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Introduction To Econometrics 4th Global Edition James H. Stock, Mark W. Watson - Solutions
9.4 What is sample selection bias? Suppose you read a study using data on college graduates of the effects of an additional year of schooling on earnings. What is the potential sample selection bias present?
9.3 What is the effect of measurement error in Y? How is this different from the effect of measurement error in X?
9.2 Key Concept 9.2 describes the problem of variable selection in terms of a trade-off between bias and variance. What is this trade-off? Why could including an additional control variable decrease bias? Increase variance?
9.1 Explain the difference between internal validity and external validity. Is it possible for an econometric study to have internal validity but not external validity?
E8.2 On the text website http://www.pearsonglobaleditions.com, you will find a data file CPS2015, which contains data for full-time, full-year workers,ages 25–34, with a high school diploma or B.A./B.S. as their highest degree. A detailed description is given in CPS2015_Description, also
E8.1 Lead is toxic, particularly for young children, and for this reason, government regulations severely restrict the amount of lead in our environment. But this was not always the case. In the early part of the 20th century, the underground water pipes in many U.S. cities contained lead, and lead
8.12 The discussion following Equation (8.28) interprets the coefficient on interacted binary variables using the conditional mean zero assumption. This exercise shows that this interpretation also applies under conditional mean independence. Consider the hypothetical experiment in Exercise 7.11.a.
8.11 Derive the expressions for the elasticities given in Appendix 8.2 for the linear and log-log models. (Hint: For the log-log model, assume that u and X are independent, as is done in Appendix 8.2 for the log-linear model.)
8.10 Consider the regression model Yi = b0 + b1X1i + b2X2i + b31X1i * X2i2 + ui.Use Key Concept 8.1 to show thata. Y> X1 = b1 + b3X2 (effect of change in X1, holding X2 constant).b. Y> X2 = b2 + b3X1 (effect of change in X2, holding X1 constant).c. If X1 changes by X1 and X2 changes by X2, then
8.9 Explain how you would use approach 2 from Section 7.3 to calculate the confidence interval discussed below Equation (8.8). [Hint: This requires estimating a new regression using a different definition of the regressors and the dependent variable. See Exercise (7.9).]
8.8 X is a continuous variable that takes on values between 5 and 100. Z is a binary variable. Sketch the following regression functions (with values of X between 5 and 100 on the horizontal axis and values of Y n on the vertical axis):a. Y n= 2.0 + 3.0 * ln1X2.b. Y n= 2.0 - 3.0 * ln1X2.c. i. Y n=
8.7 This problem is inspired by a study of the gender gap in earnings in top corporate jobs (Bertrand and Hallock, 2001). The study compares total compensation among top executives in a large set of U.S. public corporations in the 1990s. (Each year these publicly traded corporations must report
8.6 Refer to Table 8.3.a. A researcher suspects that the effect of %Eligible for subsidized lunch has a nonlinear effect on test scores. In particular, he conjectures that increases in this variable from 10% to 20% have little effect on test scores but that changes from 50% to 60% have a much
8.5 Read the box “The Demand for Economics Journals” in Section 8.3.a. The box reaches three conclusions. Looking at the results in the table, what is the basis for each of these conclusions?b. Using the results in regression (4), the box reports that the elasticity of demand for an 80-year-old
8.4 Read the box “The Effect of Ageing on Healthcare Expenditures: A Red Herring?” in Section 8.3.a. Consider a male aged 60 years. Use the results from column (1) of Table 8.1 and the method in Key Concept 8.1 to estimate the expected change in the logarithm of health care expenditures (HCE)
8.3 After reading this chapter’s analysis of test scores and class size, an educator comments, “In my experience, student performance depends on class size, but not in the way your regressions say. Rather, students do well when class size is less than 20 students and do very poorly when class
8.2 Suppose a researcher collects data on houses that have sold in a particular neighborhood over the past year and obtains the regression results in the following table.a. Using the results in column (1), what is the expected change in price of building a 1500-square-foot addition to a house?
