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Introduction To Econometrics 4th Global Edition James H. Stock, Mark W. Watson - Solutions
14.9 You have a sample of size n = 1 with data y1 = 2 and x1 = 1. You are interested in the value of b in the regression Y = X b + u. (Note there is no intercept.)a. Plot the sum of squared residuals 1y1 - bx12 2 as function of b.b. Show that the least squares estimate of b is b nOLS = 2.c. Using
14.8 Let X and Y be two random variables. Denote the mean of Y given X = x by m(x) and the variance of Y by s2(x).a. Show that the best (minimum MSPE) prediction of Y given X = x is m(x) and the resulting MSPE is s2(x). (Hint: Review Appendix 2.2.)b. Suppose X is chosen at random. Use the result in
14.7 In Exercise 14.5(b), suppose you predict Y using Y - 1 instead of Y.a. Compute the bias of the prediction.b. Compute the mean of the prediction error.c. Compute the variance of the prediction error.d. Compute the MSPE of the prediction.e. Does Y - 1 produce a prediction with a lower MSPE than
14.6 In Exercise 14.5(b), suppose you predict Y using Y>2 instead of Y.a. Compute the bias of the prediction.b. Compute the mean of the prediction error.c. Compute the variance of the prediction error.d. Compute the MSPE of the prediction.e. Does Y>2 produce a prediction with a lower MSPE than the
14.5 Y is a random variable with mean m = 2 and variance s2 = 25.a. Suppose you know the value of m.i. What is the best (lowest MSPE) prediction of the value of Y? That is, what is the oracle prediction of Y?ii. What is the MSPE of this prediction?b. Suppose you don’t know the value of μ but you
14.4 A large online retailer sells thousands of products. The retailer has detailed data on the products purchased by each of its customers. Explain how you would use these data to predict the next product purchased by a randomly selected customer.
14.3 Describe the relationship, if any, between the standard error of a regression and the square root of the MSPE of the regression’s out-of-sample predictions.
14.2 A school principal is trying to raise funds so that all her students will receive reduced-price meals; currently, only 40% qualify for reduced-priced meals.Can she use the regression in Exercise 14.1 to estimate the effect of the new policy on test scores? Explain why or why not.
14.1 A researcher is interested in predicting average test scores for elementary schools in Arizona. She collects data on three variables from 200 randomly chosen Arizona elementary schools: average test scores (TestScore) on a standardized test, the fraction of students who qualify for
14.5 Suppose a data set with 10 variables produces a scree plot that is flat. What does this tell you about the correlation of the variables? What does this suggest about the usefulness of using the first few principal components of these variables in a predictive regression?
14.4 Ridge regression and Lasso are two regression estimators based on penalization.Explain how they are similar and how they differ.
14.3 Regression coefficients estimated using shrinkage estimators are biased. Why might these biased estimators yield more accurate predictions than an unbiased estimator?
14.2 Cross-validation uses in-sample observations. How does it estimate the MSPE for out-of-sample observations, even though the econometrician does not have those observations?
14.1 Using data from a random sample of elementary schools, a researcher regresses average test scores on the fraction of students who qualify for reduced-price meals. The regression indicates a negative coefficient that is highly statistically significant and yields a high R2. Is this regression
E13.1 A prospective employer receives two resumes: a resume from a white job applicant and a similar resume from an African American applicant. Is the employer more likely to call back the white applicant to arrange an interview? Marianne Bertrand and Sendhil Mullainathan carried out a randomized
13.12 Consider the potential outcomes framework from Appendix 13.3. Suppose Xi is a binary treatment that is independent of the potential outcomes Yi(1) and Yi(0). Let TEi = Yi112 - Yi (0) denote the treatment effect for individual i.a. Can you consistently estimate E [Yi(1)] and E[Yi(0)]? If yes,
13.11 Results of a study by McClelan, McNeill, and Newhouse are reported in Chapter 12. They estimate the effect of cardiac catheterization on patient survival times. They instrument the use of cardiac catheterization by the distance between a patient’s home and a hospital that offers the
13.10 Consider the regression model with heterogeneous regression coefficients Yi = b0 + b1iXi + vi, where (vi, Xi, b1i) are i.i.d. random variables with b1 = E1b1i2.a. Show that the model can be written as Yi = b0 + b1Xi + ui, where ui = 1b1i - b12Xi + vi.b. Suppose Xi is randomly assigned, so
13.9 Derive the final equality in Equation (13.10). (Hint: Use the definition of the covariance, and remember that, because the actual treatment Xi is random, b1i and Xi are independently distributed.)
