CHAPTER 1 TEACHING NOTES You have substantial latitude about what to emphasize in Chapter 1. I find it useful to talk about the economics of crime example (Example 1.1) and the wage example (Example 1.2) so that students see, at the outset, that econometrics is linked to economic reasoning, if not economic theory. I like to familiarize students with the important data structures that empirical economists use, focusing primarily on cross-sectional and time series data sets, as these are what I cover in a first-semester course. It is probably a good idea to mention the growing importance of data sets that have both a cross-sectional and time dimension. I spend almost an entire lecture talking about the problems inherent in drawing causal inferences in the social sciences. I do this mostly through the agricultural yield, return to education, and crime examples. These examples also contrast experimental and nonexperimental data. Students studying business and finance tend to find the term structure of interest rates example more relevant, although the issue there is testing the implication of a simple theory, as opposed to inferring causality. I have found that spending time talking about these examples, in place of a formal review of probability and statistics, is more successful (and more enjoyable for the students and me). 3 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.3 and subsequently take my word about how conditioning on the independent variables in the sample is harmless. (If you prefer, the appendix to Chapter 3 does the conditioning argument carefully.) Because statistical inference is no more difficult in multiple regression than in simple regression, I postpone inference until Chapter 4. (This reduces redundancy and allows you to focus on the interpretive differences between simple and multiple regression.) You might notice how, compared with most other texts, I use relatively few assumptions to derive the unbiasedness of the OLS slope estimator, followed by the formula for its variance. This is because I do not introduce redundant or unnecessary assumptions. For example, once SLR.3 is assumed, nothing further about the relationship between u and x is needed to obtain the unbiasedness of OLS under random sampling. 4 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.3 fails. 2.2 In the equation y = 0 + 1x + u, add and subtract 0 from the right hand side to get y = (0 + 0) + 1x + (u 0). Call the new error e = u 0, so that E(e) = 0. The new intercept is 0 + 0, but the slope is still 1. 2.3 (i) Let yi = GPAi, xi = ACTi, and n = 8. Then x= 25.875, y = 3.2125, (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 = 5.8125/56.875 .1022, rounded to four places after the decimal. From (2.17), 0 = y 1 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 u 12.8 2.7143.0857 23.4 3.0209.3791 33.0 3.2253.2253 43.5 3.3275.1725 53.6 3.5319.0681 63.0 3.1231.1231 72.7 3.1231.4231 83.7 3.6341.0659 You can verify that the residuals, as reported in the table, sum to .0002, which is pretty close to zero given the inherent rounding error. 5 (iii) When ACT = 20, = .5681 + .1022(20) GPA 2.61. (iv) The sum of squared residuals, 2 1 n i i u = , is about .4347 (rounded to four decimal places), and the total sum of squar