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

Solution Manual For Introductory Econometrics: A Modern Approach, 6th Edition is the ultimate guide to solving textbook questions, offering easy-to-follow solutions.

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    1 CHAPTER 1 The Nature of Econometrics and Economic Data Table of Contents Teaching notes 2 Solutions to Problems 3 Solutions to Computer Exercises 4

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    2 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 a 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 in teaching the students how econometrics can be used. (And, it is more enjoyable for the students and me.) I do not use counterfactual notation as in the modern “treatment effects” literature, but I 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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    3 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 a larger class size is associated with lower performance. We might find a negative correlation because a 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 might 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 to be placed in 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

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    4 constraint that there are only 168 hours in a week. We can then use statistical methods to measure the association between studying and working, including regression analysis, which 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. 1.4 (i) Experimental data have to be collected to undertake a statistical analysis. (ii) Yes, it is feasible to do a controlled experiment. The factors such as consumption, investment, net exports, and so on, would be required for a controlled experiments. (iii) No, the correlation analysis between GSP growth and tax rates is not likely to be convincing as the tax rates have a significant negative effect on gross state products even after controlling factors like expenditure, fluctuations in the business, control in the supply of money, and so on. 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 in the sample is about $5.90, which seems low. (iii) Using Table B-60 in the 2004 Economic Report of the President , the CPI was 56.9 in 1976 and 233 in 2013. (iv) To convert 1976 dollars into 2013 dollars, we use the ratio of the CPIs, which is 233 / 56.9   4.09. Therefore, the average hourly wage in 2013 dollars is roughly 4.09($5.90)   $24.13, 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 1,388 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

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    5 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) 289 out of 1,823, or about 15.85 percent of the sample. (iii) 17 (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.] (vii) The percentage by which school A outspends school B is (vii) The percentage by which school A outspends school B is ( 6,000 − 5,500) 100 ∙ 5,500 ≈ 9.09%.

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    6 When we use the approximation based on the difference in 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 education have fewer children and are more likely to have electricity in the home (the latter due to an income effect). C1.6 (i) There are 2,197 counties in the dataset. Of these, 1051 counties have zero murders. The percentage of counties having zero executions is 98.6%. (ii) The largest number of murders is 1403. The largest number of executions is 3. The average number of executions is 0.0159, which is small because most of the counties have zero executions. (iii) The correlation between murders and execs is 0.21. There is very low positive relationship

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

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    8 CHAPTER 2 The Simple Regression Model Table of Contents Teaching notes 9 Solutions to Problems 10 Solutions to Computer Exercises 17

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    9 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( 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 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. 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 𝑥𝑥 𝑖𝑖 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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    10 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 = β 0 + β 1 x + u , add and subtract α 0 from the right hand side to get y = ( α 0 + β 0 ) + β 1 x + ( u − α 0 ). Call the new error e = u − α 0 , so that E( e ) = 0. The new intercept is α 0 + β 0 , but the slope is still β 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 (2.19), we obtain the slope as 1 ˆ β = 5.8125/56.875 ≈ .1022, rounded to four places after the decimal. From (2.17), 0 ˆ β = y – 1 ˆ β 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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    11 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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    12 (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 the 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 chooses 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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    13 (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 σ inc because Var( e | inc ) = 2 e σ . (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 β  = 1 n i i i x y =       ∑ / 2 1 n i i x =       ∑ . Plugging in y i = β 0 + β 1 x i + u i gives 1 β  = 0 1 1 ( ) n i i i i x x u β β =   + +     ∑ / 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 β β = = = + + ∑ ∑ ∑ . Putting this over the denominator we can write 1 β  as 1 β  = β 0 1 n i i x =       ∑ / 2 1 n i i x =       ∑ + β 1 + 1 n i i i x u =       ∑ / 2 1 n i i x =       ∑ . Conditional on the x i , we have E( 1 β  ) = β 0 1 n i i x =       ∑ / 2 1 n i i x =       ∑ + β 1

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    14 because E( u i ) = 0 for all i . Therefore, the bias in 1 β  is given by the first term in this equation. This bias is obviously zero when β 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 β  in part (i) we have, conditional on the x i , Var( 1 β  ) = 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 σ =       ∑ = 2 σ / 2 1 n i i x =       ∑ . (iii) From (2.57), Var( 1 ˆ β ) =  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 β  ) ≤ Var( 1 ˆ β ). 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 β  increases as x increases (holding the sum of the 2 i x fixed). But as x increases, the variance of 1 ˆ β increases relative to Var( 1 β  ). The bias in 1 β  is also small when 0 β is small. Therefore, whether we prefer 1 β  or 1 ˆ β on a mean squared error basis depends on the sizes of 0 β , 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:

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    15 From (2.17), we obtain the intercept as 0 β  = (c 1 y ) – 1 β  ( c 2 x ) = ( c 1 y ) – [( c 1 / c 2 ) 1 ˆ β ]( c 2 x ) = c 1 ( y – 1 ˆ β x ) = c 1 0 ˆ β ) because the intercept from regressing y i on x i is ( y – 1 ˆ β 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 β  = 1 ˆ β . The intercept is 0 β  = 1 ( ) c y + – 1 β  2 ( ) c x + = ( c 1 + y ) – 1 ˆ β ( c 2 + x ) = ( 1 ˆ y x β − ) + c 1 – c 2 1 ˆ β = 0 ˆ β + c 1 – c 2 1 ˆ β , 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 ), replace 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 β β are the original intercept and slope, then 1 1 ˆ β β =  and 0 0 2 1 ˆ ˆ log( ) c β β β = −  . 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 β β β = − 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 β β = =   − =     ∑ ∑ Because the i u are pairwise uncorrelated (they are independent), 2 2 E( ) E( / ) / i i u u u n n σ = = (because E( ) 0, i h u u i h = ≠ ). Therefore, (iii) The formula for the OLS intercept is and plugging in 0 1 y x u β β = + + gives 0 0 1 1 0 1 1 ˆ ˆ ˆ ( ) ( ) . x u x u x β β β β β β β = + + − = + − − (iv) Because 1 ˆ and u β 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 β β σ σ σ σ = + = + = + , 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 β σ = + ( ) ( ) 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 σ σ − − = =   = − + =     ∑ ∑ 2 2 1 1 1 E( ) ( / ) ( / ) 0. n n n i i i i i i i w u u w n n w σ σ = = = = = = ∑ ∑ ∑

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