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introductory econometrics modern

Introductory Econometrics A Modern Approach 5th Edition Jeffrey M. Wooldridge - Solutions

5 In Example 7.2, let noPC be a dummy variable equal to one if the student does not own a PC, and zero otherwise. (i) If noPC is used in place of PC in equation (7.6), what happens to the intercept in the estimated equation? What will be the coefficient on noPC? (Hint: Write PC 5 1 2 noPC and plug
4 An equation explaining chief executive officer salary is log(salary) 5 4.59 1 .257 log(sales) 1 .011 roe 1 .158 finance (.30) (.032) (.004) (.089) 1 .181 consprod 2 .283 utility (.085) (.099) n 5 209, R2 5 .357.The data used are in CEOSAL1.RAW, where finance, consprod, and utility are binary
3 Using the data in GPA2.RAW, the following equation was estimated: sat  51,028.10 1 19.30 hsize 2 2.19 hsize2 2 45.09 female (6.29) (3.83) (.53) (4.29) 2 169.81 black 1 62.31 female?black (12.71) (18.15) n 5 4,137, R2 5 .0858. The variable sat is the combined SAT score, hsize is size of the
2 The following equations were estimated using the data in BWGHT.RAW: log(bwght) 54.66 2 .0044 cigs 1 .0093 log( faminc) 1 .016 parity (.22) (.0009) (.0059) (.006) 1 .027 male 1 .055 white (.010) (.013) n 5 1,388, R2 5 .0472 and log(bwght) 54.65 2 .0052 cigs 1 .0110 log( faminc) 1 .017 parity
1 Using the data in SLEEP75.RAW (see also Problem 3 in Chapter 3), we obtain the estimated equation sleep 53,840.83 2 .163 totwrk 2 11.71 educ 2 8.70 age (235.11) (.018) (5.86) (11.21) 1 .128 age2 1 87.75 male (.134) (34.33) n 5 706, R2 5 .123, - R2 5 .117. The variable sleep is total minutes
C14 Use the data in BENEFITS.RAW to answer this question. It is a school-level data set at the K–5 level on average teacher salary and benefits. See Example 4.10 for background. (i) Regress lavgsal on bs and report the results in the usual form. Can you reject H0: bbs 5 0 against a two-sided
C13 Use the data in MEAP00.RAW to answer this questio (i) Estimate the model math4 5 b0 1 b2lexppp 1 b2lenroll 1 b3lunch 1 u by OLS, and report the results in the usual form. Is each explanatory variable statistically significant at the 5% level? (ii) Obtain the fitted values from the regression in
C1 Use the subset of 401KSUBS.RAW with fsize 5 1; this restricts the analysis to singleperson households; see also Computer Exercise C8 in Chapter (i) What is the youngest age of people in this sample? How many people are at that age?(ii) In the model nettfa 5 b0 1 b1inc 1 b2age 1 b3age2 1 u, what
C11 Use APPLE.RAW to verify some of the claims made in Section 6. (i) Run the regression ecolbs on ecoprc, regprc and report the results in the usual form, including the R-squared and adjusted R-squared. Interpret the coefficients on the price variables and comment on their signs and magnitudes.
