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PREFACE iii SUGGESTED COURSE OUTLINES iv Chapter 1 The Nature of Econometrics and Economic Data 1 Chapter 2 The Simple Regression Model 5 Chapter 3 Multiple Regression Analysis: Estimation 15 ... Chapter 4 Multiple Regression Analysis: Inference 28 Chapter 5 Multiple Regression Analysis: OLS Asymptotics 39 Chapter 6 Multiple Regression Analysis: Further Issues 44 Chapter 7 Multiple Regression Analysis With Qualitative 59 Information: Binary (or Dummy) Variables Chapter 8 Heteroskedasticity 75 Chapter 9 More on Specification and Data Problems 86 Chapter 10 Basic Regression Analysis With Time Series Data 95 Chapter 11 Further Issues in Using OLS With Time Series Data 106 Chapter 12 Serial Correlation and Heteroskedasticity in 117 Time Series Regressions Chapter 13 Pooling Cross Sections Across Time. Simple 127 Panel Data Methods Chapter 14 Advanced Panel Data Methods 140 Chapter 15 Instrumental Variables Estimation and Two Stage 152 Least Squares Chapter 16 Simultaneous Equations Models 168 Chapter 17 Limited Dependent Variable Models and Sample 181 Selection Corrections i Chapter 18 Advanced Time Series Topics 200 Chapter 19 Carrying Out an Empirical Project 215 Appendix A Basic Mathematical Tools 216 Appendix B Fundamentals of Probability 218 Appendix C Fundamentals of Mathematical Statistics 220 Appendix D Summary of Matrix Algebra 224 Appendix E The Linear Regression Model in Matrix Form 226 ii 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, 2nd edition. For several problems I have added additional notes to the instructor 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 replace 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. This way, if you prefer, you can distribute my answers to some of the even-numbered problems directly to the students. (The student study guide contains answers to all odd-numbered problems.) The solutions to the computer exercises were obtained using Stata, starting with version 4.0 and running through version 7.0. 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 keep a list of any substantive errors on the Web site for the book, http://wooldridge.swcollege.com. I heard from many of you regarding the first edition 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 via email at wooldri1@.msu.edu. I hope you find this instructor’s manual useful, and I look forward to hearing your reactions to the second edition. Jeffrey M. Wooldridge Department of Economics Michigan State University East Lansing, MI 48824-1038 iii 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 important practical 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 crosssectional 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 crosssectional 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 long and the payoff is minimal. (Students tend to think that they are taking another statistics course, and start to drift.) Instead, when I need a tool (such as the summation or expectations operator), I briefly review the necessary definitions and key properties. Statistical inference is not more difficult to describe in terms of multiple regression than in tests of a population mean, 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 matrix 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 masters level, could begin with some of the material in Chapter 9, particularly with the issues of proxy variables and measurement error. The advanced chapters, starting with Chapter 13, are useful for students who have an interest in policy analysis. The pooled cross section and panel data chapters (Chapters 13 and 14) emphasize how these data sets can be used, in conjunction with econometric methods, 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 error. I have intentionally separated out the conceptually more difficult topic of simultaneous equations models in Chapter 16. iv 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 series 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. v CHAPTER 1 TEACHING NOTES You have substantial latitude about what to emphasize in Chapter 1. I find it useful to talk about the economics of crime example (Example 1.1) and the wage example (Example 1.2) so that students see, at the outset, that econometrics is linked to economic reasoning, 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 cover in a first-semester course. It is probably a good idea to mention the growing importance of data sets that have both a cross-sectional and time dimension. I spend almost an entire lecture talking about the problems inherent in drawing causal inferences in the social sciences. I do this mostly through the agricultural yield, return to education, and crime examples. These examples also contrast experimental and nonexperimental (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 opposed to inferring causality. I have found that spending time talking about these examples, in place of a formal review of probability and statistics, is more successful (and more enjoyable for the students and me). 