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Introduction To Econometrics 3rd Global Edition James Stock, Mark Watson - Solutions
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 economy, using data from a sample of countries. The researcher plans to regress national income per capita on whether the country has a strong legal system or not (an indicator variable taking the value 1 or 0, based
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 $1000), 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.The data set consists of information on 7440 full-time,
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 degrees? How much more?b. Do men earn more than women, on average? How much more?The data set consists of information on 7440 full-time, full-year workers. The
6.1 Compute R 2 for each of the regressions.The data set consists of information on 7440 full-time, full-year workers. The highest educational achievement for each worker was either a high school diploma or a bachelor’s degree. The workers’ages ranged from 25 to 34 years. The data set also
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 and how is it related to multicollinearity of regressors? What is the solution for this form of multicollinearity?
6.3 What are the measures of fit that are commonly used for multiple regressions?How can an adjusted R 2 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, www.pearsonglobaleditions.com/Stock_Watson, 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,
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? This 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 (Yi, Xi). To be specific, suppose that 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
5.14 Suppose that Yi = bXi + ui, where (ui, Xi) satisfy the Gauss–Markov conditions given in Equation (5.31).a. Derive the least squares estimator of b 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 that (Yi, Xi) satisfy the least squares assumptions in Key Concept 4.3 and, in addition, ui is N(0, s2 u) and is independent of Xi.a. Is b n1 conditionally unbiased?b. Is b n1 the best linear conditionally unbiased estimator of b1?c. How would your answers to (a) and (b) change if you
5.12 Starting from Equation (4.22), 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 Ym = $565.89, and the sample standard deviation is sm = $75.62. The corresponding values for women are Yw = $502.37 and sw = $53.40. Let Women denote an indicator variable that is
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 b n0 = 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 that (Yi, Xi) satisfy the least squares assumptions in Key Concept 4.3 and, in addition, ui is N(0, s2 u) and is independent of Xi. A sample of size n = 30 yieldswhere the numbers in parentheses are the homoskedastic-only standard errors for the regression coefficients.a. Construct a
5.7 Suppose that (Yi, Xi) satisfy the least squares assumptions in Key Concept 4.3. A random sample of size n = 250 is drawn and yieldsa. Test H0 : b1 = 0 vs. H1 : b1 0 at the 5% level.b. Construct a 95% confidence interval for b1.c. Suppose you learned that Yi and Xi were independent. Would you
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 that 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 20-year-old men is selected from a population, and that these men’s height and weight are recorded. A regression of weight on height yieldswhere weight is measured in pounds and height is measured in inches. A man has a late growth spurt and grows 2 inches over
5.2 Suppose a researcher, using wage data on 200 randomly selected male workers and 240 female workers, estimates the OLS regressionwhere Wage is measured in dollars per hour and Male is a binary variable that is equal to 1 if the person is a male and 0 if the person is a female.Define the
5.1 Suppose that a researcher, using data on class size (CS) and average test scores from 50 third-grade classes, estimates the OLS regressiona. Construct a 95% confidence interval for b1, the regression slope coefficient.b. Calculate the p-value for the two-sided test of the null hypothesis H0 :
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. How are the construction of confidence intervals and
5.3 Describe the important characteristics of the variance of a conditional distribution of an 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 (Yi, Xi), i =
E4.2 On the text website, www.pearsonglobaleditions.com/Stock_Watson, 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
E4.1 On the text website, www.pearsonglobaleditions.com/Stock_Watson, 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. A detailed description is given in
4.14 Show that the sample regression line passes through the point (X, Y).
4.13 Suppose that 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(Xi - mX)ui4 3var(Xi)24 . [Hint: This equation is the variance given in Equation (4.21) 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 that Yi = b0 + b1Xi + ui, where (Xi, ui) are i.i.d., and Xi is a Bernoulli random variable with Pr(X = 1) = 0.20. When X = 1, ui is N(0, 4); when X = 0, ui is N(0, 1).a. Show that the regression assumptions in Key Concept 4.3 are satisfied.b. Derive an expression for the large-sample
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 that 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 a short nap has on memory. 200 participants in the sample are allowed to take a short nap of either 60 minutes or 75 minutes. After waking up, each of the participants takes a short test on short-term recall. Each participant is randomly
4.4 Read the box “The ‘Beta’ of a Stock” in Section 4.2.a. Suppose that the value of b is greater than 1 for a particular stock.Show that the variance of (R - Rf) for this stock is greater than the variance of (Rm - Rt).b. Suppose that the value of b is less than 1 for a particular stock.