13.8 Suppose you have the same data as in Exercise 13.7 (panel data with two periods, n observations), but ignore the W regressor. Consider the alternative regression model Yit = b0 + b1Xit + b2Gi + b3Dt + uit, where Gi = 1 if the individual is in the treatment group and Gi = 0 if the individual is
13.7 Suppose you have panel data from an experiment with T = 2 periods (so t = 1, 2). Consider the panel data regression model with fixed individual and time effects and individual characteristics Wi that do not change over time. Let the treatment be binary, so that Xit = 1 for t = 2 for the
13.6 Suppose there are panel data for T = 2 time periods for a randomized controlled experiment, where the first observation 1t = 12 is taken before the experiment and the second observation 1t = 22 is for the posttreatment period. Suppose the treatment is binary; that is, suppose Xit = 1 if the
13.5 Consider a study to evaluate the effect on college student grades of dorm room Internet connections. In a large dorm, half the rooms are randomly wired for high-speed Internet connections (the treatment group), and final course grades are collected for all residents. Which of the following
13.4 A new law will increase minimum wages in City A next year but not in City B, a city much like City A. You collect employment data from a random selected sample of restaurants in cities A and B this year, and you plan to return and collect data at restaurants next year. Let Yit denote the
13.3 Suppose that, in a randomized controlled experiment of the effect of an SAT preparatory course on SAT scores, the following results are reported:Treatment Group Control Group Average SAT score (X) 1348 1395 Standard deviation of SAT score (sX) 87.3 82.1 Number of men 60 40 Number of women 40
13.2 For the following calculations, use the results in column (3) of Table 13.2.Consider two classrooms, A and B, which have identical values of the regressors in column (3) of Table 13.2, except that:a. Classroom A is a small class, and classroom B is a regular-sized class.Construct a 90%
13.1 How would you calculate the small class treatment effect from the results in Table 13.1? Can you distinguish this treatment effect from the aide treatment effect? How would you have to change the program to correctly estimate both effects?
13.5 Consider the quasi-experiment described in Section 13.4 involving the draft lottery, military service, and civilian earnings. Explain why there might be heterogeneous effects of military service on civilian earnings; that is, explain why b1i in Equation (13.9) depends on i. Explain why there
13.4 What are experimental effects? How can such effects create bias in treatment effects? What can a researcher do to reduce the bias?