8.1 Sales in a company are $243 million in 2018 and increase to $250 million in 2019.a. Compute the percentage increase in sales, using the usual formula 100 * (Sales2019 - Sales2018)Sales2013. Compare this value to the approximation 100 * 3ln1Sales20192 - ln1Sales201824.b. Repeat (a), assuming
8.6 What types of independent variables—binary or continuous—might interact with one another in a regression? Explain how you would interpret the coefficient on the interaction between two continuous regressors and between two binary regressors.
8.5 Suppose that in Exercise 8.2 you thought that the value of b2 was not constant but rather increased when K increased. How could you use an interaction term to capture this effect?
8.4 Suppose the regression in Equation (8.30) is estimated using LoSTR and LoEL in place of HiSTR and HiEL, where LoSTR = 1 - HiSTR is an indicator for a low-class-size district and LoEL = 1 - HiEL is an indicator for a district with a low percentage of English learners. What are the values of the
8.3 How is the slope coefficient interpreted in a log-linear model, where the independent variable is in logarithms but the dependent variable is not? In a linear-log model? In a log-log model?
8.2 A Cobb–Douglas production function relates production (Q) to factors of production—capital (K), labor (L), and raw materials (M)—and an error term u using the equation Q = lKb1Lb2Mb3eu, where l, b1, b2, and b3 are production parameters. Suppose you have data on production and the factors
8.1 A researcher states that there are nonlinearities in the relationship between wages and years of schooling. What does this mean? How would you test for nonlinearities in the relationship between wages and schooling? How would you estimate the rate of change of wages with respect to years of
E7.2 In the empirical exercises on earning and height in Chapters 4 and 5, you estimated a relatively large and statistically significant effect of a worker’s height on his or her earnings. One explanation for this result is omitted variable bias: Height is correlated with an omitted factor that
E7.1 Use the Birthweight_Smoking data set introduced in Empirical Exercise E5.3 to answer the following questions. To begin, run three regressions:(1) Birthweight on Smoker (2) Birthweight on Smoker, Alcohol, and Nprevist (3) Birthweight on Smoker, Alcohol, Nprevist, and Unmarrieda. What is the
7.10 Equations (7 .13) and (7 .14) show two formulas for the homoskedasticity-only F-statistic. Show that the two formulas are equivalent.
7.9 Consider the regression model Yi = b0 + b1X1i + b2X2i + ui. Use approach 2 from Section 7 .3 to transform the regression so that you can use a t-statistic to testa. b1 = b2.b. b1 + 2b2 = 0.c. b1 + b2 = 1. (Hint: You must redefine the dependent variable in the regression.)
7.8 Referring to the Table on page 266 used for Exercises 7.1 to 7.6:a. Construct the R2 for each of the regressions.b. Show how to construct the homoskedasticity-only F-statistic for testing b4 = b5 = b6 = 0 in the regression shown in column (3). Is the statistic significant at the 1% level?c.
7.7 Question 6.5 reported the following regression (where standard errors have been added):Price = 109.7 + 0.567BDR + 26.9Bath + 0.239Hsize + 0.005Lsize 122.12 11.232 19.762 10.0212 10.000722+ 0.1Age - 56.9Poor, R2 = 0.85, SER = 45.8.10.232 112.232a. Is the coefficient on BDR statistically
7.6 In all of the regressions in the previous Exercises, the coefficient of High school is positive, large, and statistically significant. Do you believe this provides strong statistical evidence of the high returns to schooling in the labor market?
7.5 The regression shown in column (2) was estimated again, this time using data from 1993 (5000 observations selected at random and converted into 2007 units using the Consumer Price Index). The results are logAWE = 9.32 + 0.301 High school + 0.562 Male + 0.011Age, 10.202 10.0192 10.0472 10.0022
7.4 Using the regression results in column (3):a. Are there any important regional differences? Use an appropriate hypothesis test to explain your answer.b. Juan is a 32-year-old male high school graduate from the North. Mel is a 32-year-old male college graduate from the West. Ari is a 32-year-old
7.3 Using the regression results in column (2):a. Is age an important determinant of earnings? Use an appropriate statistical test and/or confidence interval to explain your answer.b. Suppose Alvo is a 30-year-old male college graduate, and Kal is a 40-year-old male college graduate. Construct a
7.2 Using the regression results in column (1):a. Is the high school earnings difference estimated from this regression statistically significant at the 5% level? Construct a 95% confidence interval of the difference.b. Is the male–female earnings difference estimated from this regression
7.1 For each of the three regressions, add * (5% level) and ** (1% level) to the table to indicate the statistical significance of the coefficients.