C10 Use the data in BWGHT2.RAW for this exercis (i) Estimate the equation log(bwght) 5 b0 1 b1npvis 1 b2npvis2 1 u by OLS, and report the results in the usual way. Is the quadratic term significant? (ii) Show that, based on the equation from part (i), the number of prenatal visits that maximizes
C The data set NBASAL.RAW contains salary information and career statistics fo 269 players in the National Basketball Association (NBA). (i) Estimate a model relating points-per-game (points) to years in the league (exper), age, and years played in college (coll). Include a quadratic in exper; the
C8 Use the data in HPRICE1.RAW for this exercis (i) Estimate the model price 5 b0 1 b1lotsize 1 b2sqrft 1 b3bdrms 1 u and report the results in the usual form, including the standard error of the regression. Obtain predicted price, when we plug in lotsize 5 10,000, sqrft 5 2,300, and bdrms 5 4;
C7 Use the data in ATTEND.RAW for this exercis (i) In the model of Example 6.3, argue that ∆stndfnl/∆priGPA < b2 1 2b4 priGPA 1 b6atndrte. Use equation (6.19) to estimate the partial effect when priGPA 5 2.59 and atndrte 5 82. Interpret your estimate. (ii) Show that the equation can be written
C6 Use the data in VOTE1.RAW for this exercis (i) Consider a model with an interaction between expenditures: voteA 5 b0 1 b1prtystrA 1 b2expendA 1 b3expendB 1 b4expendA?expendB 1 u What is the partial effect of expendB on voteA, holding prtystrA and expendA fixed? What is the partial effect of
C5 Use the housing price data in HPRICE1.RAW for this exercis (i) Estimate the model log(price) 5 b0 1 b1log(lotsize) 1 b2log(sqrft) 1 b3bdrms 1 u and report the results in the usual OLS format. (ii) Find the predicted value of log(price), when lotsize 5 20,000, sqrft 5 2,500, and bdrms 5 4. Using
C4 Use the data in GPA2.RAW for this exercis (i) Estimate the model sat 5 b0 1 b1hsize 1 b2hsize2 1 u, where hsize is the size of the graduating class (in hundreds), and write the results in the usual form. Is the quadratic term statistically significant? (ii) Using the estimated equation from part
C3 Consider a model where the return to education depends upon the amount of work expe‑ rience (and vice versa): log(wage) 5 b0 1 b1educ 1 b2exper 1 b3educ?exper 1 u.(i) Show that the return to another year of education (in decimal form), holding exper fixed, is b1 1 b3exper. (ii) State the null
C2 Use the data in WAGE1.RAW for this exercise. (i) Use OLS to estimate the equation log(wage) 5 b0 1 b1educ 1 b2exper 1 b3exper2 1 u and report the results using the usual format. (ii) Is exper2 statistically significant at the 1% level? (iii) Using the approximation %∆wage < 100(b ˆ 2 1 2b
C1 Use the data in KIELMC.RAW, only for the year 1981, to answer the following ques‑ tions. The data are for houses that sold during 1981 in North Andover, Massachusetts; 1981 was the year construction began on a local garbage incinerator. (i) To study the effects of the incinerator location on
9 If we start with (6.38) under the CLM assumptions, assume large n, and ignore the estimation error in the b ˆ j , a 95% prediction interval for y0 is [exp(21.96s ˆ) exp(logy0 ), exp (1.96s ˆ) exp(logy0 )]. The point prediction for y0 is yˆ 0 5 exp(s ˆ 2 /2) exp(logy0 ). (i) For what
8 Suppose we want to estimate the effects of alcohol consumption (alcohol) on college grade point average (colGPA). In addition to collecting information on grade point averages and alcohol usage, we also obtain attendance information (say, percentage of lectures attended, called attend). A
7 The following three equations were estimated using the 1,534 observations in 401K.RAW: prate 5 80.29 1 5.44 mrate 1 .269 age 2 .00013 totemp (.78) (.52) (.045) (.00004) R2 5 .100, R2 5 .098. prate 5 97.32 1 5.02 mrate 1 .314 age 2 2.66 log(totemp) (1.95) (0.51) (.044) (.28) R2 5 .144, R2 5
6 When atndrte2 and ACT?atndrte are added to the equation estimated in (6.19), the R‑squared becomes .232. Are these additional terms jointly significant at the 10% level? Would you include them in the model?
5 In Example 4.2, where the percentage of students receiving a passing score on a tenth-grade math exam (math10) is the dependent variable, does it make sense to include sci11—the per‑ centage of eleventh graders passing a science exam—as an additional explanatory variable?