13.3 Researchers studying the STAR data report anecdotal evidence that school principals were pressured by some parents to place their children in the small classes. Suppose some principals succumbed to this pressure and transferred some children into the small classes. How would such transfers
13.2 A clinical trial is carried out for a new cholesterol-lowering drug. The drug is given to 500 patients, and a placebo is given to another 500 patients, using random assignment of the patients. How would you estimate the treatment effect of the drug? Suppose you had data on the weight, age, and
13.1 A researcher studying the effects of a new fertilizer on crop yields plans to carry out an experiment in which different amounts of the fertilizer are applied to 100 different one-acre parcels of land. There will be four treatment levels. Treatment level 1 is no fertilizer, treatment level 2
E12.3 (This requires Appendix 12.5) On the text website, http://www.pearsonglobaleditions.com, you will find the data set WeakInstrument, which contains 200 observations on (Yi, Xi, Zi) for the instrumental regression Yi = b0 + b1Xi + ui.a. Construct b nTSLS 1 , its standard error, and the usual
E12.2 Does viewing a violent movie lead to violent behavior? If so, the incidence of violent crimes, such as assaults, should rise following the release of a violent movie that attracts many viewers. Alternatively, movie viewing may substitute for other activities (such as alcohol consumption) that
E12.1 How does fertility affect labor supply? That is, how much does a woman’s labor supply fall when she has an additional child? In this exercise, you will estimate this effect using data for married women from the 1980 U.S. Census.6 The data are available on the text website,
12.10 Two classmates are comparing their answers to an assignment. One classmate has specified an instrumental variable regression model Yi = b0 + b1Xi + b2Wi + ui, using Zi as an instrument. The other student has specified the same model, but has omitted Wi.a. The first student says that if Zi and
12.9 A researcher is interested in the effect of more secure property rights on income across countries. He collects recent data from 60 countries and runs the OLS regression Yi = b0 + b1Xi + ui, where Yi is a country’s GDP per capita and Xi is an index taking values between 0 and 10 reflecting
12.8 Consider a product market with a supply function Qsi= b0 + b1Pi + usi, a demand function Qdi= g0 + udi, and a market equilibrium condition Qsi= Qdi, where usi and usi are mutually independent i.i.d. random variables, both with a mean of 0.a. Show that Pi and usi are correlated.b. Show that the
12.7 A classmate has developed an IV regression model with one regressor, Xi, and two instruments, Z1i and Z2i. She has a strong theoretical basis as to why corr1Z1i, ui2 = 0, namely that Z1i is the result of a random lottery. Preliminary work, however, showed that the first-stage F-statistic from
12.6 Suppose a researcher is considering developing an IV regression model with one regressor, Xi, and one instrument, Zi. If she has a sample of n = 113, what range must the correlation coefficient be between Xi and Zi in order for Zi to be considered a strong instrument? [Hint: See Equation
12.5 Consider the IV regression model Yi = b0 + b1Xi + b2Wi + ui,where Xi is correlated with ui and Zi is an instrument. Suppose that the first three assumptions in Key Concept 12.4 are satisfied. Which IV assumption is not satisfied whena. Zi is independent of (Yi, Xi, Wi)?b. Zi = Wi ?c. Wi = 1
12.4 Consider TSLS estimation of the effect of a single included endogenous variable, Xi, on Yi using one binary instrument, Zi, which takes values of either 0 or 1. Noting that gn i = 11Yi - Y21Zi - Z2 = gn i = 1Zi1Yi - Y2, show that the Wald estimator can be derived from the TSLS estimator in
12.3 A classmate is interested in estimating the variance of the error term in Equation(12.1).a. Suppose she uses the estimator from the second-stage regression of TSLS: sn 2a= 1 n-2gn i = 11Yi - b nTSLS 0 - b nTSLS 1 X ni2 2, where X ni is the fitted value from the first-stage regression. Is this
12.2 Consider the regression model with a single regressor: Yi = b0 + b1Xi + ui.Suppose the least squares assumptions in Key Concept 4.3 are satisfied.a. Show that Xi is a valid instrument. That is, show that Key Concept 12.3 is satisfied with Zi = Xi.b. Show that the IV regression assumptions in
12.1 This question refers to the panel data IV regressions summarized in Table 12.1.a. Suppose the federal government is considering a new tax on cigarettes that is estimated to increase the retail price by $0.25 per pack. If the current price per pack is $6.75, use the IV regression in column (1)
12.4 In their study of the effectiveness of cardiac catheterization, McClellan, McNeil, and Newhouse (1994) used as an instrument the difference in distances to cardiac catheterization and regular hospitals. How could you determine whether this instrument is relevant? How could you determine
12.3 In their study of the effect of institutions on economic development, suppose Acemoglu et al. had used the prevalence of malaria as an instrument.Would this instrument be relevant? Would it be exogenous? Would it be a valid instrument?
12.2 Describe the key characteristics of a valid instrument. If you were a researcher, how would you determine if the variable you have selected for an endogenous regressor is a valid instrument or not?