7.3 What is a control variable, and how does it differ from a variable of interest?Looking at Table 7.1, for what factors are the control variables controlling?Do coefficients on control variables measure causal effects? Explain.
7.2 Describe the recommended approach towards determining model specification.How does the R2 help in determining an appropriate model? Is the ideal model the one with the highest R2? Should a regressor be included in the model if it increases the model R2?
7.1 What is a joint hypothesis? Explain how an F-statistic is constructed to test a joint hypothesis. What is the hypothesis that is tested by constructing the overall regression F-statistic in the multiple regression model Yi = b0 + b1X1i + b2X2i + ui? Explain using the concepts of restricted and
E6.2 Using the data set Growth described in Empirical Exercise E4.1, but excluding the data for Malta, carry out the following exercises.a. Construct a table that shows the sample mean, standard deviation, and minimum and maximum values for the series Growth, TradeShare, YearsSchool, Oil,
E6.1 Use the Birthweight_Smoking data set introduced in Empirical Exercise E5.3 to answer the following questions.a. Regress Birthweight on Smoker. What is the estimated effect of smoking on birth weight?b. Regress Birthweight on Smoker, Alcohol, and Nprevist.i. Using the two conditions in Key
6.12 A school district undertakes an experiment to estimate the effect of class size on test scores in second-grade classes. The district assigns 50% of its previous year’s first graders to small second-grade classes (18 students per classroom)and 50% to regular-size classes (21 students per
6.11 (Requires calculus) Consider the regression model Yi = b1X1i + b2X2i + ui for i = 1,c, n. (Notice that there is no constant term in the regression.)Following analysis like that used in Appendix 4.2:a. Specify the least squares function that is minimized by OLS.b. Compute the partial
6.10 1Yi, X1i, X2i2 satisfy the assumptions in Key Concept 6.4; in addition, var1ui X1i, X2i2 = 4 and var1X1i2 = 6. A random sample of size n = 400 is drawn from the population.a. Assume that X1 and X2 are uncorrelated. Compute the variance of b n1.[Hint: Look at Equation (6.20) in Appendix
6.9 1Yi, X1i, X2i2 satisfy the assumptions in Key Concept 6.4. You are interested in b1, the causal effect of X1 on Y. Suppose X1 and X2 are uncorrelated. You estimate b1 by regressing Y onto X1 (so that X2 is not included in the regression).Does this estimator suffer from omitted variable bias?
6.8 A government study found that people who eat chocolate frequently weigh less than people who don’t. Researchers questioned 1000 individuals from Cairo between the ages of 20 and 85 about their eating habits, and measured their weight and height. On average, participants ate chocolate twice a
6.7 Critique each of the following proposed research plans. Your critique should explain any problems with the proposed research and describe how the research plan might be improved. Include a discussion of any additional data that need to be collected and the appropriate statistical techniques for
6.6 A researcher plans to study the causal effect of a strong legal system on the number of scandals in a country, using data from a random sample of countries in Asia. The researcher plans to regress the number of scandals on how strong a legal system is in the countries (an indicator variable
6.5 Data were collected from a random sample of 200 home sales from a community in 2013. Let Price denote the selling price (in $1000s), BDR denote the number of bedrooms, Bath denote the number of bathrooms, Hsize denote the size of the house (in square feet), Lsize denote the lot size (in square
6.4 Using the regression results in column (3):a. Do there appear to be important regional differences?b. Why is the regressor West omitted from the regression? What would happen if it were included?c. Juanita is a 28-year-old female college graduate from the South. Jennifer is a 28-year-old female
6.3 Using the regression results in column (2):a. Is age an important determinant of earnings? Explain.b. Sally is a 29-year-old female college graduate. Betsy is a 34-year-old female college graduate. Predict Sally’s and Betsy’s earnings.
