Questions and Answers of Econometrics

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,
8.1 A researcher tells you that there are non-linearities in the relationship between wages and years of schooling. What does this mean? How would you test for non-linearities in the relationship
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
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,
7.11 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
7.10 Equations (7.13) and (7.14) show two formulas for the homoskedasticityonly 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 +
7.8 Referring to the table on page 292 used for exercises 7.1–7.6:a. Construct the R2 for each of the regressions.b. Show how to construct the homoskedasticity-only F-statistic for testing b4 = b5
7.7 Question 6.5 reported the following regression (where standard errors have been added):a. Is the coefficient on BDR statistically significantly different from zero?b. Typically four-bedroom
7.6 In all the regressions, 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
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
7.4 Using the regression results in column (3):a. Are there important regional differences? Use an appropriate hypothesis test to explain your answer.b. Juan is a 32-year-old male high school
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. Alvo is a
7.2 Using the regression results in column (1):a. Is the college–high school earnings difference, estimated from this regression, statistically significant at the 5% level? Construct a
7.1 Add * (5%) and ** (1%) to the table to indicate the statistical significance of the coefficients.The data set consists of information on over 10000 full-time, full-year workers. The highest
7.3 What is a control variable, and how does it differ from a variable of interest?Looking at Table 7.1, which variables are control variables? What is the variable of interest? Do coefficients on
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
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
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
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
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
6.10 (Yi, X1i, X2i) satisfy the assumptions in Key Concept 6.4; in addition, var(ui X1i, X2i) = 4 and var(X1i) = 6. A random sample of size n = 400 is drawn from the population.a. Assume that X1 and
6.9 (Yi, X1i, X2i) satisfy the assumptions in Key Concept 6.4. You are interested in b1, the causal effect of X1 on Y. Suppose that X1 and X2 are uncorrelated.You estimate b1 by regressing Y onto X1
6.8 A government study found that people who eat chocolate frequently weigh less than people who don’t. Researchers questioned 1000 individuals from California between the ages of 20 and 85 about
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
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
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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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?
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
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
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.
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
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
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
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
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
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
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
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
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
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
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,
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 =
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
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
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,
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
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
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
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
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
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
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
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.

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