12.1 In the demand curve model of Equation (12.3), is ln 1Pbutter i 2 positively or negatively correlated with the error, ui? If b1 is estimated by OLS, would you expect the estimated value to be larger or smaller than the true value of b1?Explain.
E11.2 Believe it or not, workers used to be able to smoke inside office buildings.Smoking bans were introduced in several areas during the 1990s. Supporters of these bans argued that in addition to eliminating the externality of secondhand smoke, they would encourage smokers to quit by reducing
E11.1 In April 2008, the unemployment rate in the United States stood at 5.0%. By April 2009, it had increased to 9.0%, and it had increased further, to 10.0%, by October 2009. Were some groups of workers more likely to lose their jobs than others during the Great Recession? For example, were young
11.11 (Requires Appendix 11.3) State which model you would use for:a. A study explaining the number of hours a person spends working in a factory during one week.b. A study explaining the level of satisfaction (0 through 5) a person gains from their job.c. A study of consumers’ choices for mode
11.10 (Requires Section 11.3 and calculus) Suppose a random variable Y has the following probability distribution: Pr1Y = 12 = p, Pr1Y = 22 = q, and Pr1Y = 32 = 1 - p - q. A random sample of size n is drawn from this distribution, and the random variables are denoted Y1, Y2,c,Yn.a. Derive the
11.9 Use the estimated linear probability model shown in column (1) of Table 11.2 to answer the following:a. Two applicants, one self-employed and one in salaried employment, apply for a mortgage. They have the same values for all the regressors other than employment status. How much more likely is
11.8 Consider the linear probability model Yi = b0 + b1Xi + ui, and assume that E1ui Xi2 = 0.a. Show that Pr1Yi = 1 Xi2 = b0 + b1Xi.b. Show that var1ui Xi2 = 1b0 + b1Xi231 - 1b0 + b1Xi24. [Hint: Review Equation (2.7).]c. Is ui heteroskedastic? Explain.d. (Requires Section 11.3) Derive the
11.7 Repeat Exercise 11.6 using the logit model in Equation (11.10). Are the logit and probit results similar? Explain.
11.6 Use the estimated probit model in Equation (11.8) to answer the following questions:a. A black mortgage applicant has a P/I ratio of 0.35. What is the probability that his application will be denied?b. Suppose the applicant reduced this ratio to 0.30. What effect would this have on his
11.5 Using the results in column (7):a. Liam Johansson is a man with 10 years of schooling. What is the probability that government will employ him?b. Anneli Karlsson is a woman with 12 years of schooling. What is the probability that government will employ her?c. Does the effect of schooling on
11.4 Using the results in columns (4) through (6):a. Compute the estimated probability of being employed by the government for men and for women.b. Are the models in (4) through (6) different? Why or why not?
11.3a. Answer (a) through (c) from Exercise 11.1 using the results in column (3).b. Sketch the predicted probabilities from the probit and linear probability in columns (1) and (3) as a function of Schooling for values of Schooling between 0 and 18. Do you think that the linear probability is
11.2a. Answer (a) through (c) from Exercise 11.1 using the results in column (2).b. Sketch the predicted probabilities from the probit and logit in columns (1)and (2) for values of Schooling between 0 and 18. Are the probit and logit models similar?
11.1 Using the results in column (1):a. Does the probability of working for the government depend on Schooling?Explain.b. Friedrich Fürnrohr has 16 years of schooling. What is the probability that he will be employed by the government?c. Hans Schneider never went to college (12 years of
11.4 What measures of fit are typically used to assess binary dependent variable regression models?
11.3 What is maximum likelihood estimation? What are the advantages of using maximum likelihood estimators such as the probit and the logit, instead of the linear probability model? How would you choose between the probit and the logit?
11.2 In Table 11.2, the estimated coefficient on black is 0.084 in column (1), 0.688 in column (2), and 0.389 in column (3). In spite of these large differences, all three models yield similar estimates of the marginal effect of race on the probability of mortgage denial. How can this be?
11.1 Suppose a linear probability model yields a predicted value of Y that is equal to 1.3. Explain why this is nonsensical.