6.2 Using the regression results in column (1):a. Do workers with college degrees earn more, on average, than workers with only high school diplomas? How much more?b. Do men earn more than women, on average? How much more?
6.1 Compute R2 for each of the regressions.
6.5 How is imperfect collinearity of regressors different from perfect collinearity?Compare the solutions for these two concerns with multiple regression estimation.
6.4 What is a dummy variable trap? Explain how it is related to multicollinearity of regressor. What is the solution for this form of multicollinearity?
6.3 What are the measures of fit commonly used for multiple regressions? How can an adjusted R2 take on negative values?
6.2 A multiple regression includes two regressors: Yi = b0 + b1X1i + b2X2i + ui.What is the expected change in Y if X1 increases by 8 units and X2 is unchanged? What is the expected change in Y if X2 decreases by 3 units and X1 is unchanged? What is the expected change in Y if X1 increases by 4
6.1 A researcher is estimating the effect of studying on the test scores of student’s from a private school. She is concerned, however, that she does not have information on the class size to include in the regression. What effect would the omission of the class size variable have on her
E5.3 On the text website, http://www.pearsonglobaleditions.com, you will find the data file Birthweight_Smoking, which contains data for a random sample of babies born in Pennsylvania in 1989. The data include the baby’s birth weight together with various characteristics of the mother, including
E5.2 Using the data set Growth described in Empirical Exercise 4.1, but excluding the data for Malta, run a regression of Growth on TradeShare.a. Is the estimated regression slope statistically significant? That is, can you reject the null hypothesis H0: b1 = 0 vs. a two-sided alternative
E5.1 Use the data set Earnings_and_Height described in Empirical Exercise 4.2 to carry out the following exercises.a. Run a regression of Earnings on Height.i. Is the estimated slope statistically significant?ii. Construct a 95% confidence interval for the slope coefficient.b. Repeat (a) for
5.15 A researcher has two independent samples of observations on 1Yi, Xi2.To be specific, suppose Yi denotes earnings, Xi denotes years of schooling, and the independent samples are for men and women. Write the regression for men as Ym,i = bm,0 + bm,1Xm,i + um,i and the regression for women as Yw,i
5.14 Suppose Yi = bXi + ui, where 1ui, Xi2 satisfy the Gauss–Markov conditions given in Equation (5.31).a. Derive the least squares estimator ofb, and show that it is a linear function of Y1,c, Yn.b. Show that the estimator is conditionally unbiased.c. Derive the conditional variance of the
5.13 Suppose 1Yi, Xi2 satisfy the least squares assumptions in Key Concept 4.3 and, in addition, ui is distributed N10, s2 u2 and is independent of Xi.a. Is b n1 conditionally unbiased?b. Is b n 1 the best linear conditionally unbiased estimator of b1?c. How would your answers to (a) and (b) change
5.12 Starting from Equation (4.20), derive the variance of b n0 under homoskedasticity given in Equation (5.28) in Appendix 5.1.
5.11 A random sample of workers contains nm = 100 men and nw = 150 women.The sample average of men’s weekly earnings 3Ym = (1>nm)gnm i = 1Ym,i4 is €565.89, and the standard deviation 3sm = 2 1 nm - 1gnm i = 1(Ym,i - Ym)24 is €75.62. The corresponding values for women are Yw = €502.37 and sw
5.10 Let Xi denote a binary variable, and consider the regression Yi = b0 +b1Xi + ui. Let Y0 denote the sample mean for observations with X = 0, and let Y1 denote the sample mean for observations with X = 1. Show that bn 0 = Y0, b n0 + b n1 = Y1, and b n1 = Y1 - Y0.