4 The following model allows the return to education to depend upon the total amount of both parents’ education, called pareduc: log(wage) 5 b0 1 b1educ 1 b2educ?pareduc 1 b3exper 1 b4tenure 1 u. (i) Show that, in decimal form, the return to another year of education in this model is
3 Using the data in RDCHEM.RAW, the following equation was obtained by OLS: rdintens 5 2.613 1 .00030 sales 2 .0000000070 sales2 (.429) (.00014) (.0000000037) n 5 32, R2 5 .1484. (i) At what point does the marginal effect of sales on rdintens become negative? (ii) Would you keep the quadratic
2 Let b ˆ 0, b ˆ 1, …, b ˆ k be the OLS estimates from the regression of yi on xi1, …, xik, i51, 2, …, n. For nonzero constants c1, …, ck, argue that the OLS intercept and slopes from the regression of c0 yi on c1xi1, …, ckxik, i 5 1, 2, …, n, are given by b ˜ 0 5 c0 b ˆ 0, b ˜ 1
1 The following equation was estimated using the data in CEOSAL1.RAW: log(salary) 5 4.322 1 .276 log(sales) 1 .0215 roe 2 .00008 roe2 (.324) (.033) (.0129) (.00026) n 5 209, R2 5 .282. This equation allows roe to have a diminishing effect on log(salary). Is this generality nec‑ essary? Explain
C5 Consider the analysis in Computer Exercise C11 in Chapter 4 using the data in HTV.RAW, where educ is the dependent variable in a regression. (i) How many different values are taken on by educ in the sample? Does educ have a continuous distribution? (ii) Plot a histogram of educ with a normal
C4 Several statistics are commonly used to detect nonnormality in underlying populatio distributions. Here we will study one that measures the amount of skewness in a distribution. Recall that any normally distributed random variable is symmetric about its mean; therefore, if we standardize a
C In equation (4.42) of Chapter 4, using the data set BWGHT.RAW, compute the LM statistic for testing whether motheduc and fatheduc are jointly significant. In obtaining th residuals for the restricted model, be sure that the restricted model is estimated using only those observations for which
C2 Use the data in GPA2.RAW for this exercise (i) Using all 4,137 observations, estimate the equation colgpa 5 b0 1 b1hsperc 1 b2sat 1 u and report the results in standard form. (ii) Reestimate the equation in part (i), using the first 2,070 observations. (iii) Find the ratio of the standard errors
C1 Use the data in WAGE1.RAW for this exercis (i) Estimate the equation wage 5 b0 1 b1educ 1 b2exper 1 b3tenure 1 u. Save the residuals and plot a histogram. (ii) Repeat part (i), but with log(wage) as the dependent variable. (iii) Would you say that Assumption MLR.6 is closer to being satisfied
4 In the simple regression model (5.16), under the first four Gauss-Markov assumptions, we showed that estimators of the form (5.17) are consistent for the slope, b1. Given such an estimator, define an estimator of b0 by b ˜ 0 5 - y 2 b ˜ 1x¯. Show that plim b ˜ 0 5 b0.
3 The data set SMOKE.RAW contains information on smoking behavior and other variables for a random sample of single adults from the United States. The variable cigs is the (average) number of cigarettes smoked per day. Do you think cigs has a normal distribution in the U.S. adult population?
2 Suppose that the model pctstck 5 b0 1 b1funds 1 b2risktol 1 u satisfies the first four Gauss-Markov assumptions, where pctstck is the percentage of a worker’s pension invested in the stock market, funds is the number of mutual funds that the worker can choose from, and risktol is some measure
1 In the simple regression model under MLR.1 through MLR.4, we argued that the slope estimator, b ˆ 1, is consistent for b1. Using b ˆ 0 5 - y 2 b ˆ 1x¯1, show that plim b ˆ 0 5 b0. [You need to use the consistency of b ˆ 1 and the law of large numbers, along with the fact that b0 5 E(y) 5
C Use the data in HTV.RAW to answer this question. See also Computer Exercise C10 i Chapter 3. (i) Estimate the regression model educ 5 0 1 1motheduc 1 2 fatheduc 1 3abil 1 4abil2 1 u by OLS and report the results in the usual form. Test the null hypothesis that educ is linearly
C Use the data in ELEM94_95 to answer this question. The findings can be compare with those in Table 4.1. The dependent variable lavgsal is the log of average teacher salary and bs is the ratio of average benefits to average salary (by school). (i) Run the simple regression of lavgsal on bs. Is
C Use the data in DISCRIM.RAW to answer this question. (See also Computer Exercise C in Chapter 3.) (i) Use OLS to estimate the model log(psoda) 5 b0 1 b1prpblck 1 b2 log(income) 1 b3prppov 1 u, and report the results in the usual form. Is bˆ1 statistically different from zero at the 5% level
C The data set 401KSUBS.RAW contains information on net financial wealth (nettfa age of the survey respondent (age), annual family income (inc), family size (fsize), and participation in certain pension plans for people in the United States. The wealth and income variables are both recorded in
C7 Refer to the example used in Section 4.4. You will use the data set TWOYEAR.RA (i) The variable phsrank is the person’s high school percentile. (A higher number is better. For example, 90 means you are ranked better than 90 percent of your graduating class.) Find the smallest, largest, and
C6 Use the data in WAGE2.RAW for this exercis (i) Consider the standard wage equation log(wage) 5 b0 1 b1educ 1 b2exper 1 b3tenure 1 u. State the null hypothesis that another year of general workforce experience has the same effect on log(wage) as another year of tenure with the current employer.