E10.2 Do citizens demand more democracy and political freedom as their incomes grow? That is, is democracy a normal good? On the text website, http://www.pearsonglobaleditions.com, you will find the data file Income_Democracy, which contains a panel data set from 195 countries for the years 1960,
E10.1 Some U.S. states have enacted laws that allow citizens to carry concealed weapons. These laws are known as “shall-issue” laws because they instruct local authorities to issue a concealed weapons permit to all applicants who are citizens, are mentally competent, and have not been convicted
10.11 Let b nDM 1 denote the entity-demeaned estimator given in Equation (10.22), and let b nBA 1 denote the “before and after” estimator without an intercept, so that b nBA 1 = 3 ni= 11Xi2 - Xi121Yi2 - Yi124 > 3 ni= 11Xi2 - Xi12 24. Show that, if T = 2, b nDM 1 = b nBA 1 . [Hint: Use the
10.10 A researcher wants to estimate the determinants of annual earnings—age, gender, schooling, union status, occupation, and sector of employment. He has been told that if he collects panel data on a large number of randomly chosen individuals over time, he will be able to regress annual
10.9a. In the fixed effects regression model, are the fixed entity effects, ai, consistently estimated as n¡ with T fixed? (Hint: Analyze the model with no X’s: Yit = ai + uit.)b. If n is large (say, n = 2000) but T is small (say, T = 4), do you think that the estimated values of ai are
10.8 Consider observations 1Yit, Xit2 from the linear panel data model Yit = Xitb1 + ai + lit + uit, where t = 1,c, T; i = 1,c, n; and ai + lit is an unobserved entity-specific time trend. How would you estimate b1?
10.7 Suppose a researcher believes that the occurrence of natural disasters such as earthquakes leads to increased activity in the construction industry. He decides to collect province-level data on employment in the construction industry of an earthquake-prone country, like Japan, and regress this
10.6 Do the fixed effects regression assumptions in Key Concept 10.3 imply that cov1v it,v is2 = 0 for t s in Equation (10.28)? Explain.
10.5 Consider the model with a single regressor. This model also can be written as Yit = b0 + b1X1,it + d2B2t + g+ dTBTt + g2D2i + g+ gnDni + uit, where B2t = 1 if t = 2 and 0 otherwise, D2i = 1 if i = 2 and 0 otherwise, and so forth. How are the coefficients 1b0, d2,c, dT, g2,c, gn2 related to the
10.4 Using the regression in Equation (10.11), what are the slope and intercept fora. Entity 1 in time period 1?b. Entity 1 in time period 3?c. Entity 3 in time period 1?d. Entity 3 in time period 3?
10.3 Section 9.2 gave a list of five potential threats to the internal validity of a regression study. Apply that list to the empirical analysis in Section 10.6 and thereby draw conclusions about its internal validity.
10.2 Consider the binary variable version of the fixed effects model in Equation(10.11) except with an additional regressor, D1i; that is, let Yit = b0 + b1Xit + g1D1i + g2D2i + g+ gnDni + uit.a. Suppose that n = 3. Show that the binary regressors and the “constant”regressor are perfectly
10.1 This exercise refers to the drunk driving panel data regression summarized in Table 10.1.a. New Jersey has a population of 8.85 million people. Suppose New Jersey increased the tax on a case of beer by $2 (in 1988 dollars). Use the results in column (5) to predict the number of lives that
10.4 In the context of the regression you suggested for Question 10.2, explain why the regression error for a given individual might be serially correlated.
10.3 Can the regression that you suggested in response to Question 10.2 be used to estimate the effect of a worker’s sex on his or her earnings? Can that regression be used to estimate the effect of the national unemployment rate on an individual’s earnings? Explain.