4.3 A regression of average weekly earnings (AWE, measured in dollars) on age(measured in years) using a random sample of college-educated full-time workers aged 25–65 yields the following:a. Explain what the coefficient values 696.7 and 9.6 mean.b. The standard error of the regression (SER) is
4.2 Suppose a random sample of 100 20-year-old men is selected from a population and that these men’s height and weight are recorded. A regression of weight on height yieldswhere Weight is measured in pounds and Height is measured in inches.a. What is the regression’s weight prediction for
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:a. A classroom has 25 students. What is the regression’s prediction for that classroom’s average test score?b. Last year a classroom had 21 students,
4.4 Distinguish between the R2 and the standard error of a regression. How do each of these measures describe the fit of a regression?
4.3 What is meant by the assumption that the paired sample observations of Yi and Xi are independently and identically distributed? Why is this an important assumption for ordinary least-squares estimation? When is this assumption likely to be violated?
4.2 Explain what is meant by an error term. What assumptions do we make about an error term when estimating an ordinary least squares regression?
4.1 What is a linear regression model? What is measured by the coefficients of a linear regression model? What is the ordinary least squares estimator?
E3.2 A consumer is given the chance to buy a baseball card for $1, but he declines the trade. If the consumer is now given the baseball card, will he be willing to sell it for $1? Standard consumer theory suggests yes, but behavioral economists have found that “ownership” tends to increase the
E3.1 On the text website, www.pearsonglobaleditions.com/Stock_Watson, you will find the data file CPS92_12, which contains an extended version of the data set used in Table 3.1 of the text for the years 1992 and 2012. It contains data on full-time workers, ages 25–34, with a high school diploma
3.21 Show that the pooled standard error 3SEpooled( Ym - Yw)4 given following Equation (3.23) equals the usual standard error for the difference in means in Equation (3.19) when the two group sizes are the same(nm = nw).
3.20 Suppose that (Xi, Yi ) are i.i.d. with finite fourth moments. Prove that the sample covariance is a consistent estimator of the population covariance, that is, sXY ¡p sXY, where sXY is defined in Equation (3.24). (Hint: Use the strategy of Appendix 3.3.)
3.19a. Y is an unbiased estimator of mY. Is Y 2 an unbiased estimator of m2 Y?b. Y is a consistent estimator of mY. Is Y 2 a consistent estimator of m2 Y?
3.18 This exercise shows that the sample variance is an unbiased estimator of the population variance when Y1,c, Yn are i.i.d. with mean mY and variance s2 Y.a. Use Equation (2.31) to show that E3(Yi - Y )24 = var(Yi ) - 2cov(Yi, Y) + var(Y).b. Use Equation (2.33) to show that cov(Y, Yi ) = s2
3.17 Read the box “The Gender Gap of Earnings of College Graduates in the United States” in Section 3.5.a. Construct a 95% confidence interval for the change in men’s average hourly earnings between 1992 and 2012.b. Construct a 95% confidence interval for the change in women’s average
3.16 Grades on a standardized test are known to have a mean of 500 for students in the United States. The test is administered to 600 randomly selected students in Florida; in this sample, the mean is 508, and the standard deviation(s) is 75.a. Construct a 95% confidence interval for the average
3.15 Let Ya and Yb denote Bernoulli random variables from two different populations, denoted a andb. Suppose that E(Ya) = pa and E(Yb) = pb. A random sample of size na is chosen from populationa, with sample average denoted pna, and a random sample of size nb is chosen from population b, with
3.14 Values of height in inches (X) and weight in pounds (Y) are recorded from a sample of 200 male college students. The resulting summary statistics are X = 71.2 in, Y = 164 lb., sX = 1.9 in, sY = 16.4 lb., sXY = 22.54 in. * lb., and rXY = 0.8. Convert these statistics to the metric system
3.13 Data on fifth-grade test scores (reading and mathematics) for 400 school districts in California yield Y = 712.1 and standard deviation sY = 23.2.a. Construct a 90% confidence interval for the mean test score in the population.b. When the districts were divided into districts with small
3.12 To investigate possible gender discrimination in a firm, a sample of 120 men and 150 women with similar job descriptions are selected at random. A summary of the resulting monthly salaries follows:a. What do the data suggest about wage differences in the firm? Do they represent statistically
3.11 Consider the estimator Y , defined in Equation (3.1). Show that(a) E(Y ) = mY and (b) var(Y ) = 1.25s2 Y>n.
3.10 Suppose a new standardized test is given to 150 randomly selected thirdgrade students in New Jersey. The sample average score Y on the test is 42 points, and the sample standard deviation, sY, is 6 points.a. The authors plan to administer the test to all third-grade students in New Jersey.