C5 Use the data in MLB1.RAW for this exerci (i) Use the model estimated in equation (4.31) and drop the variable rbisyr. What happens to the statistical significance of hrunsyr? What about the size of the coefficient on hrunsyr? (ii) Add the variables runsyr (runs per year), fldperc (fielding
C In Example 4.9, the restricted version of the model can be estimated using all 1,388 observations in the sample. Compute the R-squared from the regression of bwght on cigs parity, and faminc using all observations. Compare this to the R-squared reported for the restricted model in Example 4.9.
C3 Refer to Computer Exercise C2 in Chapter 3. Now, use the log of the housing price as th dependent variable: log(price) 5 b0 1 b1sqrft 1 b2bdrms 1 u. (i) You are interested in estimating and obtaining a confidence interval for the percentage change in price when a 150-square-foot bedroom is
C2 Use the data in LAWSCH85.RAW for this exercis (i) Using the same model as in Problem 4 in Chapter 3, state and test the null hypothesis that the rank of law schools has no ceteris paribus effect on median starting salary. (ii) Are features of the incoming class of students—namely, LSAT and
C1 The following model can be used to study whether campaign expenditures affect electio outcomes: voteA 5 b0 1 b1log(expendA) 1 b2log(expendB) 1 b3 prtystrA 1 u, where voteA is the percentage of the vote received by Candidate A, expendA and expendB are campaign expenditures by Candidates A and B,
12 The following analysis was obtained using data in MEAP93.RAW, which contains schoollevel pass rates (as a percent) on a 10th grade math test. (i) The variable expend is expenditures per student, in dollars, and math10 is the pass rate on the exam. The following simple regression relates math10
11 The following table was created using the data in CEOSAL2.RAW, where standard errors are in parentheses below the coefficients:The variable mktval is market value of the firm, profmarg is profit as a percentage of sales, ceoten is years as CEO with the current company, and comten is total years
10 Regression analysis can be used to test whether the market efficiently uses information in valuing stocks. For concreteness, let return be the total return from holding a firm’s stock over the four-year period from the end of 1990 to the end of 1994. The efficient markets hypothesis says that
9 In Problem 3 in Chapter 3, we estimated the equation sleep  5 3,638.25 2 .148 totwrk 2 11.13 educ 1 2.20 age (112.28) (.017) (5.88) (1.45) n 5 706, R2 5 .113, where we now report standard errors along with the estimates. (i) Is either educ or age individually significant at the 5% level
8 Consider the multiple regression model with three independent variables, under the classical linear model assumptions MLR.1 through MLR.6: y 5 b0 1 b1x1 1 b2x2 1 b3x3 1 u. You would like to test the null hypothesis H0: b1 2 3b2 5 1. (i) Let bˆ1 and bˆ2 denote the OLS estimators of b1 and b2.