5.9 Consider the regression model Yi = bXi + ui, where ui and Xi satisfy the least squares assumptions in Key Concept 4.3. Let b denote an estimator of b that is constructed as b = Y>X, where Y and X are the sample means of Yi and Xi, respectively.a. Show that b is a linear function of Y1, Y2,c,
5.8 Suppose 1Yi, Xi2 satisfy the least squares assumptions in Key Concept 4.3 and, in addition, ui is N10, s2 u2 and is independent of Xi. A sample of size n = 30 yields Yn= 43.2 + 61.5X, R2 = 0.54, SER = 1.52,(10.2) (7.4)where the numbers in parentheses are the homoskedastic-only standard errors
5.7 Suppose (Yi, Xi) satisfy the least squares assumptions in Key Concept 4.3.A random sample of size n = 250 is drawn and yields Yn= 5.4 + 3.2X, R2 = 0.26, SER = 6.2.(3.1) (1.5)a. Test H0 : b1 = 0 vs. H1 : b1 0 at the 5% level.b. Construct a 95% confidence interval for b1.c. Suppose you learned
5.6 Refer to the regression described in Exercise 5.5.a. Do you think that the regression errors are plausibly homoskedastic? Explain.b. SE(b n1) was computed using Equation (5.3). Suppose the regression errors were homoskedastic. Would this affect the validity of the confidence interval
5.5 In the 1980s, Tennessee conducted an experiment in which kindergarten students were randomly assigned to “regular” and “small” classes and given standardized tests at the end of the year. (Regular classes contained approximately 24 students, and small classes contained approximately 15
5.4 Read the box “The Economic Value of a Year of Education: Homoskedasticity or Heteroskedasticity?” in Section 5.4. Use the regression reported in Equation(5.23) to answer the following.a. A randomly selected 30-year-old worker reports an education level of 16 years. What is the worker’s
5.3 Suppose a random sample of 100 25-year-old men is selected from a population and their heights and weights are recorded. A regression of weight on height yields Weight = -79.24 + 4.16 * Height, R2 = 0.72, SER = 12.6, 13.422 1 .422 where Weight is measured in pounds and Height is measured in
5.2 Suppose that a researcher, using wage data on 200 randomly selected male workers and 240 female workers, estimates the OLS regression Wage = 10.73 + 1.78 * Male, R2 = 0.09, SER = 3.8, 10.162 10.292 where Wage is measured in dollars per hour and Male is a binary variable that is equal to 1 if
5.1 Suppose a researcher, using data on class size (CS) and average test scores from 50 third-grade classes, estimates the OLS regression TestScore = 640.3 - 4.93 * CS, R2 = 0.11, SER = 8.7.123.52 12.022a. Construct a 95% confidence interval for b1, the regression slope coefficient.b. Calculate the
5.4 What is a dummy variable or an indicator variable? Describe the differences in interpretation of the coefficients of a linear regression when the independent variable is continuous and when it is binary. Give an example of each case. Explain how the construction of confidence intervals and
5.3 Describe the important characteristics of the variance of the conditional distribution of the error term in a linear regression? What are the implications for OLS estimation?
5.2 When are one-sided hypothesis tests constructed for estimated regression coefficients as opposed to two-sided hypothesis tests? When are confidence intervals constructed instead of hypothesis tests?
5.1 Outline the procedures for computing the p-value of a two-sided test of H0 : mY = 0 using an i.i.d. set of observations Yi, i = 1,c, n. Outline the procedures for computing the p-value of a two-sided test of H0 : b1 = 0 in a regression model using an i.i.d. set of observations 1Yi, Xi2, i =
E4.2 On the text website, http://www.pearsonglobaleditions.com, you will find the data file Earnings_and_Height, which contains data on earnings, height, and other characteristics of a random sample of U.S. workers.2 A detailed description is given in Earnings_and_Height_Description, also available
E4.1 On the text website, http://www.pearsonglobaleditions.com, you will find the data file Growth, which contains data on average growth rates from 1960 through 1995 for 65 countries, along with variables that are potentially related to growth.1 A detailed description is given in
1. You are interested in predicting the value of Yoos from a randomly chosen out-of-sample observation with Xoos = xoos.a. Suppose the out-of-sample observation is from the same population as the in-sample observations 1Xi, Yi2 and is chosen independently of the in-sample observations.i. Explain
4.15 (Requires Appendix 4.4) A sample (Xi,Yi), i = 1,c, n, is collected from a population with E(Y X) = b0 + b1X and used to compute the least squares estimators b n0 and b n