3.9 Suppose a light bulb manufacturing plant produces bulbs with a mean life of 1000 hours, and a standard deviation of 100 hours. An inventor claims to have developed an improved process that produces bulbs with a longer mean life and the same standard deviation. The plant manager randomly selects
3.8 A new version of the SAT is given to 1500 randomly selected high school seniors. The sample mean test score is 1230, and the sample standard deviation is 145. Construct a 95% confidence interval for the population mean test score for high school seniors.
3.7 In a given population, 50% of the likely voters are women. A survey using a simple random sample of 1000 landline telephone numbers finds the percentage of female voters to be 55%. Is there evidence that this survey is biased? Explain.
3.6 Let Y1,c, Yn be i.i.d. draws from a distribution with mean m. A test of H0: m = 10 vs. H1: m 10 using the usual t-statistic yields a p-value of 0.07.a. Does the 90% confidence interval contain m = 10? Explain.b. Can you determine if m = 8 is contained in the 95% confidence interval? Briefly
3.5 A survey is conducted using 1000 registered voters, who are asked to choose between candidate A and candidate B. Let p denote the fraction of voters in the population who prefer candidate A, and let pn denote the fraction of voters who prefer Candidate B.a. You are interested in the competing
3.4 Using the data in Exercise 3.3:a. Construct a 95% confidence interval for p.b. Construct a 99% confidence interval for p.c. Why is the interval in (b) wider than the interval in (a)?a. Without doing any additional calculations, test the hypothesis H0: p = 0.50 vs. H1: p 0.50 at the 5%
3.3 In a survey of 400 likely voters, 215 responded that they would vote for the incumbent, and 185 responded that they would vote for the challenger. Let p denote the fraction of all likely voters who preferred the incumbent at the time of the survey, and let pn be the fraction of survey
3.2 Let Y be a Bernoulli random variable with success probability Pr(Y = 1) =p, and let Y1,c, Yn be i.i.d. draws from this distribution. Let pn be the fraction of successes (1s) in this sample.a. Show that pn = Y.b. Show that pn is an unbiased estimator of p.c. Show that var(pn) = p(1 - p)>n.
3.1 In a population, mY = 75 and s2 Y = 45. Use the central limit theorem to answer the following questions:a. In a random sample of size n = 50, find Pr(Y 6 73).b. In a random sample of size n = 90, find Pr(76 6 Y 6 78).c. In a random sample of size n = 120, find Pr( Y 7 68).
3.8 Sketch a hypothetical scatterplot for a sample of size 10 for two random variables with a population correlation of (a) 1.0; (b) −1.0; (c) 0.9; (d) −0.5;(e) 0.0.
3.7 What is a scatterplot? What statistical features of a dataset can be represented using a scatterplot diagram?
3.6 Why does a confidence interval contain more information than the result of a single hypothesis test?
3.5 What is the difference between a null hypothesis and an alternative hypothesis? Among size, significance level, and power? Between a onesided alternative hypothesis and a two-sided alternative hypothesis?
3.4 Differentiate between standard error and standard deviation. How is the standard error of a sample mean calculated?
3.3 A population distribution has a mean of 15 and a variance of 10. Determine the mean and variance of Y from an i.i.d. sample from this population for(a) n = 50; (b) n = 500; and (c) n = 5000. Relate your answers to the law of large numbers.