7 In Example 4.7, we used data on nonunionized manufacturing firms to estimate the relationship between the scrap rate and other firm characteristics. We now look at this example more closely and use all available firms. (i) The population model estimated in Example 4.7 can be written as
6 In Section 4.5, we used as an example testing the rationality of assessments of housing prices. There, we used a log-log model in price and assess [see equation (4.47)]. Here, we use a level-level formulation. (i) In the simple regression model price 5 b0 1 b1assess 1 u,the assessment is rational
5 Consider the estimated equation from Example 4.3, which can be used to study the effects of skipping class on college GPA: colGPA  5 1.39 1 .412 hsGPA 1 .015 ACT 2 .083 skipped (.33) (.094) (.011) (.026) n 5 141, R2 5 .234. (i) Using the standard normal approximation, find the 95% confidence
4 Are rent rates influenced by the student population in a college town? Let rent be the average monthly rent paid on rental units in a college town in the United States. Let pop denote the total city population, avginc the average city income, and pctstu the student population as a percentage of
3 The variable rdintens is expenditures on research and development (R&D) as a percentage of sales. Sales are measured in millions of dollars. The variable profmarg is profits as a percentage of sales.Using the data in RDCHEM.RAW for 32 firms in the chemical industry, the following equation is
2 Consider an equation to explain salaries of CEOs in terms of annual firm sales, return on equity (roe, in percentage form), and return on the firm’s stock (ros, in percentage form): log(salary) 5 b0 1 b1log(sales) 1 b2roe 1 b3ros 1 u. (i) In terms of the model parameters, state the null
1 Which of the following can cause the usual OLS t statistics to be invalid (that is, not to have t distributions under H0)? (i) Heteroskedasticity. (ii) A sample correlation coefficient of .95 between two independent variables that are in the model. (iii) Omitting an important explanatory variable.
C9 Use the data in CHARITY.RAW to answer the following question (i) Estimate the equation gift 5 b0 1 b1mailsyear 1 b2giftlast 1 b3propresp 1 u by OLS and report the results in the usual way, including the sample size and R-squared. How does the R-squared compare with that from the simple
C8 Use the data in DISCRIM.RAW to answer this question. These ar zip code–level on prices for various items at fast-food restaurants, along with characteristics of the zip code population, in New Jersey and Pennsylvania. The idea is to see whether fast-food restaurants charge higher prices in
C7 Use the data in MEAP93.RAW to answer this questio (i) Estimate the model math10 5 b0 1 b1log(expend) 1 b2lnchprg 1 u, and report the results in the usual form, including the sample size and R-squared. Are the signs of the slope coefficients what you expected? Explain.(ii) What do you make of the
C6 Use the data set in WAGE2.RAW for this problem. As usual, be sure all of the following regressions contain an intercep (i) Run a simple regression of IQ on educ to obtain the slope coefficient, say, ˜ δ1. (ii) Run the simple regression of log(wage) on educ, and obtain the slope coefficient,
C5 Confirm the partialling out interpretation of the OLS estimates by explicitly doing th partialling out for Example 3.2. This first requires regressing educ on exper and tenure and saving the residuals, r ˆ1. Then, regress log(wage) on r ˆ1. Compare the coefficient on ˆr1 with the coefficient
C4 Use the data in ATTEND.RAW for this exercis (i) Obtain the minimum, maximum, and average values for the variables atndrte, priGPA, and ACT. (ii) Estimate the model atndrte 5 b0 1 b1priGPA 1 b2ACT 1 u, and write the results in equation form. Interpret the intercept. Does it have a useful meaning?
C The file CEOSAL2.RAW contains data on 177 chief executive officers and can be use to examine the effects of firm performance on CEO salary. (i) Estimate a model relating annual salary to firm sales and market value. Make the model of the constant elasticity variety for both independent variables.
C2 Use the data in HPRICE1.RAW to estimate the mode price 5 b0 1 b1sqrft 1 b2bdrms 1 u, where price is the house price measured in thousands of dollars. (i) Write out the results in equation form. (ii) What is the estimated increase in price for a house with one more bedroom, holding square
C1 A problem of interest to health officials (and others) is to determine the effects of smoking during pregnancy on infant health. One measure of infant health is birth weight; a birth weight that is too low can put an infant at risk for contracting various illnesses. Since factors other than
13 (i) Consider the simple regression model y 5 b0 1 b1x 1 u under the first four GaussMarkov assumptions. For some function g(x), for example g(x) 5 x2 or g(x) 5 log(1 1 x2 ), define zi 5 g(xi ). Define a slope estimator as b˜ 1 5 ∑ i51 n (zi 2 _ z)yi /∑ i51 n (zi 2 _ z)xi  .
