ) E( ) n i i i zx z z u s    = − = + =  because E( u i ) = 0 for all i . (ii) From the fourth equation in part (i) we have (again conditional on the z i and x i in the sample), 2 1 1 1 2 2 2 2 1 2 Var ( ) ( ) Var( ) Var( ) ( ) n n i i i i i i zx zx n i i zx z z u z z u s s z z s   = = =   − −     = = − =    because of the homoskedasticity assumption [Var( u i ) =  2 for all i ]. Given the definition of s zx , this is what we wanted to show.

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