4.14 Show that the sample regression line passes through the point (X, Y).
4.13 Suppose Yi = b0 + b1Xi + kui, where k is a nonzero constant and (Yi, Xi) satisfy the three least squares assumptions. Show that the large-sample variance of b n1 is given by s 2b1 = k2 1n var3 1Xi - mX2ui4 3var1Xi2 24 . [Hint: This equation is the variance given in Equation (4.19) multiplied
4.12a. Show that the regression R2 in the regression of Y on X is the squared value of the sample correlation between X and Y. That is, show that R2 = r2 XY.b. Show that the R2 from the regression of Y on X is the same as the R2 from the regression of X on Y.c. Show that b n1 = rXY(sY > sX), where
4.11 Consider the regression model Yi = b0 + b1Xi + ui.a. Suppose you know that b0 = 0. Derive a formula for the least squares estimator of b1.b. Suppose you know that b0 = 4. Derive a formula for the least squares estimator of b1.
4.10 Suppose Yi = b0 + b1Xi + ui, where (Xi, ui) are i.i.d. and Xi is a Bernoulli random variable with Pr(X = 1) = 0.30. When X = 1, ui is N(0, 3); when X = 0, ui is N(0, 2).a. Show that the regression assumptions in Key Concept 4.3 are satisfied.b. Derive an expression for large-sample variance of
4.9a. A linear regression yields b n1 = 0. Show that R2 = 0.b. A linear regression yields R2 = 0. Does this imply that b n1 = 0?
4.8 Suppose all of the regression assumptions in Key Concept 4.3 are satisfied except that the first assumption is replaced with E(ui Xi) = 2. Which parts of Key Concept 4.4 continue to hold? Which change? Why? (Is b n1 normally distributed in large samples with mean and variance given in Key
4.7 Show that b n0 is an unbiased estimator of b0. (Hint: Use the fact that b n1 is unbiased, which is shown in Appendix 4.3.)
4.6 Show that the first least squares assumption, E(ui Xi) = 0, implies that E(Yi Xi) = b0 + b1Xi.
4.5 A researcher runs an experiment to measure the impact of a short nap on memory. There are 200 participants and they can take a short nap of either 60 minutes or 75 minutes. After waking up, each participant takes a short test for short-term recall. Each participant is randomly assigned one of
4.4 Your class is asked to investigate the effect of average temperature on average weekly earnings (AWE, measured in dollars) across countries, using the following general regression approach:AWE = b n0 + b n1 * temperature One of your classmates, Rachel, is an American and decides to analyze the
4.3 A regression of average monthly expenditure (AME, measured in dollars) on average monthly income (AMI, measured in dollars) using a random sample of collegeeducated full-time workers earning €100 to €1.5 million yields the following:AME = 710.7 + 8.8 * AMI, R2 = 0.030, SER = 540.30a.
4.2 A random sample of 100 20-year-old men is selected from a population and these men’s height and weight are recorded. A regression of weight on height yields Weight = -79.24 + 4.16 * Height, R2 = 0.72, SER = 12.6, where Weight is measured in pounds and Height is measured in inches.a. What is
4.1 Suppose that a researcher, using data on class size (CS) and average test scores from 50 third-grade classes, estimates the OLS regression:TestScore = 640.3 - 4.93 * CS, R2 = 0.11, SER = 8.7.a. A classroom has 25 students. What is the regression’s prediction for that classroom’s average
4.4 Distinguish between R2 and SER. How do each of these measures describe the fit of a regression?
4.3 What is meant by the assumption that a paired sample observations of Yi and Xi are independently and identically distributed? Why is this an important assumption for OLS estimation? When is this assumption likely to be violated?
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