12 The following equation represents the effects of tax revenue mix on subsequent employment growth for the population of counties in the United States: growth 5 b0 1 b1shareP 1 b2shareI 1 b3shareS 1 other factors, where growth is the percentage change in employment from 1980 to 1990, shareP is
11 Suppose that the population model determining y is y 5 b0 1 b1x1 1 b2x2 1 b3x3 1 u, and this model satisifies Assumptions MLR.1 through MLR.4. However, we estimate the model that omits x3. Let b˜ 0, b˜ 1, and b˜ 2 be the OLS estimators from the regression of y on x1 and x2. Show that the
10 Suppose that you are interested in estimating the ceteris paribus relationship between y and x1. For this purpose, you can collect data on two control variables, x2 and x3. (For concreteness, you might think of y as final exam score, x1 as class attendance, x2 as GPA up through the previous
9 The following equation describes the median housing price in a community in terms of amount of pollution (nox for nitrous oxide) and the average number of rooms in houses in the community (rooms): log(price) 5 b0 1 b1log(nox) 1 b2rooms 1 u. (i) What are the probable signs of b1 and b2? What is
8 Suppose that average worker productivity at manufacturing firms (avgprod ) depends on two factors, average hours of training (avgtrain) and average worker ability (avgabil): avgprod 5 b0 1 b1avgtrain 1 b2avgabil 1 u. Assume that this equation satisfies the Gauss-Markov assumptions. If grants have
7 Which of the following can cause OLS estimators to be biased? (i) Heteroskedasticity. (ii) Omitting an important variable. (iii) A sample correlation coefficient of .95 between two independent variables both included in the model.
6 Consider the multiple regression model containing three independent variables, under Assumptions MLR.1 through MLR.4: y 5 b0 1 b1x1 1 b2x2 1 b3x3 1 u. You are interested in estimating the sum of the parameters on x1 and x2; call this u1 5 b1 1 b2. (i) Show that ˆ u1 5 bˆ 1 1 bˆ 2 is an
5 In a study relating college grade point average to time spent in various activities, you distribute a survey to several students. The students are asked how many hours they spend each week in four activities: studying, sleeping, working, and leisure. Any activity is put into one of the four
4 The median starting salary for new law school graduates is determined by log(salary) 5 b0 1 b1LSAT 1 b2GPA 1 b3log(libvol) 1 b4log(cost) 1 b5rank 1 u, where LSAT is the median LSAT score for the graduating class, GPA is the median college GPA for the class, libvol is the number of volumes in the
3 The following model is a simplified version of the multiple regression model used by Biddle and Hamermesh (1990) to study the tradeoff between time spent sleeping and working and to look at other factors affecting sleep: sleep 5 b0 1 b1totwrk 1 b2educ 1 b3age 1 u, where sleep and totwrk (total
2 The data in WAGE2.RAW on working men was used to estimate the following equation: educ 5 10.36 2 .094 sibs 1 .131 meduc 1 .210 feduc n 5 722, R2 5 .214, where educ is years of schooling, sibs is number of siblings, meduc is mother’s years of schooling, and feduc is father’s years of
1 Using the data in GPA2.RAW on 4,137 college students, the following equation was estimated by OLS: colgpa 5 1.392 2 .0135 hsperc 1 .00148 sat n 5 4,137, R2 5 .273, where colgpa is measured on a four-point scale, hsperc is the percentile in the high school graduating class (defined so that,
C8 To complete this exercise you need a software package that allows you to generate data from the uniform and normal distributions. (i) Start by generating 500 observations xi – the explanatory variable – from the uniform distribution with range [0,10]. (Most statistical packages have a
C7 Use the data in CHARITY.RAW [obtained from Franses and Paap (2001)] to answer the following questions: (i) What is the average gift in the sample of 4,268 people (in Dutch guilders)? What percentage of people gave no gift? (ii) What is the average mailings per year? What are the minimum and
C6 We used the data in MEAP93.RAW for Example 2.12. Now we want to explore the relationship between the math pass rate (math10) and spending per student (expend). (i) Do you think each additional dollar spent has the same effect on the pass rate, or does a diminishing effect seem more appropriate?
C5 For the population of firms in the chemical industry, let rd denote annual expenditures on research and development, and let sales denote annual sales (both are in millions of dollars). (i) Write down a model (not an estimated equation) that implies a constant elasticity between rd and sales.