1 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 substantial 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 is 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 firms 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 abilities 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 workers would also affect output. So, two firms with exactly the same kinds of employees would generally have different outputs if they use different amounts 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 listed 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 behavior, such as maximizing utility subject to the constraint that there are only 168 hours in a week. We can then use statistical methods to 2 measure the association between studying and working, including regression analysis that we 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 2005. (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 . Therefore, the average hourly wage in 2003 dollars is roughly , which is a reasonable figure. 184 / 56.9 3.23 ≈ 3.23($5.90) ≈ $19.06 (v) The sample contains 252 women (the number of observations with female = 1) and 274 men. C1.2 (i) There are 1,388 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 percent of the women did not smoke. 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 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 29.027 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 largest is 100, the smallest is 0. (ii) 38 out of 1,823, or about 2.1 percent of the sample. (iii) 17 3 (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 rates on one test have a strong tendency to have high pass rates on the other test. (vi) The average of exppp is about $5,194.87. The standard deviation is $1,091.89, which shows rather wide variation in spending per pupil. [The minimum is $1,206.88 and the maximum is $11,957.64.] 4 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 usually I derive the variance. At a minimum, I talk about the factors affecting the variance. To simplify the notation, after I emphasize the assumptions in the population model, and assume random sampling, I just condition on the values of the explanatory variables in the sample. Technically, this is justified by random sampling because, for example, E(ui |x1,x2,…,xn) = E(ui |xi) by independent sampling. I find that students are able to focus on the key assumption SLR.4 and subsequently take my word about how conditioning on the independent variables in the sample is harmless. (If you prefer, the appendix to Chapter 3 does the conditioning argument carefully.) Because statistical inference is no more difficult in multiple regression than in simple regression, I postpone inference until Chapter 4. (This reduces redundancy and allows you to focus on the interpretive differences between simple and multiple regression.) You might notice how, compared with most other texts, I use relatively few assumptions to derive the unbiasedness of the OLS slope estimator, followed by the formula for its variance. This is because I do not introduce redundant or unnecessary assumptions. For example, once SLR.4 is assumed, nothing further about the relationship between u and x is needed to obtain the unbiasedness of OLS under random sampling. 5 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) z 0, and so SLR.4 fails. 2.2 In the equation y = E0 + E1x + u, add and subtract D0 from the right hand side to get y = (D0 + E0) + E1x + (u  D0). Call the new error e = u  D0, so that E(e) = 0. The new intercept is D0 + E0, but the slope is still E1. 2.3 (i) Let yi = GPAi, xi = ACTi, and n = 8. Then x = 25.875, y = 3.2125, (x 1 n i ¦ i – x )(yi – y ) = 5.8125, and (x 1 n i ¦ i – x ) 2 = 56.875. From equation (2.9), we obtain the slope as 1 E ˆ = 5.8125/56.875 ≈ .1022, rounded to four places after the decimal. From (2.17), 0 E ˆ = y – 1 E ˆ x 3.2125 – (.1022)25.875 .5681. So we can write ≈ ≈ = .5681 + .1022 ACT GPA ฀ n = 8. The intercept does not have a useful interpretation because ACT is not close to zero for the population of interest. If ACT is 5 points higher, increases by .1022(5) = .511. GPA ฀ (ii) The fitted values and residuals — rounded to four decimal places — are given along with the observation number i and GPA in the following table: i GPA GPA ฀ uˆ 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. 6 (iii) When ACT = 20, = .5681 + .1022(20) GPA ฀ ≈ 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, (y 1 n i ¦ 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, = 109.49. This is about an 8.6% drop. ฀bwght (ii) Not necessarily. There are many other factors that can affect birth weight, particularly overall health of the mother and quality of prenatal care. These could be correlated with cigarette smoking during birth. Also, something such as caffeine consumption can affect birth weight, and might also be correlated with cigarette smoking. (iii) If we want a predicted bwght of 125, then cigs = (125 – 119.77)/( –.524) –10.18, or about –10 cigarettes! This is nonsense, of course, and it shows what happens when we are trying to predict something as complicated as birth weight with only a single explanatory variable. The largest predicted birth weight is necessarily 119.77. Yet almost 700 of the births in the sample had a birth weight higher than 119.77. ≈ (iv) 1,176 out of 1,388 women did not smoke while pregnant, or about 84.7%. 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: = –124.84 + .853(30,000) = 25,465.16 dollars. ฀cons (iii) The MPC and the APC are shown in the following graph. Even though the intercept is negative, the smallest APC in the sample is positive. The graph starts at an annual income level of $1,000 (in 1970 dollars). 7 inc 1000 10000 20000 30000 .7 .728 .853 APC MPC .9 APC MPC 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. (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 ) 2Var(e|inc) = 2 V e inc because Var(e|inc) = 2 V 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. 8 [Show More]

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