3.2 What is meant by the efficiency of an estimator? Which estimator is known as BLUE?
3.1 What is the difference between an unbiased estimator and a consistent estimator?
E2.1 On the text website, http://www.pearsonglobaleditions.com/Stock_Watson, you will find the spreadsheet Age_HourlyEarnings, which contains the joint distribution of age (Age) and average hourly earnings (AHE) for 25- to 34-year-old full-time workers in 2012 with an education level that exceeds a
2.27 X and Z are two jointly distributed random variables. Suppose you know the value of Z, but not the value of X. Let X= E(X Z) denote a guess of the value of X using the information on Z, and let W = X - X denote the error associated with this guess.a. Show that E(W ) = 0. (Hint: Use the law
2.26 Suppose that Y1, Y2,c, Yn are random variables with a common mean mY, a common variance s2Y, and the same correlation r (so that the correlation between Yi and Yj is equal to r for all pairs i and j, where i j).a. Show that cov(Yi, Yj ) = rs2 Y for i j.b. Suppose that n = 2. Show that E(Y
2.25 (Review of summation notation) Let x1,c, xn denote a sequence of numbers, y1,c, yn denote another sequence of numbers, anda, b, and c denote three constants. Show thata. a ni = 1 axi = aa ni = 1 xib. a ni = 1(xi + yi) = a ni = 1 xi + a ni = 1 yic. a ni = 1 a = nad. a ni = 1(a + bxi + cyi)2 =
2.24 Suppose Yi is distributed i.i.d. N(0, s2) for i = 1, 2,c, n.a. Show that E(Y 2i > s2) = 1.b. Show that W = (1>s2)gni= 1Y 2 i is distributed x2 n.c. Show that E(W) = n. [Hint: Use your answer to (a).]d. Show that V = Y1n Bgni= 2Yi 2n - 1 is distributed tn - 1.
2.23 This exercise provides an example of a pair of random variables X and Y for which the conditional mean of Y given X depends on X but corr(X, Y) = 0. Let X and Z be two independently distributed standard normal random variables, and let Y = X2 + Z.a. Show that E(Y 0 X ) = X2.b. Show that mY =
2.22 Suppose you have some money to invest—for simplicity, $1—and you are planning to put a fraction w into a stock market mutual fund and the rest, 1 - w, into a bond mutual fund. Suppose that $1 invested in a stock fund yields Rs after 1 year and that $1 invested in a bond fund yields Rb,
2.21 X is a random variable with moments E(X), E(X2), E(X3), and so forth.a. Show E(X - m)3 = E(X3) - 3[E(X2)][E(X)] + 2[E(X)]3.b. Show E(X - m)4 = E(X4) - 4[E(X)][E(X3)] + 6[E(X)]2[E(X2)] -3[E(X)]4.
2.20 Consider three random variables X, Y, and Z. Suppose that Y takes on k values y1,c, yk, that X takes on l values x1,c, xl, and that Z takes on m values z1,c, zm. The joint probability distribution of X, Y, Z is Pr(X = x, Y = y, Z = z), and the conditional probability distribution of Y given X
2.19 Consider two random variables X and Y. Suppose that Y takes on k values y1,c, yk and that X takes on l values x1,c, xl.a. Show that Pr(Y = yj) = gli= 1Pr(Y = yj X = xi) Pr(X = xi). [Hint:Use the definition of Pr(Y = yj X = xi).]b. Use your answer to (a) to verify Equation (2.19).c. Suppose
2.18 In any year, the weather may cause damages to a home. On a year-to-year basis, the damage is random. Let Y denote the dollar value of damages in any given year. Suppose that during 95% of the year Y = $0, but during the other 5% Y = $30,000.a. What are the mean and standard deviation of
2.17 Yi, i = 1,c, n, are i.i.d. Bernoulli random variables with p = 0.6. Let Y denote the sample mean.a. Use the central limit to compute approximations for i. Pr(Y 7 0.64) when n = 50.ii. Pr(Y 6 0.56) when n = 200.b. How large would n need to be to ensure that Pr(0.65 7 Y 7 0.55) = 0.95?(Hint: Use
2.16 Y is distributed N(5, 100) and you want to calculate Pr(Y 6 3.6). Unfortunately, you do not have your textbook, and do not have access to a normal probability table like Appendix Table 1. However, you do have your computer and a computer program that can generate i.i.d. draws from the N(5,
2.15 Suppose Yi, i = 1, 2,c, n, are i.i.d. random variables, each distributed N(10, 4).a. Compute Pr(9.6 … Y … 10.4) when (i) n = 20, (ii) n = 100, and(iii) n = 1000.b. Suppose c is a positive number. Show that Pr(10 - c … Y … 10 + c)becomes close to 1.0 as n grows large.c. Use your answer
2.14 In a population mY = 50 and J2 Y = 21. Use the central limit theorem to answer the following questions:a. In a random sample of size n = 50, find Pr(Y … 51).b. In a random sample of size n = 150, find Pr(Y 7 49).c. In a random sample of size n = 45, find Pr(50.5 … Y … 51).
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