C4 Use the data in WAGE2.RAW to estimate a simple regression explaining monthly salary (wage) in terms of IQ score (IQ). (i) Find the average salary and average IQ in the sample. What is the sample standard deviation of IQ? (IQ scores are standardized so that the average in the population is 100
C3 Use the data in SLEEP75.RAW from Biddle and Hamermesh (1990) to study whether there is a tradeoff between the time spent sleeping per week and the time spent in paid work. We could use either variable as the dependent variable. For concreteness, estimate the model sleep 5 b0 1 b1totwrk 1 u,
C2 The data set in CEOSAL2.RAW contains information on chief executive officers for U.S. corporations. The variable salary is annual compensation, in thousands of dollars, and ceoten is prior number of years as company CEO. (i) Find the average salary and the average tenure in the sample. (ii) How
C1 The data in 401K.RAW are a subset of data analyzed by Papke (1995) to study the relationship between participation in a 401(k) pension plan and the generosity of the plan. The variable prate is the percentage of eligible workers with an active account; this is the variable we would like to
12 Consider the problem described at the end of Section 2.6: running a regression and only estimating an intercept. (i) Given a sample {yi : i 5 1, 2, . . . , n}, let b ˜ 0 be the solution to min b0 ∑ i51 n (yi 2 b0) 2 . Show that b ˜ 0 5 y –, that is, the sample average minimizes the sum of
11 Suppose you are interested in estimating the effect of hours spent in an SAT preparation course (hours) on total SAT score (sat). The population is all college-bound high school seniors for a particular year. (i) Suppose you are given a grant to run a controlled experiment. Explain how you would
10 Let bˆ 0 and bˆ 1 be the OLS intercept and slope estimators, respectively, and let u¯ be the sample average of the errors (not the residuals!). (i) Show that bˆ 1 can be written as bˆ 1 5 b1 1 ∑i n wiui where wi 5 di /SSTx and di 5 xi 2 x¯. (ii) Use part (i), along with ∑i n wi 5
9 (i) Let bˆ 0 and bˆ 1 be the intercept and slope from the regression of yi on xi, using n observations. Let c1 and c2, with c2  0, be constants. Let b ˜ 0 and b ˜ 1 be the intercept and slope from the regression of c1yi on c2xi. Show that b ˜ 1 5 (c1/c2)bˆ 0 and b ˜ 0 5 c1bˆ 0,
8 Consider the standard simple regression model y 5 b0 1 b1x 1 u under the Gauss-Markov Assumptions SLR.1 through SLR.5. The usual OLS estimators bˆ 0 and bˆ 1 are unbiased for their respective population parameters. Let b ˜ 1 be the estimator of b1 obtained by assuming the intercept is zero
7 Consider the savings function sav 5 b0 1 b1inc 1 u, u 5 √ ___ inc ·e, where e is a random variable with E(e) 5 0 and Var(e) 5 se 2 . Assume that e is independent of inc. (i) Show that E(uuinc) 5 0, so that the key zero conditional mean assumption (Assumption SLR.4) is satisfied. [Hint: If e is
6 Using data from 1988 for houses sold in Andover, Massachusetts, from Kiel and McClain (1995), the following equation relates housing price (price) to the distance from a recently built garbage incinerator (dist): log(price ) 5 9.40 1 0.312 log(dist) n 5 135, R2 5 0.162. (i) Interpret the
5 In the linear consumption function cons 5 bˆ 0 1 bˆ 1inc, the (estimated) marginal propensity to consume (MPC) out of income is simply the slope, bˆ 1, while the average propensity to consume (APC) is cons/inc 5 bˆ 0/inc 1 bˆ 1. Using observations for 100 families on annual income and
4 The data set BWGHT.RAW contains data on births to women in the United States. Two variables of interest are the dependent variable, infant birth weight in ounces (bwght), and an explanatory variable, average number of cigarettes the mother smoked per day during pregnancy (cigs). The following
3 The following table contains the ACT scores and the GPA (grade point average) for eight college students. Grade point average is based on a four-point scale and has been rounded to one digit after the decimal.(i) Estimate the relationship between GPA and ACT using OLS; that is, obtain the

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