10.2 A researcher is using a panel data set on n = 1000 workers over T = 10 years(from 2008 through 2017) that contains the workers’ earnings, sex, education, and age. The researcher is interested in the effect of education on earnings.Give some examples of unobserved person-specific variables
10.1 What is meant by panel data? What is the advantage of using such data to make statistical and economic inferences?
E9.2 Use the data set Birthweight_Smoking introduced in Empirical Exercise 5.3 to answer the following questions.a. In Empirical Exercise 7.1(f), you estimated several regressions and were asked: “What is a reasonable 95% confidence interval for the effect of smoking on birth weight?”i. In
E9.1 Use the data set CPS2015, described in Empirical Exercise 8.2, to answer the following questions.a. Discuss the internal validity of the regressions that you used to answer Empirical Exercise 8.2(l). Include a discussion of possible omitted variable bias, misspecification of the functional
9.13 Assume that the regression model Yi = b0 + b1Xi + ui satisfies the least squares assumptions in Key Concept 4.3. You and a friend collect a random sample of 300 observations on Y and X.a. Your friend reports that he inadvertently scrambled the X observations for 20% of the sample. For these
9.12 Consider the one-variable regression model Yi = b0 + b1Xi + ui, and suppose it satisfies the least squares assumptions in Key Concept 4.3. The regressor Xi is missing, but data on a related variable, Zi, are available, and the value of Xi is estimated usingX i = E1Xi Zi2. Let wi = X i - Xi.a.
9.11 Read the box “The Demand for Economics Journals” in Section 8.3. Discuss the internal and external validity of the estimated effect of price per citation on subscriptions.
9.10 Read the box “The Effect of Ageing on Healthcare Expenditures: A Red Herring?”in Section 8.3. Discuss the internal and external validity as a causal effect of the relationship between age and healthcare expenditures, considering both models 1 and 3.
9.9 Consider the linear regression of TestScore on Income shown in Figure 8.2 and the nonlinear regression in Equation (8.18). Would either of these regressions provide a reliable estimate of the causal effect of income on test scores? Would either of these regressions provide a reliable method for
9.8 Would the regression in Equation (4.9) in chapter 4 be useful for predicting test scores in a school district in Massachusetts? Why or why not?
9.7 Are the following statements true or false? Explain your answer.a. “An ordinary least squares regression of Y onto X will not be internally valid if Y is correlated with the error term.”b. “If the error term exhibits heteroskedasticity, then the estimates of X will always be biased.”
9.6 Suppose that n = 50 i.i.d. observations for 1Yi, Xi2 yield the following regression results:Y n= 49.2 + 73.9X, SER = 13.4, R2 = 0.78.123.52 116.42 Another researcher is interested in the same regression, but he makes an error when he enters the data into his regression program: He enters each
9.5 The demand for a commodity is given by Q = b0 + b1P + u, where Q denotes quantity, P denotes price, and u denotes factors other than price that determine demand. Supply for the commodity is given by Q = g0 + g1P + v, where v denotes factors other than price that determine supply. Suppose u and
9.4 Using the regressions shown in columns (2) of Tables 8.3 and 9.3, and column(2) of Table 9.2, construct a table like Table 9.3 and compare the estimated effects of a 10 percentage point increase in the students eligible for free lunch on test scores in California and Massachusetts.
9.3 Labor economists studying the determinants of women’s earnings discovered a puzzling empirical result. Using randomly selected employed women, they regressed earnings on the women’s number of children and a set of control variables (age, education, occupation, and so forth). They found that
9.2 Consider the one-variable regression model Yi = b0 + b1Xi + ui, and suppose it satisfies the least squares assumptions in Key Concept 4.3. Suppose Yi is measured with error, so the data are Y i = Yi + wi, where wi is the measurement error, which is i.i.d. and independent of Yi and Xi. Consider
9.1 Suppose that you have just read a careful statistical study of the effect of improved health of children on their test scores at school. Using data from a project in a West African district in 2000, the study concluded that students who received multivitamin supplements performed substantially
9.6 A researcher estimates a regression using two different software packages.The first uses the homoskedasticity-only formula for standard errors. The second uses the heteroskedasticity-robust formula. The standard errors are very different. Which should the researcher use? Why?
9.5 What is simultaneous causality bias? Explain the potential for simultaneous causality in a study of the effects of high levels of bureaucratic corruption on national income.
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